nexus® technical reference guide - halliburton

Nexus® Technical Reference Guide

© 2009 Halliburton

December 2009

© 2009 HalliburtonAll Rights Reserved

This publication has been provided pursuant to an agreement containing restrictions on its use. The publication is also protected by Federal copyright law. No part of this publication may be copied or distributed, transmitted, transcribed, stored in a retrieval system, or translated into any human or computer language, in any form or by any means, electronic, magnetic, manual, or otherwise, or disclosed to third parties without the express written permission of:

Halliburton | Landmark Software & Services2107 CityWest Blvd, Building 2, Houston, Texas 77042-3051, USA

P.O. Box 42806, Houston, Texas 77242, USAPhone:713-839-2000, FAX: 713-839-2015Internet: www.halliburton.com/landmark

Trademarks3D Drill View, 3D Drill View KM, 3D Surveillance, 3DFS, 3DView, Active Field Surveillance, Active Reservoir Surveillance, Adaptive Mesh Refining, ADC, Advanced Data Transfer, Analysis Model Layering, ARIES, ARIES DecisionSuite, Asset Data Mining, Asset Decision Solutions, Asset Development Center, Asset Development Centre, Asset Journal, Asset Performance, AssetConnect, AssetConnect Enterprise, AssetConnect Enterprise Express, AssetConnect Expert, AssetDirector, AssetJournal, AssetLink, AssetLink Advisor, AssetLink Director, AssetLink Observer, AssetObserver, AssetObserver Advisor, AssetOptimizer, AssetPlanner, AssetPredictor, AssetSolver, AssetSolver Online, AssetView, AssetView 2D, AssetView 3D, BLITZPAK, CasingLife, CasingSeat, CDS Connect, Channel Trim, COMPASS, Contract Generation, Corporate Data Archiver, Corporate Data Store, Data Analyzer, DataManager, DataStar, DBPlot, Decision Management System, DecisionSpace, DecisionSpace 3D Drill View KM, DecisionSpace AssetLink, DecisionSpace AssetPlanner, DecisionSpace AssetSolver, DecisionSpace Atomic Meshing, DecisionSpace Nexus, DecisionSpace Reservoir, DecisionSuite, Deeper Knowledge. Broader Understanding., Depth Team, Depth Team Explorer, Depth Team Express, Depth Team Extreme, Depth Team Interpreter, DepthTeam, DepthTeam Explorer, DepthTeam Express, DepthTeam Extreme, DepthTeam Interpreter, Desktop Navigator, DESKTOP-PVT, DESKTOP-VIP, DEX, DIMS, Discovery, Discovery Asset, Discovery Framebuilder, Discovery PowerStation, DMS, Drillability Suite, Drilling Desktop, DrillModel, Drill-to-the-Earth-Model, Drillworks, Drillworks ConnectML,DSS, Dynamic Reservoir Management, Dynamic Surveillance System, EarthCube, EDM, EDM AutoSyn, EDT, eLandmark, Engineer’s Data Model, Engineer’s Desktop, Engineer’s Link, ESP, Event Similarity Prediction, ezFault, ezModel, ezSurface, ezTracker, ezTracker2D, FastTrack, Field Scenario Planner, FieldPlan, For Production, FZAP!, GeoAtlas, GeoDataLoad, GeoGraphix, GeoGraphix Exploration System, GeoLink, Geometric Kernel, GeoProbe, GeoProbe GF DataServer, GeoSmith, GES, GES97, GESXplorer, GMAplus, GMI Imager, Grid3D, GRIDGENR, H. Clean, Handheld Field Operator, HHFO, High Science Simplified, Horizon Generation, i WellFile, I2 Enterprise, iDIMS, Infrastructure, Iso Core, IsoMap, iWellFile, KnowledgeSource, Landmark (as a service), Landmark (as software), Landmark Decision Center, Landmark Logo and Design, Landscape, Large Model, Lattix, LeaseMap, LogEdit, LogM, LogPrep, Magic Earth, Make Great Decisions, MathPack, MDS Connect, MicroTopology, MIMIC, MIMIC+, Model Builder, Nexus, Nexus, Nexus View, Object MP, OpenBooks, OpenJournal, OpenSGM, OpenVision, OpenWells, OpenWire, OpenWire Client, OpenWire Direct, OpenWire Server, OpenWorks, OpenWorks Development Kit, OpenWorks Production, OpenWorks Well File, PAL, Parallel-VIP, Parametric Modeling, PetroBank, PetroBank Explorer, PetroBank Master Data Store, PetroStor, PetroWorks, PetroWorks Asset, PetroWorks Pro, PetroWorks ULTRA, PlotView, Point Gridding Plus, Pointing Dispatcher, PostStack, PostStack ESP, PostStack Family, Power Interpretation, PowerCalculator, PowerExplorer, PowerExplorer Connect, PowerGrid, PowerHub, PowerModel, PowerView, PrecisionTarget, Presgraf, Pressworks, PRIZM, Production, Production Asset Manager, PROFILE, Project Administrator, ProMAGIC, ProMAGIC Connect, ProMAGIC Server, ProMAX, ProMAX 2D, ProMax 3D, ProMAX 3DPSDM, ProMAX 4D, ProMAX Family, ProMAX MVA, ProMAX VSP, pSTAx, Query Builder, Quick, Quick+, QUICKDIF, Quickwell, Quickwell+, QUIKRAY, QUIKSHOT, QUIKVSP, RAVE, RAYMAP, RAYMAP+, Real Freedom, Real Time Asset Management Center, Real Time Decision Center, Real Time Operations Center, Real Time Production Surveillance, Real Time Surveillance, Real-time View, Reference Data Manager, Reservoir, Reservoir Framework Builder, RESev, ResMap, RTOC, SCAN, SeisCube, SeisMap, SeisModel, SeisSpace, SeisVision, SeisWell, SeisWorks, SeisWorks 2D, SeisWorks 3D, SeisWorks PowerCalculator, SeisWorks PowerJournal, SeisWorks PowerSection, SeisWorks PowerView, SeisXchange, Semblance Computation and Analysis, Sierra Family, SigmaView, SimConnect, SimConvert, SimDataStudio, SimResults, SimResults+, SimResults+3D, SIVA+, SLAM, SmartFlow, smartSECTION, Spatializer, SpecDecomp, StrataAmp, StrataMap, StrataModel, StrataSim, StratWorks, StratWorks 3D, StreamCalc, StressCheck, STRUCT, Structure Cube, Surf & Connect, SynTool, System Start for Servers, SystemStart, SystemStart for Clients, SystemStart for Servers, SystemStart for Storage, Tanks & Tubes, TDQ, Team Workspace, TERAS, T-Grid, The Engineer’s DeskTop, Total Drilling Performance, TOW/cs, TOW/cs Revenue Interface, TracPlanner, TracPlanner Xpress, Trend Form Gridding, Trimmed Grid, Turbo Synthetics, VESPA, VESPA+, VIP, VIP-COMP, VIP-CORE, VIPDataStudio, VIP-DUAL, VIP-ENCORE, VIP-EXECUTIVE, VIP-Local Grid Refinement, VIP-THERM, WavX, Web Editor, Well Cost, Well H. Clean, Well Seismic Fusion, Wellbase, Wellbore Planner, Wellbore Planner Connect, WELLCAT, WELLPLAN, WellSolver, WellXchange, WOW, Xsection, You’re in Control. Experience the difference, ZAP!, and Z-MAP Plus

are trademarks registered trademarks or service marks of Halliburton.

All other trademarks, service marks and product or service names are the trademarks or names of their respective owners.

NoteThe information contained in this document is subject to change without notice and should not be construed as a commitment by Halliburton. Halliburton assumes no responsibility for any error that may appear in this manual. Some states or jurisdictions do not allow disclaimer of expressed or implied warranties in certain transactions; therefore, this statement may not apply to you.

Halliburton acknowledges that certain third party code has been bundled with, or embedded in, its software. The licensors of this third party code, and the terms and conditions of their respective licenses, may be found at the following location:

..\Nexus-Nexus-VIP5000.0.2\help\com\lgc\dspx\ThirdParty.pdf

Landmark Nexus® Technical Reference Manual

Table of Contents

Chapter 1: Parallel ProcessingParallel Solver Implementation ................................................................................. 1-13

ILUT, Gauss-Seidel, and Half-Fast ..................................................................... 1-13AMG, AMG-RS ................................................................................................... 1-15ILU Full System Preconditioner .......................................................................... 1-16

Chapter 2: SolverIntroduction................................................................................................................ 2-21Overview.................................................................................................................... 2-22Red-Black Reduction................................................................................................. 2-23Network Coupling and Solution ................................................................................ 2-26CPR............................................................................................................................ 2-27Pressure Solution ....................................................................................................... 2-28

Incomplete LU factorization ................................................................................ 2-28Algebraic Multigrid ............................................................................................. 2-29

Full System Preconditioner........................................................................................ 2-30Multiple Subgrids and Parallel Processing ................................................................ 2-31Tolerances .................................................................................................................. 2-32Miscellaneous ............................................................................................................ 2-33

Direct Solution ..................................................................................................... 2-33Action on Solver Failure ................................................................................ 2-33Dual Porosity Feature .................................................................................... 2-33

Chapter 3: PVT RepresentationIntroduction................................................................................................................ 3-35Water Properties ........................................................................................................ 3-36Black Oil Models ....................................................................................................... 3-37

Standard Black Oil Calculations .......................................................................... 3-37The Physical Model ....................................................................................... 3-37The Nexus Internal Representation ................................................................ 3-40The Nexus Black Oil Flash ............................................................................ 3-41Nexus Volumes and Volume Derivatives ...................................................... 3-42

Enhanced Black Oil Calculations ........................................................................ 3-44The Physical Model ....................................................................................... 3-44The Nexus Internal Representation ................................................................ 3-45The Nexus Enhanced Black Oil Flash ........................................................... 3-46Nexus Volumes and Volume Derivatives ...................................................... 3-48

Conversion from VIP Black Oil Model ............................................................... 3-50Calculation of Surface Volumes .......................................................................... 3-57The API Interpolation Model ............................................................................... 3-58The Water-Oil Model and the Gas-Water Model ................................................ 3-59

R5000.0.2 Table of Contents v

Nexus® Technical Reference Manual Landmark

Equation of State (EOS) Model ................................................................................. 3-60A General EOS Model ......................................................................................... 3-60Calculation of Z-factors and Molar Density ........................................................ 3-62Calculation of Fugacity Coefficient and Fugacities ............................................ 3-64Other Thermodynamic Properties ........................................................................ 3-65Hydrocarbon Phase Viscosities ........................................................................... 3-66

Lohrenz-Bray-Clark Correlation .................................................................... 3-66Pedersen Correlation ...................................................................................... 3-68

Principal Algorithms ............................................................................................ 3-72Two Isothermal Phase Flash Calculation ....................................................... 3-72Saturation Pressure Calculation ..................................................................... 3-78Gibbs Stability Analysis ................................................................................ 3-78

The Search for a Two-Phase State ....................................................................... 3-81Transition Test Technique ............................................................................. 3-81

Single Phase Identification .................................................................................. 3-88Surface Separator Calculations ............................................................................ 3-89

Multi-stage Separator ..................................................................................... 3-90Standard Conditions ....................................................................................... 3-93Standing-Katz Correlation ............................................................................. 3-95Gas Plants .................................................................................................... 3-100

Calculation of Volume Derivatives ................................................................... 3-102Three Phase Equilibrium ......................................................................................... 3-104

Calculation of Water Properties ......................................................................... 3-104Conversion of Solubility Data to Fugacity Coefficient Data ............................. 3-105Flash Procedure .................................................................................................. 3-106

References................................................................................................................ 3-107

Chapter 4: Network CalculationsIntroduction.............................................................................................................. 4-109Network Solution Caused By Procedures................................................................ 4-111Network Solutions Caused By Targeting ................................................................ 4-112Simultaneous Reservoir/Network Solution ............................................................. 4-113Targeting .................................................................................................................. 4-115

Targeting Input .................................................................................................. 4-116Target Rate Calculations .................................................................................... 4-118Net Voidage/Fillup Algorithm ........................................................................... 4-120Pressure Maintenance Algorithm ....................................................................... 4-121Control of the target rate .................................................................................... 4-123Tree Structured Guide Rates .............................................................................. 4-126Multiple Targets and the Order of Target Calculations ..................................... 4-128

Remote Control of Constraints ................................................................................ 4-130Hydraulic Calculations ............................................................................................ 4-133

General Description ........................................................................................... 4-133Hydrostatic Gradient .......................................................................................... 4-133Hydraulic Tables ................................................................................................ 4-135

vi Table of Contents R5000.0.2


Hydraulic Correlations ....................................................................................... 4-138Use of Correlations ...................................................................................... 4-138Model Description ....................................................................................... 4-140Algorithm of Pressure Drop Calculations .................................................... 4-142

Pump Model ....................................................................................................... 4-144Calculation Method ...................................................................................... 4-145

Valve Model ...................................................................................................... 4-147Solution Algorithm ...................................................................................... 4-148

Choke Model ...................................................................................................... 4-149Wells ........................................................................................................................ 4-151

Gridded Wells .................................................................................................... 4-152Lumped Wells .................................................................................................... 4-155Branched Wells .................................................................................................. 4-157Complex Wells .................................................................................................. 4-159Connecting Wells to the Network ...................................................................... 4-163Well Constraints ................................................................................................ 4-164

Treatment of Datum Depth .......................................................................... 4-165Crossflow ........................................................................................................... 4-166Perforation Equations ........................................................................................ 4-167

WAG (Water-Alternating-Gas) Injection ................................................................ 4-169Separation Nodes ..................................................................................................... 4-172Treatment of ONTIME factors ................................................................................ 4-176Network Heat Loss Calculations ............................................................................. 4-177Drilling Queues........................................................................................................ 4-179

The Drill Procedure ..................................................................................... 4-179The Redrill Procedure .................................................................................. 4-181

Gaslift Optimization ................................................................................................ 4-184Optimal Gaslift Table Method ........................................................................... 4-185Optimization Using a Nonlinear Optimizer ....................................................... 4-186

Network PVT Data .................................................................................................. 4-191PVT and Separator Method Assignments .......................................................... 4-191

Multifield networks with Mixed PVT ..................................................................... 4-192Producing Perforations ...................................................................................... 4-192Injecting Perforations ......................................................................................... 4-194IMPES Elimination and Elimination of Water Mass for Implicit Runs ............ 4-194Reservoir Equations ........................................................................................... 4-194Mixing of Fluid in the Network ......................................................................... 4-195

Fuel and Shrinkage .................................................................................................. 4-196Calculation of Connection Length and Depth Change ............................................ 4-197

Depth Change (DDEPTH) ................................................................................. 4-197Connection Length (CON_LENGTH) ............................................................... 4-198Consistency Checks ........................................................................................... 4-198

Network Reports ...................................................................................................... 4-199References................................................................................................................ 4-200

R5000.0.2 Table of Contents vii


Chapter 5: Well DataCalculation of the Wellbore Constant for Each Perforation .................................... 5-201Effective Grid Block Radius RADB........................................................................ 5-203

One-Dimensional Flow ...................................................................................... 5-203Linear Case .................................................................................................. 5-203

Two-Dimensional Flow ..................................................................................... 5-205Well in Center of Square Gridblock ............................................................ 5-205General Definition of rb ............................................................................... 5-207An Approximate Derivation of rb for Square Well Block ........................... 5-208Well in Center of Rectangular Gridblock .................................................... 5-210Well in Center of Block in Anisotropic Rectangular Grid .......................... 5-212Single Well Arbitrarily Located in Isolated Well Block ............................. 5-215Multiple Wells in Same Isolated Well Block .............................................. 5-216Two Wells With Same Rate in Adjacent Blocks ......................................... 5-218Single Well in Edge Block ........................................................................... 5-220Single Well Exactly on Edge of Grid .......................................................... 5-221Single Well Exactly at Corner of Grid ......................................................... 5-222Single Well Arbitrarily Located in Corner Block ........................................ 5-224Horizontal Well ............................................................................................ 5-225

Inclined Well ...................................................................................................... 5-227Incorporating Skin into Well Model ........................................................................ 5-232

Derivation of Skin Due to Altered Permeability ............................................... 5-232Including Mechanical Skin In Well Index ......................................................... 5-233Skin Due to Restricted Entry ............................................................................. 5-235Effects of Restricted Entry on Well Index ......................................................... 5-238

References................................................................................................................ 5-240

Chapter 6: EquilibriumIntroduction.............................................................................................................. 6-241Mainstream Path ...................................................................................................... 6-242Supercritical Initialization........................................................................................ 6-252Gibbs Sedimentation................................................................................................ 6-254Search for GOC ....................................................................................................... 6-256

Chapter 7: CompactionIntroduction.............................................................................................................. 7-257Pore Volume Adjustment......................................................................................... 7-258

Reversible Cases ................................................................................................ 7-258Irreversible Cases ............................................................................................... 7-259Water-Induced Compaction Modeling .............................................................. 7-260

Transmissibility and Well KH Adjustment ............................................................. 7-262Water-Induced Compaction Table Effects ......................................................... 7-263

Implicitness of Calculations..................................................................................... 7-265

viii Table of Contents R5000.0.2


Chapter 8: Relative Permeability MethodsIntroduction.............................................................................................................. 8-267Three-Phase Oil Relative Permeability Models....................................................... 8-268

Stone's Model I .................................................................................................. 8-268Stone's Model II ................................................................................................. 8-269Saturation Weighted Interpolation Model ......................................................... 8-269Guidelines for Selecting the Models .................................................................. 8-272

Hysteresis................................................................................................................. 8-273Introduction ........................................................................................................ 8-273Background ........................................................................................................ 8-273Linear Method .................................................................................................... 8-277Scaling of Primary Drainage Curve .................................................................. 8-278Carlson Method .................................................................................................. 8-279User-Supplied Bounding Method (Non-Wetting Hysteresis) ............................ 8-280User-Supplied Bounding Method (Wetting Hysteresis) .................................... 8-282

End-Point Scaling .................................................................................................... 8-284Introduction ........................................................................................................ 8-284Normalized Saturation End Points ..................................................................... 8-284Rules for Missing or Incomplete End-Point Scaling Arrays and Data Errors ... 8-287Two-Point Scaling Option ................................................................................. 8-288

Two-Point Scaling of Capillary Pressure Functions .................................... 8-288Two-Point Scaling of Relative Permeability Functions .............................. 8-289

Three-Point Scaling Option (Default Option) ................................................... 8-290Four-Point Scaling of Capillary Pressure Functions .................................... 8-290Three-Point Scaling of Relative Permeability Functions ............................ 8-290

Normalized End-Point Relative Permeability.......................................................... 8-291End-Point Scaling for Oil (Two-Point Scaling) ................................................. 8-292End-Point Scaling for Water and Gas (Two-Point Scaling) .............................. 8-293End-Point for Oil (Three-Point Scaling) ............................................................ 8-294End-Point Scaling for Water and Gas (Three-Point Scaling) ............................ 8-295End-Point Scaling for Gas-Water Cases (Two-Point Scaling) .......................... 8-296Data Defaulting and Errors Applied to End-Point Relative Permeability ......... 8-297

Three-Point Scaling Defaulting Rules ......................................................... 8-297Three-Point Scaling Defaulting Rules (Gas-Water) .................................... 8-300

Directional Relative Permeability............................................................................ 8-302IFT Adjustment of Capillary Pressure and Gas/Oil Relative Permeability ............. 8-303References................................................................................................................ 8-306

Chapter 9: TracersTracer Calculations .................................................................................................. 9-307

Chapter 10: GridIntroduction............................................................................................................ 10-309Unstructured Grid Form......................................................................................... 10-310

R5000.0.2 Table of Contents ix


Map Output ...................................................................................................... 10-310Structured Grid Form............................................................................................. 10-311

Grid Geometry Data ......................................................................................... 10-311Fault Specification ........................................................................................... 10-311Calculations of Cell Bulk Volume and Center Depth ...................................... 10-312Calculations of Cell Dimensions ..................................................................... 10-312Calculations of Intercell Transmissibility ........................................................ 10-313

Standard Transmissibility Option .............................................................. 10-313NEWTRAN Transmissibility Option ........................................................ 10-314

Local Grid Refinement (LGR) ......................................................................... 10-316LGR Keyword Parameters ............................................................................... 10-317Propagation of Reservoir Properties to LGR Cells .......................................... 10-318Grid Coarsening ............................................................................................... 10-318Coarse Cell Properties ..................................................................................... 10-318

Chapter 11: ProceduresIntroduction............................................................................................................ 11-323Procedure Functions .............................................................................................. 11-325Debugging Procedures ........................................................................................... 11-328

Chapter 12: AquifersIntroduction............................................................................................................ 12-329Analytic Model — The Carter-Tracy Aquifer ....................................................... 12-330Analytical Model - The Fetkovich Aquifer ........................................................... 12-334Numerical Aquifer ................................................................................................. 12-336References.............................................................................................................. 12-337

Appendix A: Corner-Point GeometryMapping of Gridblock to Unit Cube....................................................................... A-339

Two-Dimensional Mapping .............................................................................. A-339Three-Dimensional Mapping ............................................................................ A-341Volumetric Calculations ................................................................................... A-343Integration by Gaussian Quadrature ................................................................. A-346Transmissibility Calculations ........................................................................... A-348Calculation of Transmissibility in 2D ............................................................... A-348

Simplest Case: Orthogonal Grid ................................................................. A-348Non-Orthogonal Grid with Parallel Sides ................................................... A-349Non-Orthogonal Grid with Non-Parallel Sides .......................................... A-350Alternate Approach for Gridblock with Non-Parallel Sides ....................... A-354

Calculation of Transmissibility in 3D (HARINT) ............................................ A-356Tubes and Slices in 3D ............................................................................... A-356Differential Area of the Slice ...................................................................... A-357Location of Centroid, in Real Space, and in the Unit Cube ........................ A-357

x Table of Contents R5000.0.2


Evaluation of Arc Length Derivative .......................................................... A-358Evaluation of cos (u,v,w) ........................................................................ A-358Half-Block Transmissibility to Other Faces ............................................... A-361

Calculation of Transmissibility in 3D by the NEWTRAN Option ................... A-361Half-Block Transmissibility to All Six Faces ............................................. A-363

Calculation of Full Transmissibility Between Gridblocks ............................... A-363Unfaulted Case ............................................................................................ A-363Faulted Case ................................................................................................ A-364

Choice Between HARINT and NEWTRAN Options ....................................... A-365References............................................................................................................... A-370

R5000.0.2 Table of Contents xi


xii Table of Contents R5000.0.2


Chapter 1

Parallel Processing

Parallel Solver Implementation

In Nexus, we provide parallel ILUT, AMG and AMG-RS solvers as the pressure preconditioner and Gauss-Seidel, Half-Fast, and ILU solvers as the full system preconditioner. All of these parallel solvers are based on domain decomposition.

ILUT, Gauss-Seidel, and Half-Fast

These three parallel solvers require grid coloring. The basic requirement for grid coloring is that no two adjacent grids (i.e., grids having connections to each other) have the same color. Each process loads multiple grids with different colors, and each color should be present on all processes. Usually the number of colors is very small in practice. As shown in Figure 1.1, it is a simple example of two-dimensional grid decomposition. Four grids are colored in two colors, and then grid 1 and grid 2 are loaded to process 1 and the remaining two grids to process 2. The coefficient matrix is reordered according as shown in Figure 1.2. All operations proceed according to the grid colors. For example, firstly grid 1 and 4 are handled simultaneously on process 1 and 2, respectively, and initiate cross-process message passing, and then treat grids 2 and 3 with local information and finally update grid 2 and 3 using cross-process information. The cross-process communication is overlapped with local computations.

R5000.0.2 Parallel Processing: Parallel Solver Implementation 13


Figure 1.1: Example of Grid Coloring

As a very effective iterative solver, the threshold-based incomplete LU factorization (ILUT) determines the potential in-fills based on the matrix entry values instead of solely on structure of the matrix. But the lower and upper triangular factors (L and U) of the coefficient matrix have to be created dynamically, which is formidably expensive in parallelization. Thus a phase of pre-factorization is designed. In this phase, all the nodes are ordered, and the potential in-fill patterns are determined and stored in a template using the coefficient matrix at a certain time step. This template is updated as necessary. Obviously performance of the solver is compromised a little bit to save cost on parallelization. Then the factorization is pretty much like the level-based incomplete factorization.

14 Parallel Processing: Parallel Solver Implementation R5000.0.2


Figure 1.2: Structure of a Reordered Coefficient Matrix

AMG, AMG-RS

The aggregation type of AMG solver1 differs from AMG-RS solver2 only in the generation of grid hierarchy as described in the chapter on the Solver. The difficulty in parallelizing AMG solver lies in the grid coarsening, which is sequential in nature. There has been extensive research on parallel AMG solver. We adopted both the decoupled and coupled aggregation approaches for the aggregation-type AMG solver 3. In the decoupled aggregation method, each process defines the aggregates on its local grid without considering the cross-boundary dependency between boundary nodes. This approach avoids a lot of message passing in the setup phase, and in both restriction and prolongation calculations in the solution phase. In the coupled aggregation approach, first all boundary nodes are processed and aggregates information is sent to adjacent processes. Then each process defines its local interior aggregates in parallel. If there is conflict on definition of boundary aggregates between two adjacent processes, the process with a smaller grid size prevails.

The grid coarsening procedure of AMG-RS includes three steps. Our approach is similar to RS3 in Henson and Yang's work4. In the first step, each process defines local C/F splitting on its own grid ignoring cross-boundary connections, and then initiates message passing of



boundary C/F splitting information to adjacent processes. In the second step, the interpolation weight for interior nodes and C boundary nodes is calculated. The final step checks the dependence between the boundary and ghost nodes and adjusts the C/F splitting to avoid strong cross-boundary F-F dependence. In the current implementation, the cross-boundary F-F dependence is eliminated by making one of F nodes to C node. To preventing from penetrating too deep into external grids on the coarser level, all boundary nodes must be interpolated by direct connecting neighboring C nodes, and only the strongest cross-boundary dependence is interpolated.

Parallelizing the solution phase is straightforward. The major components in this phase are update of residual, restriction of residual, pre-smoothing, prolongation of solution and post-smoothing. We use block Jacobi smoothing, in which point Gauss-Seidel or ILU(0) is applied to each grid with old values on cross-process nodes used. A sparse direct solver is applied on the coarsest grid level. Communication involved in calculation and restriction of residuals and prolongation of solution is readily overlapped with local computation.

ILU Full System Preconditioner

A parallel ILU solver is implemented as a preconditioner for the full system5. The algorithm for incomplete factorization is:

1. Divide all nodes into interior nodes (I) and boundary nodes (C).

2. Color the boundary nodes connected to multiple external grids. If more than two colors are needed, all colors but color 1 and 2 are made color 3. Local connections among boundary nodes are not included in the coloring procedure.

3. Color all other boundary nodes with color 1 and color 2. Only the cross-grid connections between boundary nodes are included for coloring. If necessary, color 3 is used to break the chain connections and switch colors from one side to the other on the boundary. The requirement is that the number of nodes with those two colors should be close and the number of breaking nodes with color 3 is as small as possible.

4. The breaking nodes with color 3 are accessible to both sides of the boundary. This can be achieved by message passing.

5. When message passing is initiated, all the local interior nodes are to be factored. This part is CPU time intensive.



6. Then factor local boundary nodes with color 1.

7. Non-blocking receive upper factorization coefficients related to color 1 nodes and those interior coefficients related to boundary color 3 nodes from adjacent processor.

8. Update local color 2 and 3 boundary nodes using local interior connections.

9. Non-blocking send upper factorization coefficients related to color 1 nodes and those interior coefficients related to boundary color 3 nodes to adjacent processor.

10. Factor color 2 and color 3 nodes with coefficients sent from color 1 nodes.

Message passing is required in the factorization (Steps 5, 7 and 9), and fortunately it can easily be overlapped with the local computation.

Figure 1.3 shows a re-ordered matrix for two processors. Each processor has only one grid. I and C represent the interior nodes and boundary nodes, respectively. As stated earlier, C3 might be empty.

The forward substitution proceeds as follows:

1. Initiate message passing of right hand side (RHS) of C3 nodes if exist.

2. Forward solution of local interior nodes (I) on each process.

3. Forward solution of local C1 nodes on each process.

4. No-blocking receive solution of C1 nodes and I-C3 nodes from adjacent process;

5. No-blocking send solution of C1 nodes and I-C3 nodes to adjacent process;

6. Update C2 and C3 nodes due to local I and C1 solutions;

7. Update C2/C3 nodes due to remote I and C1 solutions;

8. Forward solution of C2/C3 nodes on each process;

The backward substitution proceeds as follows:

1. Backward solution of C2 nodes on each process;



2. No-blocking receive solution of C2 nodes from adjacent process;

3. No-blocking send solution of C2 nodes to adjacent process;

4. Update C3/C1/I nodes due to local C2 solutions;

5. Update C3/C1/I nodes due to remote C2 solutions;

6. Backward solution of C3 nodes on each processor;

7. Backward solution of C1 nodes on each process;

8. Backward solution of I nodes on each process;

In the algorithm, I-C3 denotes the interior nodes that have connections to local C3 nodes if C3 nodes exist in this grid.

Figure 1.3: Incomplete Factorization of Coefficient Matrixwith Re-ordered Boundary Nodes

1. P. Vanek, J. Mandel, and M. Brezina, Algebraic multigrid based on smoothed aggregation for second arnd fourth order problems, Computing, Vol. 56, pp. 179-196, 1996.



2. K. Stuben, An introduction to algebraic multigrid, in Multigrid, authors: U. Trottenberg, C. Oosterlee, A. Schuller, Elsevier Ltd, 2001.

3. C. H. Tong and R. S. Tuminaro, Parallel soothed aggregation multigrid: aggregation strategies on massively parallel machines, Proceedings of Supercomputing, Nov. 2000.

4. V. Henson and U. Yang, BoomerAMG, A parallel algebraic multigrid solver and preconditioner, Applied Numerical Mathematics, Vol. 41, pp. 155-177, 2002.

5. Q. Wang and J.W. Watts, A parallel incomplete LU factorization algorithm for distributed linear system.

6. Amdahl, G.M.. Validity of single-processor approach to achieving large-scale computing capability, Proceedings of AFIPS Conference, Reston, VA. 1967. pp. 483-485.



Chapter 2

Solver

Introduction

At each Newton iteration, it is necessary to solve a matrix equation. Depending on the number of cells used to represent the reservoir, the reservoir part of this equation must be solved iteratively. For the simulation to be successful, this iteration must converge in a reasonable number of iterations.

In general, the overall convergence rate will be limited by convergence of a particular part of the overall process. Potentially limiting the convergence rate are the following

• Convergence problems related to the coupling between the facility network and the reservoir.

• In implicit computations, convergence of the CPR computation.

• Convergence of the reservoir pressure solution.

• In multiple subgrid runs, which typically use multiple processors, convergence problems related to coupling between neighboring subgrids.

In selecting solver parameters, we are generally attempting to deal with issues related to the potential problem areas listed above. The following is intended to help in doing so.

R5000.0.2 Solver: Introduction 21


Overview

As noted above, at each Newton iteration it is necessary to solve a global matrix equation. It takes the following form.

2-1

Ann and Anr contain the network equation coefficients that multiply network and reservoir unknowns, respectively. Arn and Arr contain the reservoir equation coefficients that multiply these same unknowns. xn and xr are the network and reservoir unknown vectors, respectively. The network unknowns are pressures and compositions at network nodes and total component mass rates on network connections, including perforations. The reservoir unknowns are cell pressures if IMPES is being used and cell pressures and component masses if the calculations are fully implicit. and are the initial residual vectors.

We normally solve Equation 2-1 iteratively. Before beginning the iteration, we perform a network solve and update the reservoir residuals. The network is solved exactly, within roundoff error, so we assume that . That brings us to the state of Equation 2-1. We begin each iteration with a reservoir solve, which generally is only approximate.

2-2

This solve can be performed in several ways, which are discussed below. As the reservoir solve proceeds, we continually update the reservoir residuals. At the end of the solve, they are current with the reservoir solution, which means that in effect the following computation has been performed.

2-3

We then compute the network residuals resulting from the change in reservoir solution, recalling that the network residuals were initially zero.

A A

A A

x

x

r

rnn nr

rn rr

n

r

n

r

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥ =

⎡

⎣⎢

⎤

⎦⎥

0

0

rn0 rr

0

rn0 0=

A x rrr rk

rk + =1

r r A xrk

rk

rr rk+ += −1 2 1/

22 Solver: Overview R5000.0.2


2-4

Then we solve the network equation.

2-5

Following this, the network residuals are again zero. We update the reservoir residuals.

2-6

Red-Black Reduction

To decrease the cost of solving Equation 2-2, we perform a red-black reduction of the reservoir grid. In a rectangular structured grid, this eliminates half of the reservoir unknowns and yields a set of equations that is often easier to solve iteratively. Whether the reduction is done is controlled by the keyword SYSTEM_REDUCED ON or OFF. The default is ON. The option not to use the reduction is provided because sometimes the AMG pressure solver works much better on the original matrix than on the reduced matrix. The line Gauss-Seidel full system preconditioner also works better on the original matrix.

On a structured grid, the reduction results in a checkerboard pattern, and we refer to the process as coloring the cells, with the colors being red and black. We determine the cell colors as follows.

1. Designate all cells to be uncolored.

2. Designate the first cell as red. This creates the first group of red cells.

3. Find all uncolored cells connected to the latest group of red cells. Designate these to be black.

4. Find all uncolored cells connected to the latest group of black cells (from Step 3). These are candidates to be red. Designate the first one to be red. Cell by cell within this group, determine whether each cell is connected to any red cell. If it is not, designate it to be

r A xnk

nr nk+ += −1 2 1/

A x rnn nk

nk + +=1 1 2/

r r A xrk

rk

rn nk+ + += −1 1 2 1/

R5000.0.2 Solver: Red-Black Reduction 23


red. If it is connected to a red cell, designate it to be black. The red cells identified in this step become the next group of red cells.

5. Repeat steps 3 and 4 until no connected uncolored cells can be found.

6. Search all cells for an uncolored cell. If one is found, designate it to be red, and return to Step 3.

On a structured grid, this procedure yields the normal red-black checkerboard pattern. The procedure results in two lists, one containing the red cells and the other containing the black cells. Given these lists, we can partition Arr and rewrite Equation 2-1 as follows.

2-7

The R and B subscripts denote red and black cells, respectively. ARR is diagonal for the IMPES pressure equation and block diagonal for the fully implicit set of equations. As a result, it can be inverted economically.

We next eliminate the red unknowns from Equation 2-7, yielding the reduced matrix equation, which is of the form

2-8

where

2-9

2-10

2-11

A A A

A A A

A A A

x

x

x

r

rRR Rn RB

nR nn nB

BR Bn BB

R

n

B

R⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=

0

nn

Br

0

0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

A A

A A

x

x

r

rnn nB

Bn BB

n

B

n

B

⎡

⎣⎢

⎤

⎦⎥

⎡

⎣⎢

⎤

⎦⎥ =

⎡

⎣⎢

⎤

⎦⎥

0

0

A A A A Ann nn nR RR Rn= − −1

A A A A ABn Bn BR RR Rn= − −1

A A A A AnB nB nR RR RB= − −1

24 Solver: Red-Black Reduction R5000.0.2


2-12

2-13

2-14

We actually solve the reduced matrix equation, Equation 2-8, rather than the original global matrix equation, Equation 2-1. We obtain xR by backsolving.

A A A A ABB BB BR RR RB= − −1

r r A A rn n nR RR R0 0 1 0= − −

r r A A rB B BR RR R0 0 1 0= − −

R5000.0.2 Solver: Red-Black Reduction 25


Network Coupling and Solution

To solve Equation 2-8, we use what the mathematicians call a multiplicative Schwartz procedure. We solve for the reservoir unknowns, update the network equations based on the new reservoir solution, solve the network equations, update the reservoir equations based on the new network solution, solve the reservoir equations, etc. We continue this process until the solution is deemed to be accurate enough.

Generally, this procedure is adequate. At times, however, it is necessary to use more complex equations. To do this, we “extend” the network into the reservoir. That is, we add some of the reservoir unknowns and accompanying equations in the network equation set, while also retaining them in the reservoir set. The result is an overlapping Schwartz procedure, where the overlap region consists of the reservoir cells included in the network set. We can extend the network one to five cells using the keyword FACILITIES EXTENDED n, where n is a value in the range 1-5. When the extended network is used, it is also necessary to specify IMPLICIT_COUPLING NONE for implicit grids or IMPES_COUPLING NONE for IMPES grids.

The meaning of n is as follows. If n = 1, the network set includes the normal network plus the cells containing the perforations. If n = 2, the network set includes the n = 1 set plus the perforated cells’ neighboring cells. The n = 3 set includes the n = 2 set plus the perforated cells’ neighbors’ neighbors, and so on for n = 4 and 5.

In practice, we have never found it best to use n larger than 2. In part this is because as n increases, the cost of the network solve increases and can become quite expensive. It may be possible to reduce this expense by performing the network solve iteratively by specifying FACILITIES ITERATIVE.

Another choice relating to the network-reservoir coupling is the IMPLICIT_COUPLING or IMPES_COUPLING. The default for these is ROWSUM. Other possibilities are COLSUM and NONE. We don’t know of a way to tell in advance which of these will be best, but recent experience has shown that sometimes using COLSUM reduces cpu time. As noted above, NONE should be used if the facilities are extended.

26 Solver: Network Coupling and Solution R5000.0.2


CPR

CPR stands for Constrained Pressure Residual. As we use the term, it refers to a method in which an implicit model is solved iteratively, with each iteration consisting of

1. A fairly accurate pressure solution

2. An inexpensive and not very accurate solution of the full set of equations

There are two ways to generate the pressure equation, selected by specifying CPR_PRESSURE_EQUATION NORMAL or IMPES. If CPR_PRESSURE_EQUATION NONE is specified, then no pressure solution is used.

The idea behind the method is as follows. Pressure changes can be transmitted rapidly over large distances. As a result, the convergence rate of the iterative solution is often controlled by how fast pressures converge. At the same time, solving for the full set of unknowns greatly increases the cost of an iteration. If we solve at the same time for all unknowns, we get the worst of both worlds – a number of iterations associated with a pressure solution at the higher cost for each iteration associated with solving for all unknowns. CPR allows us to determine an improved pressure solution at a cost associated with pressure only and then to use this to improve the solution for the other unknowns with a relatively inexpensive calculation.

The default pressure equation is NORMAL, and this generally works at least reasonably well. There are cases, typically associated with the large transmissibilities and small pore volumes of fractures, where the IMPES pressure equation is much better. Only in very rare cases is the omission of the pressure equation (CPR_PRESSURE_EQUATION NONE) best.

CPR requires a pressure solution and an iterative method for the full unknown set. The next two sections discuss the methods used for these.

R5000.0.2 Solver: CPR 27


Pressure Solution

If the reservoir equations are IMPES, we need to solve the IMPES pressure equation. If they are fully implicit, we normally use CPR, which also requires solving the pressure equation. In either event, we need to solve for pressures, and for this we have three choices of preconditioner, one incomplete factorization method and two algebraic multigrid methods. (A preconditioner computes an inexpensive, approximate solution. It is used as part of an iterative solution method. The more accurate the preconditioner’s solution is, the faster the iteration will converge.)

Incomplete LU factorization

Incomplete LU factorization is specified by SOLVER PRECON_ILU. To improve performance of the preconditioner, we do two things:

1. Reorder the unknowns and equations in a way that reduces the amount of infill. In deciding which unknown to eliminate next, we determine the infill that would be generated by each possibility, and eliminate the unknown that generates the smallest infill. Here, infill is computed as the sum of the magnitudes of all infill coefficients that would be introduced into the matrix during the elimination of this particular unknown.

2. Determine which infill coefficients to retain based on their sizes. The procedure is similar to Saad’s ILUT. We retain a possible infill coefficient if its magnitude is larger than 0.007 times the magnitude of the diagonal coefficient in the row in which it falls.

The process of reordering the unknowns and determining which potential infill to retain is quite expensive computationally. As a result, we create a template containing the information needed to perform and use the factorization. The template is constructed at the beginning of a run based on the matrix coefficients for the first Newton iteration of the first timestep. It can be updated from time to time later in the run if needed by specifying OUTPUT TEMPLATE ENDOUTPUT in the Run Control file.

28 Solver: Pressure Solution R5000.0.2


Algebraic Multigrid

The multigrid method begins with a matrix equation on the original fine grid and from this generates a hierarchy of grids and corresponding matrices on these coarse grids. A simple relaxing iteration (e.g.. Gauss-Seidel or ILU(0)) is carried out on each grid level to reduce the different wave-length error components. In Nexus, the grid hierarchy is constructed solely according to the fine matrix equation with two different approaches, i.e., aggregation and the classic Ruge-Stueben method.

In the aggregation AMG solver, the coarse grids are generated by grouping the fine elements/cells according to their connection strength, which is defined as the ratio of non-diagonal component to the main diagonal ones in the matrix. The operators on the coarse grids are obtained using the Galerkin approximation. Both the restriction and the prolongation operators are piece-wise constants. This solver can be improved by rescaling or smoothing the prolongation operator. But in our numerical experiments, smoothing does not help in most cases. The solver can be activated using SOLVER PRECON_AMG. Three types of cycle are supported, V, F, and W. Only one cycle is applied to the pressure preconditioning.

The classic Ruge-Stueben method (AMG_RS) defines a coarse/fine splitting in which nodes in the fine set are interpolated by coarse nodes directly or indirectly and nodes in the coarse set occur in the next coarse grid level. The interpolation/prolongation operators are calculated based on the finding that algebraic smooth (large scale) error varies slowly in the direction of the strong connection, and the restriction operator is taken to be the transpose of the prolongation operator. The Solver can be activated using SOLVER PRECON_AMG_RS. The user can specify the cycle type (V/F/W) and the interpolation method (standard or classic).

Both AMG solvers work very well and have shown superior performances in large problems if the pressure matrix equation is an M-matrix or close to it. Aggregation AMG is much cheaper in setup and solution, particularly as a pressure preconditioner with a coarse convergence criterion, but Ruge_Stueben AMG seems more robust with a much higher cost in setup phase.

R5000.0.2 Solver: Pressure Solution 29


Full System Preconditioner

As noted above, a preconditioner computes an inexpensive, approximate solution, and is used as part of an iterative solution method. In an implicit model, at each iteration a preconditioning (i.e., an approximate solution) of the full set of unknowns is performed. If CPR is being used, this preconditioning is part of an overall CPR iteration (or preconditioning, where CPR is a two-step preconditioner). If CPR is not being used, it is the entire preconditioning.

Nexus provides six full system preconditioners. The simplest of these is point Gauss-Seidel, which is selected by specifying IMPLICIT_PRECON GAUSS_SEIDEL. It solves one cell at a time, using the latest solution at the cell’s neighbors as boundary conditions for the solve. It tends not to work well when the transmissibilities in one direction are much larger than in the other directions. The next simplest preconditioner is selected by specifying IMPLICIT_PRECON HALF_FAST. It is an incomplete factorization method, in which only diagonal blocks are modified during the factorization process. It provides some of the advantage of an incomplete factorization while requiring less computational cost to generate the factorization and less memory to store it. The most powerful general purpose preconditioner is ILU.

The other three full system preconditioners are specialized. They can only be used on structured grids, and they are most useful when the transmissibilities in one direction are much, much larger than in the other two. These are selected by specifying X_LINE_GAUSS_SEIDEL, Y_LINE_GAUSS_SEIDEL, or Z_LINE_GAUSS_SEIDEL. They perform Gauss-Seidel by lines in the x-, y-, or z-direction, respectively. Line Gauss-Seidel solves a line at a time, using the latest solution at the cells in the neighboring lines as boundary conditions. This method works well when the transmissibilities in the direction of the lines are very large.

30 Solver: Full System Preconditioner R5000.0.2


Multiple Subgrids and Parallel Processing

The description up to this point has assumed execution on a single processor. If computations are to be performed in parallel, the reservoir is divided into grids. Depending on the solution used, only one grid may be required for each process, or more may be required. Specifically, PRECON_AMG, PRECON_AMG_RS, and IMPLICIT_PRECON ILU require only one grid per process, while PRECON_ILU and all IMPLICIT_PRECON’s other than ILU require that the grids be colored as described below. When a method is based on grid colors, effective parallelization requires that each process have at least one grid of each color; hence, there will be more than one grid on each process.

The grids are colored using a procedure similar to that described above for coloring cells. There is one difference: grids of any given color must not communicate with other grids of the same color. Thus, as necessary, a group of black grids can be subdivided into two colors. This process is repeated as needed to ensure the required non-communication condition. The idea, of course, is to perform computations in parallel on all grids of a given color, pass messages, and then perform computations on grids of the next color, etc.

Overall, the procedure is to first do a global coloring of the cells. Then, the red unknowns are eliminated as indicated by Equations 2-7 to 2-14. Then the grids are colored based on the connectivity in the matrix

and then assigned to processors.ABB

R5000.0.2 Solver: Multiple Subgrids and Parallel Processing 31


Tolerances

Iterative solutions require tolerances to use in deciding when to stop the iteration. Nexus uses as its stopping criterion a specified reduction in the L2 norm of the residual. The L2 norm is the square root of the sum of the squares. The residual is the error in the equation being solved at a given point in the solution. For example, if we are solving the matrix equation Ax = b, where A is a matrix, x is the solution vector, and b is the right-hand side vector, the residual vector after k iterations will be rk = b – Axk. The L2 norm of this residual vector will be

As mentioned above, the tolerance is specified as a reduction in the residual L2 norm from its value at the beginning of the solution.

Nexus uses two such tolerances. The GLOBAL parameter GLOBALTOL is the overall convergence criterion for the coupled network-reservoir system. It is used regardless of whether the run uses IMPES or implicit calculations or one or more grids.

If the global tolerance is too loose, the Newton iteration in an implicit solution may converge slowly or not at all. Thus, on rare occasions tightening the global tolerance may reduce the number of Newton iterations. In IMPES runs, too loose a global tolerance results in reduced timestep sizes, either through repeats or timestep restriction due to large changes. Again, on rare occasions tightening the global tolerance may improve performance.

The GRIDSOLVER IMPLICIT_RED and PRESS_RED parameters are used only in single-grid implicit models. They relate to the preconditioning of the full set of unknowns (as opposed to only pressure) performed at each global solver iteration. If IMPLICIT_RED is zero (the default), the preconditioner is applied once. If IMPLICIT_RED is a specified fraction, the preconditioner is applied repeatedly until the residual L2 norm has been reduced by the specified fraction from its value before the preconditioning. At each preconditioning the required pressure residual L2 norm reduction is no smaller than PRESS_RED.

r rkik

i2

2= ( )∑

32 Solver: Tolerances R5000.0.2


Miscellaneous

The following miscellaneous options are also available.

Direct Solution

Direct solution can be used for either the global system or the reservoir system by specifying either GLOBAL DIRECT or RESERVOIR DIRECT. This can be useful in small models that the iterative methods have difficulty with. In large models, direct solution is prohibitively expensive.

Action on Solver Failure

Unfortunately, on rare occasions the iterative solver can fail to converge in the allowed number of iterations. In this case, two actions are possible. SOLVER CUT will cause the timestep size to be reduced, and then the solution will be attempted again. SOLVER NOCUT allows the solution to proceed as if convergence had occurred. Somewhat surprisingly, this often does not cause any problems.

Dual Porosity Feature

Multiple-porosity, single-permeability models have a special matrix structure that can be exploited by specifying DUAL_SOLVER ON. The result is a significant improvement in solver efficiency.

R5000.0.2 Solver: Miscellaneous 33


34 Solver: Miscellaneous R5000.0.2


Chapter 3

PVT Representation

Introduction

In this chapter, the calculations involving fluid properties and fluid phase equilibrium are discussed.

The major fluid types discussed are:

• Standard black oil• Extended black oil• API interpolation• Water-oil• Gas-water• Equation of state for oil and gas phases in equilibrium• Equation of state for oil, gas, and water phases in equilibrium.

For each of the fluid types the major calculations discussed are

• Calculation of fluid phase equilibrium• Conversion to surface fluid volumes• Calculation of volume derivatives• Calculation of oil and gas phase properties including density and

viscosity.

Water phase properties are calculated the same way for all fluid systems, except for the case where water is in equilibrium with oil and gas phases.

R5000.0.2 PVT Representation: Introduction 35


Water Properties

Multiple water PVT data methods can be used in Nexus. Different gridblocks or connections in the surface network can be assigned different water property methods.

Note that this discussion in this section does not apply when gas solubility is allowed in the water phase.

The simplest water property method consists of specifying a reference density , water compressibility cw, a reference water formation volume factor Bw, a reference pressure pref, and a reference water viscosity . Optionally, a derivative of viscosity with pressure

can be specified. Otherwise, it is assumed that viscosity does not vary with pressure.

Additionally, this set of properties can be entered for different values of temperature and water salinity.

For any single value of temperature and salinity, water density is calculated by

3-1

For any single value of temperature and salinity, water viscosity is calculated by

3-2

If the salinity is between the minimum and maximum tables values, water properties will be linearly interpolated between properties calculated from the appropriate pair of tables. If the salinity is below the minimum, the properties will be the same as those calculated using the minimum salinity. If the salinity is greater than the maximum, the properties will be the same as those calculated using the maximum salinity. No extrapolation beyond the table data is done.

Interpolation and extrapolation rules for temperature are the same as those for salinity.

Temperature and salinity can both vary at the same time.

w ref

w refwp

w

w ref

Bw-------------- 1 cw p pref– + =

w w ref 1 wp p pref– + =

36 PVT Representation: Water Properties R5000.0.2


Black Oil Models

Standard Black Oil Calculations

The Physical Model

Tables of saturated data and undersaturated data can be entered for multiple fluid models. Different fluid models can be assigned to different gridblocks in the reservoir and to connections in the surface network. Additionally, PVT tables can be grouped together to be used with the API interpolation option.

Nexus is not a traditional black oil simulator that uses as its primary variables pressure and two of the three-phase saturations. Traditional black oil simulations typically use variable substitution for the undersaturated state, with Rs, the solution gas-oil ratio substituted for one of the phase saturations.

Nexus is a compositional model that in black oil mode has three components: stock tank oil, surface gas, and water. There is limited solubility of the components in the three phases which we will call liquid, vapor, and aqueous to avoid confusion. The water component can only exist in the aqueous phase. In the traditional black oil model, the oil component can only exist in the liquid phase. (See the extended black oil model for a lifting of this restriction.) The gas component can exist in both the liquid and vapor phases. As its primary variables, Nexus uses pressure and the three component masses. The water component mass is eliminated so that only three simultaneous equations are solved in black oil mode.

Black oil data is typically available as tables of the following properties with respect to pressure

• the formation volume factor of oil (Bo) • the formation volume factor of gas (Bg), • the solution gas-oil ratio (Rs).• oil phase viscosity• gas phase viscosity

The definition of Bo is the ratio of the volume of the liquid phase at reservoir conditions and the volume of the liquid phase at stock tank conditions.

R5000.0.2 PVT Representation: Black Oil Models 37


The definition of Bg is the ratio of the volume of the vapor phase at reservoir conditions and the volume of the vapor phase at surface conditions.

The definition of Rs is the ratio of the volume of surface gas dissolved in reservoir oil and the volume of stock tank oil.

Figure 3.1 illustrates how volumes of phases at reservoir conditions is related to volumes at surface standard conditions. In a standard black oil, the reservoir gas will expand to a surface gas with the same composition. In the reservoir, the liquid phase contains both the stock tank oil component and the surface gas component. As the liquid reaches the surface, the surface gas will evolve from the liquid phase leaving both a liquid and a vapor phase at the surface.

Figure 3.1: Relationship of Volumes at Reservoir Conditions to Volumesat Surface Conditions for Standard Black Oil Model

The black oil data is usually provided by a laboratory experiment called a differential liberation expansion (DLE) test. In this test, a fluid at bubble point pressure is placed in a PVT cell. The pressure is lowered in a series of steps, which results in the evolution of free gas. At each stage, after equilibrium has been achieved, the evolved gas is removed from the cell. The gas volume and gas gravity are measured, which in turn provides the moles of gas evolved. The pressure is eventually reduced to atmospheric pressure. A residual oil volume is measured at atmospheric pressure and reservoir temperature. The residual oil volume, Vor, becomes the basis for the tabular data of the DLE test. Oil formation volume factor, Bod, and solution gas oil ratio, Rsd are

Reservoir gas

Surface gas

Reservoir oil with dissolved gas

Surface separator

Surface gas fromreservoir oil

Surface oil

38 PVT Representation: Black Oil Models R5000.0.2


both based on Vor.. In summary, for each pressure stage, the DLE test yields:

Gas volume, gas gravity, and moles of gas evolved

Rsd = (total standard volumes of gas evolved for this stage and all subsequent stages)/ Vor

Bod = Vo / Vor

Bg = Vg / Vgs where Vgs is the volume of gas at standard temperature and pressure.

Like most other black oil models (except VIP), Nexus expects black oil data to be based on stock tank oil volume, and not residual oil volume. To accommodate this, the laboratory will usually perform a surface separator flash with the bubble point fluid for at least one set of separator operating conditions. The separator train may have multiple stages, but the last stage will be at standard temperature and pressure.

The separator flash test yields values of solution gas oil ratio, Rsf, and formation volume factor, Bof. To correct the DLE data so that Rsf and Bof are honored, the following equations are usually utilized

3-3

3-4

However, these equations are only approximate. They do a poor job when the produced gas gravity is not constant. Additionally, at atmospheric conditions, they lead to strange errors. Rs often becomes negative, while Bo may drop below 1. These errors are usually ignored in use. A better method would be to use an equation of state program to simulate the effect of the separator flash.

The original black oil model (BOTAB) is based on the results of an unadulterated DLE test. It uses a sophisticated material balance calculation to take the original DLE table data and then uses it to calculate reservoir condition oil and gas properties. The results of the flash are entered separately and used to calculate the effects of the surface flash as they are required during the simulation.

Rs Rsf Rsdb Rsd– Bof

Bodb-----------–=

Bo Bod

Bof

Bodb-----------=



Knowledge of the internal workings of the VIP black oil model (in the traditional BOTAB input format) have been used to convert data to the more traditional Nexus black oil table form.

The Nexus Internal Representation

Internally, Nexus must convert the black oil engineering quantities to tables that are functions of pressure, p, the mass of the oil component,

and the mass of the gas component, , where the symbol .represents the oil component as opposed to the oil phase, and the

symbol represents the gas component as opposed to the gas phase.

The original black oil data is converted and stored internally as tables. In the following discussion, we will only discuss field units, but calculations in different unit systems will of course use different conversion constants.

(RB/MSCF in field units) is converted to dry gas density (lb/SCF in field units) by

3-5

where is the density of gas at standard conditions.

(MSCF/STB in field units) is converted to a mass ratio (lbs of gas/lbs of oil in field units) by

3-6

where is the density of oil at standard conditions.

(RB/STB in field units) is converted to oil density by

3-7

Oil and gas viscosity data are stored in their original state.

mo mgo

g

Bg

g 178.0939g std

Bg-------------=

g std

Rs

Rgo 178.0939gstd

ostd-----------Rs=

o std

Bo

oostd 1 Rgo+

Bo-----------------------------------=



The Nexus Black Oil Flash

The black oil flash consists of a test comparing the value of tabulated at the flash pressure and the actual mass ratio given by . If

the actual mass ratio is greater than the tabulated one, a free gas phase will evolve. Otherwise, the fluid is in an undersaturated state.

In a saturated state, the oil phase mass fractions are given by

3-8

3-9

3-10

In an undersaturated state, the oil phase mass fractions are the same as the overall mass fractions

3-11

3-12

Saturated densities and viscosities are acquired through interpolation of the tabulated data. For undersaturated oil, density and viscosity factors are tabulated are respected with respect to the difference between the pressure and saturation pressure.

Rgo mg mo

o o1

1 Rgo+------------------=

g o 1 o o–Rgo

1 Rgo+------------------= =

w o 0=

o omo

mo mg+--------------------=

g o 1 o o–mg

mo mg+--------------------= =



Nexus Volumes and Volume Derivatives

In the Nexus model, volumes and volume derivatives of each phase are required. The volume of each phase is equal to the mass of the phase divided by the density of the phase.

The mass of the gas phase in a saturated block is given by:

3-13

The mass of the oil phase in a saturated block is given by

3-14

The oil phase volume derivatives are given by

3-15

3-16

3-17

3-18

The gas phase volume derivatives are given by

3-19

mg mg Rgomo–=

mo mo 1 Rgo+ =

moVo 1 Rgo+

o------------------=

mgVo 0=

mwVo 0=

pVo mo pd

d 1 Rgo+

o------------------ –=

moVg Rgo

g--------–=



3-20

3-21

3-22

In the undersaturated state, the oil volume is given by

3-23

The volume derivatives are given by

3-24

3-25

3-26

The density of the undersaturated oil is calculated as the product of density of the saturated oil and an undersaturated multiplication factor. Both of these quantities are tabulated with respect to which is the ratio of . In addition, the undersaturated multiplication factor is tabulated as a function of pressure.

mgVg 1

g-----=

mwVg 0=

pVg

mo pd

dRgo Vg pd

dg+ –

g-------------------------------------------------=

Vo

mo mg+

o--------------------=

moVo 1

o----- 1 Vo mo

o–

=

mgVo 1

o----- 1 Vo mg

o–

=

pVo 1

o----- 1 Vo p

o– =

Rgomg mo



Enhanced Black Oil Calculations

The Physical Model

A modification of the traditional black oil formulation exists in Nexus to account for the surface liquid content in reservoir gases. Figure 3.2 is a schematic of the relationship between reservoir and surface phase volumes for the enhanced black oil model. A new value, is introduced which is the ratio of the surface oil from reservoir gas and the surface gas from reservoir gas.

Figure 3.2 Enhanced Black Oil Model

The values cannot be obtained directly from the differential liberation test. For a condensate, a constant volume depletion (CVD) experiment is usually performed in the laboratory. The constant volume depletion experiment is meant to simulate the depletion of a condensate fluid. Starting with a fluid sample at the dewpoint, the pressure is dropped in increments. At each stage, a separate liquid phase will form. After equilibrium, the volume of the cell is restored to its value at the dewpoint by removing the appropriate amount of gas. The oil and gas volumes, and the removed gas composition and density are measured. In the Whitson-Torp method1, the experiment is simulated by an equation of state package and the resulting fluid characterization is used to conduct surface separator experiments to yield the value of .

Rv

Reservoir gas with dissolved oil

Surface gas

Reservoir oil with dissolved gas

Surface separator Surface oil

Surface gas fromreservoir oil

Surface oil fromreservoir gas

Rv

Rv



The Nexus Internal Representation

Analogous to the conversion of to , the solution ratio is converted to a mass ratio.

The following equation is used for field units where units are in STB/MSCF, and units are in lbs of oil/lbs of gas.

3-27

Equation 3-7 is used for liquid density.

The vapor phase liquid density becomes similarly complicated and is defined by

3-28

Implicit in these equations is the assumption that passing the reservoir oil and gas fluids through the same separation systems would yield identical standard oil and gas densities, and that these densities are constant. For some fluids, especially gases, the standard density is far from constant. This would be a good assumption to remove for future development.

1. Whitson, C. H. and Torp, S.B. "Evaluating Constant Volume Depletion Data", JPT, (March 1983) 610; Trans., AIME, 275.

Rs Rgo

RvRog

Rog 0.005615ostd

gstd-----------Rv=

g 178.0939gstd 1 Rog+

Bg-----------------------------------=



The Nexus Enhanced Black Oil Flash

As in the basic black oil model, the input to the flash is pressure, temperature, and the masses of the two components. The flash must yield the mass fraction of each phase, and the composition of each phase.

The mass balance for the oil component is

3-29

The mass balance for the gas component is

3-30

From these equations, the mass fractions of the phases can be derived. In a saturated case, the mass fractions in the oil phase are

3-31

3-32

3-33

3-34

Combining the above six equations leads to the equations that yield the mass of each phase

3-35

mo o o mo o g mg+=

mg g o mo g g mg+=

o o1

1 Rgo+------------------=

g oRgo

1 Rgo+------------------=

o gRog

1 Rog+------------------=

g g1

1 Rog+------------------=

mg

1 Rog+ 1 RgoRog–

------------------------------- mg moRgo– =



3-36

Where both and must be greater than 0 in a saturated state.

Equivalently, it can be stated that the saturated state exists when

3-37

mo

1 Rgo+ 1 RgoRog–

------------------------------- mo mgRog– =

mo mg

Rgo

mg

mo------ 1

Rog--------



From these equations it is evident that the input data must be restricted so that Equation 3-37 applies for each value of saturation pressure. Additionally

3-38

Nexus Volumes and Volume Derivatives

Volumes are calculated in the same way that are done for the standard black oil model. Volume derivatives are somewhat more complicated.

In a saturated state, densities, viscosities, and mass ratios are strictly functions of pressure and can be determined from the table look-up. The partial volume derivatives of the oil phase are given by

3-39

3-40

3-41

The partial volume derivatives of the gas phase are given by

3-42

3-43

RgoRog 1

moVo 1 Rgo+

o 1 RgoRog– -------------------------------------=

mgVo Rog 1 Rgo+

o 1 RgoRog– -------------------------------------–=

pVo mo pd

d 1 Rgo+

o 1 RgoRog– ------------------------------------- –=

moVg Rgo 1 Rog+

g 1 RgoRog– -------------------------------------–=

mgVg 1 Rog+

g 1 RgoRog– -------------------------------------=



3-44

When only undersaturated oil exists, the oil volume derivatives are

3-45

3-46

3-47

When an undersaturated gas exists, the volume derivatives are

3-48

3-49

3-50

pVg

1g----- mo

1 Rog+

1 RgoRog– -------------------------------

pd

dRgo Vg pd

dg+

–

mg moRgo– pd

d 1 Rog+

1 RgoRog– -------------------------------

–

=

moVo 1

o----- 1

VoRgo

mo---------------

Rgoo+

=

mgVo 1

o----- 1

Vo

mo------

Rgoo–

=

pVo Vo

o------

po–=

moVg 1

g----- 1

Vg

mg------

Rogg–

=

mgVg 1

g----- 1

VgRog

mg---------------

Rogg+

=

pVg Vg

g------

pg–=



Conversion from VIP Black Oil Model

The VIP black oil model is in many ways more sophisticated than either of the Nexus black oil models. It is also a two component compositional model, but it allows for both oil and gas components to exist in either hydrocarbon phase. It allows for variable produced gas gravity, and it allows for complex surface separator calculations. Its major weakness is that it is strongly predicated on the DLE test, and so cannot easily be used for condensate systems.

Conversion of VIP black oil data to the Nexus enhanced black oil model requires an intimate understanding of the VIP internal representation, so that we can calculate the properties of the fluid at input conditions. The Nexus black oil input is manipulated so that the phase splits and the phase properties at reservoir conditions is matched at conversion points. Additionally, while the Nexus model does not require the application of separator calculations, in this case they are necessary to match the surface volumes calculated by VIP.

In the first stage of the conversion process, the original DLE data and surface separator data are converted to the same internal representation as VIP.

From the DLE experiment, tabular values of the solution gas-oil ratio ( , in SCF/STB of residual oil), the formation volume factor (Bo, in RB/STB of residual oil), gas compressibility factor ( ), gas formation volume factor (Bg, in RB/MSCF), gas density relative to air at standard conditions (Gr), and oil and gas viscosities (o and g) as functions of the saturation pressure. (Note that both the solution gas-oil ratio and the oil formation volume factor are based on the standard state of the residual oil, rather than on the stock tank oil.) In this form, the formation volume factor is more frequently referred to as the relative oil volume in typical laboratory reports. Such form allows the user to enter the laboratory data directly, without first converting them to stock tank oil conditions. The values of Bo and o at pressures above the saturation pressure also are required for undersaturated conditions.

Since the simulator uses a compositional formulation internally, these black oil data are converted into compositional parameters (e.g., fugacity coefficients, Zo, and o are as functions of pressure and gas component mole fraction in the oil phase). The compositional parameters then are reconstructed into equal-spaced tables. For pressures beyond the range of the tabular data, VIP uses constant slope extrapolation to derive values at those conditions. The conversion procedure is described in the following sections.

RsZg



The gas phase viscosity and compressibility factor are assumed to be a function of pressure only (no compositional dependency). An equal-space viscosity versus pressure table is constructed from the input viscosity versus saturation pressure table. A Zg versus pressure table is likewise constructed. If Bg instead of Zg is entered as a function of pressure, then the following equation is used to convert Bg to Zg:

3-51

In deriving this equation, gas compressibility at the surface temperature (Ts) and surface pressure (Ps) is assumed to be one.

Oil data are converted to equal-space tables of K-values (needed for fugacity equality equations), Zo, and o.

Data conversion requires oil and gas component molecular weights, which are set internally by the simulator. To ensure non-zero component K-values, the simulator assumes that the oil component molecular weight, , equals the maximum among all residual oil and stock tank oil molecular weights plus two (2.0). Additionally, the gas component molecular weight, , is assumed to equal the minimum among all gas phase molecular weights (29Gr) minus one (1.0). Consequently, the gas phase has at least a trace of the oil component at any pressure level (including the surface pressure) and vice versa.

The residual oil molecular weight is either directly entered by the user or derived by the simulator. Normally the user is required to enter either the density and molecular weight of residual oil ( , ) or those of bubble-point oil ( , ) following the BOTAB card. These properties can be derived from each other based on the differential liberation data. The procedure to derive and from and

is explained below.

The differential liberation data are entered in monotonically decreasing pressure steps. The last pressure must be the surface pressure; i.e., p(i) > p(i+1) and p(n) =ps, where n is the entry number of the last stage. Based on 1 cc of the residual oil (equivalent to Bob cc or x Bob gm of residual oil plus gas at the bubble-point), the overall gas liberated in gm for the differential liberation is

3-52

Zg

0.0056146 Bg Po Ts

Ps T-------------------------------------------------=

Mo

Mg

or Morob Mob

or Mor obMob

ob

Gg29.

5.6146 62.428---------------------------------------

ps

RTs--------- Rs i Rs i 1+ – Gr i 1+

i 1=

n 1–

=



Thus, the density of the residual oil is given by

3-53

where and are given in gm/cc, and Bob is the Bo (RB/STB of residual oil) for the bubble-point oil. Again, based on 1 cc of the residual oil, the overall gas liberated in gmole at the end of the differential liberation for the bubble-point oil is

3-54

The amounts of bubble-point oil in gmole given 1 cc of residual oil is

3-55

The molecular weight of the residual oil then is

3-56

Following a similar procedure, the density and molecular weight of the stock tank oil can be derived from and if the Rs, Bo (RB/STB of stock tank oil), and Gr were known from the separator flash test. This is the case when the SEPTEST card is entered in the VIP input deck.

VIP provides two additional separator options that warrant special attention since they may create an overly defined fluid system. The first option is use of the BOSEP card. This option requires that the user enter the API or density of the stock tank oil. Since the same quantity also can be derived from and , we have an overly defined fluid system. In this case, VIP honors the stock tank oil density entered by the user and disregards the Gr in deriving all the relevant stock tank oil properties.

The second option is the use of the SEPARATOR card, where K-values for the separator are entered. The stock tank oil molecular weight can be defined from the separator K-values. However, the user may enter the stock tank oil molecular weight (MWL) which, if entered, produces

or ob Bob Gg–=

or ob

Gm1

5.6146 652.428------------------------------------------

ps

RTs--------- Rs 1 Rs n – =

zo

Bob obMob

-----------------------=

Moror

zo Gm–------------------=

ob Mob

ob Mob



an overly defined fluid system. In this case, VIP honors the user-entered MWL and disregards the molecular weight derived from the separator K-values in deriving the stock tank oil compressibility (ZLSEP).

The molecular weight of the liberated gas, , at each liberation stage (pressure entry No. I-1 to I) is

3-57

Having previously established the molecular weights of the oil and gas components, the mole fraction of either component in a phase can be calculated from knowing the molecular weight of that phase from the relationship

3-58

and

3-59

We also make use of the constraints for a two component system that

3-60

3-61

The gas component mole fraction in the liberated gas phase is then

3-62

Based on 1 cc of residual oil, the amounts of the overall gas liberated from stage I to the standard conditions in gm is

Mgd

Mgd I 29 Gr I =

Mgd I yoMo ygMg+=

Mod I xoMo xgMg+=

xo xg+ 1=

yo yg+ 1=

yg I Mo Mgd I –

Mo Mg–------------------------------=



Thus, the density of the oil at stage I is

3-63

Since the amounts of total gas liberated from stage I to the standard conditions are given by

3-64

the molecular weight of the oil at stage I can be calculated as

3-65

Gg I 29.5.6146 62.428---------------------------------------

ps

RTs--------- Rs i Rs i 1+ – Gr i 1+

i 1=

n 1–

=

o I or Gg I +

Bo I ----------------------------=

Gm I 15.6146 62.428---------------------------------------

ps

RTs--------- Rs I Rs n – =

Mo I o I Bo I

or Mor Gm I +------------------------------------------=



Making use of Equation 3-59, the gas component mole fraction in the oil at stage I can be calculated from

3-66

The equilibrium K-values of the gas component and the oil component at stage I are calculated according to

3-67

and

3-68

respectively. Oil compressibility then is calculated according to

3-69

The above procedure establishes K-values as a function of pressure, and Zo as a function of (or the saturation pressure). The relationship between and saturation pressure, which is needed in constructing the full and tables for undersaturated conditions, also is established.

For the undersaturated oil phase, the user must enter ratios of the oil formation volume factor at pressures above the saturation pressure to the oil formation volume factor at the saturation pressure. Ratios of oil viscosity at pressures above the saturation pressure to viscosity at the saturation pressure also are required. To convert the undersaturated Bo values to undersaturated Zo values, the following relationship is recognized from Equation 3-69:

xg I Mo Mod I –

Mo Mg–------------------------------=

Kg I yg I xg I ------------=

Ko I 1 yg I –

1 xg I –---------------------=

Zo I p I Mo I

5.6146 62.428 o I RT ------------------------------------------------------------------------=

xgxg

Zo p xg o p xg



3-70

where the superscripts denote saturated conditions. Note that the right side of Equation 3-70 represents the input ratios of formation volume factors. This equation suggests that the same ratios can be used directly if the undersaturated Zo/p table is constructed. The independent variables in the two-dimensional table are and p. For a given , the saturation pressure is given by the -pressure relationship established earlier in this section. For any pressure which is greater than the saturation pressure, the right side of Equation 3-70 is determined from the input undersaturated table through interpolation. The resulting value is then multiplied by to yield Zo/p for and p. The two-dimensional oil viscosity table is constructed likewise.

Finally, the K-values are used to establish the following fugactiy equality equations:

3-71

3-72

After the K-values have been calculated for each of the data entries in the differential liberation data, an equal-spaced table is constructed through a linear interpolation process involving ln(Kijpi) versus ln(pi).

To convert this internal VIP representation to one that can be stored in the Nexus format, a single value of standard oil density and standard gas density for each set of tables is chosen. At each input saturation pressure, the mass fraction of the oil and gas components in each phase are determined. (These are intermediate values generated for the K-value calculation). Thus, we have mass ratios and . Using Equation 3-6 and Equation 3-27, respectively, the engineering quantities and are calculated. The Z-factors and thus the densities of each phase are also known at each saturation pressure. To preserve the values of densities, is calculated by applying Equation 3-7, and is calculated from Equation 3-29. The viscosities are copied without any modification. Therefore, at each input saturation pressure, Nexus and VIP will have identical oil and gas

Zo

p-----

Zos

ps

----- -----------

Bo p

Bo ps

----------------=

xg xgxg

Zos

ps xg

o p xg

oi Ki p i o g= =

gi 1 i 1 2= =

Rgo Rog

Rs Rv

BoBg



phase densities, viscosities, and compositions. The same calculations are performed for each undersaturated table entry. Values will differ in between tabulated values because of differences in interpolation. Both VIP and Nexus employ linear interpolation, but on the internally stored values, which we have seen are quite different. Problems may arise when there are large gaps in pressure in VIP tabulated data.

While the above procedure ensures that reservoir properties will be the same for oil and gas phases in the two models, it does not ensure that surface volume calculations will yield the same result. How this match is accomplished is discussed in the next section.

Calculation of Surface Volumes

In Nexus, conversion to surface volumes is very simple. Internally, all calculations in the network yield mass flow rates of the oil, gas, and water components. The surface volume is calculated by dividing the mass flow rate by the surface density of the corresponding phase. These surface densities are constant input quantities for each PVT table.

In VIP, surface separator calculations are applied. Regardless of the separator type, VIP internally converts the data to constant K-values for each separator stage. The total hydrocarbon molar rate is passed through a series of separator stages. The split of oil and gas phases for each separator stage is determined by a flash with constant K-values. The final output gas stream molar rate is converted to a volume by application of the gas law with gas compressibility factor set equal to 1, and the pressure and temperature at standard conditions. The final output oil stream molar rate is similarly converted to a surface volume, but using a non-unit Z-factor. The oil phase Z-factor is internally determined in VIP from separator input data and is a constant for each separator train.

To ensure that Nexus will produce the same surface volumes as VIP, in the conversion process, the molecular weights, the K-values for each separator stage, and the oil phase surface Z-factors are saved. As stated in the previous section, Nexus will have the same phase mass compositions at each point. Application of the molecular weights, and use of the same surface separator calculations as VIP uses will ensure that Nexus will yield the same surface volumes.

Even when conversion from VIP is not required, the availability of the Nexus surface separator input allows for additional flexibility. Thus, Nexus can retain the same black oil table, but just change separator data for the cases where multiple separator trains exist, or for a change in



separator output conditions. Otherwise, a different PVT table would have to be input for these circumstances.

The API Interpolation Model

In the API interpolation model, three hydrocarbon components are utilized instead of just two for the normal black oil model. There are two oil components, a heavy component, and a light component. Additionally, there is a gas component. The two oil components have the same K-values. In other words, they will apportion in the same ratio in each phase as they exist in the overall system.

Within a group of input black oil tables, each table has a distinct value of oil surface density. This is converted to the mass fraction of the heavy oil component by the calculation

3-73

Whenever any calculation is performed with an API fluid, the value of the heavy component mass fraction is calculated by

3-74

where is the mass of the heavy component and is the mass of the light component.

The value of is used to determine which pair of black oil tables are used for calculation. If is less than or equal to the minimum , then the data corresponding to the minimum is used for the fluid properties calculations. If is greater than or equal to the maximum

, the data corresponding to the maximum is used. Otherwise, the data is determined by linear interpolation between tables.

The API interpolation is used on a step-by-step basis of the fluid property calculation, rather than just being applied to the final result. For example, to calculate the phase equilibrium, and will be determined in each table and interpolated linearly by to use in the calculation of the oil and gas phase fractions. This is done instead of

h

1o std------------- 1

o std min,----------------------–

1o std max,----------------------- 1

o std min,----------------------–

----------------------------------------------------=

hmh

mh ml+-------------------=

mh ml

hh h

hh

Rgo Rogh



doing each phase equilibrium calculation with a separate table and then interpolating the result.

The Water-Oil Model and the Gas-Water Model

Nexus can use a variation of the standard black oil model where only two components exist -- an oil component and a water component. The system is treated as an undersaturated oil system so only data that is appropriate for such a system is input.

Not only are the fluid calculations much simplified since the flash calculation is unnecessary, but only two component flow equations are solved. In implicit mode, this reduces the number of simultaneous equations solved to 2 instead of 3 for the standard black oil model.

A constant non-zero gas-oil ratio (Rs) can be specified. In such a case, when the fluid is converted to surface conditions, a surface oil volume and a surface gas volume will be calculated even though only mass flow rates of the oil component are calculated.

Similarly, Nexus can use a gas-water model, which has only two components -- a gas component and a water component. Computational advantages are similar to those of the water-oil model.

A constant non-zero oil-gas ratio (Rv) can be specified. In such a case, when the fluid is converted to surface conditions, a surface oil volume and a surface gas volume will be calculated even though only mass flow rates of the gas component are calculated.



Equation of State (EOS) Model

A General EOS Model

Nexus uses a generalized form of a cubic EOS as follows:

3-75

where p is the absolute pressure, T is the absolute temperature, and v is the molar volume. C1 and C2 are constants that depend on the equation of state, while a and b are parameters that are compositionally dependent.

a for a single component i is defined as

3-76

b for a single component i is defined as

3-77

where and are equation of state constants that have different defaults for different equations of state, is the critical temperature of component i, is the critical pressure of component i, and R is the universal gas constant.

For the Redlich-Kwong equation of state (RK-EOS), and the Soave-Redlich-Kwong2 equation of state (SRK-EOS), the default value of

is 0.4274802 and the default value of is 08664035. The value of constant C1 is 1 and value of constant C2 is 0.

For the RK-EOS,

3-78

p RTv b–----------- a

v C1b+ v C2b+ -----------------------------------------------–=

ai aii

R2Tci

2

pci-------------=

bi bi

RTci

pci-----------=

ai biTci

pci

ai bi

i

Tci

T-------=

60 PVT Representation: Equation of State (EOS) Model R5000.0.2


For the SRK-EOS,

3-79

where

3-80

and is the acentric factor.

For the Peng-Robinson3 equation of state (PR-EOS), the default value of is 0.457235529 and the default value of is 077796074. The value of constant C1 is and value of constant C2 is .

Equation 3-79 also applies to the PR-EOS, but the definition of m differs.

For the original PR-EOS (specified by the PR_OLD keyword),

3-81

For the newer version of the PR-EOS4 (specified by the PR keyword)

3-82

and

3-83

For mixtures, the value of a is given by

3-84

where is the binary interaction coefficient between components i and j, and xi is the mole fraction of component i.

i 1 mi 1 Tri– + 2

=

mi i 0.176i– 1.574+ 0.48+=

i

ai bi1 2+ 1 2–

mi i 0.26992i– 1.54226+ 0.37464 for all i+=

mi i 0.26992i– 1.54226+ 0.37464 if i 0.49+=

mi 0.379642 i 1.48503 i 0.164423– 0.016666i+ + +=

if i 0.49

a xixj 1 ij– aiaj

j 1=

nc

i 1=

nc

=

ij

R5000.0.2 PVT Representation: Equation of State (EOS) Model 61


For the mixture value of b, two possible mixing rules are available. The default one is

3-85

Alternatively, a mixing rule involving a binary interaction coefficient for the b parameter can also be invoked.

3-86

The above mixing rule is only used if the BINB table is input.

Calculation of Z-factors and Molar Density

The general EOS can be rearranged to become a cubic equation for the Z-factor which is defined as

3-87

The general cubic EOS can be written for each hydrocarbon phase as

3-88

where

3-89

and3-90

b xibi

i 1=

nc

=

b xixj 1 dij– bibj

j 1=

nc

i 1=

nc

=

Z pvRT-------=

Z3

C3B 1– Z2A C4B– C5B

2– Z C6B B 1+ AB–+ + + 0=

A ap

RT 2--------------=

B bpRT-------=



The constants are defined as follows

3-91

3-92

3-93

3-94

The cubic equation (Equation 3-88) may yield three real roots. Normally, the largest root is chosen when evaluating the compressibility factor for the gas phase, while the smallest positive root is chosen as the compressibility factor for the oil phase. However, if the user invokes option ZGIBBS, the Z-factor that yields the lowest Gibbs free energy will be chosen regardless of the phase.

Equation 3-87 can be rearranged to provide the molar volume

3-95

When the volume shift correction5,6 is used (caused by inputting the VSHIFT column in the EOS data table), the molar volume is corrected as follows

3-96

The correction parameter ci is related to the data input in the VSHIFT column (si) by

3-97

C3 C1 C2 1–+=

C4 C1 C2+=

C5 C1 C2 C1C2–+=

C6 C1C2–=

v ZRTp

-----------=

vcorrected v xici

i 1=

nc

–=

ci bisi=



Calculation of Fugacity Coefficient and Fugacities

Fugacities play the key role in determining fluid phase equilibrium. When the oil phase is in equilibrium with the gas phase, the following equation applies for each component

3-98

where is the fugacity of component i in the gas phase, and is the fugacity of component i in the oil phase.

The fugacity coefficient of component i in phase j, , is defined as

3-99

The fugacity coefficient can be calculated from an EOS (dropping the j phase index) by using

3-100

Substituting the EOS into the above equation yields:

3-101

fg i fo i=

fg i fo i

j i

j ifj i

xj i p-----------=

RT ilnni

p RTV

-------– Vd

V

RT Zln–=

lni ln Z B– –Z 1–

b-----------------

ni nb C7

p

B RT 2------------------ 1

n---

ni an

2 ARTB

-----------ni nb –

lnZ C2B+

Z C1B+--------------------+ +=



Other Thermodynamic Properties

The general EOS has been used to generate many other thermodynamic properties.7

Second and third orders of derivatives of Helmholtz free energy with respect to mole numbers are required for calculation of critical points. The Helmholtz free energy (A) is calculated by substituting the EOS into

3-102

where is the ideal chemical potential for component i and is the ideal entropy for component i.

The EOS is also used to calculate enthalpy which is used in energy balance equations. The EOS is substituted into

3-103

where H* is the ideal state enthalpy.

The EOS has been used to calculate heat capacities that are used in compressor calculations.The EOS has been substituted into the following equations:

3-104

where is the ideal state isochoric heat capacity.

3-105

A p nRTV

----------– Vd

V

RT niV

niRT------------ ln

i 1=

nc

– ni i

Tsi–

i 1=

nc

+=

i

si

H H*

–RT

----------------- Z 1–1

RT------- T

Tp

v

p– vd

v

+=

Cv Cv*

–

R------------------

TR---

T2

2

p

v

vd

v

=

Cv*

Cp Cp*

–

R------------------

Cv Cv*

–

R------------------

TR---

Tp

2

v

vp

T

------------------– 1–=



where Cp* is the ideal state isobaric heat capacity.

Hydrocarbon Phase Viscosities

Lohrenz-Bray-Clark Correlation

The viscosities of the oil and gas phases are calculated using the Lohrenz-Bray-Clark (LBC) correlation.8

3-106

where

3-107

3-108

3-109

3-110

and

3-111

j 0.18383Mj

3pcj

4

Tcj--------------

0.16667

LBC1 r{LBC2 r LBC3++=

r LBC4 +LBC5r 4 10

4– –U3

xi Mi

i 1=

nc

-----------------------------+ +

r jVcj=

Aj xiAi A

i 1=

Nc

M Pc Tc Vc = =

Vci10.73615 ZciTci

5.6146 pci---------------------------------------=

U3 0.18383 xi Mi

Mi3Pci

4

Tci---------------

0.16667

U5i

i 1=

nc

=

U5i 3.4 104–Tri

0.94if Tri 1.5=



or

3-112

The symbol Mi denotes the molecular weight of component i. In Equations through 3-112, j, Vci, pci, Tci, and j are in units of lbmol/bbl, bbls/lbmol, psi, °R, and cp, respectively.

The default values for the LBC constants are

Constant Default

LBC1 0.1023

LBC2 0.023364

LBC3 0.058533

LBC4 -0.040758

LBC5 0.0093324

U5i 1.778 104–

4.58Tri 1.67– 0.625 if Tri 1.5=



Pedersen Correlation

The Pedersen correlation9,10 is useful for heavy oils where the LBC correlation fails to give a proper viscosity prediction for some cases.

The Pedersen correction is based on the corresponding states principle. A group of substances obeys the corresponding states principle if these substances have the same reduced viscosity at the same reduced density and reduced temperature. In such a case, only comprehensive viscosity data for one component (the reference component) in the group are needed. Other data can be calculated from the reduced viscosity. The Pedersen correlation uses methane as a reference substance.

The viscosity of a hydrocarbon phase is calculated as

3-113

where Tcm is the mixture critical temperature, Pcm is the mixture critical pressure, and Mm is the mixture molecular weight. The subscripts o and m refer to the reference substance and mixture, respectively. They are calculated from the mixing rules:

3-114

3-115

The coefficients Xij are introduced in Equation 3-114 to account for the binary component interaction.

Mm is calculated by the following equation:

p T Tco

Tcm---------

1 6 pcm

pco--------

2 3 Mm

Mo--------

1 2 m

o------- o

p pcoopcmm

---------------------T TcooTcmm

---------------------- =

Tcm

xixj

Tci

pci-------

1 3 Tcj

pcj-------

1 3+

31 Xij– TciTcj 1 2

j 1=

nc

i 1=

nc

xixj

Tci

pci-------

1 3 Tcj

pcj-------

1 3+

3

j 1=

nc

i 1=

nc

-------------------------------------------------------------------------------------------------------------------------------------=

pcm

8 xixj

Tci

pci-------

1 3 Tcj

pcj-------

1 3+

3TciTcj 1 2

j 1=

n

i 1=

n

xixj

Tci

pci-------

1 3 Tcj

pcj-------

1 3+

3

j 1=

n

i 1=

n

2

--------------------------------------------------------------------------------------------------------------------=



3-116

where

3-117

3-118

Parameters m and o in Equation 3-113 account for the density and molecular weight effects:

3-119

3-120

The symbol r denotes the reduced density for the reference component:

3-121

The density of the reference component (methane) o is calculated from the three-parameter Peng-Robinson equation of state.

Mm 1.304 104–

Mw2.303

Mn2.303

– =

Mw

xiMi2

i 1=

nc

xiMi

i 1=

nc

---------------------=

Mn xiMi

i 1=

nc

=

m 1 7.378 103– r

1.847Mm

0.5173+=

o 1 7.378 103– r

1.847Mo

0.5173+=

r

o

P PcoPcm

----------------T Tco

Tcm---------------

co-----------------------------------------------=



The viscosity of the reference component is based on the model of Hanley et al.11

3-122

where

3-123

and

b1 = -0.2090975x105

b2 = 2.647269x105

b3 = -1.472818x105

b4 = 4.716740x104

b5 = -9.491872x103

b6 = 1.219979x103

b7 = -9.627993x10b8 = 4.274152b9 = -8.14153x10-2

3-124

3-125

By default:

j1 = 1.035060586x10j2 = 1.7571599671x10j3 = 3.0193918656x103

j4 = 1.8873011594x102

j5 = 4.2903609488x10-2

o T o T 1 T o F1 o T F2 o T + + +=

o T b1T1–

b2T2 3–

b3T1 3–

b4 b5T1 3

b6T2 3

+ + + + +=

b7T b8T4 3

b9T5 3

+ + +

1 T 1.6969859 0.13337235 1.4 T168---------ln–

2–=

o T exp j1–j4

T----+

=

exp o0.1

j2

j3

T3 2

-----------–

o0.5

j5

j6

T----

j7

T----+ +

+ 1.0–



j6 = 1.4529023444x102

j7 = 6.1276818706x103

3-126

By default:

k1 = 9.74602k2 = 18.0834k3 = 4126.66k4 = 44.6055k5 = 0.976544k6 = 81.8134k7 = 15649.9

3-127

Coefficients j1 to j7 in Equation 3-125 are obtained from a least-squares fit to the methane viscosity data at temperatures above the freezing temperature of methane (set ”= 0), while k1 to k7 in Equation 3-126 are determined from temperature below the freezing point (set ’= 0). To ensure continuity between viscosities above and below the freezing point, two weighting factors F1 and F2 were introduced:

3-128

3-129

3-130

o T exp k1–k4

T-----+

=

exp o0.1

k2k3

T3 2

-----------–

o0.5

j5k6

T-----

k7

T2

-----+ +

+ 1.0–

o co–

co-------------------=

F11 HTAN+

2-------------------------=

F21 HTAN–

2-------------------------=

HTAN exp T exp T– –exp T exp T– +----------------------------------------------------=



3-131

where TF is the freezing point of methane. The units in Equations 3-122 through 3-130 are gm/cc for density, °K for temperature, and micropoise for viscosity.

To invoke the Pedersen viscosity option, the user enters keyword VISPE in the PVT methods file. The user may optionally specify binary interacting coefficients Xij in Equation 3-114 (by using the VISKKIJ table) to calculate the mixture critical temperature. If the user does not use the VISKKIJ table, Xij are taken to be zero. Additionally, the user optionally may override the default values of j1 to j7 in Equation 3-125 (by entering the VISKJ table) and k1 to k7 in Equation 3-126 (by entering the VISKK table).

Principal Algorithms

Nexus employs several major EOS algorithms which are at the heart of the simulator calculations. The following sections review the major algorithms used.

Two Isothermal Phase Flash Calculation

Given a particular temperature, pressure, and overall composition, the two-phase isothermal flash yields the phase split of the two phases, and the composition of each phase. The flash calculation works very well when two phases exist and there is a good initial estimate of the compositions in each phase. It is however unreliable as a predictor of whether two phases or a single phase exist when the initial composition estimates are poor.

When two phases are in equilibrium, the fugacities of each component in each phase are equal.

3-132

In Nexus, the flash calculation is converged when

3-133

T T TF–=

fo i fg i= i 1 nc =

1fg i

fo i-------–

2

i 1=

nf

TOL



where TOL is some tolerance. The number nf is by default equal to nc, the number of hydrocarbon components. However, the user can specify nf to be less than nc.

The Successive Substitution Flash

In Nexus, when a flash calculation is solved (except for a gridblock flash that is initially in a two-phase state) a two-step process is followed. First, the flash is solved using the successive substitution algorithm to a tolerance that is greater than the final tolerance. The successive substitution flash is more robust and has a better change of converging even when the initial guess is poor.

For the successive substitution calculation, initial K-values must be set. The K-values are defined as

3-134

Then, the Rachford-Rice 12 equation is solved for the vapor phase mole fraction (fv)

3-135

The individual phase compositions are updated during each iteration by

3-136

3-137

Then, the fugacities are updated by the combination of Equation 3-99 and Equation 3-101. The K-values are then updated by

3-138

Kiyi

xi----= i 1 nc =

g fv yi xi–

i 1=

nc

zi Ki 1–

1 fv Ki 1– +--------------------------------

i 1–

nc

0= = =

xizi

fv Ki 1– 1+--------------------------------=

yi Kixi=

Kin 1+

Kin fo i

fg i-------

n=



The Newton-Raphson Flash

After the the successive substitution algorithm reaches its tolerance, the flash is solved by the Newton-Raphson method.

The following matrix system is solved iteratively.

3-139

where

3-140

Compositions are updated as follows

3-141

3-142

3-143

Then, g and its partial derivatives are updated, and the Newton-Raphson iteration is repeated until convergence is reached.

y1g1

y2g1

ync 1–g1

fvg1

y1g2

y2g2

ync 1–g2

fvg2

: : : :

. . . .

y1gnc

y2gnc

ync 1–gnc

fvgnc

y1

y2

:

ync 1–

fv

g1

g2

:

.

gnc

–=

gifo i fg i–

1 fv–--------------------------=

yin 1+

yin yi+= i 1 nc 1– =

yncn 1+

1 yin 1+

i 1=

nc 1–

–=

xin 1+ zi fv

n 1+yi

n 1+–

1 fvn 1+

–---------------------------------= i 1 nc =



Solution in Two-Phase Gridblocks

In gridblocks that begin an outer iteration already in a two-phase state, a special flash calculation is performed. In the course of the construction of the Jacobian, the following matrix system is formed:

3-144

where g is defined by Equation 3-140.

Forward elimination is carried out on the above system in the process of forming the Jacobian that is passed to the solver. The solver returns with solutions for . These are back-substituted into the above equation to yield

.

If the resulting solution meets the flash convergence criteria, the flash is avoided altogether. Moreover, the partial derivatives of g which are utilized in the construction of the matrix system sent to the solver are left unchanged.

y1g1

y2g1

ync 1–g1

fvg1

z1g1

z2g1

znc 1–g1

pg1

y1g2

y2g2

ync 1–g2

fvg2

z1g2

z2g2

znc 1–g2

pg2

: : : : : : : :

. . . . . . . .

y1gnc

y2gnc

ync 1–gnc

fvgnc

z1gnc

z2gnc

znc 1–gnc

pgnc

y1

y2

:

ync 1–

fv

z1

z2

:

znc 1–

p

g1

g2

:

.

gnc

–=

z1 z2 znc 1– and p,,,

y1 y2 ync 1– and fv,,,



Variable Reduction

In solving the Newton-Raphson flash, it is evident that as the number of components grows, so does the size of the Jacobian and of the computational effort. Within Nexus, the variable reduction method13 has been applied to avoid this problem. Instead of using phase mole fractions as the primary variables of the flash calculation, a smaller set of variables are defined that will yield a solution that provides unique phase mole fractions.

The new variable set is derived from consideration of the mixing rule for EOS parameter a, as stated in Equation 3-84. The key is to recognize that the term is symmetric. That means that the matrix can be expressed as

3-145

where is a diagonal matrix given by

3-146

whose diagonal elements are the eigenvalues of , and is the matrix of the eigenvectors of .

If we define

3-147

where is the k’th eigenvector of , then it can be shown that

3-148

This is useful because only a few (usually 3 to 5) of the values of dk are of significant size. If m is the number of significant eigenvalues, then

ij 1 ij– =

SDS1–

=

D

D

d1 0

: .di :

0 dnc

=

S

Qk xiqk i

i 1=

nc

=

qk

a dkQk2

k 1=

nc

=



3-149

It can be further shown that the Z-factors and the fugacity coefficients can also be expressed as functions of Qk (k = 1,...,m), and the EOS parameter b (when the simple mixing rule for b is used) so that

3-150

and

3-151

Using this method, the number of primary variables for the Newton-Raphson flash calculation is reduced from nc to m+1.

Unfortunately, there is some overhead cost in the conversion of the Q variables to the usual compositional ones. At this time, this calculation is only implemented for the flash calculations of the surface network calculations of the simulator. Obviously, this reduction factor could be extended for use with the calculation of any thermodynamic quantity.

a dkQk2

k 1=

m

Z Z Q1 Qm b

i i Q1 Qm b



Saturation Pressure Calculation

Given a fluid composition and a temperature, the saturation pressure calculation finds the pressure(s) where the fluid mixture is at equilibrium with an infinitesimal amount of an incipient phase. If the fluid is initially a liquid, a bubble point is found. If it is initially a vapor, a dew point is found. When a phase boundary is drawn on a pressure-temperature diagram, the saturation pressure is the pressure corresponding to each temperature on the two-phase envelope.

The conditions for the solution of a saturation pressure are the equilibrium of fugacities, Equation 3-132, and the sum of the mole fractions of the incipient phase sum to 1.

Unfortunately, the trivial solution, a solution where an incipient phase has the same composition as the original fluid, has the strongest attraction. Saturation pressure calculations often fail because of convergence to the trivial solution. This is especially true around the critical point, because the differences in equilibrium phase compositions are very small. Initiating a saturation pressure algorithm with K-values that are close to the true ones provides the best chance at finding the true solution. In Nexus, the initial K-values may be provided by a Gibbs stability analysis (See next section).

In Nexus, the saturation pressure calculation may be calculated by a successive substitution algorithm accelerated by GDEM14 (general dominant eigenvalue method) proposed by Michelsen15 or by a Newton-Raphson algorithm.

Gibbs Stability Analysis

Nexus employs the Gibbs tangent plane analysis proposed by Michelsen16 to determine whether a hydrocarbon mixture at a particular temperature and pressure is more stable in a single phase state or in a two-phase state.

This calculation is more reliable than the saturation pressure calculation around the critical region, and is not as dependent on having good initial K-value estimates.

Simply put, the calculation determines the tangent plane of the Gibbs energy surface at the mixture composition and parallel tangent planes at possible incipient phase compositions. If any of the parallel tangent planes lie below the tangent plane of the mixture composition, a two-phase state will exist. To avoid testing every possible composition for tangent plane location, the stationary points of the tangent plane distance are calculated.



These stationary points are located at compositions x satisfying

3-152

where k is a constant that is equal to tangent plane distance (TPD), is the overall composition, and is the composition of the incipient phase.

In Nexus, Equation 3-152 is solved for two branches. In one branch, the initial starting point assumes the fluid is a liquid, and the search is for an incipient vapor phase. The phases are flipped in the second branch. The Gibbs analysis will yield different results depending on the position of the phase envelope.

Referring to Figure 3.3, point 1 is within the two-phase envelope. The mixture composition is unstable. Equation 3-152 will have either one or two solutions in which k is negative. One of the solutions may be a trivial solution. Point 2 is right on the phase boundary and one solution will be trivial and one will have a solution with k equal to 0. Point 3 is in the single phase region, between the phase boundary, and the dotted line, which will be in close proximity to the phase boundary. We could call this region the near phase boundary region. A stability test at this point will yield one branch with a trivial solution and one branch with a positive value of k. Point 4 is in the single phase region and outside of the near phase boundary region. Both branches of the stability test will yield a trivial solution. With our example, all the points are on the bubble point side, so trivial solutions will exist in the branch that starts with a liquid-like estimate for the incipient phase composition.

It is evident that while the Gibbs stability test can identify whether the phase state is single- phase or two-phase, in the vast majority of cases that are in a single phase state, the outcome will be two trivial solutions. When this happens, no useful information can be gleaned to indicate whether the mixture is a liquid or a vapor phase, and no estimates of K-values will be produced for a saturation pressure or flash calculation.

xi i x zi i z ln–ln–ln+ln k=

zx



Figure 3.3: Phase Diagram for a Typical Oil

T or x

1

23

4

x

x

x

x



The Search for a Two-Phase State

Significant flexibility exists in Nexus in the search for the transition of a single phase state to a two-phase state. The most important choice is the detection method used. Additionally, for reservoir calculations, there are different methods available for choosing which gridblocks to test.

Transition Test Technique

There are three possible methods available in Nexus to perform the transition test. The default one is called the increasing pressure, which uses a series of flash calculations. A second method is the saturation pressure method, which is similar to the standard technique used in VIP. A third method is the Gibbs stability analysis which is similar to using the Gibbs option in VIP.

Increasing Pressure Method

The increasing pressure method performs a sequence of flash calculations at the same temperature and compositions but at different pressures. As discussed above, the two-phase isothermal flash is not very reliable unless the initial K-values are good. It is easy for the flash to converge to a single phase state even when two phases should exist. As one moves away from the phase boundary within the two-phase region, the flash calculation becomes more and more reliable. Also, at low pressures, K-values can be estimated well by the Wilson correlation:

3-153

Therefore, the increasing pressure starts by steadily decreasing the pressure and performing a flash until a two-phase region can be found. The initial K-values are obtained either from a good saved value of K-values (from previous iterations) or from the Wilson K-value estimate. If no two-phase region can be found when pressure reaches standard pressure, it is concluded that the system is in a single phase state. If a two-phase region is found, the pressure is steadily increased back towards the original pressure. At each step, K-values are estimated from the previous step by

Ki

pci 5.37 1 i+ 1Tci

T-------–

exp

p---------------------------------------------------------------------------=



3-154

where

3-155

Of course, the trick is to not increase the pressure too quickly. If that happens, the extrapolation may yield a poor result. This is especially a problem in the critical region where K-values can change very rapidly with small changes in pressure.

To help take smaller increments around the phase envelope, a parameter PHASEFRAC is defined. If the flash finds a two-phase state at 0.999 times the pressure, while the vapor phase mole fraction is either equal to PHASEFRAC or (1 - PHASEFRAC), and then in the next step finds a single phase state at the gridblock pressure, then the increasing pressure calculation is retried starting at 0.9993 times the pressure with another 4 increments to reach the gridblock pressure. The default value of PHASEFRAC is 0.05.

We see that the increasing pressure method can be performed in a manner that is very robust, but the cost can be excessive when we are near a phase boundary, and especially around the critical region.

Figure 3.4 illustrates the increasing pressure mechanism. A series of flashes is conducted along the vertical line. The first flash is performed at a pressure below the initial pressure. If it does not yield a two-phase state, the pressure is decreased until an eventual solution is found that yields a two-phase state. This pressure may be much lower than the initial flash pressure. Then, the pressure is built back up to the target, which is the gridblock pressure. On the way up, good K-values are estimated by the extrapolation. In the figure, it would eventually find that conditions are in a single phase state after all.

Kin 1+

Kin p

pn

-----n 1+

r

=

r Kn 1–

Kn ln

pn 1–

pn ln

----------------------------------=



Figure 3.4: Illustration of Increasing Pressure Method

Saturation Pressure Method

An obvious way to use the saturation pressure is to compare it with the gridblock pressure. If the gridblock pressure is below the saturation pressure and the saturation pressure is a bubble point, then a two-phase state exists. If the saturation pressure is a dew point, we really can’t be sure which state we are in without knowing whether we have an upper or lower dew point. Usually the lower dew point is much below reservoir pressure, and we can safely assume that we are in a two-phase state if the pressure is below the dew point pressure. This is the assumption made in both Nexus and VIP.

For any fluid, a region will exist near the critical region in which it is nearly impossible to find the saturation pressure because of the attractiveness of the trivial solution.. This region is illustrated in Figure 3.5. In Nexus, when this occurs, the simulator switches to an increasing pressure method.

The saturation pressure can be more computationally efficient than performing a series of flashes when it can be found. Also, it provides a reliable mechanism for determining whether a single phase fluid is a liquid (because it finds a bubble point) or a vapor (because it finds a dew point).

first pressure that yields two phases

target pressure

initial flash pressure

T or x



The success of the saturation pressure calculation depends on starting with good K-values. Sometimes the stability test can provide K-values near the critical region.

Figure 3.5: Illustration of Region Where No Saturation Pressure Can Be Found

Gibbs Stability Analysis

As discussed above, the Gibbs stability analysis is a good mechanism for determining whether we have a two-phase state or a single phase state. It is more reliable than either of the other two tests, but often doesn’t yield any information about the whether a single phase is a liquid or a vapor. When both branches of the stability test yield a trivial solution, Nexus conducts an increasing pressure calculation to identify the single phase state.

Choice of Gridblocks for Application of Transition Test

The transition test for a gridblock going from a single phase to a two-phase state is one of the most expensive in reservoir simulation using an EOS. Thus, for computational efficiency, it is advantageous to avoid testing every gridlbock in every timestep.

There are five possible mechanisms that control the gridblock choice. These are all specified by the TRANSITION keyword under the EOSOPTIONS keyword.

T or x

region where saturation pressure cannot be found



The default mechanism is referred to by the keyword TEST. The TEST algorithm will only test for a transition in gridblocks that contain perforations or in single phase gridblocks that are adjacent to two-phase gridblocks or single phase blocks that are of a different phase. This mechanism may miss many blocks that should undergo phase transition if large timesteps are taken in implicit mode or if there is a general drop in pressures below the fluid saturation pressure.

A slight variation of this method is activated by the keyword NEIGHBOR. The NEIGHBOR method tests the same gridblocks that the TEST method would test. However, it uses the K-values of neighboring two-phase gridblocks to initiate the transition test.

The DELTA method will result in more gridblocks being tested than the default method, and will be more costly computationally. It may be more appropriate depending on the operational mechanism. It is also much more similar to the criteria used in VIP. In addition to all blocks that would be included under the criteria established by TEST, the DELTA method evaluates several criteria in each gridblock:

1. If a saturation pressure is available, a transition test will be performed in gridblocks where the pressure has dropped below the saturation pressure.

2. A transition test will be performed in gridblocks if pressure has dropped by more than input parameter TDELP (default value of 0.05) multiplied by the gridblock pressure.

3. A transition test will be performed in gridblocks if the overall mole fraction change of any component in the gridblock exceeds the input parameter TDELZ (default value of 0.001).

4. A transition test will be performed in gridblocks that judged to be near critical. The test for criticality is

3-156

where TCRIT is a user defined parameter that has a default value of 0.15.

xi yi– xi

i 1=

nc

xi2

i 1=

nc

--------------------------------- TCRIT



The B method17 is similar to the DELTA method but additionally it uses a parameter called the "b-parameter" to adjust the tests for their location with respect to the phase envelope. In the section describing the Gibbs stability test, the existence of a near phase boundary region that exists just outside of the phase boundary in the single phase region was discussed. This near phase boundary region approximately follows the contours of the smallest eigenvalue of a matrix of parameters B, whose elements are defined as

3-157

Let’s denote the smallest eigenvalue of B as b. At the limit of intrinsic stability and at the critical point, b equals 0. Inside the phase envelope, b is mostly negative. If b is negative, we are in the two-phase region. Outside the phase envelope, b is always positive. The value of b becomes progressively more positive as the distance above the phase envelope increases. The absolute value of b is proportional to the square of the distance from the critical point. Thus it is possible to use b to scale the changes required to trigger a phase transition test.

In Nexus, b is calculated and then used in the next outer iteration. It can be seen as a smarter variation of the DELTA method. The drawback is the additional overhead required to calculate b, and the explicit use of b.

The transition test is triggered in any gridblock where any of the following criteria are met:

1. Any gridblock that would be tested according to the criteria of the TEST or NEIGHBOR method.

2. A gridblock that has a negative value of b.

3. If a saturation pressure is available, a gridblock where the pressure has dropped below the saturation pressure.

4. A gridblock where pressure has dropped by more than input parameter TDELP (default value of 0.1) multiplied by the gridblock pressure and multiplied by b. This means that near the critical point and the phase boundary, smaller changes in pressure will trigger the transition test.

5. A gridblock where the change in the overall mole fraction of any component exceeds TDELZ (default value of 0.1) multiplied by b.

Bij ij ninj nj iln +=



Near the critical point and the phase boundary, smaller changes in composition will trigger the transition test. The default value of TDELZ is larger than when used in the DELTA criteria, because near the phase envelope b is usually less than 0.1.

Finally, the most extreme selection method available is the ALL criteria. This forces every single phase gridblock to undergo a phase transition test. While this of course is very costly, it ensures that all gridblocks that should switch to a two-phase state do so. It provides a standard by which the effectiveness of the other criteria can be judged.



Single Phase Identification

As noted above, in the course of a two-phase transition test, a single phase state may be detected, and the single phase will be labelled as either a liquid or a vapor. However, there are some cases where the phase labelling may fail or be wrong. For example, if the temperature is greater than the cricondentherm, no saturation pressure can be found, the Gibbs stability will yield a trivial solution, and the flash will not converge.

Nexus allows the user to use the PHASID keyword (which is a sub-keyword to the EOSOPTIONS keyword) to give the simulator some guidance in single phase labelling. The possible options are PREVIOUS, DENSITY, FLASH, OIL_INITIAL, GAS_INITIAL, OIL, GAS, or PSAT. Another alternative is to use the Li pseudocritical temperature method18 which is triggered by entering the Li factor in the equilibrium data.

Using the PREVIOUS option, the gridblock retains the same single phase label that was in previous existence.

Using the DENSITY option, a density value is specified. If the phase mass density is less than the specified density, the phase is labelled as a gas. Otherwise it is labelled as an oil.

Using the FLASH option, if an increasing pressure was not already applied, the increasing pressure method will be employed.

The OIL_INITIAL option only applies during initialization. All single phase gridblocks initially will be labelled as oil.

The GAS_INITIAL option only applies during initialization. All single phase gridblocks initially will be labelled as gas.

Using the OIL option will force all single phase gridblocks to be labelled as OIL.

Using the GAS option will force all single phase gridblocks to be labelled as gas.

Using the PSAT option, if a saturation pressure calculation has not already been performed, one will be done to detect a dew point or bubble point.

Using the Li method, the Li pseudo-critical temperature (Tpc) is calculated in each gridblock using the following formula:



3-158

where Vci is the component critical volume.

If the reservoir temperature is less than Tpc, the fluid is labelled as an oil. Otherwise, it is labelled as a gas. Obviously, using a true critical temperature would be preferable, but the Li pseudocritical temperature consumes only a small fraction of the effort required to calculate the true critical temperature.

The true critical temperature may differ significantly from the actual critical temperature. The discrepancy can be mitigated by the introduction of a Li multiplication factor. The multiplication factor is applied to the pseudo-critical temperature before its comparison with the reservoir temperature. Therefore, if the multiplication factor is less than 1, Tpc will be shifted to the left, and single phase blocks will be more likely to be labelled as a gas. If the multiplication factor is greater than 1, Tpc will be shifted to the right, and single phase blocks will be more likely to be labelled as oil.

Surface Separator Calculations

Surface separator calculations use one of two methods. In the usual method, a multistage configuration is specified with the temperature and pressure set for each stage, and a flash calculation performed for each stage to determine the phase splits and the phase compositions. The liquid phase density can be calculated from performing an EOS calculation with the composition of the output liquid at standard temperature and pressure. Alternatively, the Standing-Katz method19,20 can be applied to calculate the surface density.

Recognizing that it may be difficult to create a single EOS that is accurate for a wide range of conditions (especially temperatures), a different EOS method may be specified for each stage.

Tpc

ziVciTci

i 1=

nc

ziVci

i 1=

nc

----------------------------=



Multi-stage Separator

The multi-stage separator capability in Nexus is similar to that found in VIP. The user can specify any number of stages. The temperature and pressure of each stage is set. The upper and lower outflow streams can be split and the split streams can be assigned to other stages or to the outlet.

VIP and Nexus allow a rather simple configuration of a separator battery consisting of several separator stages connected in sequence. A sample configuration is shown in Figure 3.6. Each stage has one feed stream and two output streams, one vapor and one liquid. Specified fractions of the vapor and liquid streams from a separator stage are sent to the battery gas and oil sales lines. The remaining fractions of the vapor and liquid streams are sent to downstream separator stages.

Figure 3.6: Sample Separator Battery Configuration

The configuration of a separator battery does not allow recycling of streams.

The feed stream (moles of hydrocarbon component i) to the first separator stage (N1,i) is known and is equal to the battery feed. Here, the first subscript is used as the notation for the stage number, while the second subscript is used for the hydrocarbon component number. The feed stream to the last stage ns is the sum of the vapor and liquid streams sent to this stage from all preceding stages:

3-159

Stage

1Stage

Ns

Gas Sales Line

Oil Sales Line

Feed

Liquid

StreamVapor

Stream

Stage

2Stage

3

Nn i Fmnl

Nmil

Fmnv

Nmiv

+

m 1=

n 1–

=

i 1 = 2 nc n 2 3 ns =



Here, Flmn and Fv

mn are fractions of the liquid and vapor streams that are sent to stage n from stage m. These fractions are defined by a user on the SEPARATOR card. The sum of the vapor and liquid streams in moles for component i, Nv

mi and Nlmi, respectively, which leaves stage

m, is equal to the feed stream to this stage:

3-160

Moles that are sent to the oil and gas sales lines, No and Ng, are equal to the sum of all streams sent to these lines from all the separator stages:

3-161

Here, Flno and Fv

ng are fractions of the liquid and vapor streams, which are sent to the sales lines. They are defined by a user on the SEPARATOR card.

The main difference between the separator treatment in the two models is the much higher level of implicitness in Nexus. The VIP separator calculation is explicit and the conversion factor from moles to standard volumes is lagged an iteration. The Nexus calculation is fully implicit.

In Nexus, the implicit separator calculation is based on forming two matrices for each stage n

3-162

Nmi Nmiv

Nmil

+ i 1 2 nc n 1 2 ns = = =

No Fnol

Nnil

Ng

n 1=

ns

i 1=

nc

Fngv

Nniv

n 1=

ns

i 1=

nc

= =

Nn 1Nn 1

v

Nn 1nc

Nn 1v

: :

Nn 1Nn nc

v

Nn nc

Nn ncv



and

3-163

For the final oil and gas output streams, the following vectors are also formed:

3-164

and

3-165

where qgas is the volumetric flow rate of produced gas in standard volumes, and qoil is the volumetric floweret of produced oil in standard volumes.

Products and sums of the above equations are combined according to the separator stage configuration to produce the following resulting vectors (which are the derivatives of output volumetric rates with respect to the molar feed rates)

3-166

Nn 1Nn 1

l

Nn 1nc

Nn 1l

: :

Nn 1Nn nc

l

Nn nc

Nn ncl

Ng 1qgas

Ng ncqgas

T

No 1qgas

No ncqgas

T

N1 1qgas

N1 ncqgas

T



and

3-167

Standard Conditions

For a multistage separator to behave properly, the last stage of the separator should be at standard conditions of temperature and pressure. As discussed in the keyword manual, if the standard conditions are different than the last stage separator conditions for any separator, the simulator will try to reconcile the differences if they are small enough. It will also try to reconcile the differences between standard conditions specified in different reservoirs.

If the standard conditions and last stage separator conditions are different in a VIP dataset, the converted dataset may have difficulty obtaining the same volumetric outputs in Nexus. That is because the simulators treat the discrepancy inconsistently. In both simulators, the gas volumes will be calculated at standard conditions, regardless of the separator conditions. In VIP, the produced oil volume is calculated using the Z-factor calculated at last stage conditions but with standard temperature and pressure. In Nexus, the produced oil volume will be calculated with the Z-factor, pressure, and temperature all evaluated at last stage separator conditions.

The following set of rules applies to both the standard temperature and standard pressure.

If a standard condition of the reservoir has been explicitly set in the options file, then that standard condition is honored for that reservoir. (Note: Use of TSTD or PSTD keywords for the individual reservoirs will set the standard conditions for that reservoir. The standard condition can also be set by the use of TSTD ALL or PSTD ALL in the master case options file.). The last stage condition of each separator belonging to that reservoir is checked against the standard condition. If the difference is within a tolerance, the last stage condition of the separator is reset to the standard condition. If the difference is greater than the tolerance, a warning message is issued.

N1 1qoil

N1 ncqoil

T



If the standard condition of the reservoir has not been set, then the condition of the last stage of each separator in the reservoir is checked. If the conditions are all identical and the difference between the common condition and the default standard condition is within a tolerance, then the standard condition is reset to the common last stage separator condition. If the conditions are all identical but the difference is not within the tolerance of the default standard condition, a warning is issued. If the conditions are not all identical, then the last stage condition of each separator is checked against the default standard condition. If the difference is within the tolerance, the last condition of the separator is reset to the default standard condition. If the difference is not within the tolerance, a warning is issued.

In addition, if for the master case, the standard condition has not been explicitly set with the keyword in the options file, and the standard condition of all the reservoirs is the same, then the standard condition of the master case is set to the common value.

In the above rules, the following set of tolerances applies:

• For standard temperature and field units, the tolerance is 1.0 degrees F.

• For standard temperature and all other unit systems, the tolerance is 0.5 degrees C.

• For standard pressure and field or lab units, the tolerance is 1.0 psia.

• For standard pressure and metric units, the possible tolerances are 7.0d0 kPa, 0.07 bars, and 0.07 kg/cm2.

• The default standard temperature is 60 degrees F in field units, and 15 degrees C for all other unit systems.

• The default standard pressures are 14.696 psia for field and lab units, and 101.325 kPa or 1.01325 bars or 1.03321 kg/cm2 for the various metric unit systems.



Standing-Katz Correlation

The user can choose to apply the Standing-Katz correlation to calculate the hydrocarbon liquid surface density instead of using the result of the equation of state. The correlations were originally developed as a graphical procedure by Standing and Katz19 to determine the liquid phase density. Various authors have fitted the curves with programmable formulas. The formulas used in VIP and Nexus are similar to but not identical to those reported by Pedersen et al.20

Nexus uses the same mathematical equations as VIP. However, the Nexus implementation is much more flexible in the way it identifies the composition of the oil stream.

The procedure involves first determining the standard mass density of a liquid which contains pseudo-components representing C3+ and H2S. Then, a correction is applied for the presence of C2. Another correction is applied for the presence of CO2. A third correction is applied for the presence of C1 and N2. This is further corrected for pressure, and then for temperature.

In VIP, the mole fractions of the C2, CO2, and C1+N2 are determined by evaluation of the molecular weights of the pseudo-components. VIP uses the following rules:

1. It assigns pseudo-components which have molecular weights of 30.07 +/- 0.01 (representing C2H6) to the C2 group.

2. The simulator will assign pseudo-components which have molecular weights of 44.01 +/- 0.01 (representing CO2) and 64.06 +/-0.01 (representing SO2) to the CO2 group.

3. The simulator will assign pseudo-components which have molecular weights of 2.016 +/- 0.01 (representing H2), 16.043 +/- 0.01 (representing C1), 28.01 +/- 0.01 (representing N2 or CO), and 32.00 +/- 0.01 (representing O2) to the C1-N2 group.

The user can allow Nexus to use the same rules, or the user can explicitly assign mole fractions of each pseudo-component to be a part of the C2 group (FC2 ), the CO2 group ( FCO2) and the C1+N2 group ( FC1N2 ), by entering columns of data in the PROPS table with headers STK_C2, STK_CO2, and STK_C1N2 respectively. These fractions should be between 0. and 1, and for any one pseudo-component must sum to be less than or equal to 1. That is,

3-1680 FT i FC2 i FCO2 i FC1N2 i 1 i=1,...,nc+ +=



When using the Standing-Katz procedure, the user can choose to either enter none of these columns or all three of these columns. If the three columns are entered, the sum of the values for a particular component must be positive and less than or equal to 1. Each entry must be between 0 and 1. For example, a pseudocomponent could represent C1 (83 mole percent), N2 (5 mole percent), and CO2 (10 mole percent), and H2S (2 mole percent). Then, the value of STK_C1N2 should be 0.88, the value of STK_CO2 should be 0.10, and the value of STK_C2 should be 0.0.

The implementation of Standing-Katz in VIP and Nexus is similar to that published by Pedersen. However, it is not identical. The exact algorithm has not been previously published, so they are outlined below:

Step 1: Uncorrected Density

In the first step, the mass density at standard conditions of the fraction comprised of C3+ and H2S components is calculated using the input values of the component specific gravities:

3-169

where

Mi is the molecular weight of component i,

xi is the mole fraction of component i,

is the specific gravity of component i from the STKATZ column of the PROPS table.

FTi is the fraction of the pseudo-component assigned to the combined C2, C1+N2, and CO2 groups.

1

1 FTi– xiMi

i 1=

nc

1 FTi– xiMi

0.9991i---------------------------------

i 1=

nc

------------------------------------------=

i



Step 2: Correction for C2

First, the weight fraction of C2 (not considering the CO2 and C1N2 groups) is calculated:

3-170

Then, density is adjusted by:

3-171

3-172

Step 3: Correction for CO2

3-173

wC2

FC2 i xiMi

i 1=

nc

1 FCO2– FC1N2– xiMi

i 1=

nc

--------------------------------------------------------------------=

2 1 wC2 0.3158 0.2583 1 2u– – 0.01457 1 6u u 1– + + –=

u 2.5 1 0.46– =

3

FCO2 i xiMi 1 F– CO2 i FC1N2 i– xiMi

i 1=

nc

+

i 1=

nc

FCO2 i xiMi

i 1=

nc

0.8213

-------------------------------------

1 F– CO2 i FC1N2 i– xiMi

i 1=

nc

2

---------------------------------------------------------------------------+

------------------------------------------------------------------------------------------------------------------------=



Step 4: Correction for C1-N2

First, the weight fraction of the C1-N2 group is calculated:

3-174

Then, the density is corrected by

3-175

where

3-176

3-177

3-178

Step 5: Correction for pressure

The correction is as follows:

3-179

where

wC1N2

FC1N2 i xiMi

i 1=

nc

xiMi

i 1=

nc

---------------------------------------=

4

3 0.088255 0.095509c1– 0.007403c2 0.000603c3–+ –

0.142079 0.150175c1– 0.006679c2 0.001163c3+ + 3 0.65– –=

c1 1 5wC1N2–=

c2 1 30wc1N2 1 5wC1N2– –=

c3 1 5wC1N2 12 150wC1N2– 500wC1N22

+ –=

5

4 0.034674 0.026806p1 0.003705p2 0.000465p3+ + + –

0.022712– 0.015148p1 0.004263p2 0.000218p3+ + + 1 2u– –

0.007692– 0.003521p1 0.002482p2 0.000397p3+ + + 1 6u u 1– + –

0.001261– 0.000294p1– 0.000941p2 0.000313p3+ + 1 12u– 30u2

20u3

–+ –

=



3-180

3-181

3-182

3-183

3-184

and p is the pressure in psia.

Step 6: Correction for temperature

The correction is as follows:

3-185

where

3-186

3-187

3-188

3-189

p1 1 2g–=

p2 1 6g 1 g– –=

p3 1 g 12 30g– 20g2

+ –=

u 2.5 4 0.48– =

g 0.0001 p 500– =

6

5 0.055846 0.060601t1– 0.005275t2 0.00075t3–+ –

0.037809 0.049262t1– 0.012043t2 0.000455t3+ + 1 2v– –

0.021769 0.032396t1– 0.011015t2 0.000247t3+ + 1 6v v 1– + –

0.009675 0.015500t1– 0.006520t2 0.000653t3–+ 1 12v– 30v2

20v3

–+ –

=

t1 1 2h–=

t2 1 6h 1 h– –=

t3 1 h 12 30h– 20h2

+ –=

v 2.5 5 0.52– =



3-190

and t is the temperature in degrees F.

Gas Plants

Gas plants can be used as an alternative to multistage separator calculations. While the actual calculation is much simpler, the result can be used to represent a complex surface gas plant that cannot be represented by the simple multistage separator configuration.

This method uses liquid molar recovery fractions for each component, input as a function of a key component plus composition in the well stream. The key component plus composition is defined as the sum of the over-all mole fractions for the components listed under the KEYCPTLIST keyword.

The interpolated values of the liquid recovery fractions multiplied by the overall composition for each component are used to obtain the

h 0.005 t 60– =



produced liquid composition. Figure 3.7 illustrates how the production stream compositions are calculated using the input recovery factor

Figure 3.7: Illustration of Recovery Calculations Used in Gas Plant



EOS parameters are then used to compute liquid densities. Liquid density plus surface total molar production rate for the liquid provides the standard surface rate. Gas composition and, thus, the densities are determined by the difference of the overall composition and liquid composition. Again, the total molar production rate of gas and the density provide the standard surface production rate.

Calculation of Volume Derivatives

When using the EOS model, volume derivatives are required with respect to total moles and pressure in the isothermal case.

In a single phase state, the volume derivative calculations are trivial. The EOS calculations yield density derivatives for each phase and volume equals moles divided by density.

In a two-phase state, the volume derivatives with respect to pressure and overall moles must be solved simultaneously taking into account how equilibrium phase compositions are affected by these changes.

For example for the oil phase, a following system of equations is constructed

3-191

where g is defined by Equation 3-140 and Vo is the oil phase volume.

Forward elimination of columns with derivatives of yi and fv will yield a last row with the derivatives of Vo with respect to zi and p. A simple transformation will yield derivatives with respect to Ni and p.

y1g1

y2g1

ync 1–g1

fvg1

z1g1

znc 1–g1

pg1

y1g2

y2g2

ync 1–g2

fvg2

z1g2

znc 1–g2

pg2

: : : : : : :

. . . . . . .

y1gnc

y2gnc

ync 1–gnc

fvgnc

z1gnc

znc 1–gnc

pgnc

y1Vo

y2Vo

ync 1–Vo

fvVo

z1Vo

znc 1–Vo

pVo



A similar system of equations is solved for the gas phase volume.

This system of equations is similar to the one described in detail by Young and Stephensen.21 The proof that the above procedure yields volume derivatives with respect to overall moles is described in detail by Wong et al.22



Three Phase Equilibrium

Nexus can be instructed to calculate the three-phase equilibrium state where hydrocarbon components are weakly soluble in the water phase.

This calculation option is activated by inputting tables of hydrocarbon solubility in water. For each hydrocarbon component that is soluble in water, a table of solubility data can be entered at each level of temperature and salinity. The tables have columns of saturation pressure (PSAT), solubility (RSW), water formation volume factor (BW), water compressibility (CW), water viscosity (VW), and water viscosity variation with pressure (CVW). The properties of pure water are entered for a RSW of 0. The properties of pure water at the same temperature and salinity must be the same for all soluble components. Water properties become a function of the amount of hydrocarbon dissolved in the water phase.

Solubility should be limited to components which exist as gases at atmospheric conditions, such as CO2, N2, H2S, and hydrocarbons lighter than pentane. The combined solubility of all non-water components in the water phase should not exceed a few percentage points.

It is also assumed that the solubility of one component has no effect on the solubility of other components. Again, this will only be approximately true at low concentration levels.

Calculation of Water Properties

First, the properties of pure water are calculated from the data input for table values of RSW = 0. For pure water, the input values are saturation pressure ( ), formation volume factor ( ), compressibility ( ), viscosity ( ), and the derivative of visocity with respect to pressure ( ).

The pure water density is calculated by

3-192

The pure water viscosity is calculated by

3-193

psat0 Bw sat0cw sat0 w sat0

wp sat0

w

w std

Bw sat0----------------- 1 cw sat0 p psat0– + =

w w sat0 1 wp sat0 p psat0– + =

104 PVT Representation: Three Phase Equilibrium R5000.0.2


Comparison of the above equations with Equation 3-1 and Equation 3-2 show that the water property calculation for pure water is identical to that of the normal water property calculation if .

For each soluble component, the solubility data are internally converted to tables of , , , , with mole fraction as the index. Density is calculated by:

3-194

where is the mass ratio of hydrocarbon to water, and

Tables can be input for different levels of temperature and salinity. Data is calculated by linear interpolation. However, data is not extrapolated.

Conversion of Solubility Data to Fugacity Coefficient Data

The two-phase equation of state calculations in Nexus use the condition of the equality of fugacities to determine the equilibrium state. To facilitate the calculation of the equilibrium state for a three-phase (oil, gas, and water) system, the tabulated gas solubility data is translated to the fugacity of the hydrocarbon component in the water phase. Chang et al.23 introduced this concept for use with CO2 solubility in water. We generalize this technique to any hydrocarbon component, while ignoring the interaction between the different components. This technique only works on the assumption of very limited solubility of hydrocarbons in water. It should not be used for hydrocarbons with molecular weights greater than pentane. When used with multiple components, the solubility and the swelling of water will be overestimated. It is best used with one or two components.

The phase equilibrium constraint for a non-aqueous component in a binary system of water and hydrocarbon can be stated as

3-195

where is the fugacity of component i in the aqueous phase, and is the fugacity of component i in the hydrocarbon rich phase.

The above equation can be expressed in terms of fugacity coefficients as follows:

3-196

psat0 pref=

psat Bw sat cw w

ww std 1 Rhw+

Bw--------------------------------------=

Rhw

1Bw------

1Bw sat-------------- 1 cw p psat– + =

fi w fi h=

fi w fi h

wii w yii h=

R5000.0.2 PVT Representation: Three Phase Equilibrium 105


where is the mole fraction of component i in the aqueous phase, is the fugacity coefficient of component i in the aqueous phase,

is the mole fraction of component in the hydrocarbon rich phase, and is the fugacity coefficient of component i in the hydrocarbon rich

phase.

If we assume that the hydrocarbon rich phase contains virtually no water, then , and

3-197

The fugacity coefficient in the hydrocarbon rich phase can be calculated from an equation of state given the pressure, temperature, and the composition. For each of the components, we assume that hydrocarbon rich phase is purely composed of that component, so that the compositional dependence is removed. The value of is calculated from the input solubility data. 3-198

Flash Procedure

For each gridblock or connection in the network, the phase equilibrium for a three-phase system must be determined. With our assumptions, no water component is allowed in the liquid or vapor hydrocarbon phases, and some of the non-water components are allowed in the aqueous phase.

In the first step, a two-phase flash with an aqueous phase and a single hydrocarbon phase is performed. This establishes a good initial estimate of the hydrocarbon content in the aqueous phase.

The composition of the hydrocarbon phase is then tested by a combination of Gibbs stability analysis and saturation pressure calculation to determine if it will split to form two hydrocarbon phases. If the hydrocarbon phase does not split, the flash calculation is considered finished.

If the hydrocarbon is determined to split into two phases, a two-phase flash is performed to set the initial compositional estimates of the hydrocarbon phases. The, Nexus employs a three-phase flash using the successive substition technique, followed a three-phase flash using Newton’s method.

Volume derivatives are then determined by solutions of several matrix systems that are extensions of the ones used in the two-phase flash procedure.

wii wyii h

yi 1

i wi h

wi----------

wi

106 PVT Representation: Three Phase Equilibrium R5000.0.2


References

1. Whitson, C. H. and Torp, S.B. "Evaluating Constant Volume Depletion Data", JPT, (March 1983) 610; Trans., AIME, 275.

2. Soave, G., "Equilibrium Constants from a Modified Redlich-Kwong Equation of State", Chem. Eng. Sci. (1972) 27, No. 6, 1197.

3. Peng, D.Y. and Robinson, D.B.: "A New Two-Constant Equation of State", Ind. & Eng. Chem. (1976) 15, No. 1, 89.

4. Robinson, D. B., Peng, D.Y., and Ng, H-Y: "Capabilities of the Peng-Robinson Programs, Part 2: Three-Phase and Hydrate Calculations", Hydrocarbon Process. (1979), 58, 269.

5. Peneloux, A., Rauzy, E., and Freze, R.: "A Consistent Correction for Redlich-Kwong-Soave Volumes", Fluid Phase Equilibria (1982), 8, 7.

6. Jhaveri, B. S. and Youngren, G. K.: "Three-Parameter Modification of the Peng-Robinson Equation of State to Improve Volumetric Predictions", SPHERE, (Aug., 1988) 1033.

7. Edmister, W. C. and Lee, B.I.: Applied Hydrocarbon Thermodynamics, Volume I, 2nd ed., Gulf Publishing Co., Houston, (1984),.

8. Lohrenz, J., Bray, B.G., and Clark, C.R.: "Calculating Viscosities of Reservoir Fluids from their Compositions", J. Pet. Tech., (1964) 16, 1171.

9. Pedersen, K. S., Fredenslund, A., and Christensen, P. L.: "Viscosity of Crude Oils", Chem Eng. Sci., (1984), 39, 1011.

10. Pedersen, K.S., Fredenslund, A., and Thomassen, P..: Properties of Oils and Natural Gases, Gulf Publishing Co., Houston (1989).

11. Hanley, H.J.M., McCarty, R.D., and Haynes, W.M.: Cryogenics (1975), 15, 413.

12. Rachford, H.H. and Rice, J.D.: "Procedure for Use of Electrical Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium", JPT (Oct., 1952), 19.

13. Pan, H. and Firoozabadi, A.: "Fast and Robust Algorithm for Compositional Modeling: Part II-Two-Phase Flash Computations" SPE 71603 presented at the SPE Annual Technical Conference and Exhibition (Sept., 2001), New Orleans.

14. Crowe, A.M. and Nishio, M.: "Convergence Promotion in the Simulation of Chemical Processes-the General Dominant Eigenvalue Method", AICHE J. (1975) 21, 528.

15. Michelsen, M.L.: "Saturation Point Calculations", Fluid Phase Equilibria (1985) 23,181.

16. Michelsen, M.L.: "The Isothermal Flash Problem. Part I. Stability", Fluid Phase Equilibria (1982), 9, 1.

17. Rasmussen, C. P. et al: "Increasing Computational Speed of Flash Calculations with Applications for Compositional, Transient Simulations", SPE 84181 presented at the SPE Annual Technical Conference and Exhibition (Oct., 2003), Denver.

18. Li, C.C.: "Critical Temperature Estimation for Simple Mixtures", Can. J. Chem.

R5000.0.2 PVT Representation: References 107


Eng. Data, (1987) 32(4), 447.

19. Standing, M. B. and Katz, D. L.: "Density of Crude Oils Saturated with Natural Gas", Am. Inst. Min. Metall. Eng., Tech Pub. No. 1397, meeting in Los Angeles,(1941), 159.

20. Pedersen, K. S., Thomassen, P. and Fredenslund, Aa.: "Thermodynamics of Petroleum Mixtures Containing Hydrocarbons. 2. Flash and PVT Calculations with the SRK Equation of Stage", Ind. Eng. Chem. Process Des. Dev., (1984), 23, 566.

21. Young, L.C. and Stephensen, R. E.: " A Generalized Compositional Approach for Reservoir Simulator", SPEJ (Oct., 1983), 727; Trans., AIME, 275.

22. Wong, T. et al.: "Relationship of the Volume Balance Method of Compositional Simulation to the Newton-Raphson Method", SPERE (Aug, 1990), 5, 3, 415.

108 PVT Representation: References R5000.0.2


Chapter 4

Network Calculations

Introduction

The network model in Nexus represents the wellbores, pipelines, and equipment in injection and production networks as a series of nodes, connections between them, connections from nodes to reservoir cells (perforations), and connections to sinks and from sources. Network flow is assumed to be steady state (i.e., no accumulation of fluid in the network), phases are treated as being in equilibrium in the connections, and all phases flow in the same direction in each connection. The network equations consist of perforation rate equations, component mass balances at network nodes, hydraulics equations, network constraint equations, and composition equations. The network constraint equations may include surface or in-situ rate constraints on connections and/or pressure constraints on nodes. The composition equations result from source composition specifications, and from outflows of multiple connections from a node, including nodes representing separator batteries. The primary network variables are the mass flow rate of each component for each perforation, the mass flow rate of each component for each connection, and the pressures at each node. With only a few exceptions, all network equations are treated fully implicitly.

At any point in time, not all of the network equations are active. For example, consider a well connection which has a maximum rate constraint. If the rate constraint is active, i.e., the well is flowing at its maximum allowed rate, the pressure drop across the connection will be greater than the hydraulic pressure drop (which is calculated using whatever hydraulics method is specified for the connection, i.e hydraulics table, pressure drop correlation, hydrostatic gradient etc), so the hydraulic equation will be inactive. Note that this implies that there is a flow control device in the connection which is restricting flow, i.e., an implied valve. On the other hand, if the hydraulic equation is active, then the well connection will be flowing at a rate less than the specified rate constraint, and the rate constraint equation is not active.

At the start of each Newton iteration (non-linear iteration of the combined reservoir/network system), the network is first solved using

R5000.0.2 Network Calculations: Introduction 109


fixed pressure and fluid mobilities at the perforated grid blocks as a boundary condition. The purpose of this “standalone” network solution is to determine which equations should be applied during the simultaneous solution of the reservoir and network equations. If there is no targeting, and either no procedures, or the procedures make no changes to the network, then this standalone solve of the network is performed only once per Newton. If the input data includes targeting, or procedures, then several network solves may be necessary.

This is illustrated in Figure 4.1 and discussed in the following two sections.

110 Network Calculations: Introduction R5000.0.2


Network Solution Caused By Procedures

If the user has specified one or more procedures that result in a change to the network, such as a change in a constraint, activating or deactivating connections, changing the hydraulic, PVT or separator methods assigned to a connection etc, the network must be solved again. Note that procedures that return information on the state of the network, such as cum (cumulative surface production or injection), q (component rates), p (node pressures) etc, return the values obtained from the first network solve. The network is not resolved after each change made by the procedures, only after all the procedures are completed. So if updated information is required by subsequent procedure statements, the user must invoke the solve_network function. Procedures are only executed on the first Newton of a timestep, unless one or more connections have been opened to flow. In this case, the estimate of network rates and pressures obtained from the first network solve can be quite inaccurate, because the network is solved assuming a fixed reservoir pressure, which is a very poor assumption when wells first start producing or injecting. To alleviate this inaccuracy, the procedures are executed again on the second Newton.

R5000.0.2 Network Calculations: Network Solution Caused By Procedures 111


Network Solutions Caused By Targeting

Targeting involves satisfying a group constraint (a target) by allocating it between the connections that make up the group. The allocation that each connection gets constitutes a constraint to be applied to that connection, and a change in constraint requires the network to be solved again. In addition, before allocating the group target amongst the connections, the program must know what each connection is capable of producing or injecting. Therefore, the first step of the targeting calculations, is to solve for the network potential. The network potential is another network solve, but with the group targets disregarded. All individual connection (rate) constraints and node (pressure) constraints are still applied. The targeting algorithms then allocate the target constraints to the individual connections, and the network is solved again. In some cases, the allocation might prove to be incorrect, usually because evaluating the hydraulic equations at the reduced flow rates imposed by the allocated constraints, can result in some connections being unable to flow. In this case, the targeting allocation must be repeated, and the network solved yet again.

112 Network Calculations: Network Solutions Caused By Targeting R5000.0.2


Simultaneous Reservoir/Network Solution

After the final standalone network solve, the correct set of active network equations has been determined, including any actions taken by the procedures and any allocated constraints from the targeting. Up to this point, the reservoir conditions have been taken to be fixed and have been applied as a boundary condition at each perforation -- i.e, as fixed grid block pressure and fixed fluid in the perforated grid blocks. To couple the network to the reservoir, the full perforation equations are used. These equations include terms for the perforated grid block pressure and for the dependence of component mobility on both the component masses in the grid block and the grid block pressure. The full system of network and reservoir equations are solved simultaneously in the linear solver. The linear solve is performed iteratively. The network and each grid are treated as separate domains, and the residuals for the equations are updated on each iteration with the contributions from the coupling terms.

Nexus provides several options for solving the network domain, which are controlled by the SOLVER FACILITIES keywords. The default options are NOGRID, DIRECT, which results in the use of an efficient direct solver for the network equations. Usually, the DIRECT solver does not generate any infill, so an ITERATIVE solve will be less efficient. Instead of NOGRID, the EXTENDED ncells option can be selected, in which case the network domain is augmented to include the reservoir terms for at least one grid block (the perforated grid block) for each perforation equation. The number of grid blocks to include is controlled by the ncells parameter, which defaults to 1. Use of the EXTENDED option can speed convergence of the linear system, but at the cost of a more expensive network solve on each iteration. In some cases, when the EXTENDED option is used, particularly with ncells > 1, the ITERATIVE solution method may be faster.

R5000.0.2 Network Calculations: Simultaneous Reservoir/Network Solution 113


Figure 4.1: Sequence of Network Calculations

114 Network Calculations: Simultaneous Reservoir/Network Solution R5000.0.2


Targeting

Targeting provides the capability to satisfy group constraints on production or injection. For example, a gathering center may have maximum water handling capacity of 100000 STBD, which requires the production from the wells feeding into the gathering center to be controlled so that this limit is not exceeded. This is illustrated below.

Figure 4.2: Illustration of Targeting

Nexus allows the input of constraints at any point in the network, so it would be possible to specify a water rate constraint of 100000 STBD on the outlet connection from the gathering center, gc_offtake. However, this implies that production is controlled by a control device located in connection gc_offtake, which would reduce the production rate from all five wells by applying a back pressure. This does not represent a reasonable operating scenario, and also results in the network solution being non-unique when the constraint is active. For example, it may be possible to produce 100000 STBD with fewer than all five wells producing, so that any one well shut in, and the four others producing may satisfy the constraint. For this reason, it is highly recommended that constraints are only used when they control only

R5000.0.2 Network Calculations: Targeting 115


one well. i.e., constraints should be used only in the linear series of connections flowing to or from a single well.

Instead of using a constraint at gc_offtake, a target could be used. For example, the water production target could be apportioned between each of the wells, and controlled by adjusting control devices at the wellhead connection for each well (connections wh1, wh2, wh3, wh4 and wh5). This would result in the program having five degrees of freedom with which to satisfy a single condition, so in addition to specifying the target rate and the controlling connections, the user must also specify a method that will uniquely determine what fraction of the target rate to allocate to each well. For example, to control water production rate, the user might specify that the highest water cut wells are shut in until the target is not exceeded.

The example above illustrates a case where control of the target conforms to the physical layout of the network, but this need not be the case. For example, the off take limit of 100000 STBD could be met by two targets, one of which is controlled by connections wh1 and wh2, and the other by wh3, wh4 and wh5, where the sum of the two targets is 100000 STBD. Nexus accommodates this by allowing targets to be controlled by arbitrary groups of connections, which are defined in connection lists (CONLIST keyword), or well lists (WELLLIST keyword).

Targeting Input

Targets are input in a TARGET table in Nexus. This input provides for a method to determine the target rate (the rate which can not be exceeded), the list of connections which will control the target, and the method which will be used to determine how each connection should be operated (how much of the target should be allocated to each controlling connection). The TARGET table keywords that specify how the target is controlled are the following:

CTRL Specifies what fluid is being controlled. i.e., oil, water gas, liquid, hydrocarbon or total fluid.

CTRLCOND The conditions used when evaluating the control rate, which could be surface conditions, or average reservoir or region conditions (pressure and temperature).

CTRLCONS The list of connections where the control devices are located.

116 Network Calculations: Targeting R5000.0.2


CTRLMETHOD The method used to determine how to allocate the target rate among the control connections. For example, the SCALE method scales back each well in proportion to it’s potential productivity/injectivity. The potential is determined by solving the network in the absence of any targets, but with all other constraints imposed. The possible CTRLMETHODs are discussed in more detail below.

QMIN The minimum control rate allowed for a control connection. Note that the targeting algorithm does not consider any minimum constraints specified for the control connections other than the QMIN value input for that target.

QGUIDE The guide rate used for allocation if the GUIDERATE CTRLMETHOD is specified. The target rate is allocated to the control connections in proportion to the input guide rate, or alternatively, a formula can be used to calculate the guide rates.

RANKDT The minimum time change between reranking of connections if the CTRLMETHOD is one which shuts in connections based on some measure such as water cut, or gas/oil ratio.

The target rate can be a specified value, or it can be computed by the program. The TARGET table keywords that specify how the target rate is calculated are the following:

CALCMETHOD The method used to calculate the target rate. The method can be a specified rate, the sum of flow rates of a specified phase in the CALCCONS list of connections, production or injection of a specified phase from or into a specified region, or a rate sufficient to maintain region pressure or replace voidage. The possible CALCMETHODs are discussed in more detail below.

CALCCOND The conditions used when evaluating the target rate, which could be surface conditions, or average reservoir or region conditions (pressure and temperature).



CALCCONS The list of connections used to calculate the target rate (if applicable to the CALCMETHOD chosen).

VALUE The target rate, if the CALCMETHOD is a specified rate, or a multiplier that is applied to the target rate calculation.

ADDVALUE An amount added to the target rate.

REGION The region used for voidage, or pressure maintenance calculations (if applicable to the CALCMETHOD chosen).

MAXDPDT The maximum rate of change of region pressure versus time if a pressure maintenance CALCMETHOD is chosen.

The target table also allows each target to be given a priority, which can be used to control the order in which the program satisfies the targets.

Target Rate Calculations

The target rate is calculated using the results of the first network solution (if there are no procedures), or the network solution following execution of the procedures. This network solution is at fixed reservoir pressure and perforation fluid mobilities. All recurrent data for the timestep has been processed, so any changes to the network and wells are incorporated, but the previous Newton iterations target allocations are still applied (on the first Newton of the first timestep there are no target allocations). This is an explicit calculation, so for example, if the target rate is calculated from the sum of production of several wells, it will be calculated at the start of the Newton iteration, and will differ to some degree from the sum of production of those wells at the end of the Newton iteration.

The methods available to calculate the target rate are as follows:

• A specified value

• The sum of the flow rates of a specified phase in the list of CALCCONS connections, evaluated at the conditions specified by CALCCOND. Note that flow rates are positive values, when flow is in the direction from the inlet node to the outlet node, so the sum of a list of connections that contains both injectors and producers will be additive. i.e., The injection rates will be added to the



production rates, assuming that all connections flow in the forward direction.

• The sum of the potential flow rates of a specified phase in the list of CALCCONS connections, evaluated at the conditions specified by CALCCOND

• The sum of the net production or injection flow rates of a specified phase in the list of CALCCONS connections, evaluated at the conditions specified by CALCCOND. For net production, the flow rates in connections which are part of the injection network are subtracted from flow rates in connections which are part of the production network. For net injection, the flow rates in connections which are part of the production network are subtracted from flow rates in connections which are part of the injection network.

• The total injection or production of a specified phase, into a specified region, evaluated at the conditions specified by CALCCOND (which could be the average region pressure and temperature).

• The injection (or production) rate required to maintain a specified pressure in a specified region. The specified pressure can be a pore volume weighted average, or a hydrocarbon pore volume weighted average. Details of the algorithm used are given below.

• The injection (or production) rate required to ensure that net voidage is replaced in a specified region. Details of the algorithm are given below.

For the methods that involve summing the flow in the CALCCONS connection list, a multiplier for each connection can be specified in a TGTCON table.



Net Voidage/Fillup Algorithm

To calculate the injection rate required to replace net voidage, the mass in place, average pressure and average temperature are calculated, and the change in mass from the previous timestep is determined. The mass in place is flashed at the average pressure and temperature, to get the partial molar (or mass) volume (dvdm), which is used to determine the change in reservoir volume that would result from a change in the mass in place of each component. The net voidage rate of each component is calculated explicitly by

4-1

where

vi is the voidage rate of component i

qji is the mass rate of component i for the jth connection in the CTRLCONS list of connections, during the previous timestep. CTRLCONS should consist of connections which inject into the region.

dmi is the change in mass in the specified region during the previous timestep

dtold is the size of the previous timestep

The rate required to replace net voidage is then given by

4-2

where

qv is the rate required to replace net voidage

dvdmij is the partial molar (or mass) volume of component i in phase j, calculated by flashing the mass in place in the region at the average region pressure and temperature.

vi is the voidage rate of component i

nc is the number of components

v q dm dti ji

j

i old= ∑ − *

q dvdm vv ij i

i

nc

j

np

===∑∑ *

11



np is the number of phases

This target rate is allocated between the connections in the CTRLCONS connection list, which should consist of connections that inject into the region, each of which will be assigned some fraction of qv as a constraint. When this constraint is applied, the rate in each connection is calculated using the expression for qv given above, but with vi replaced with the mass flow rate in the connection.

The production rate required to balance net fillup for a region is similar, except that the first equation is replaced by

4-3

and the connections in the CTRLCONS connection list should consist of connections that produce from the region.

Pressure Maintenance Algorithm

The net voidage algorithm will approximately maintain reservoir pressure, but because it is an explicit calculation, the average region pressure may drift over time. The pressure maintenance algorithm provides a method to maintain the pressure of a region at a specified value. If the pressure is initially below the specified pressure, the algorithm will over inject until the average pressure is reached. If the pressure is initially above the specified pressure, the algorithm will reduce injection until the average pressure is reached. The pressure maintenance algorithm calculates the target injection rate as follows:

4-4

where

qv is the net voidage rate

v q dm dti ji

j

i old= +∑ *

q q qt v m= +



qm is the make up rate to raise (or lower) the reservoir pressure from the current pressure to the specified pressure

where

pv is the total pore volume of the region

cravg is the average rock compressibility of the region

dvdpj is the derivative of volume of phase j wrt pressure

dt is the current timestep size

pm is the target pressure for the region

pavg is the current average pressure of the region

dpdtmx is the maximum allowed rate of change in the pressure (MAXDPDT)

qv is the net voidage rate

In the above equations, is the make up rate, restricted only by the user specified maximum allowed rate of change in region pressure. qlim is an additional restriction on the maximum allowed make up rate, to prevent unintentionally large injection rates. This limitation primarily restricts the make up rate to be no larger than the calculated net voidage rate. Once the region reaches the specified pressure, the make up rate becomes very small, just accounting for the small drift in pressure caused by the explicit nature of the net voidage calculation.

q q q q

q q q q

m m m

m m m

= ≥

= − <

min( , ),

max( , ),

*lim

*

*lim

*

0

0

q pv cr dvdp dp dtm avg j

j

np* ( * )* /= −

=∑

1

dp p p dpdt dt p p

dp p p dpdt dt

m avg mx m avg

m avg mx

= − ≥= − −

min( , * ),

max( , * )), p pm avg<

q abs q abs q dtv mlim*max( ( ), ( )*min( . * , . ))= 0 02 0 5

qm*



Note that the Nexus pressure maintenance algorithm is significantly different from that used in VIP.

Control of the target rate

In order to allocate the target rate among the controlling connections, the maximum flow rate that each connection is capable of making must be known. To determine this, the network is solved with all targets removed, but with all constraints still active (see the section on Network Solutions caused by Targeting). This potential solution gives the maximum mass flow rates in every connection in the network. The potential solution is then summed over all connections in the CTRLCONS connection list, for the phase specified by CTRL, and at the conditions specified by CTRLCOND to get the potential control rate. If the potential control rate is less than the target rate, then no control is needed. If the potential control rate is greater than the target rate, then the connections in the CTRLCONS connection list are constrained, so that the sum of the constrained production equals the target rate. These allocated constraints are fixed values, which are honored implicitly, so the target rate will be exactly honored as long as all the connections remain capable of producing the rate that they are allocated when the full reservoir/network solution is computed.

If there is more than one target, the reductions in flow rates necessary to meet a target may reduce the potential flow rate for subsequent targets, and therefore the allocations that the subsequent target calculates. Therefore, the order in which the targets are processed can affect the result. This order can be set using PRIORITY input in the TARGET table.

Nexus provides 10 methods of allocating the target rate, plus a gas lift optimization method that is discussed separately.

The SCALE, SCALEQP, GUIDERATE, AVG, AVGGOR, and AVGWCUT all use the same basic algorithm to allocate the target rate, differing only in the way in which the proportioning factor (the guide rate) is calculated. The potential flow rate for each connection, for the phase specified by CTRL and at the conditions specified by CTRLCOND, is first reduced by a decline factor, given by

4-5

where is the adjusted potential for connection i in connection list CTRLCONS

q e qpidt

pi* . *. * *= −0 95 0 0005

qpi*



qpi is the potential flow rate of the phase specified by CTRL, at the conditions specified by CTRLCOND, for connection i in connection list CTRLCONS

dt is the timestep size.

The purpose of the decline factor is to allow for decline in production rate over the timestep, so that when a connection is allocated a fraction of the target rate, it will be able to make the allocated rate. The program attempts to allocate each connection a rate (which will be imposed as a constraint) given by

4-6

where

qi is the allocated rate for connection i, in connection list CTRLCONS

qt is the target rate

qgi is the guide rate for connection i, in connection list CTRLCONS, which is calculated from the specified CTRLMETHOD method.

qmini is the minimum flow rate for connection i, specified in the TARGET table, or by connection in the TGTCON table. The default value for qmini is zero.

nctrlcons is the number of connections in connection list CTRLCONS

It is possible that the application of this formula will result in the sum of the allocated rates being greater than the target rate, for example if all connections can flow at qmini, and is greater than the target rate. In this case, connections with the lowest potential flow rates are shut in, until the target rate is not exceeded.

The guide rate, qgi, used in the above equations, is calculated as follows:

For the SCALE CTRLMETHOD, the guide rate is the potential flow rate of the phase specified by CTRL, at the conditions specified by CTRLCOND, reduced by previously applied targets.

q q q q q

q q q q

i t gi gj

j

nctrlcons

j

i i i p

* * / max( , )

max( , ),

min

*min

=

==

∑1

ii i

i pi i

q

q q q

*min

*min,

<

= <0

q imin∑



For the SCALEQP CTRLMETHOD, the guide rate is the potential flow rate of the phase specified by CTRL, at the conditions specified by CTRLCOND, without any reduction due to previously applied targets.

For the GUIDERATE CTRLMETHOD, the guide rate is input by the user, or calculated by the guide rate formula.

For the AVG CTRLMETHOD, the guide rate is 1 for all connections, so they will all be allocated the same rate, unless this rate is less than qmini.

For the AVGGOR CTRLMETHOD, the guide rate is the oil/gas ratio of each connection, which results in the highest GOR connections being allocated the least rate.

For the AVGWCUT CTRLMETHOD, the guide rate is the oil/(water + oil) ratio of each connection, which results in the highest WCUT connections being allocated the least rate.

The WCUT, GOR, WRATE and GRATE methods also all use the same basic algorithm to allocate the target rate, differing only in the criteria used to determine which wells to shut in, or scale back. In each case, the criteria is calculated using the potential network solution, without any reduction due to the effect of other targets. The criteria are: the water cut of each connection for the WCUT method, the gas/oil ratio for the GOR method, the water rate for the WRATE method, and the gas rate for the GRATE method. The connections are ordered from the lowest value of the criteria, to the highest (this ordering is only performed every RANKDT days (default 0) so that frequent changes in the wells to be shut in are avoided). Starting with the connection with the lowest value for the criteria, the wells are allocated a rate equal to their potential multiplied by a decline factor. When the sum of the allocated rates exceeds the target rate, all connections with higher value for the criteria are shut in, and the last connection still flowing (the swing connection) is allocated a rate which is just sufficient for the target rate to be met. In the event that this would result in the swing connection being allocated less than qmini, it will instead be allocated qmini, and the allocation for the previous connection will be reduced so that the target rate is still met. Note that all of the connections that are not shut in are allocated a fraction of the target rate, which will be used as a constraint during the subsequent network solve. This means that the target rate should be exactly met, but that each of the active connections is not flowing at its full potential.



Tree Structured Guide Rates

The TGTCON table PARENT column can be used to specify a hierarchical structure for allocating target rates. Tree structured guide rates can be used for targeting using the GUIDERATE, SCALE, SCALEQP, WRATE, GRATE, AVG, AVGGOR, and AVGWCUT options for the CTRLMETHOD. For each parent connection in the tree, the calculated guide rates of the connections that have that parent are normalized by dividing by their sum, and multiplying by the normalized guide rate of the parent. The targeting logic attempts to preserve the ratio of the guide rates at each level in the tree. This is illustrated by the following example.

TGTCON ptarg PARENT CGATH1

CON QGUIDE

PRD1 0.6

PRD2 0.4

ENDTGTCON


CON QGUIDE

PRD3 0.5

PRD4 0.5

ENDTGTCON


CON QGUIDE

CGATH1 0.6

CGATH2 0.4

ENDTGTCON

TGTCON ptarg

CON QGUIDE

PRD5 0.2

CGATH4 0.8

ENDTGTCON



This example corresponds to the following network.

Figure 4.3: Illustration of Tree Structured Guide Rates

In this example, the normalized guide rates of connections CGATH1 and CGATH2 are determined by dividing each guide rate by the sum of the guide rates (1.0 in this case), and multiplying by the parent guide rate (0.8), to obtain a normalized guide rate of 0.48 for CGATH1 and 0.32 for CGATH2. Similarly, the normalized guide rates for connections PRD1 and PRD2 are obtained by dividing each guide rate by the sum of their guide rates, and multiplying by 0.48, the normalized guide rate of their parent. This results in PRD1 having a normalized guide rate of 0.288, and PRD2 having a normalized guide rate of 0.192. This calculation is repeated for all the connections, resulting in the normalized guide rates for PRD1, PRD2, PRD3, PRD4 and PRD5 of 0.288, 0.192, 0.16, 0.16 and 0.2. The targeting logic will attempt to preserve the ratio of the guide rates at each level in the tree. So, if for example, PRD1 is not capable of making its fraction of the target, but PRD2 can make more than its fraction, then the production from PRD2 would be increased so that CGATH1 could make it’s fraction of the target. Likewise, if CGATH1 could not make its fraction of the target, then production from CGATH2 will be increased (if possible) to maintain the ratio between CGATH4 and PRD5.

This example used a specified guide rate, but the method works the same if the guide rates are actually calculated using one of the other methods, such as SCALE. In this case, the TGTCON table would not need to include the QGUIDE column.



Multiple Targets and the Order of Target Calculations

There are no restrictions on the number of targets that can be input for Nexus. If the same connections are used to control multiple targets, then the most restrictive target will be imposed. For example, if a CTRLCONS connection list is given a pressure maintenance target, and the same connection list is given a net voidage target (with a multiplier of 1.0), and the initial region pressure is less than the target pressure, then the target pressure will never be reached, because the net voidage target will limit injection to net voidage, and will not allow the make up injection required to raise the reservoir pressure. However, if the net voidage target was given a multiplier greater than 1, then both targets could work in concert, with the net voidage target controlling the maximum amount of injection during the pressure up phase, and the pressure maintenance target taking over to maintain pressure once the target pressure had been reached.

Targets are processed according to the PRIORITY assigned in the TARGET table, in the order of increasing priority number. Targets which have the same priority are processed in the order in which they are input. For multifield runs, target input at the same time for different reservoirs is read in the order that the reservoirs are specified in the fcs file, with the multifield network file being read last. The order of processing the targets can have a large effect on the result.

For example, if two gathering centers flow into a flow station, and each gathering center has an maximum oil handling capacity, and the flow station also has a maximum oil handling capacity, then if the flow station target is processed first, and as a result of imposing the flow station target, the target for gathering center 1 is still exceeded, while oil production for gathering center 2 is less than the target, then imposing the target for gathering center 1 will further reduce production and the flow station target will not be met. If on the other hand, the two gathering center targets are imposed first, then the imposition of the flow station target will only result in further reduction in production if the flow station target is exceeded, which is a more desirable result.

As another example, if a connection list has both a water production target, which uses the WCUT method of control, and an oil production target, which uses the SCALE method of control, then if the water production target is processed first, connections will be shut in until the water production target rate is met. If at that point, the oil production target rate is exceeded, the connections that are not shut in will have their production further reduced in order to satisfy the oil target rate. As a result, water production will be less than the water production target rate. On the other hand, if the oil target is processed first, the



connections will be scaled back to meet the oil production target rate. If at that point, the water production target is exceeded, then some of the connections will be shut in to meet the water production target rate, and the oil production will be less than the oil production target rate.



Remote Control of Constraints

By default, rate constraints input in a CONSTRAINTS table are controlled by an implied valve located in the connection where the constraint is imposed, PMIN constraints are controlled by an implied valve located in the connection immediately downstream of the node where the PMIN constraint is imposed, and PMAX constraints are controlled by an implied valve in the connection immediately upstream of the node where the PMAX constraint is imposed. When the constraint is active, there is an additional pressure drop in the control connection (over and above the normal hydraulic pressure drop), which is sufficient to restrict the rate or pressure to the specified constraint value. This additional pressure drop is the result of the implied valve being adjusted. Note that the valve need not be explicitly modeled. It is sufficient for the program to know that a device is present which is capable of imposing an additional pressure drop. The location of the control device (the implied valve), can be specified in a CONTROL table, or by using WELLCONTROL WELLHEAD keywords if the control is at the wellhead connection. In this case, the additional pressure drop necessary to satisfy the constraint, is imposed in the connection specified in the CONTROL table, rather than in the default location.

The control device must be located at a point in the network where the constraint can be controlled. If the control connection is downstream of the constrained connection, then there can be no stream split between a constrained connection and the controlling connection, and all connections feeding into a gathering node between the constrained connection and the controlling connection must not allow backflow.

Figure 4.4: Control Downstream of Constrained Connection With Stream Split

Figure 4.4 is an example of an invalid control location. In this case, control_con can not control constrained_con, because fluid from constrained_con could flow out split_con, bypassing the attempted control.

split_con

control_conconstrained_con

130 Network Calculations: Remote Control of Constraints R5000.0.2


Figure 4.5: Control Downstream of Constrained Connection With Gathering Node

Figure 4.5 is an example of a valid control location. In this case, control_con can control constrained_con, as long as inflow_con is not allowed to backflow.

If the control connection is upstream of the constrained connection, then there can be no gathering node between the control connection and the constrained connection, and any stream split must not allow backflow in the outflowing connections

Figure 4.6: Control Upstream of Constrained Connection with Stream Split

Figure 4.6 is an example of a valid control location. In this case, control_con can control constrained_con, as long as split_con is not allowed to backflow.

Figure 4.7: Control Upstream of Constrained Connection with Gathering Node

Figure 4.7 is an example of an invalid control location. In this case, control_con can not control constrained_con, because fluid from inflow_con can flow through constrained_con, bypassing the attempted control.

There are several other restrictions on remote control.

• A connection (or node) can only be controlled by one other connection.

inflow_con

control_conconstrained_con

split_con

constrained_concontrol_con

inflow_con

constrained_concontrol_con

R5000.0.2 Network Calculations: Remote Control of Constraints 131


• Connections below the bottom hole node can not be specified as a control connection.

• Connections below the bottom hole connection and nodes below the bottom hole node can not be remotely controlled.

• If a connection is the control connection for multiple nodes (or connections), then only one of the constraints specified for the nodes (or connections) can be exactly satisfied, and the constraints at the remaining nodes (or connections) will not be violated. E.g., if the production from three wells are gathered into a single node, with one outflow connection, and the outflow connection is specified as the PMIN control connection for the bottom hole pressure of all three wells, then (in the absence of other constraints), the bottom hole pressure of one of the wells will be PMIN, and the bottom hole pressure of the two other wells will be greater than PMIN.

• If a connection is the control connection for multiple connections, then only one of the constraints specified for the connections can be exactly satisfied, and the constraints at the remaining connections will not be violated. E.g., if the production from three wells are gathered into a single node, with one outflow connection, and the outflow connection is specified as the rate control connection for the bottom hole well connection of all three wells, then (in the absence of other constraints), the rate of one of the wells will exactly satisfy its most restrictive rate constraint, and the rate at the other wells will be less than any of their rate constraints.

• Remote control of rate constraints also applies to allocated targets -- i.e., if a connection has specified a controlcon in a CONTROL table, then if that connection is part of a CTRLCONS connection list in a TARGET table, the fraction of the target allocated to the connection will be a rate constraint controlled by the controlcon connection.

132 Network Calculations: Remote Control of Constraints R5000.0.2


Hydraulic Calculations

General Description

Hydraulic models of a multi-phase fluid flow in different surface network devices (tubings, pipelines, separators, valves, etc.) are basic elements of the surface pipeline network system. They are “building blocks” from which the surface pipeline network system is constructed.

The hydraulic model of the flow device determines relationships between a pressure at an inlet of the device, a pressure at its outlet, and flow rates of multi-phase fluids in the device. It is assumed that each device has one inlet and one outlet.

Mass/molar rates of hydrocarbon components are assumed to be known in the models of the flow devices. However, volumetric rates and PVT properties of hydrocarbon phases are required for the determination of a pressure drop in the flow device. For this reason, phase-equilibrium computations are applied for the calculations of the number of the hydrocarbon phases, their compositions, volumetric rates, and PVT properties.

Hydrostatic Gradient

In this option, the fluid density gradient is used to calculate the pressure drop over a pipe (connection). The pressure loss due to friction and kinetic change are ignored. The value of the fluid density gradient in a pipe can be either input or calculated. See the description of keyword METHOD in the CONFEDAULTS table or in the NODECON table in Nexus Keyword Document1.

If the fluid density gradient is specified by input, the pressure drop over the pips is evaluated as follows,

4-7

where:

pressure drop over the pipe

input fluid density gradient in the pipe

Δp pL= −∇ sin

Δp

∇p

R5000.0.2 Network Calculations: Hydraulic Calculations 133


L pipe length

inclination angle from horizontal of the pipe

If the fluid density gradient is to be calculated, the pressure drop over the pipe is evaluated as follows,

4-8

where:

g gravitational constant

gc unit conversion factor

average density of fluid mixture in the pipe

The average fluid density is calculated as follows,

4-9

where:

average water phase density in the pipe

average oil phase density in the pipe

average gas phase density in the pipe

Sw average water saturation in the pipe

So average oil saturation in the pipe

Sg average gas saturation in the pipe

The average fluid densities and saturations are evaluated with the phase-equilibrium computations based on the current average pressure and temperature in the pipe, assuming the gas and liquid travel at the same velocity (no-slippage).

Δpg

gL

c

= − sin

= + +S S Sw w o o g g

w

o

g

134 Network Calculations: Hydraulic Calculations R5000.0.2


Hydraulic Tables

Hydraulics tables may be used to calculate the pressure change between the inlet and outlet nodes of a connection. Hydraulic tables specify the pressure at one end of a pipe (usually the inlet end for producers, and outlet end for injectors), as a function of the pressure at the other end (usually the outlet end for producers, and inlet end for injectors), and as a function of a stock tank phase rate such as oil rate, gas rate, water rate, or liquid rate, a gas ratio such as GLR, GOR, OGR or GWR, a water ratio such as Water cut, WOR, WGR or Oil cut, and possibly an artificial lift quantity which could represent gas lift, or a pump setting.

Ideally, the hydraulic table assigned to a connection would be calculated using the exact geometry of the connection i.e length, elevation change, deviation profile, pipe inner diameter, roughness etc. However, this would lead to a very large number of tables, and often many connections have the same diameter and roughness, but differ in length and elevation change. For example, vertical tubings may have the same properties, but because the reservoir depth is not constant, the length and elevation change will not be the same. To approximately account for this, the length used to calculate the hydraulics table may be specified. The program will then multiply the pressure change obtained by looking up the table by the ratio of the length of the connection divided by the length in the table. This is illustrated below.

Figure 4.8: Illustration of Hydraulic Table Correction for Length



Note that this is different from the treatment in VIP. To the pressure change obtained from the table, VIP adds a hydrostatic correction term to account for the difference between the elevation change used to calculate the table, and the actual elevation change for the connection. To mimic the VIP treatment in Nexus, it would be necessary to add an extra node and connection, so that the connection that has the hydraulic table assigned has the same geometry as that used to calculate the table, and the added connection fills the gap created by truncating, or extending the original connection. The extra connection should then be assigned a hydraulics method of GRADCALC so that it will use a hydrostatic gradient to calculate the pressure change across it. This is illustrated below.

Figure 4.9: Illustration of Using Additional Connection to Mimic VIP Hydraulic Table Correction

It is also possible to specify a datum in the table input. If a datum is input, then the program will account for any difference between the datum depth of the table, and the datum depth of the outlet node if the table is a function of the inlet pressure, or the inlet node if the table is a function of the outlet pressure, using a hydrostatic correction. The



method used for the hydrostatic correction can be chosen to be either the flowing fluid gradient evaluated at the outlet (or inlet) node, the initial average oil gradient in the reservoir, the initial average gas gradient in the reservoir, or the initial average water gradient in the reservoir. Note that the only network node that can have a datum specified, is the bottom hole node (input in the WELLS table). For any other node, the hydrostatic correction will account for the difference between the datum depth of the table, and the depth of the outlet (or inlet) node.

In some cases, the hydraulics table may not encompass the range of flow rates or pressures encountered during the simulation. This is particularly true during the Newton iterations performed to solve the network, when at any given iteration, the flow rates may not be representative of the converged flow solution. It is good practice to input a wide range of flow rates in the hydraulic table input, because large extrapolations of the table can cause unrealistic and numerically difficult behavior. But in the event that the range of the table is exceeded, the program will do the following. Up to the limits specified by the LIMITS table in the hydraulic table input (if any), the program will linearly extrapolate the table. The default minimum limits are 0 for flow rates and fluid ratios, and the maximum values input for the maximum limit. For PIN or POUT, the default limits are the minimum and maximum values input. Beyond the extrapolation limits, the program will smoothly asymptote to the result that would be obtained by looking up the table at 90% of the minimum limit, or 110% of the maximum limit. For example, an extremely large flow rate would return the pressure change calculated from a flow rate of 110% of the maximum flow rate limit. The equation used to perform the asymptotic lookup (using flow rate q as an example) is

4-10

where q is the actual flow rate, q* is the value used to lookup the table, and

4-11

where qlim is the extrapolation limit, and d = 0.9 for a minimum limit and 1.1 for a maximum limit

q a b q c* / = + ( )−

a d q = * lim



4-12

4-13

This equation results in a lookup that is both continuous, and continuously differentiable at the table limit.

Hydraulic Correlations

Analytical hydraulic correlations can be applied to calculate the pressure drop of gas-liquid two-phase flow in a pipe. The correlations predict pressure gradient, liquid holdup, and possibly flow patterns (regimes) in pipes. The flow may be vertical, inclined, or horizontal and a temperature distribution along the pipe may be specified.

The description of PDCORR table in the Nexus Keyword Document lists all correlations implemented in Nexus. A correlation can be specified through keyword METHOD in a CONFEDAULTS table or in a NODECON table.

The NOSLIP correlation assumes the gas and liquid travel at the same velocity (no-slippage), while all other correlations take into account the effect of gas-liquid slip. Some of the correlations use different formulations for different flow regimes, such as Bubble Flow, Slug Flow, Mist Flow, Annular Flow and Transition Flow, etc.

The detailed description and formulations of the correlations (except for correlation SUPRIB) can be found in Reference 2. The formulations of correlation SUPRIB (the drift flux slip model) are described in Reference 3.

Use of Correlations

The correlations were developed based on the experiments of two-phase flow in pipes of different inclination angles. Although all correlations implemented in Nexus have been adjusted with inclination angle so that they can be theoretically applied to any inclination angles, a correlation may be most accurate at a limited range of inclination angles on which they were originally developed for. The following is the recommendation of the correlation usage according to the pipe inclinations.

c q a = 2* lim −

b q c = − ( )lim − 2



Horizontal or Inclined Pipe Correlations

• NOSLIP - Without slip effect.

• DUKLER - Dukler II with Flanigan correction for elevation.

• DUKEAT - Dukler II with Eaton holdup and Flanigan correction for elevation.

• BEGGS - Beggs and Brill.

• HAG_BEG - Use Beggs and Brill correlation if the angle is in the range -45 to 45 degrees, otherwise use Hagedorn and Brown.

Vertical or Inclined Pipe Correlations

• NOSLIP - Without slip effect.

• HAGEDORN - Hagedorn and Brown.

• DUNROS - Dunns and Ross.

• BEGGS - Beggs and Brill.

• AZIZ - Aziz, Govier, Fogarasi.

• ORKISZ - Orkiszewski.

• GRIFFITH - Griffith, Lau, Hon, and Pearson.

• HAG_BEG - Use Beggs and Brill correlation is the angle is in the range -45 to 45 degrees, otherwise use Hagedorn and Brown.

• SUPRIB - Drift flux slip model.



Model Description

The pipe hydraulic models determine the pressure distribution (p(L)) along the pipe assuming that the following parameters are known:

• pressure pout at the pipe outlet L = LENGTH (where LENGTH is the total pipe length), p(LENGTH) = pout

• molar rates of hydrocarbon components and water component

• temperature profile along the pipe

• inclination angle profile along the pipe

• pipe parameters such as diameter, thickness, length, roughness, etc.

In steady-state conditions, the following energy conservation equation2 is applied in the models for the determination of the pressure gradient at different pipe locations:

4-14

where:

L variable pipe length from the inlet to the current location

total pressure gradient at the current pipe location

component of the pressure gradient due to a potential energy or elevation change

g gravitational constant

gc unit conversion factor

fluid density

inclination angle from horizontal of the pipe, which can change along the pipe.

dp

dLL

g

gL

v

g

dv

dL

f v

g dc c c

( ) = − − −

sin2

2

dp

dLL( )

dp

dL

g

gL

elev c

⎛⎝⎜

⎞⎠⎟

= − sin



component of the pressure gradient due to kinetic energy change or convective acceleration

v fluid velocity

fluid velocity gradient

component of the pressure gradient due to friction losses

f Moody friction factor

d pipe diameter.

Figure 4.10: Pressure Gradient in Pipes

Analytical correlations can be applied to evaluate all components of the pressure gradient, , at location L, using the known parameters.

All the correlations were originally developed for flow in pipes. However, they can be applied to casing-tubing annular using the hydraulic diameter, dhyd, evaluated as follows:

4-15

dp

dL

v

g

dv

dLacc c

⎛⎝⎜

⎞⎠⎟

= −

dv

dL

dp

dL

f v

g dfricc

⎛⎝⎜

⎞⎠⎟

= − 2

2

dp

dLL( )

d d dhyd outer inner= −



where douter and dinnerr refer to the outer and inner diameters of the annulus, respectively. They are specified through keywords DIAM and INNERDIAM, respectively. The hydraulic diameter works as the pipe diameter in the correlations. The area used in velocity calculation is the area of the annulus, which is .

Algorithm of Pressure Drop Calculations

Since the pressure gradient of gas-liquid flow may vary significantly over a connection as the fluid properties change with pressure and temperature, the connection needs to be gridded so that it has multiple integration intervals (or length increments). In general, the pressure drop across an interval should be no larger than one-tenth of the average pressure in the interval. The fluid properties and pressure gradient of an interval are evaluated using a correlation based on the average conditions of pressure, temperature and pipe inclination in the interval. The overall pressure drop over the connection is the sum of the pressure drops of all intervals.

An automatic selection of integration intervals is performed as follows:

1. Start with the known pressure, temperature and component mass flow rates at the pipe outlet L = Length, calculate the phase volumetric flow rates using phase-equilibrium computations; determine the viscosities of all phases using Lohrenz-Bray-Clark or Pederson correlations for EOS fluids, or table look up for black oil fluids; calculate gas-oil and gas-water surface tensions. These properties are necessary for the correlation calculations.

2. Calculate a pressure gradient, , at current pipe location, L, using the correlation.

3. Select an integration interval, , to assure that the pressure drop in this interval is not larger than the user specified value PRESIN and that is not smaller than zero:

4

2 2d douter inner−( )

dp

dLL( )

ΔL

L L L1 = − Δ

dp

dLL L PRESIN L L( ) ,Δ Δ≤ − ≥ 0



4. Calculate pressure, , at the middle of the interval as

, where

5. Calculate a pressure gradient, , at the middle of the

interval, , using the correlation.

6. Calculate pressure, , at the location L1 as

7. Set the current pipe location L to L1 and repeat Steps 1 – 6 if the current location is not smaller than zero.

Once the integration intervals are determined, an equation of the pressure drop at an integration interval n is obtained:

4-16

where N is the total number of intervals at the connection, Ln and Ln+1 are the positions of the outlet node and inlet node of the interval n, respectively. The equations of all intervals can be solved for the pressures at all interval end nodes (except for pressure at the outlet node, which is known) using Newton iterations. Since the computations of phase-equilibrium, the pressure gradient and their derivatives with respect to node pressure and component mass flow rates are repeated for each Newton iteration at each interval end node, these computations may require a significant amount of CPU time (especially for compositional models).

The overall pressure drop over the connection, as well as the derivatives of the overall pressure drop with respect to the pressure and component mass flow rates at the outlet, can be obtained through integration. These values will be used in construction of the hydraulics equation of the connection in the network solve.

p L( ).0 5

p L p Ldp

dLL L0 5 0 5. ( ) . ( )( ) = − Δ L L L0 5 0 5. .= − Δ

dp

dLL( ).0 5

L0 5.

p L( )1

p L p Ldp

dLL L1 0 5( ) = −( ) ( ). Δ

p L p Ldp

dLL

dp

dLL

L Lnn n n n

n n+ +

+( ) − ( ) = − +⎡⎣⎢

⎤⎦⎥

−( ) =1 11

21( ) ( ) , ....NN



Pump Model

The pump is assumed to be of the centrifugal kind with variable speed.

A pump increases the pressure of the liquid flowing through it. It is supplied a specified level of power, and the increase in the pressure that it achieves is dependent on the rates and fluid properties. The specified power level is treated as a maximum: If it exceeds the power that the pump can consume at the current fluid flow rate, then the pump will operate at the maximum speed, consuming a lower level of power that corresponds to that speed and the rate.

The pump characteristics are defined though the water performance curves that are input in tabular from. The curve for a given speed consists of the variation of pressure rise (head) generated by the pump with regard to flow of water, along with the pump efficiency and the Net Positive Suction Head required for avoiding cavitation (NPSHR). The last item defines the minimum pressure at the inlet of the pump for a given rate and speed. The performance curves for different speed are assumed to not intersect, and the pressure head declines with increasing rate, starting with a maximum at zero rate and reaching a zero at the maximum rate. The pressure head curves for different speeds form a nested set, with the highest speed curve on the outer side as viewed from the origin, and the speed reducing inwards. Figure 4.11 shows an example of the head vs. rate curves. The top curve defines the maximum speed and the innermost curve defines the minimum speed of pump operation. If the input power level is below the power required to operate the pump at the lowest speed at the current rate, then the pump will be shut off, and no pressure rise will occur from inlet to outlet. Similarly, if the fluid flow rate exceeds the maximum flow rate for the highest-speed curve, the pump will be shut off. Once shut off, the pump may be restarted periodically through the “TEST” option.



Calculation Method

Since the pump performance characteristics are available for input for water as the flowing fluid, a calculation method is needed to translate the pump performance for a fluid other than water. This calculation is carried out using the correlation published by the Hydraulic Institute (Reference 6). This correlation provides the formulas to calculate the non-water viscous fluid rate ( ), the pressure head ( ) and the efficiency ( ), given a water rate and the pump speed, the fluid specific gravity( ), the fluid viscosity and the water performance curve for the pump. From the output the power can be calculated by,

4-17

The correlation is used iteratively to obtain the specified power level and the viscous (non-water) fluid flow rate; the output of this calculation is the speed of operation of the pump that achieves these. The specified power level is achieved subject to the limitations mentioned above on the minimum and maximum power that can be actually consumed by the pump. Suitable interpolation is used to determine the performance at speeds intermediate between the curves.

If the actual pressure at the pump inlet is smaller than the NPSHR value at current operating point, then the pump will be shut off as cavitation will be assumed to occur.

In multiphase flows, the flow cannot have significant volumetric fraction of gas phase for successful pump operation. Hence if the gas fraction in the flow exceeds a set fraction (4%), the pump will be shut off. The liquid viscosity for a mixture of two or more liquid phases is obtained by volumetric fraction weighting of the individual phase viscosities.

qvis hvis

vis

p q hvis vis vis vis= /



Figure 4.11: Pump Performance for Water Flow

Pump Performance Curves

0

20

40

60

80

100

120

0 10 20 30 40

Water flow rate, m3/hr

Hea

d,

m o

f w

ater

Speed=500

Speed=400

Speed=300



Valve Model

The valve model determines the pressure drop from inlet to the outlet of a valve assuming that the

following parameters are known:

• molar rates of hydrocarbon components,• water rate at stock tank conditions,• temperature at the outlet of the valve (used in flash calculations),• valve control,• valve coefficient profile.

The model (VALVEC valve model) predicts subcritical pressure drop across the valve using the following equation:

4-18

where:

pout pressure at the outlet of the valve in psi,

pin pressure at the inlet of the valve in psi,

qsp The mass flow rate of fluid of the specified type in lb per second. The mass rate used for valve calculations may be one of the following as selected by the user: total fluid, oil, gas, water or liquid mass rate.

density of the mixture in lb. per cu.ft.

valve coefficient which depends on a valve setting x.

The determination of the valve coefficients for different types of valves is described in References 2 and 9. The user must input the valve coefficients in a valve table, and the valve setting may be specified in a CONSTRAINT table in the surface network file. The setting can also be modified by procedures.

The setting of the valve is used to look up (with interpolation) the coefficient and used in the above equation to determine the pressure drop across the valve. If the valve setting is outside the range of the input table of valve coefficient data, the valve is set to CLOSED.

p p c x q abs qout in vx sp sp− = − ( ) ( ) /

c xvx( )



Solution Algorithm

The solution procedure for determining the pressure drop in a valve is as follows:

1. Determine the number of the hydrocarbon phases, their compositions, and compressibility factors using the phase-equilibrium computations, or the black oil model. Determine the densities of oil, gas, and water phases. (Flash calculations.)

2. Determine the specified type of mass rate.

3. Calculate density of the fluid mixture.

4. Determine a valve coefficient as described in the previous section.

5. Determine the pressure differential across the valve using the above equation.



Choke Model

The choke flow is modeled using the Perkins model7. The Perkins model accounts for multiphase flow through the choke. The liquid phase compressibility is ignored, and no phase changes are assumed to occur.

The flow is first examined to determine if it is critical, wherein the fluid velocity in the throat is equal to the sonic velocity, and fluid mass flow is independent of the downstream pressure when the upstream pressure is held constant. For relating the flow conditions at the orifice throat to those at the inlet, the flow is assumed to be adiabatic and with no friction loss (isentropic). The energy conservation and continuity conditions are used to determine the pressure ratio. The critical pressure ratio is determined from the maximum mass flow rate condition. (See Equation A-30 in Reference 7.) Perkins also provides the method to determine whether the flow through a choke is critical or subcritical. This method depends on comparing the choke throat pressure against the calculated pressure (p3) several diameters downstream of the throat in assumed subcritical flow. Pressure p3 is related to the overall pressure drop across the choke (far-field upstream pressure minus far-field downstream pressure) by an approximated Perry relationship8. This relationship accounts for pressure recovery downstream of the orifice including the energy loss due to turbulence in the flow expansion, when liquid is present in the flow stream. If the critical pressure at the throat is higher than p3, then it is concluded that the flow is critical, otherwise the flow is deemed subcritical.

Once the throat flow conditions are determined, the mass flow rate is calculated by multiplying the isentropic flow rate with a discharge coefficient. The value of the discharge coefficient used in this calculation is 0.8562. This value is based on analysis of experimental results at Tulsa University by A. Pilehvari; both critical and subcritical data points were included in this analysis.

Two special settings of choke are treated separately:

• When the choke throat diameter is greater than or equal to the inlet pipe diameter, no pressure drop is assumed to occur in the choke (full open choke).

• When the choke is fully closed, no flow is assumed to occur in this case.

In implementation of the Perkins choke model, the total mass flow rate through the choke is calculated given the pressures upstream and downstream of the choke and the phase mass fractions. The flow rate is



then incorporated in the network equations that are solved iteratively such that the total flow rate in the network branch with the choke equals the choke flow. Thus, in the absence of other flow constraints, the choke will either limit (critical flow) or determine (subcritical flow) the actual flow through a network path containing it.



Wells

Nexus treats wells as part of the network, and represents each well with nodes, connections between nodes, and perforations which connect nodes to grid blocks. The network representation for each well is automatically generated from the data input in a WELLSPEC table (see the Well Specification Table in the Keyword document), and the well connection data input in a WELLS table (see the Well connection Data in the Keyword document). In addition, the CONDEFAULTS and/or WELLBORE input can be used to specify whether the well should be represented with as few nodes as possible (the LUMPED option, which causes all perforations assigned to a single flow section to flow to the same node), or gridded, with one node assigned to each perforation. The LUMPED option is computationally less expensive, and also more similar to the well treatment usually employed in the prior generation of reservoir simulators.

The pressure distribution in gridded wells can be calculated by using the hydrostatic pressure gradient calculated by flashing the fluid flowing in each connection at the average pressure in the connection (GRADCALC method), or by using the hydrostatic pressure gradient calculated assuming that the fluids in the wellbore are fully mixed and flashed at the average pressure in the connection (MIXED method), or by using a pressure drop correlation. Hydrostatic gradients, and the pressure drop calculated by a correlation in the wellbore, are treated explicitly in Nexus, meaning that the derivatives of these terms are not used during the Newton iteration to solve the network. This is because these terms are discontinuous when flow directions change, as they can in the wellbore, and this frequently would result in the Newton iterations failing to converge. The pressure calculations described here can all be selected by flow section on a well by well basis. A flow section (keyword FLOWSECT in the WELLSPEC table) specifies a contiguous section of the wellbore with common flow properties (such as tubing diameter). For example, each lateral in a multilateral well would be a separate flow section. However, there is a special option that applies to all wells in a reservoir, which can be invoked with the GRADIENT VIP input. This specifies that the fluid gradient will be a mobility weighted average of the grid block fluid densities for producers, and the density of the injected stream at wellbore pressure for injectors. This option most closely approximates the treatment in VIP.

R5000.0.2 Network Calculations: Wells 151


Gridded Wells

For a gridded well, each entry in the WELLSPEC table usually represents a perforation, or a connection from the specified reservoir grid cell to a network node at the center of the (optionally specified) wellbore section (keyword SECT in the WELLSPEC table). An entry in the WELLSPEC table may also represent an unperforated node in the wellbore connected to one or more perforated nodes, but we still refer to these entries here as ‘perforations’. The default section number is the perforation number in the order specified. This optionally specified section number is used when several grid blocks flow to the same section of the wellbore, such as in a radial case with multiple angular increments. The section number could also be used to generate a coarse wellbore model relative to the reservoir grid, such that groups of grid cells singly penetrated by a section of wellbore are connected to a single network node.

Since as many perforations as desired may be defined in the WELLSPEC table for any given grid cell, the amount of refinement in a gridded wellbore representation is completely user controlled by the number of defined perforations and sections. A single network node is created for each section, at the depth specified in the WELLSPEC table for the first perforation in the section (all perforation depths in a given section should be the same and equal to the depth at the center of the wellbore section). All perforations in the section are attached to the section node. These ‘perfnodes’ are automatically connected to one another sequentially in the order of their section number. The default flow directions of these perfnode connections are inferred from the well type specified in the WELLS data (producer or injector). For other than simple linear, single flow channel wellbores (like branching wellbores with more than two branches, or wells with annular/tubular communication), the user must define nodes in the wellbore in a WELLNODE table, then connect these nodes appropriately for the desired well configuration.

The automatically generated node corresponding to section n is named wellname%k, where k is the first perf in the section, in the order defined in the WELLSPEC table. The automatically generated wellbore connection between section n-1 and section n is wellname%k, with a default direction depending on whether the well is a producer or injector. A connection named wellname%bh is automatically made between the withdrawal/injection node wellname%1 to a bottom hole node, named wellname. If a bottom hole depth is not entered in the WELLS table, then it is defaulted to the depth of the first perf. If the bottom hole node is at the same location as the first perf, the connection wellname%bh is a ‘ghost’ connection and there is no pressure change

152 Network Calculations: Wells R5000.0.2


across the connection. A default sink or source connection, named wellname, is attached to node wellname. The well sink and source connections may then be optionally redefined to connect to user-defined nodes to form networks.

The following example illustrates the well sub-network created for a simple, gridded, single flow section well, that is not connected to a network. The nodes are represented by solid circles, with the default node names in italics. The connections are represented by an arrow connecting nodes, with the direction of the arrow going from the inlet node to the outlet node. And the perforations (connections to grid blocks) are represented by the cell number and a dashed line. Note that wellbore connections are allowed to flow in the reverse direction, so the direction of the arrow does not necessarily represent the direction of flow at all times. Since the depth of the bottom hole node is not specified in the WELLS table, it defaults to the depth of the first perf, so node well and node well%1 are actually coincident, and connection well1%bh has no length or depth change. It is illustrated as a very short connection in the figure.

WELLSPEC well1CELL KH SKIN RADW RADB18 432.1 0.0 0.5 3.71 23l 364.2 0.0 0.5 3.71 468 295.4 0.0 0.5 3.71 92 489.9 0.0 0.5 3.71 359 395.5 0.0 0.5 3.71 592 934.5 0.0 0.5 3.71 WELLSNAME STREAMWell1 PRODUCERENDWELLS



Figure 4.12: Network Representation of Gridded Well



Lumped Wells

If the wellbore treatment is specified as LUMPED, then one node is created for all perforations assigned to the same flow section. This perf node is located at the average depth of all the perforations that connect to it, and is given the name wellname%k, where k is the first perforation (in the order input in the WELLSPEC table) that is connected to it. If there are multiple flow sections, then by default, each perf node is connected to the preceding perf node with a connection also named wellname%k. As for a gridded well, a connection named wellname%bh is automatically made between the first perf node and the bottom hole node, which, as for a gridded well, is located at the depth of the first perf, and is called wellname.

The following example illustrates the well sub-network created for a simple, lumped, well with two flow sections. The well is not connected to a network. In this example, the perf node well%1 is located at the average depth of the first three perfs, and the perf node well%4 is located at the average depth of perfs 4 through 6. Unlike the gridded well, node well1%1 and node well1 are no longer coincident, and connection well1%bh will usually have a non-zero length and depth change. This representation of the well is computationally more efficient, because with fewer nodes and connections, there are many fewer variables to be solved.

WELLSPEC well1CELL KH SKIN RADW RADB FLOWSECT 18 432.1 0.0 0.5 3.71 123l 364.2 0.0 0.5 3.71 1468 295.4 0.0 0.5 3.71 192 489.9 0.0 0.5 3.71 2359 395.5 0.0 0.5 3.71 2592 934.5 0.0 0.5 3.71 2WELLSNAME STREAMWell1 PRODUCERENDWELLS



Figure 4.13: Network Representation of Lumped Well



Branched Wells

Simple branched wells can be specified using the PARENT, MD and MDCON input in the WELLSPEC table. The PARENT table column specifies the parent flow section, to which the first completion in the current flow section will connect. This input is only applicable to the first perforation in a flow section. The MD table column specifies the measured depth for each perforation. Measured depth is the length measured from a reference point to the location of the perforation in the wellbore, and should increase monotonically in each flow section. The combination of flow section and measured depth identifies a unique location in the wellbore. The MDCON table column, like the PARENT column, is only applicable to the first perforation in a flow section, and specifies the measured depth in the parent flow section where the current flow section will connect.

The automatic generation of the wellbore configuration for a branched production well with 2 branches is illustrated below. The perforated length of each branch passes through three grid cells, with a single perforation defined in each grid cell, each connected to a separate node. The perforation in each flow section must be ordered starting with the perforation closest to the offtake point.

WELLSPEC well1CELL KH SKIN RADW RADB DEPTH FLOWSECT PARENT MD MDCON 18 432.1 0.0 0.5 3.71 # 1 NA 5150 NA 23l 364.2 0.0 0.5 3.71 # 1 NA 5250 NA 468 295.4 0.0 0.5 3.71 # 1 NA 5350 NA 92 489.9 0.0 0.5 3.71 # 2 1 5175 5010 359 395.5 0.0 0.5 3.71 # 2 NA 5275 NA 592 934.5 0.0 0.5 3.71 # 2 NA 5375 NA

The # in the depth column indicates that the midpoint of each perforated section is at reservoir cell depth.

WELLSNAME STREAM BHDEPTH BHMDWell1 PRODUCER 4950 5000ENDWELLS

The WELLS table indicates that the bottomhole node is at a depth of 4950, and a measured depth of 5000. A node is created at the point where the second flow section joins the first flow section, and this is named well1%b1.



Figure 4.14: Network Representation of Branched Well



Complex Wells

More complex well configurations, including wells with controllable connections (connections where constraints can be specified), and flow control devices such as valves or chokes, can be specified using WELLNODE input. A WELLNODE table provides for the specification of user named nodes in the wellbore, which can then be connected to any other node, in the network, including nodes in the same wellbore. Well nodes are located by specifying a measured depth and flow section, and will be inserted between two existing nodes. When this occurs, the connection between the two existing nodes must be replaced by a connection between the first existing node and the new well node, and optionally (depending on the CONNECT data in the WELLNODE table), between the new well node and the second existing node. This is illustrated in the following example. The input

WELLSPEC well1

CELL KH SKIN RADW RADB MD

18 432.1 0.0 0.5 3.71 5150

23l 364.2 0.0 0.5 3.71 5250

468 295.4 0.0 0.5 3.71 5350

592 934.5 0.0 0.5 3.71 5375

would result in the normal well sub-network shown below.

Figure 4.15: Default Network Representation of Well



The input

WELLNODE

NODE WELL MD CONNECT

new_node well1 5360 NO

ENDWELLNODE

would result in the node new_node being inserted between nodes well1%4 and well1%3. To accomplish this, the existing connection between nodes well1%4 and well1%3 is modified by the program, to be a connection between well1%4 and new_node. The input in the connect table column indicates that there should be no connection generated between new_node and well1%3. This is shown below.

Figure 4.16: Well Sub-network with Inserted Well Node

If YES was input in the CONNECT column, then a new connection, also named new_node, would be created to connect new_node to node well1%3

The following example illustrates how well nodes can be used to model a smart well. In this case, the reservoir produces through perforated casing into the annulus, and valves (sleeves) control flow from the annulus into the tubing contained inside the casing. Annular flow in the casing is blocked by packers above and between three perforated sections of the wellbore.



WELLSPEC P-1CELL KH SKIN RADW RADB MD 1080 500.0 0.0 3.5 117.9126 50501264 500.0 0.0 3.5 117.9126 51501440 500.0 0.0 3.5 117.9126 52501772 500.0 0.0 3.5 117.9126 53501956 500.0 0.0 3.5 117.9126 54502140 500.0 0.0 3.5 117.9126 5550!! ADDTUBING entry of 1 adds 1 tubing string from the bottom ! hole node down through the perforated section of the wellbore.! Because there is only one flow section in the WELLSPEC data, the! tubing is given a flowsection index of 2.!WELLSNAME STREAM TYPE METHOD BHMD ADDTUBING BHDEPTH DIAM ROUGHNESSP-1 PRODUCER PIPE HAGEDORN 4950 1 4900 3.5 0.0005ENDWELLS

WELLNODE NODE WELL MD FLOWSECT CONNECT P-1-T1 P-1 5000 2 YES ! Point in tubing, just ! below packer 1 P-1-A1 P-1 5000 1 NO ! Point in annulus, just ! below packer 1 P-1-T2 P-1 5200 2 YES ! Point in tubing, just ! below packer 2 P-1-A2 P-1 5200 1 NO ! Point in annulus, just ! below packer 2 P-1-T3 P-1 5400 2 YES ! Point in tubing, just ! below packer 3 P-1-A3 P-1 5400 1 NO ! Point in annulus, just ! below packer 3ENDWELLNODENODECONNAME NODEIN NODEOUT TYPE METHODVALVE1 P-1-A1 P-1-T1 VALVE 1 ! valve connecting annulus to ! tubingVALVE2 P-1-A2 P-1-T2 VALVE 1 ! valve connecting annulus to ! tubingVALVE3 P-1-A3 P-1-T3 VALVE 1 ! valve connecting annulus to ! tubingENDNODECON



Figure 4.17: Network Representation of Smart Well



Connecting Wells to the Network

By default, the well connection (defined in a WELLS table) connects the bottom hole node to a sink if the well is a producer, and to an injection source, if the well is an injector. It is not required for the wells to be connected to a network, and the simulator can be run in this mode, with all wells flowing directly to a sink, or from a source. This might be appropriate for a history run, where the historical flow rates and/or bottom hole pressures are known, and the conditions in the network are not required to model the reservoir behavior. However, for prediction runs, it will usually be necessary include at least the production/injection tubing to the wellhead, and often also the surface pipeline network and other facilities. To connect the wells to the network, the default well connection is reconnected to a node (rather than the sink) in a NODECON table.

For example, production well P-1 and injection well I-1 can be connected to a simple network as follows:

NODESNAME TYPE DEPTHWH-P-1 WELLHEAD 0NODE1 NA 0WH-I-1 WELLHEAD 0ENDNODES

NODECONNAME NODEIN NODEOUT TYPE METHOD DIAM LENGTHP-1 P-1 WH-P-1 PIPE HAGEDORN 3.5 #WH-P-1 WH-P-1 NODE1 PIPE BEGGS 3.5 1000SINK NODE1 SINK NA NA NA NASOURCE WATER WH-I-1 NA NA NA NAI-1 WH-I-1 I-1 PIPE NOSLIP 3.5 #ENDNODECON

Here well P-1 is connected to a well head node named WH-P-1, and the Hagedorn and Brown correlation is used to calculate the hydraulic pressure drop. Therefore, connection P-1 represents the tubing, and has a diameter (the inner diameter of the tubing) of 3.5 inches. The wellhead node is then connected by the wellhead connection (a 1000 ft pipe) to another node named NODE1, and from there flows to a sink. The wellhead connection uses the Beggs and Brill pressure drop correlation. Note that the length and depth change of the tubing, connection P-1, is defaulted, and will be calculated by the program from the difference in depth between the well head node (depth 0) and the bottom hole node.



The injection well connection I-1 is similarly reconnected to flow from the wellhead node, WH-I-1, to the bottom hole node, using the NOSLIP pressure drop correlation.

In the NODES table, the wellhead nodes are identified in the TYPE column. The well head node need not be the first node after the bottom hole node for a producer (or before the bottom hole node for an injector), but there must not be any branches in the network between the well head node and the bottom hole node, except for gas lift in the case of a producer. The designation of a wellhead node is important, because it is used for several purposes. In particular, remote control of wells at the wellhead (WELLCONTROL keyword), requires that the location of the well head is known. Also, the procedures automatically create a list of well head nodes and connections, which also requires that the well heads are identified.

Well Constraints

Rate and pressure constraints can be applied to any user named connection or node, and in addition, to the automatically named well bottom hole connection, which has %bh appended to the well name. The WELLS table names the bottom hole node, and the well connection, which is the offtake from the bottom hole node in the case of a producer, or the intake to the bottom hole node in the case of an injector. So if a production well is named well1, inputting a PMIN constraint for well1 specifies a minimum bottom hole pressure, and inputting a rate constraint for well1 specifies a maximum rate in the well connection. Both the pressure constraint and the rate constraint are controlled by an implied control device (valve) located in the well connection, unless remote control is in use (see the section on remote control of constraints). The location of the implied control device means that an additional pressure drop will be applied to that connection, over and above that obtained by the hydraulic calculation, so that the constraint is not violated. If one of the well constraints is active, the network pressures downstream of the well connection are then determined by the flow conditions downstream of the well connection, i.e., the hydraulic pressure drops and constraints. For example, if a well has a 100 psi minimum pressure specified at the well head node, and a rate constraint of 1000 STB/Day of oil, and the rate constraint is active, then the pressure at the well head node will be 100 psi, regardless of the pressure at the bottom hole node, and the implied control device located in the well connection sets the bottom hole pressure to the value required for the well to flow at 1000 STB/Day of oil.



Rate constraints can also be applied to the well bottom hole connection. This is particularly useful in the case of gas lift injection, because the rates downstream of the gas lift injection node, which is often the bottom hole node, will include the gas lift gas.

A drawdown/buildup constraint can also be specified for wells (DPBHMX keyword). This constrains the maximum drawdown (or buildup) at any perf to be less than or equal to the specified value. The equation used to calculate drawdown/buildup is given in Equation 4-20 in the perf equations section below. Note that this treatment of drawdown is different from that used by VIP, which uses an average drawdown for the well as a whole, rather than the maximum drawdown at any individual perforation.

Treatment of Datum Depth

The datum may be specified for the bottom hole pressure constraint in the WELLS table. By default, the datum is set to the same depth as the bottom hole node depth. If it is different from the bottom hole node depth, then a datum correction is made when applying a bottom hole pressure constraint. If the difference between the datum depth and the bottom hole node depth is d, then a bottom hole pressure constraint of pmin will result in the actual pressure at the bottom hole node being constrained to not less than pmin – d * fluid_gradient, where the fluid_gradient is calculated by one of five methods. The OILGRAD method uses the initial average density gradient of the oil phase in the reservoir, the GASGRAD method uses the initial average density of the gas phase, the WATGRAD method uses the initial average density of the water phase, the WELLGRAD method uses the density gradient of the fluids flowing in the bottom hole connection, evaluated at the bottom hole pressure, and the MOBGRAD method uses the mobility weighted reservoir density gradient in the perforated grid blocks. The MOBGRAD method is the method used by VIP. The datum correction term is treated explicitly in the simulator, and if the WELLGRAD or MOBGRAD methods are used, can lead to unexpected behavior, such as well shut ins, if the difference in depth between the bottom hole node and the datum depth is large. The default method is OILGRAD if oil is present in-situ, otherwise GASGRAD if there is free gas, otherwise WATGRAD, and for stability of the calculations, one of these three methods is recommended.

If the datum depth is deeper than the bottom hole node, and the depth difference is large, then it is possible that a small value for the PMIN constraint at the bottom hole node, might translate to a negative pressure at the bottom hole node. This is physically impossible, and indicates that the value input for PMIN should be increased. But in the



event this occurs, the simulator will make an automatic adjustment to the constraint to prevent the pressure from falling below the minimum allowed pressure, issue a warning message, and continue execution.

Crossflow

By default, flowing wells in Nexus are allowed to crossflow, and shut in wells are not allowed to crossflow. This means that for a producing well, individual perfs can either produce or inject, depending on the pressure calculated for the perf nodes, and the pressures in the grid blocks. In addition, the connections between perf nodes can flow in either direction. Crossflow can be enabled or disabled for individual wells by input in the WELLS table. The default behavior can be changed for all wells using the CROSSFLOW keyword. If crossflow is disabled, then on each network solve, Nexus first opens all active perfs to flow if the well is not shut in, and solves the network equations allowing the perfs to backflow. When the network calculations converge, all back flowing perfs are plugged, and additional iterations are taken until convergence is reached again. This is repeated until no perfs are back flowing. Note that within the network solve, the wells are solved individually to speed convergence. From one timestep to the next, or even from one global Newton iteration to the next, the perfs which are open to flow may change, and this may slow convergence of the timestep, and possibly cause smaller timesteps. If crossflow is allowed, perfs may change flow direction during the timestep, or from one timestep to the next, but this is usually a less abrupt change than the opening and closing of perfs when cross flow is off. For this reason, it is recommended that cross flow be enabled for flowing wells in Nexus.



Perforation Equations

The mass production rate of component i from reservoir cell r through each producing perforation p to network node n is given by

4-19

4-20

4-21

4-22

where C is the wellbore constant (see Equation 5-1), equal to the well index times the permeability-thickness product, kr is the volumetric mobility (relative permeability/viscosity) of phase k in cell r, kr is the density of phase k in cell r, and zikr is the mass fraction of component i in phase k in cell r. is the perforation reservoir potential, obtained by correcting the reservoir cell pressure to perforation depth Dp using an explicit hydrostatic pressure gradient in the reservoir cell, r. is the perforation wellbore potential, obtained by correcting the node pressure Pn to perforation depth using the wellbore gradient . Capillary pressure is neglected.

For injecting perforation p contained in reservoir cell r, the component mass rate equations are not independent, and the single equation representing the total mass perforation rate (negative for injection) is

4-23

where Tr is the total reservoir cell mobility (sum of kr) and is the perforation injected fluid density. Optionally, endpoint mobility can be used instead of the total cell mobility, using the water endpoint or gas endpoint mobility depending on the injection source. The injected fluid density is treated implicitly or explicitly and is computed from phase

Q C zip p p krk

N

kr ikr

phases

==

∑ΔΦ 1

ΔΦ Φ Φp rp wp= −

Φrp r r p rP D D= + − ( )

Φwp n wp p nP D D= + − ( )

Φrp

Φwp

wp

Q CTp p Tr pinj

p= ΔΦ

pinj



saturations and densities obtained by flashing the perforation fluid composition at perforation node pressure.

For gridded wellbores, the node to which each perforation connects is placed at perforation depth, so the wellbore gradient correction appearing in Equation 4-22 is zero. If the wellbore is lumped, then the perf node depths will be different from the grid block depths, and an approximate explicit mixed wellbore gradient is computed for each perf from the sum of phase densities weighted by their saturations, which are obtained from a flash of an average perforated grid cell composition (weighted by the product of phase mobilities and wellbore constant) at the node pressure. Optionally, if GRADIENT VIP is specified, the wellbore gradient for producers is computed from the mobility weighted average of the grid block fluid densities.

Usually, the perforation depth and grid block depth are coincident, so there is no gradient correction in Equation 4-21. However, if the perforation depth can be optionally input in the WELLSPEC table, in which case the gradient used in Equation 4-21 is obtained from the saturation weighted average fluid density in the grid block.



WAG (Water-Alternating-Gas) Injection

In order for a well to be a WAG injector, it must be connected to two sources, one of which is a water source, and the other a hydrocarbon source. This is illustrated in Figure 4.18. Injection is switched from water to gas by constraining the water injection rate (QWSMAX) in Connection 1 to zero, while removing any constraints on Connection 2. Similarly, to switch from gas to water, the gas injection rate (QGSMAX) would be constrained to zero in Connection 2, and all constraints would be removed on Connection 1. The WAG procedures do this switching automatically. After cycling starts, any rate constraint specified by the user for these connections will be ignored. The maximum injection rate of gas or water is controlled by one of the connections in the linear series leading to the well, which in this example would be Connection 3, the Wellhead Connection, or the Well connection. The user specifies these constraints (QWSMAX and QGSMAX) in a CONSTRAINTS table, just like any other constraint data. The switching connections (Connections 1 and 2) must be the last branch in the network before the well is reached. I.e downstream of the inlet node to Connection 3, there can be no connections that either remove fluid, or inject additional fluid, otherwise the amount of fluid injected into the well can not be controlled.

Figure 4.18: Source Connections for WAG Wells

R5000.0.2 Network Calculations: WAG (Water-Alternating-Gas) Injection 169


The WAG procedure is used to control the cycling automatically. Automatic WAG cycling is activated by inputting a PROCS section in the surface network file that includes one or more variations of the WAG procedure. (See the Procedures section of the Keyword document.) For example

WAGWELL WSCUM GSCUMI1 300 300ENDWAG

CONSTRAINTSI1 PMAX 8000.0 QGSMAX 2000 QWSMAX 1000ENDCONSTRAINTS

PROCS NAME WAG_1INTEGER NCYCLE = 5WAG("WW", NCYCLE, I1)ENDPROCS

Here the WAG procedure takes three arguments, WW, to indicate that the first cycle is water, and that water should be injected after cycling is finished, the integer variable NCYCLE, which in this case is set to 5, indicating that the procedure should execute five WAG cycles (where a cycle means injection of both a water slug and a gas slug), and I1, which is the name of a WAG injection well, which is connected to both a water and hydrocarbon source, as discussed above. The size of the slugs, is determined for each WAG well by data input in a WAG table (see Water-Alternating-Gas (WAG) Wells in the keyword document). In the above example, the WAG table specifies a water cycle slug size of 300 MSTB, and a gas cycle slug size of 300 MMSCF. The WAG procedure does not attempt to exactly honor the slug sizes specified in the WAG table, because in order to do so might severely restrict the timestep size, particularly if there were many WAG injectors. Instead, the procedure switches from water to gas, or vice versa, as soon as the specified slug size is exceeded. The size of the next slug of the same kind (i.e., water or hydrocarbon), is reduced by the amount by which the current slug is exceeded, so that the over injection does not accumulate with each cycle. The WAG procedure controls the cumulative amount of fluid injected in each cycle only. The injection well is still subject to whatever other constraints are specified (including targets). In the above example, the WAG injection well is constrained to a maximum gas injection rate of 2000 MSCF/Day, and a maximum water injection rate of 1000 STB/Day, and the bottom hole node has a maximum injection pressure of 8000 psia.

170 Network Calculations: WAG (Water-Alternating-Gas) Injection R5000.0.2


The other variations of the WAG procedure provide for initiating WAG on a group of wells (specified in a WELLLIST), and allow for multipliers to be applied to the slug sizes specified in the WAG table. These are documented in the Procedures section of the keyword document.

R5000.0.2 Network Calculations: WAG (Water-Alternating-Gas) Injection 171


Separation Nodes

A separator can be defined as a separation node (SEPNODES) in Nexus1. It is not necessary to define terminal nodes as separation nodes if the separator outflows may all be considered as sink connections – fluids produced to a sink are always automatically flashed through the separator battery defined for the sink connection. However, if internal separations are desired (for example, for modeling gas reinjection or gaslift), separation node data are required. Separation nodes may have an arbitrary number of outflow connections, but are restricted to having a single inflow connection, and inflow and outflow connections may not be allowed to backflow. A typical separation node and its inflow and outflow connections are illustrated below.

Figure 4.19: A Separator with Three Outflow Connections.

For compositional fluids, separation nodes may represent any of the separator batteries or gas plants described in the separator method files. Total phase rates obtained by flashing the inflow through a separator battery (at the stage pressures and temperatures specified in the separator method file, which is referred to by the separator battery number assigned through keyword IBAT in a CONDEFAULTS table or in a NODECON table), or through a gas plant, are implicitly split among the outflow connections using user-defined volumetric phase fractions for the arbitrarily defined outflow connections (these fractions are essentially separation efficiency factors). Therefore, the mass/mole flow rate of component i of outflow connection n, qni, is evaluated as:

4-24

q F Q z i n n Nni nk fkk

n

k ik c

p

= = ==

∑1

1 1 , ... , ...

172 Network Calculations: Separation Nodes R5000.0.2


Here, np is the number of phases, nc is the number of components, N is the number of separator outflow connections, Qfk is the volumetric rate of phase k obtained by flashing the flow of the separator inflow connection f, k is the density of phase k, zik is the mass/mole fraction of component i in phase k, Fnk is the user-defined volumetric fraction of the phase k that is sent to outflow connection n, which must satisfied,

4-25

Fnk are assigned through keywords FOIL, FGAS and FWATER in a SEPNODES table.

Network node pressures computed by the model are independent of separator battery stage pressures; rather they are determined by active network constraints, reservoir behavior, and hydraulics. In other words, it implies that there may be hydraulic devices (such as valves or pumps) in a separation node to make up the pressure difference between the separator battery stage pressures and the separation node pressure, although these devices are not explicitly simulated in the network model.

For black-oil or compositional fluids, separation nodes may also represent separator stages which can be sequentially connected to form separator batteries of arbitrary configuration. For each stage node, the volumetric phase splits among the defined outflow connections are specified, and for compositional fluids, the EOS method (IPVT) to be used for the flash calculation (phase-equilibrium computations). The flash calculation at each separation node is performed implicitly at network node conditions of temperature and pressure. Node pressures are always determined by some combination of active network constraints, network hydraulics, and reservoir conditions. Appropriate constraints, targets, and hydraulics must be specified in order to, for example, set up a separator battery system that operates at fixed stage pressures. Node temperatures may be input by (in order of increasing precedence):

1. The temperature associated with the EOS method, for compositional fluids.

2. Specification of NETTEMP in the surface network file, defining a single temperature for all non-perforation network nodes. Required for black oil, gas-water, oil-water, and gas-water fluid types when using the temperature dependency option. Perforation

F k nnkn

N

p

=∑ = =

1

1 1, ...

R5000.0.2 Network Calculations: Separation Nodes 173


nodes are internally assigned average connecting reservoir grid cell temperatures.

3. Specification of node temperatures in the NODES table of the surface network file.

Alternatively, node temperatures may be approximately calculated using the Network Heat Loss Option.

The use of separation nodes introduces physical and corresponding numerical restrictions upon placement of constraints. Of the N outflow connections of a separation node, N-1 of them are considered to be controlled by proportioning devices such as level controllers. The remaining connection is referred to as the control connection, which may contain a device that can be used to regulate minimum pressure at the separation node. The model takes the control connection of a given separation node as that outflow connection with the highest specified gas fraction. Separation systems are defined as groups of interconnected separation nodes and their connections flowing only fluids originating from a single common upstream separator inflow connection. Since, with respect to rates, all connections within the system downstream of any proportioned connection are considered proportioned, separation systems have a single control path (or path of possible hydraulic communication) from the common upstream connection into the first (furthest upstream) separation node to a junction, a sink, or a reinjection system. Rate constraints in separation systems are allowed to be applied only in connections along the control path of the system. Constraints on rate-proportioned connections of a separation system must currently be targeted to devices in connections upstream of the inlet separation node, or remotely controlled by connections upstream of the inlet separation node or by connections in the control path. Maximum pressure constraints cannot be specified for nodes that are not in the system control path (such as for pumps for water disposal wells). These constraints must be handled through the predictive logic with constraint modifications or flow rerouting. Applying a prohibited constraint within a proportioned connection while keeping other controls fixed would physically result in flooding of the upstream separator stages and system shutdown, and would numerically result in a singular system of equations. There are no limitations on specifications of minimum pressure constraints on nodes. If a node is a separation node, its minimum pressure constraint is automatically applied within the control connection for that node.

Stream splitting nodes limit the split of the total flow rate into a node amongst the outflow connections by specified fractions. No phase separation occurs. Specification of the split implies the presence of

174 Network Calculations: Separation Nodes R5000.0.2


flow controlling devices in each defined outflow connection of the node. Each device is assigned its specified maximum split as a maximum split constraint. If the sum of the specified splits is unity, and at any given time, no other constraints are user-specified for these outflow devices, then N-1 of the N outflow devices will be actively constraining the specified split. Which of the N-1 devices are controlling is dependent upon hydraulics behavior and other active network constraints, and is automatically determined by the program. If the sum of the fractions is unity, then the specified fractions are honored unless other constraints specified by the user in at least two of the outflow connections become violated. In this case, the devices in the other connections (those without other violated constraints) will continue to constrain their maximum split to the specified value.

R5000.0.2 Network Calculations: Separation Nodes 175


Treatment of ONTIME factors

ONTIME factors can be specified for wells in the WELLS table. ONTIME is the fraction of the time that the well is actually producing or injecting, and can be used to approximately account for periods when the well is not on line. When an ONTIME factor is input for a well, all connections in linear series with the well are assigned the same ONTIME factor. At gathering nodes, the outlet connections (and the inflowing gaslift connection if there is one) are given a total mass rate weighted average of the ontime factors of the inflowing connections, not including gaslift connections. This means that a gas lift connection injecting into the tubing string for a single well, will get the same ontime factor as the well, while a gas lift connection injecting into a riser that several wells produce into, will get the average ontime factor of the contributing wells. Connections from an external source, which inject into a production network, will be given an ontime factor of 1.0.

ONTIME factors modify the network equations as follows. Wellbore constants for the perforations and rate constraints are multiplied by the ontime factor. The rates used in all hydraulics calculations are the full on rates, i.e., the calculated connection rate divided by the connection ontime factor. This results in the network pressures being calculated as if the wells were fully on. To illustrate this, consider a well with a single perforation, with a grid block pressure of 1000 psia, flowing against a bottom hole pressure constraint of 900 psia, and say this results in a total mass flow rate of 10000 lb/day if the ontime factor is 1.0. If the ontime factor is reduced to 0.9, then the total mass flow rate will be 9000 lb/day, and because the hydraulics are evaluated at the full on rate of 9000/0.9 lb/day, and the wellbore constant has been multiplied by 0.9, the pressure drop from the grid block to the bottom hole node is still 100 psia.

Ontime factors significantly less than unity are not recommended, because the assumption that pressures downstream of gathering nodes should be evaluated using the full on rate becomes less and less valid as the ontime factors are reduced. For example, if two wells are gathered at a node, each has an ontime factor of 0.5 and are flowing at the same total mass flow rate, and if the periods that the wells are on are random, then the offtake connection from the gathering node would expect to flow at the full rate 25% of the time, have one well flowing 50% of the time, and have no flow 25% of the time, where as the treatment in Nexus would assume that the wells were on and off simultaneously, so that the offtake connection would see all of the flow for 50% of the time, and no flow for the remaining 50% of the time.

176 Network Calculations: Treatment of ONTIME factors R5000.0.2


Network Heat Loss Calculations

The simplified heat transfer calculations in the surface network are used to approximately determine the fluid temperature at all points in the network. This option is activated by the HEATTR keyword. Fluid temperature affects the in-situ phase flow rates and fluid densities, and these are used in imposing in-situ rate constraints and hydraulics calculations involving hydrostatic fluid gradients, pressure drop correlations, valves, chokes and pumps. The fluid temperature does not affect hydraulic tables, because these are input as functions of stock tank phase rates; however, the table should have been generated using representative fluid temperatures. Network node temperatures can be input in NODES tables, and connections can be assigned temperature profiles in NODECON tables. Perforation temperatures are determined by the temperatures input for the reservoir grid. If the heat transfer option is not activated, then these input temperatures are taken to represent the temperatures in the network. If the heat transfer option is activated, then these temperatures represent the ambient temperature.

The simplified heat transfer calculations make the following assumptions:

• The calculations are steady state. That is, there is no heat capacity associated with the network components or the surroundings.

• Latent heat as a result of phase change is neglected

• Conductive heat flow in pipes and other network components is neglected

• A simple approximation of fluid heat capacities is used

At each node in the network, a heat balance is imposed.

4-26

4-27

where

hi is the enthalpy of the fluid flowing in connection i at the node.

h SHCOIL q SHCGAS q SHCWAT q Ti oil gas wat i= + +( * * * )*

h hi

i

ninflow

i

i

noutflow

== =∑ ∑

1 1

R5000.0.2 Network Calculations: Network Heat Loss Calculations 177


Ti is the temperature, either the outlet temperature of the connection for connections flowing into the node, or the inlet temperature for connections flowing out of the node (which is the same as the node temperature).

SHCOIL, SHCGAS and SHCWAT are the heat capacities of the oil, gas and water phases

qoil, qgas and qwat are the in-situ flow rates, evaluated at temperature Ti and the node pressure.

ninflow and noutflow are the number of inflowing and outflowing connections at the node.

Heat loss from a connection to the surroundings are calculated using

4-28

where

x is the distance from the inlet of the connection

h is the enthalpy at distance x

htc is the input heat transfer coefficient

d is the connection diameter

T is the temperature at distance x

Tamb is the ambient temperature of the surroundings at distance x

Integrating the above equation from the inlet to outlet of a connection gives the enthalpy at the outlet, as a function of the enthalpy of the inlet.

The boundary conditions are that fluid flowing into the network from the perforations is at the temperature of the grid block, and the fluid temperature at the injection source is the ambient temperature of the source node.

These equations are used to determine network temperatures explicitly. i.e., The network temperatures are treated as known, invariant quantities for the combined reservoir/network solution.

dh dx htc d T Tamb/ * * *( )= − −

178 Network Calculations: Network Heat Loss Calculations R5000.0.2


Drilling Queues

Drilling queues can be used to automatically add or replace wells over time, in order to meet production or injection requirements. A drill queue is started by specifying the drill or redrill function in a procedure. These procedures require that data has been input to define drill sites (DRILLSITE table), rigs (RIG table) and the time to drill and complete each well (DRILL table). Drill site data consists of the names of the drill sites, and optionally the maximum number of rigs allowed at the drill site (the default is 1). Rigs can be assigned to an initial drill site, or the program can decide the initial location. The time to move a rig from one drill site to any other drill site, and the default time required for a rig to drill a well (which can be superseded on a well by well basis) is also input for each rig. For each well, the drill site from which it will be drilled, the rigs that can be used to drill the well, the time required to drill the well and the time required to complete the well can be specified. Also, for use with the redrill procedure, the wells which can replace an existing well can be specified. All wells in the drill queue must be fully defined in the surface network input, along with their connections to the network, but initially deactivated.

The Drill Procedure

There are two ways to invoke the drill procedure. The first method mimics the options available in VIP for determining when wells should be drilled. The second allows for arbitrary user defined criteria to be used. For the first method, well(s) are drilled if an adjusted target rate (specified in a TARGET table), can not be met. The drill procedure is as follows:

drill(key1, key2, target, adjust, wl, nw, mask)

key1 can be either RIGS or NORIGS. If NORIGS is specified, then drilling is not subject to rig availability, but only nw wells can be in progress simultaneously. If RIGS is specified, then a well can only be drilled if there is a rig available, and the drill site(s) from which the well can be drilled has less than the maximum number of rigs operating. The procedure first checks the first nw inactive wells in the drill queue well list, wl, to see if this criteria can be met without moving a rig. If not, it will check all the inactive wells in the drill queue (in the order specified in the drill queue well list), allowing rigs to move between drill sites.

key2 can be either MULT, ADDVALUE or VALUE. This specifies how the argument adjust is used to adjust the target rate. The target

R5000.0.2 Network Calculations: Drilling Queues 179


must have been input in a TARGET table, and when the adjusted target rate can not be met, the program will attempt to drill a well(s). To determine whether the adjusted target rate can be met, the program will solve the network potential (with all targets removed), then compare the flow rates from this potential solution with the adjusted target. If key2 is MULT, then the target rate will be multiplied by the argument adjust. If key2 is ADDVALUE, the amount adjust will be added to the target rate, and if key2 is VALUE, then the argument adjust will be used instead of the target rate.

adjust is a real number or variable, which is used to adjust the target rate, as discussed above.

wl is a well list, containing all the wells in the drill queue, in the order that the procedure will attempt to drill them. These wells should initially be inactive.

nw is an integer number or variable. If key1 is NORIGS, then this is the number of wells that can be drilling simultaneously. If key1 is RIGS, then this is the number of inactive wells in the drill queue well list that the procedure will check (to see if a rig is available at the drill site from which the well would be drilled), before allowing rigs to move between drill sites. If the target rate can not be met, and RIGS is specified, the program will start drilling wells until all rigs are in use (subject to the limitations on the number of rigs that can drill at each drill site).

mask is an array of logicals (true or false), which if false, will prevent the corresponding well in the drill queue well list from being drilled. This provides a means for the user to override the automatic drilling algorithm, but will usually be input as ALL1D, which specifies a value of true for all members of the well list.

When an inactive well on the drill queue is drilled, the well will be available to start production or injection after the sum of the drill time, and completion time, specified in the DRILL table, plus, the move time specified in the RIG table if the rig had to be moved from one drill site to another. The rig is available to drill another well after the sum of the drill time and the move time (if any). Timesteps are not adjusted to coincide with start of production or injection, but rather, the well will start operations as soon as the simulation time equals or exceeds the availability time for the well.

The second form of the drill procedure is more general, and allows users to specify their own criteria to determine when wells should start drilling. This form assumes that drilling will be subject to RIG availability, but if the rigs are given zero move time, are made available

180 Network Calculations: Drilling Queues R5000.0.2


to all drill sites, and the drill sites have no limitations on the number of rigs that can drill, then this will act the same as the NORIGS option in the first form of the drill procedure. The arguments are as follows:

drill(wl, nw, mask). Here the arguments are:

wl is a well list, containing all the wells in the drill queue, in the order that the procedure will attempt to drill them. These wells should initially be inactive.

nw is integer number or variable. This is the number of inactive wells in the drill queue well list that the procedure will check (to see if a rig is available at the drill site from which the well would be drilled), before allowing rigs to move between drill sites. The program will start drilling wells until all rigs are in use (subject to the limitations on the number of rigs that can drill at each drill site).

mask is an array of logicals (true or false), which if false, will prevent the corresponding well in the drill queue well list from being drilled. This provides a means for the user to override the automatic drilling algorithm, but will usually be input as ALL1D, which specifies a value of true for all members of the well list.

The following example illustrates the use of this form of the drill procedure. In this example, inactive wells on the drill queue will start drilling, subject to rig availability, if the field potential oil rate falls below 100000 STBD and the field water rate is less than 300000 STBD. Here, the wells in the drill queue have been specified in a well list called wl.

PROCSREAL field_potential_oil_rate, field_water_ratefield_potential_oil_rate = sum(qp(“OS”,prodbhcons,ALL1D))field_water_rate = sum(q(“WS”,prodbhcons,ALL1D))if(field_potential_oil_rate < 100000 and &field_water_rate < 300000)thendrill(wl,3,ALL1D)endifENDPROCS

The Redrill Procedure

The redrill procedure provides a means to automatically drill replacement wells when the production (or injection) from existing active wells is no longer sufficient. There are two ways to invoke the redrill procedure. The first method mimics the options available in VIP for determining when replacement wells should be drilled. The second



allows for arbitrary user defined criteria to be used. For the first method, replacement well(s) are drilled if a trigger criterion is violated. The redrill procedure is as follows:

redrill(key1, key2, wl, value, mask)

key1 can be either RIGS or NORIGS. If NORIGS is specified, then drilling is not subject to rig availability. If RIGS is specified, then a well can only be drilled if there is a rig available, and the drill site(s) from which the well can be drilled has less than the maximum number of rigs operating.

key2 can be any one of 14 trigger keywords, such as MINQOS, which would trigger the drilling of a replacement well for any well whose oil rate fell below the specified value. See the keyword document for a list of all the possible triggers.

wl is a well list, containing all the wells which should be checked for replacement. For each well in this well list, the possible replacements, must be specified in a DRILL table. The possible replacement wells should initially be inactive. If the DRILL table specifies a well list as the replacement, rather than a single well, then the first well in that list that is inactive and not currently drilling, and that can be drilled by an inactive rig, will become the replacement. The same well can be listed as a possible replacement for more than one well, in which case, once it is chosen as a replacement for one of the wells, and starts drilling, it is no longer available as a replacement for any other well. Only one replacement well is drilled for each well in the well list wl, even if the replacement well subsequently violates the trigger. However, the replacement well could itself be replaced, if it is listed in well list wl, and has possible replacement wells specified in a DRILL table.

value is the value used when determining whether the trigger has been violated.

mask is an array of logicals (true or false), which if false, will prevent the corresponding well in the well list from being replaced. This provides a means for the user to override the automatic drilling algorithm, but will usually be input as ALL1D, which specifies a value of true for all members of the well list.

When a replacement well is drilled, the well will be available to start production or injection after the sum of the drill time, and completion time, specified in the DRILL table, plus the move time specified in the RIG table if the rig had to be moved from one drill site to another. The rig is available to drill another well after the sum of the drill time and the move time (if any). Timesteps are not adjusted to coincide with

182 Network Calculations: Drilling Queues R5000.0.2


start of production or injection, but rather, the well will start operations as soon as the simulation time equals or exceeds the availability time for the well. The well which is being replaced is immediately deactivated at the time that the trigger is violated. If desired, the redrill procedure can be specified multiple times in a PROCS block, using different triggers for the same well list, in which case the replacement will be drilled as soon as the first trigger is violated.

The second form of the redrill procedure is more general, and allows users to specify their own criteria to determine when replacements are needed. The arguments are as follows:

redrill(key1, wl, mask). Here the arguments are:

key1 can be either RIGS or NORIGS. If NORIGS is specified, then drilling is not subject to rig availability. If RIGS is specified, then a well can only be drilled if there is a rig available, and the drill site(s) from which the well can be drilled has less than the maximum number of rigs operating.

wl is a well list, containing all the wells which should be checked for replacement. For each well in this well list, a replacement or replacements, must be specified in a DRILL table. The replacement wells should initially be inactive.

mask is an array of logicals (true or false), which if true, will cause a replacement well to be drilled for the corresponding well in the well list wl.

The following example illustrates the use of this form of the redrill procedure. In this example, producers in the well list PROD_WELLS, which have an oil rate less than 60 STBD, and a water cut greater than 0.95 will be replaced.

PROCSLOGICAL_1D redrill_maskredrill_mask = q("OS",PROD_WELLS,ALL1D) < 60.0 and &wcut(PROD_WELLS,ALL1D) > 0.95REDRILL("NORIGS",PROD_WELLS,redrill_mask)ENDPROCS



Gaslift Optimization

Gaslift is used for gas-liquid two-phase producers to boost the production. Increasing gas-liquid ratio (GLR) reduces the average fluid density in the well connection, and therefore reduce the hydrostatic pressure drop (pressure drop due to elevation change) and hence reducing the bottomhole pressure (assuming tubing head pressure is fixed), resulting in a higher production rate. However, as the fluid flow rate further increases, the increase in the frictional pressure drop will offset the decrease in the hydrostatic pressure drop, therefore an optimal gaslift gas rate exists for a maximum production rate, or more generally, a maximum production profit.

In Nexus, the gaslift gas rate can be specified by a QGSMAX constraint at the gaslift connection. It can also be automatically allocated using either an optimal gaslift table, or an optimization algorithm using a nonlinear optimizer, while the QGSMAX constraint will be the upper bound of the gaslift flow rate.

184 Network Calculations: Gaslift Optimization R5000.0.2


Optimal Gaslift Table Method

The optimal gaslift table is used to calculate the gas rate for the gaslift connection to which it assigned (see the TYPE keyword in the NODECON table, described in the Surface Network section of Nexus Keyword Document). The gaslift rate is calculated by interpolating the optimal GLR or GOR from the gas lift table at the oil rate (or liquid rate) and water cut in the connection into which the gas is injected, and the mobility weighted average wellbore pressure (if it is in a linear series of connections that start at a production well). Referring to Figure 4.20, for a table of optimal GLR, the gas rate in connection 2 would be given by:

4-29

where

is the surface gas rate for connection 2

is the surface liquid rate for connection 3

is the surface gas rate for connection 1

GLR is the GLR value interpolated from the table at the oil (or liquid) rate, and water cut in connection 3, and the mobility weighted average wellbore pressure of the well that the production is flowing from.

And for a table of optimal GOR, the gas rate in connection 2 would be given by:

4-30

where

GOR is the GOR value interpolated from the table at the oil (or liquid) rate, and water cut in connection 3, and the mobility weighted average wellbore pressure of the well that the production is flowing from.

The gas rate in the gaslift connection can also not exceed the maximum gas rate assigned to the gaslift connection in a CONSTRAINT table.

q GLR q qgs liqs gs2 3 10= −max( , * )

qgs2

qliqs3

qgs1

q GOR q qgs liqs gs2 3 10= −max( , * )

R5000.0.2 Network Calculations: Gaslift Optimization 185


Figure 4.20: A Well with a Gaslift Connection.

Optimization Using a Nonlinear Optimizer

The objective of this option is to automatically allocate gaslift gas rates so that the profit of the production operation can be maximized. This option achieves the "instantaneous" optimization at the each time step. Unlike the conventional method that uses a performance curve4, there is no requirement that the tubing head pressure must be specified; in other words, the interaction between wells has been taken into account. Besides, a gaslift connection can be connected to any node of the production network.

Mathematically, the automatic gaslift gas allocation problem can be formulated to a constrained nonlinear optimization problem, or a Nonlinear Programming (NLP) problem in the following form,

Maximize F(x), 4-31

subject to constraints,

hi(x) = 0, for i = 1, ..., m1, 4-32

lgi gi(x) ugi , for i = 1, ..., m2, 4-33

lx x ux, 4-34

where F(x) is the benefit (objective) function, x is the vector of decision variables; hi and gi are the constraint functions; lgi and ugi are the lower

Connection 2(gas liftconnection)

Connection3(towards well heador sink)

Connection 1(from well)



and upper bounds of the inequality constraints, respectively; m1 and m2 are the numbers of equality and inequality constraints, respectively; lx and ux are the vectors of lower and upper bounds of the decision variable, respectively.

We define the benefit function as the profit of the production,

4-35

where Qoil is the oil phase volumetric production rate, Qgas is the gas phase volumetric production rate, Qwat is water phase volumetric production rate, Qglift is the gaslift gas volumetric injection rate, Roil is the unit value of produced oil, Rgas is the unit value of produced gas, Rwat is the unit processing cost of produced water, Rglift is the unit operating cost of gaslift gas. The unit values and costs can be input using the following keywords under the keyword GLIFTOPT.

The decision variable x is the vector of the gaslift gas volumetric rates at the gaslift connections. The lower bound of a decision variable is zero, and the upper bound is the value of the QGSMAX constraint at the corresponding gaslift connection. The constraints of the optimization problem are constructed based on the user-specified targets in TARGET tables, such as the maximum total oil phase production rate of a list of production wells, and the maximum total gaslift gas rate. (A target is the total rate of a group of connections.) Normally all the constraints are inequality constraints.

The optimization problem is solved using an optimizer with Generalized-Reduced-Gradient (GRG) method10, which is highly efficient for solving nonlinearly constrained optimization problems.

This option is invoked by specifying the keyword GASLIFTOPT as the CTRLMETHOD in a TARGET table, the maximum total gaslift gas rate is specified through the VALUE in the TARGET table, and the name of the gaslift connections list that is controlled by the

Parameter Keyword

Roil OILVALUE

Rgas OILVALUE

Rwat WATCOST

Rglift GLIFTCOST

F x Q R Q R Q R Q Roil oil gas gas wat wat glift glift( ) = + − −



optimization algorithm is specified through the CTRLCONS in the TARGET table. The maximum gaslift rate at a gaslift connection is specified by the QGSMAX keyword in a CONSTRAINT table.

Since the optimization calculation may require multiple network solves, which can be expensive, it is only performed at the first network/reservoir global Newton iteration, and normally it is not performed at every time step. The maximum time interval between optimization calculations can be specified by keyword MAXNOGLDT under keyword GLIFTOPT, and the minimum time interval between optimization calculations can be specified by keyword MINNOGLDT under keyword GLIFTOPT.

The method works in the following steps,

1. At the time step when the gaslift optimization is on, solve the optimization problem using the GRG optimizer. The function evaluations of the benefit function and the constraint functions are provided by the "potential solve" of the network. A potential solve is a network solve with all user-specified constraints (such as QOSMAX and PMIN constraints, etc.) being imposed, and all user-specified targets being relaxed.

2. Allocate the gaslift gas rates according to the solution of the decision variables. Allocate all other targets according to the solution of potential solve as usual. (See the "Targeting Input" section in this chapter.)

3. Solve the network with the allocated gaslift rates and other allocated targets, finish the reservoir/network global Newton iterations of current time step.

4. March to the next time step and repeat from Step 1.

The gaslift gas rates from the previous time step are taken as the initial guess of the decision variables of optimization problem. The initial gaslift gas rates may need to be adjusted so that they are in the feasible region -- that is, no constraints of the optimization problem should be violated with the initial gaslift gas rates.

The derivatives of the benefit function and constraint functions with respect to the decision variables are necessary for the GRG optimizer. They can be evaluated by solving the perturbed network Jacobian system, described as follows.



The equation system of the network potential solve is,

4-36

where u is the vector of the primary unknowns of the network solve (the component flow rates in the connections and the pressures at nodes), is the vector of the parameters (the gaslift gas rates) to be perturbed. With a given value of , Equation 4-36 has been solved for u using Newton iterations.

Now we impose a perturbation to , the corresponding changes of u would be u,

4-37

Expanding Equation 4-37 in a Taylor series about (u, ) and including only the first-order terms yields,

4-38

Substituting Equation 4-36 into Equation 4-38 yields,

4-39

The Jacobian matrix of the system is defined as,

4-40

Equation 4-38 can be rewritten as,

4-41

F u( , ) = 0

F u u( ,+ + = ) 0

F uF

uu

F( ,

) +∂

∂+ ∂

∂= 0

∂∂

+ ∂∂

=F

uu

F

0

J uF

u( ) = ∂

∂

J uu F F u F u F u( ) = − ∂

∂≈ − + − = − +

( , ) ( , ) ( , )



The can be solved by a linear solver after the on the

right-hand-side has been calculated by the perturbations.

The solution can be used to calculate the derivatives of the benefit function and the constraint functions with respect to gaslift gas rates (the decision variables) through train rule. Note that the Jacobian matrix J(u) has already been factorized in the network potential solve with a set of gaslift gas rates. Equation 4-41 can be solved with a forward-substitution and a back-substitution. Since (the gaslift gas rates) is a linear item in the connection equations at the gaslift connections, the right-hand-side of Equation 4-41 is -1 in the row containing ; the right-hand-sides of other rows are zero.

u −

F



Network PVT Data

PVT and Separator Method Assignments

All connections in the network are assigned a PVT method, and for EOS runs, a separator method. Connections in non-EOS runs may also be optionally assigned a separator. The default PVT method (and separator method for EOS runs) is the first method specified in the case file (the .fcs file). The PVT method is used to calculate in-situ fluid flow rates, densities, and possibly viscosities, as a function of the mass flow rates, pressure and temperature. The in-situ flow rates are used in evaluating in-situ constraints, and hydraulics calculations using pressure drop correlations, hydrostatic gradients, valves, chokes and pumps.

The separator method is used to calculate the stock tank flow rates that would result if the fluid flowing in the connection was put through a test separator. Stock tank rate constraints, such as QOSMAX, are evaluated using this test separator calculation, as are hydraulic table lookups. Note that in EOS runs, and in some non-EOS runs, the sum of the stock tank phase rates for a group of connections will be different from that calculated if flow from all the connections are first combined, then fed through a separator. The COMMINGLE option provides an alternative way to treat the test separators. In this case, the test separation is performed, using the partial mass/molar volumes obtained from a separation calculation using all the connections flowing to a real separator. The real separator must be identified as a SEPNODE. If this option is active, then the sum of the stock tank phase rates for a group of connections that flow to a common SEPNODE will sum to the same rates as that obtained by separating the combined flow.

Note that the PVT and separator methods are treated as a function of location in the network, and not as a property of the fluid. This is the same as PVT assignments for reservoir grid blocks, where the PVT is treated as a function of the rock. This is of course, not correct, so the user must be careful to avoid inconsistencies in PVT and separator assignments, which could lead to apparent changes in the calculated flow rates due to a difference in the PVT or separator method used by two connections that actually have the same fluid flowing.

R5000.0.2 Network Calculations: Network PVT Data 191


Multifield networks with Mixed PVT

For multifield simulations, Nexus allows each reservoir to have a different “Black Oil” fluid system, with commingled production/injection in the network. Here, “Black Oil” fluid systems include Black Oil, API, Gas/Water and Water/Oil. It is assumed that all fluid systems have three phases at surface conditions, (although one of the phases may have zero volume) and that the network has at least as many components as the largest number of components in any of the grids. The network has one fluid system (one set of components), and the fluid is transformed from one fluid system to another at the perforations.

Fluid properties in the network can be evaluated using PVT methods for any of the fluid systems. If the network fluid system is different than the PVT method, then the fluid is converted to the same fluid system as the PVT method before the flash is performed, then converted back to the network fluid system after the flash.

The current implementation assumes no composition or pressure dependence for the conversion from one fluid system to another. This is appropriate for mixing gas-water, water-oil, black oil and API fluid systems.

The fluid transformation at the perforations is handled as follows:

Producing Perforations

The grid perforation equations are

4-42

Where the subscript g means grid, and n means network, and i is the component.

We also have an equation relating the grid components to the network components

4-43

Q C zgi k k ik g nk

nphases

= −=

∑ ( )1

Q A Qn gn g=

192 Network Calculations: Multifield networks with Mixed PVT R5000.0.2


Where Agn is a matrix with ncn rows and ncg columns.

For gas-water/black oil system, if we choose = {Qggas, Qgwater}, and = {Qngas, Qnwater, Qnoil}, then

4-44

where

4-45

where

OGR is the oil/gas ratio, and and are the oil and gas densities at standard conditions.

Equation 4-43 can be used to eliminate the grid component rates.

4-46

4-47

where Ang is a matrix with ncg rows and columns, consists of ncg of the network component rates, consists of the remaining ncn - ncg network component rates, and Ann is a matrix with ncn - ncg rows and ncg columns.

Qg

Qn

A

cgn =

1 0

0 1

0

c OGR oilstd gasstd= /

oilstd gasstd

Q A Qg ng np=

Q A Qns

nn np=

Qnp

Qns

R5000.0.2 Network Calculations: Multifield networks with Mixed PVT 193


For the gas-water/black oil system, we chose = {Qngas, Qnwater} and = {Qnoil}, then

4-48

4-49

We substitute Equation 4-48 into the grid perforation equations (Equation 4-42), to obtain ncg perforation equations relating reservoir variables (masses and pressure), to network variables (network component rates), and Equation 4-49 provides the remaining ncn - ncg equations.

Injecting Perforations

We have a single perforation equation for total mass rate, plus ncn – 1 composition equations, determined by the network. We use Equation 4-46 to eliminate the grid component mass rates.

IMPES Elimination and Elimination of Water Mass for Implicit Runs

The Impes elimination, and the elimination of water mass for Implicit runs is performed exactly the same regardless of whether the network and grid use the same component set or not. The perforation equations are formulated in terms of component rates for the grid fluid system, and all the elimination is performed using these variables. Then the variable substitutions outlined above are performed when the perforation equations are loaded into the solver.

Reservoir Equations

The source/sink term in the reservoir equations is converted from the reservoir component rates to network component rates using Equation 4-48.

Qnp

Qns

Ang =1 0

0 1

A cnn = 0

194 Network Calculations: Multifield networks with Mixed PVT R5000.0.2


Mixing of Fluid in the Network

The properties of fluids in the network are calculated using the PVT method (and separator method for stock tank rates) assigned to each connection. These are a property of the connection and not the fluid, so when fluids mix in the network, the user must select PVT methods appropriate for the mixture. The only “mixing” model currently available in Nexus is the API fluid system. In this model, the fluid properties are a function of the standard density of the oil. Oil is modeled using two components, oil1 and oil2, and the relative proportion of these two components determines the standard density used to look up the PVT table.

R5000.0.2 Network Calculations: Multifield networks with Mixed PVT 195


Fuel and Shrinkage

Gas used as fuel, or lost to shrinkage, can be specified for connections. Either the stock tank rate of fuel or shrinkage can be specified, or the fraction of the gas flowing in the connection that will be used as fuel or lost to shrinkage can be specified. This can only be specified if the connection is the only outflow connection from a node. Therefore, fuel and shrinkage can not be specified for the outflows of a SEPNODE or from a stream splitting node. The fuel and or shrinkage are removed at the inlet (upstream) node to the connection, so the rates reported for the connection are the rates after fuel and shrinkage have been removed. Fuel and shrinkage are only calculated if the connection is flowing in the normal direction, from the inlet to the outlet node. If the connection reverses flow direction, the fuel and shrinkage are set to zero.

196 Network Calculations: Fuel and Shrinkage R5000.0.2


Calculation of Connection Length and Depth Change

The length of a connection and the depth change from the inlet to the outlet node can potentially be determined from several different inputs, and these inputs may be inconsistent. Nexus uses the following procedure to calculate these values.

Depth Change (DDEPTH)

1. If DDEPTH is input in the NODECON table for this connection, this value is used

2. If DDEPTH is not input, and node depths at both ends of the connection have been input in a NODES table, then the difference in the inlet and outlet node depths is used

3. If both node depths have not been input, and no elevation profile is specified, then any node depth that has not been input it defaulted to zero depth (except for the bottom hole node, which is either input on the WELLS table in the BHDEPTH column, or defaulted to the depth of the first perforation). These depths are then used to calculate DDEPTH.

4. If there is an elevation profile, DDEPTH is calculated from the difference in the depth of the start and end of the profile.

5. If DDEPTH is obtained by 1 or 2 above, and an elevation profile is available, and ALLOW_BHMOVE is specified, then for the well connection only, if DDEPTH is greater than the depth change in the profile, the depth change from the profile is used, the bottom hole node is moved to be consistent, and the connection from the first perforation to the bottom hole node is extended to make up the difference. If DDEPTH is less than the depth change from the profile, then the profile is truncated.

R5000.0.2 Network Calculations: Calculation of Connection Length and Depth Change 197


Connection Length (CON_LENGTH)

1. If MDIN and MDOUT are specified in the NODECON table, then the difference in these values gives the connection length.

2. Else if an elevation profile is present, CON_LENGTH is calculated from the length of the profile. Note that if ALLOW_BHMOVE is specified, then the profile is truncated if the DDEPTH is less than the depth change in the profile

3. Else if a temperature profile is present, CON_LENGTH is calculated from the length of the temperature profile

4. Else if LENGTH is input in the NODECON table, that length is used

5. Else if the X, Y and DEPTH values are all available for the inlet and outlet nodes (input in a NODES table), then the length is calculated from the straight line distance between the two nodes.

6. Else CON_LENGTH is set to the same value as DDEPTH.

Consistency Checks

Nexus performs the following consistency checks.

1. DDEPTH <= CON_LENGTH

2. DDEPTH <= Elevation Profile Depth Change

Nexus does not attempt to reconcile differences between node depths input in the NODES table, and DDEPTH specified in the NODECON table..

198 Network Calculations: Calculation of Connection Length and Depth Change R5000.0.2


Network Reports

The rates for connections are calculated by flashing the mass flow rates for the connection through the separator assigned to that connection. If the run uses a black oil, API, gas/water or water/oil fluid, and no separator is assigned, then the fluid is converted to stock tank conditions using the standard densities from the PVT method.

Well rates are calculated by summing the mass flow rates from all perforations, and separated using the separator (or PVT method if no separator is assigned) assigned to the well. This separator (or PVT method) can be specified in a WELLS table. The default separator is the first separator specified in the case file, and the default PVT method is the PVT method used by the grid block for the first perforation.

The total reservoir rate is obtained by summing the well rates. An optional report, activated by the NETSUM keyword in the run control file, outputs a total field rate obtained by summing production and injection at the network sources and sinks. These rates can be different than the sum of the wells, due to commingling of fluids in the network, the separation system being different from the separator assigned to the wells, or inconsistencies in the separator, or PVT methods in the network.

Perforation rates are calculated using the partial molar/mass volumes obtained from the separation of the total well flow rates. If the well is shut in and crossflowing, then the separation is performed on the sum of the producing perforations. The reported perforation rates represent the contribution of each perforation to the total (commingled) well production, and sums to the total well flow rate.

R5000.0.2 Network Calculations: Network Reports 199


References

1. Nexus Keyword Document, Landmark Graphics Corp., 2007.

2. Brill, J.P. and Beggs, H.D., Two-Phase Flow in Pipes, University of Tulsa, Sixth Edition, January, 1991.

3. ECLIPSE Technical Description, Chapter 34, Schlumberger, 2003.

4. VIP-EXECUTIVE Technical Reference, Chapter 39, Landmark Graphics Corporation, 1997.

5. Fletcher, R., Practical Methods of Optimization, Chapter 8, John Wiley & Sons, Ltd, Second Edition, 2004.

6. Hydraulic Institute standard HI 9.6.7-2004. “American National Standard for Effects of Liquid Viscosity on Rotodynamic (Centrifugal and Vertical) Pump Performance.”

7. “Critical and Subcritical Flow of Multiphase Mixtures Through Chokes”, by T. K. Perkins, SPE Drilling and Completion, Dec. 1993, pp. 271-276

8. “Chemical Engineers’ Handbook”, third edition, John H. Perry (ed.), McGraw-Hill Book Co. Inc., New York City (1950), p. 404.

9. Beggs, H. D., Production Optimization Using NODAL Analysis, OGCI Publications, 1991.

10. Fletcher, R., Practical Methods of Optimization, Chapter 8, John Wiley & Sons, Ltd, Second Edition, 2004.

200 Network Calculations: References R5000.0.2


Chapter 5

Well Data

Calculation of the Wellbore Constant for Each Perforation

The wellbore constant is calculated as follows:

5-1

whereC is the wellbore constant.

kh is the permeability*(wellbore length) product for the grid block

pperf is a partial perforation factor (which will plug the perforation if it is set to 0)

wi is the wellbore index

D is a constant that results in the correct units for flow rate.

Wellbore Conversion Constants

The wellbore index wi is calculated as follows:5-2

Units Volumetric Flow Rate Units Conversion Constant D

ENGLISH ft3/d 6.328286*10-3

METRIC m3/d 8.5270171*10-5

METBAR m3/d 8.5270171*10-3

METKG/CM2 m3/d 8.3622802*10-3

LAB cc/hr 7.4665476

C D kh pperf wi= * * *

wi r r skinb w= +2 360 *( / ) / (ln( / ) )

R5000.0.2 Well Data: Calculation of the Wellbore Constant for Each Perforation 201


where

is the angle open to flow (in degrees)

rb is the effective grid block radius

rw is the wellbore radius

skin is the skin factor

However, wi can be input in the WELLSPEC table, in which case the input value is used instead of the calculated value. radb, radw and skin must still be input in the WELLSPEC table, but are not used unless radw or skin are modified. If modified values of radw or skin are input, then Equation 5-2 is used to recalculate wi.

The wellbore constant is used in the perforation equations, given in the "Network Calculations" chapter, to calculate the mass flow rates for production and injection perforations.

202 Well Data: Calculation of the Wellbore Constant for Each Perforation R5000.0.2


Effective Grid Block Radius RADB

One-Dimensional Flow

Linear Case

The first, and simplest, example of the calculation of WI is one-dimensional, linear (i.e., rectilinear) flow, where the grid is as shown in Figure 5.1.

Figure 5.1: One-Dimensional, Linear, Model

The boundary condition, pwf, is to be imposed at the left end of the model, which is at a distance x/2 from the nearest node (also called the “pressure center”), located at x1. Assuming single-phase flow (and ignoring the dimensional constant 0.0001127), from Darcy’s law we have 5-3

y

z

x2

P

x

1 5

qkyz x 2 ---------------------- p1 pwf– =

R5000.0.2 Well Data: Effective Grid Block Radius RADB 203


so the “correction” that adjusts p1 to the correct value at the boundary is5-4

For single phase flow, the well index is defined by5-5

Comparison of Equations 5-3 and 5-5 yields5-6

(where WI is dimensionless)

And comparison with Equation 5-2 yields an effective grid block radius given by

5-7

p1 pwf– q x 2 kyz

-------------------------=

q WIkz

------------- p1 pwf– =

WI yx 2

------------------=

r r x yb w= ( ) /

204 Well Data: Effective Grid Block Radius RADB R5000.0.2


Two-Dimensional Flow

Well in Center of Square Gridblock

The simplest two-dimensional situation is shown in Figure 5.2.

Figure 5.2: Well in Square Grid

The figure depicts an areal model with uniform square gridblocks and one well at the center of one of the blocks, arbitrarily designated by (i,j) = (0,0). To determine the well index for the well, one may carry out a simple finite-difference calculation for a situation where the exact solution satisfies the steady-state single-phase radial flow equation:

5-8

p0,0 is not expected to be equal to pwf, but how well do the pressures at the other nodes satisfy Equation 5-8? If we plot the difference between pi,j and p0,0 versus the log of the radial distance from node (i,j) to node (0,0), then a straight line with the expected slope of q/2kz is obtained1. By plotting the dimensionless pressure drop:

(pij - p0,0)/(q/kz)

against the log of the dimensionless radius , the straight line has a slope of 1/(2), as shown in Figure 5.3. The pressure at node (1,0) is slightly off from the straight line, but all the other pressures lie extremely well on the line. This figure may be considered to be a

p pwfq

2kz--------------------ln

rrw-----+=

r x i2

j2

+=



universal plot, applicable to any square grid with a single well located far from any boundaries.

The extrapolation of the straight line to the horizontal axis is extremely significant. This is the point where the exact radial solution is equal to the well block pressure, p0,0, and occurs when the radius equals 0.2x. Thus the straight line of Figure 5.3 has the equation:

5-9

Figure 5.3: Numerical Solutions for Pressure Plotted versus Radius

Comparison with Equation 5-8 shows that the well block looks like a well with a wellbore radius of 0.2x. We call this radius the equivalent radius of the well block, rb.

pij p0 0q

2kZ---------------------ln

rij

0.2x--------------+=



To obtain the well index, we proceed as follows. From Equation 5-8 we can write:

5-10

Subtracting Equation 5-10 from Equation 5-9, and rearranging, yields:5-11

Comparison of Equation 5-11 (in which the well block pressure is p0,0) with Equation 5-5 (in which the well block pressure is p1) gives:

5-12

General Definition of rb

The following values apply for the square grid of the preceding section:

0.2x = rb, and p0,0 = pb (the well block pressure)

Therefore, Equation 5-11 can be written:5-13

We shall take this as a general definition of rb, for any geometry of the well block.

Similarly, Equation 5-12 can be written:5-14

We express the well index in this form, since rb in many cases is easy to compute. Note that Equation 5-14 does not take skin into account.

We shall show how to incorporate skin in the "Incorporating Skin into Well Model" section below.

pij pwfq

2kz--------------------ln

rij

rw-----+=

q2kz

ln 0.2x rw --------------------------------------- p0 0 pwf– =

WI 2ln 0.2x rw ---------------------------------=

pb pwf–q

2kz--------------------ln

rb

rw-----=

WI 2ln rb rw ------------------------=



An Approximate Derivation of rb for Square Well Block

One can take advantage of the fact that the pressure at (i,j) = (1,0) almost satisfies the radial flow equation, to derive an approximate formula for rb. The difference equation satisfied by the gridblock pressures (for steady state) is:

5-15

For the case shown in Figure 5.2 above, the pressures are symmetric about (0,0), so that p0,1 = p-1,0 = p0,-1 = p1,0. Also x = y. Then:

5-16

Assuming that p1,0 satisfies the radial flow Equation 5-8 exactly gives:5-17

since r1,0 = x. From the definition for rb, Equation 5-13 yields:5-18

Subtracting Equation 5-18 from Equation 5-17 yields:

and combining with Equation 5-16, yields

or5-19

kz

-------------yx------ pi 1 j+ 2pij– pi 1 j–+

kz

-------------xy------ pi j 1+ 2pij– pi j 1–+ qij–+ 0=

4 p1 0 p0 0– qkz-------------– 0=

p1 0 pwfq

2kz--------------------ln

xrw------+=

p0 0 pwfq

2kz--------------------ln

rb

rw-----+=

p1 0 p0 0–q

2kz--------------------ln

xrb------=

lnxrb------ 2

---=

rb

x------ e

– 20.208= =



We shall show later that an exact value for rb for a well far from the boundaries or from any other well (i.e., an “isolated” well) in a uniform square grid is 0.1985 x. Thus, a good rule of thumb to use for a square well block is rb = 0.2 x.

The approximate method described in this section can be extended to nonsquare grids, wherexy¼. It gives adequate results when the aspect ratio y/x lies between 0.5 and 2.0. Outside that range, it gives poor results2, because it is based on the assumption that the pressures in all the blocks adjacent to the well block satisfy the radial flow equation exactly, and that assumption breaks down badly for long skinny gridblocks.



Well in Center of Rectangular Gridblock

Numerical experiments similar to that described in the "Well in Center of Rectangular Gridblock" discussion can also be carried out for uniform rectangular grids, where y is not equal to x. When the gridblocks are not square, then rb is no longer a simple function of x but depends also on the grid aspect ratio, = y/x. The following table lists values of rb/x and rb/y for a single well at the center of a large rectangular grid for various aspect ratios3.

Neither rb/x nor rb/y are particularly constant with , but it can be seen that rb/x converges to a limit for small y, and rb/y converges to the same limit for small x. This leads to the possibility that a better quantity to divide rb by is the length of the diagonal of each gridblock, because:

looks like rb/x as y goes to zero, and looks like rb/y as x goes to zero. That ratio is very constant with as can be seen from the last column of the table. So now we have the more general rule of thumb, that:

5-20

= y/x rb/x rb/y

1/16 0.1406 2.2502 0.140365

1/8 0.1415 1.1317 0.140365

1/4 0.1447 0.5787 0.140365

1/2 0.1569 0.3139 0.140365

1 0.1985 0.1985 0.140365

2 0.3139 0.1569 0.140365

4 0.5787 0.1447 0.140365

8 1.1317 0.1415 0.140365

16 2.2502 0.1406 0.140365

rb x2 y

2+

rb

x2 y

2+

----------------------------

rb 0.14 x2 y

2+=



In fact, Equation 5-20 can be derived mathematically. It is shown in Reference 3 that:

5-21

where is Euler’s constant, one definition of which is the limit of the difference between the finite harmonic series and the natural log, i.e.,

That gives, for the ratio of Equation 5-21, the value 0.1403649.

Note that, for a square grid with x = y, rb/x is times this constant, or 0.198506, so that gives us the complete mathematical derivation of the first rule of thumb, that rb = 0.2x for a square grid.

rb

x2 y

2+

----------------------------NX NY

lim e–

4-------=

1k---

1

n

ln n–

n lim 0.5772157...= =

2



Well in Center of Block in Anisotropic Rectangular Grid

In an anisotropic medium, where kx ky , the finite-difference equation satisfied by the gridblock pressures is:

5-22

There is an equivalent isotropic problem that is satisfied by the same gridblock pressures, namely:

5-23

where

5-24

5-25

5-26

Corresponding to Equation 5-20, we have:5-27

or

5-28

kxz

---------------y

x------ pi 1 j+ 2pij– pi 1 j–+

kyz

---------------x

y------ pi j 1+ 2pij– pi j 1–+ qij–+ 0=

kez

---------------

ye

xe-------- pi 1 j+ 2pij– pi 1 j–+

kez

---------------

xe

ye-------- pi j 1+ 2pij– pi j 1–+ qij–+ 0=

ke kxky 1 2=

xe ky kx 1 4 x=

ye kx ky 1 4 y=

rbe 0.14 xe2 ye

2+=

rbe 0.14 ky kx 1 2 x2

kx ky 1 2 y2

+=



In this case rbe is the radius of the (almost) circular isobar in the transformed problem (i.e., in the xe-ye plane) that has the same pressure as the well block.

However, in the transformed problem, the wellbore is elliptical, not circular. It is shown in Reference 3 that the pressure solution to the exact differential problem in the xe-ye plane essentially satisfies the equation:

5-29

where5-30

and5-31

We see that Equation 5-31 is a correction to the wellbore radius that accounts for the fact that the wellbore is elliptical in the xe-ye plane.

Now, from Equation 5-29, we can write:5-32

Corresponding to Equation 5-13, which defines rb, we have:

5-33

Comparing Equations 5-33 and 5-32 yields:5-34

p pwf–q

2kez----------------------ln

re

rwe-------=

re xe2

ye2

+=

rwe12---rw ky kx 1 4

kx ky 1 4+ =

pb pwf–q

2kez----------------------ln

rbe

rwe-------=

pb pwf–q

2kez----------------------ln

rb

rw-----=

rb rbe rw rwe =



Combining Equations 5-34, 5-28 and 5-31 then yields the final result for rb, that is:

5-35

which can be used in Equation 5-14 for the well index.

General Extension of Isotropic Results to Anisotropic Grids

In the following sections, we shall be presenting formulas for equivalent well block radius for various well geometries. These will be derived for isotropic grids, presenting rbe in terms ofxe and ye, which may be obtained from Equations 5-25 and 5-26. To obtain rb, this rbe can be substituted into:

5-36

This equation is obtained by combining Equations 5-31 and 5-34.

rb 0.28ky kx 1 2 x

2kx ky 1 2 y

2+

ky kx 1 4kx ky 1 4

+----------------------------------------------------------------------------------=

rbrbe

0.5 ky kx 1 4kx ky 1 4

+ -----------------------------------------------------------------------=



Single Well Arbitrarily Located in Isolated Well Block

By an isolated well block, we mean one that is not near another well block nor near the boundary of the grid, as shown in Figure 5.4.

Figure 5.4: Single Well in Isolated Well Block

A conservative requirement for an isolated well is that it be no closer than 10x or 10y from any other well and no closer than 5x or 5y from any grid boundary.

In Reference 2, it is shown that rbe is independent of the location of the well within such an isolated well block, as in the above figure. Then Equation 5-27 still holds:

5-37

This result, i.e., independence of position, is somewhat surprising, but it does depend on the assumption that the well block is isolated.

rbe 0.14 xe2 ye

2+=



Multiple Wells in Same Isolated Well Block

Two Wells with Same Rate

Figure 5.5: shows two wells in the same well block.

Figure 5.5: Two Wells in the Same Well Block

In Reference 2, it is shown that if two wells with the same rate are placed in the same well block (which is otherwise isolated from any other well block), then:

5-38

where reAB is the distance in the xe-ye plane between the two wells. In terms of the actual coordinates of the two wells, this distance is given by:

5-39

Note that rbe is independent of the actual location of the two wells; the only thing that matters is the scaled distance between them (provided they are isolated from other wells).

rbe

0.14 2 xe2 ye

2+

reAB-----------------------------------------------=

reAB ky kx 1 2xA xB– 2 kx ky 1 2

yA yB– 2+=



Two Wells with Different Rates

If the two wells in the same block have different rates, qA and qB, then Equation 5-38 must be modified as follows:

5-40

Note that qA and qB do not have to be absolute rates; relative rates will do. Also note that if qA qB, then the two wells will have different values of rbe.

Three Wells with Same Rate

Figure 5.6 shows three wells in the same well block.

Figure 5.6: Three Wells in the Same Well Block

Similarly, it can be shown for three wells with the same rate (A, B, and C), that the following equation applies:

5-41

Note that when there are two wells and they have the same rate, they have the same rbe. But when there are three wells, even with equal rates, in general they do not have the same rbe (unless reAB = reBC = reAC).

rbe AqA 0.14 xe

2 ye2

+ qA qB+

reAB qB

----------------------------------------------------------=

rbe A0.14 3 xe

2 ye2

+ 3 2

reABreAC-------------------------------------------------------=



Multiple Wells with Different Rates

The above results can be generalized to take into account any number of wells, A, B, C, D, ... through the following equation:

5-42

where k = B, C, D, ..., is the product over all k, and qt = qA + .

Two Wells With Same Rate in Adjacent Blocks

Figure 5.7 shows two wells in adjacent well blocks.

Figure 5.7: Two Wells in Adjacent Well Blocks

In Reference 4, it is shown that if two wells with the same rate are placed in adjacent blocks in the same row, as in the above figure, then:

5-43

On the other hand, if the two adjacent blocks are in the same column, then x andy should be interchanged, to give:

Note again the isolation requirement: the pair of wells should be sufficiently far from any other wells or from the grid boundaries. Also note again that the scaled distance between the two wells is the important quantity, rather than the actual location of the two wells.

rbe AqA reAk

qk

k 0.14 xe

2 ye2

+ qt

=

k qk

k

rbereAB 0.14 2 xe2 ye

2+ exp 2

xe

ye-------- 1– ye

xe-------- tan=

rbereAB 0.14 2 xe2 ye

2+ exp 2

ye

xe-------- 1– xe

ye-------- tan=



If is the value of rbe when the two wells are at the centers of their respective blocks, and is the distance between the block centers (in the xe-ye plane), then, if the wells are not centered, we can write:

5-44

Note that is equal to either xe or ye.

Wells at Centers of Adjacent Blocks in Isotropic Square Grid

If the medium is isotropic, the gridblocks are square and the wells are at the centers of two adjacent blocks, then Equation 5-43 reduces to:

or

This is not far from the rule of thumb of 0.2 x for a single isolated well in a square grid. If the two wells are at the centers of two blocks that are not adjacent, then this rule of thumb is even better. If the wells are not at the centers of two blocks that are not adjacent, then Equation 5-44 can be used to find the correct rb. But we still require that the pair of wells be isolated from other wells or from the grid boundary.

rbec

reABc

rbe rbec

reABc

reAB =

reABc

rbx 0.14 2 2x2 exp

2--- =

rb 0.190x=



Single Well in Edge Block

In Figure 5.8 shows a single well in a block at the left edge of the grid.

Figure 5.8: Well in Edge Block

Because of the reflection boundary condition, the well and its block have images. Thus, this situation is equivalent to two wells in adjacent blocks, and Equation 5-43 is applicable, provided we take:

Note that rb will be independent of the vertical location of the well.

If the edge block is at the top or bottom boundary, then we should take:

where d is the vertical distance to the boundary.

Again we note that if the medium is isotropic, the grid is square, and the well is at the center of the edge block, then:

reAB 2 ky kx 1 4d=

reAB 2 kx ky 1 4d=

rb 0.190x=



Single Well Exactly on Edge of Grid

We have already looked at cases of a single well near the edge of the grid. It is perhaps less likely that the well will be exactly on the edge of the grid, as shown in Figure 5.9.

Figure 5.9: Well on Edge of Grid

Equation 5-43 is not applicable here. Instead, as Reference 4 shows, the applicable equation is:

5-45

Note that rbe is independent of the vertical location of the well on the edge. If the well is exactly on the top or bottom edge of the grid, then xe and ye should be interchanged, to give:

rbe 0.14 xe2 ye

2+ exp

xe

ye-------- 1–

tanye

xe-------- =

rbe 0.14 xe2 ye

2+ exp

ye

xe-------- 1–

tanxe

ye-------- =



Well Exactly on Edge of Isotropic, Square Grid

If the medium is isotropic, and the grid is square, then:

This is quite different from the value of 0.2 x that one might expect.

Single Well Exactly at Corner of Grid

Figure 5.10 shows the situation where a well is located exactly at the corner of the grid.

Figure 5.10: Well at Corner of Grid

This is an important case, as it occurs frequently in five-spot calculations. In Reference 4, it is shown that:

5-46

where5-47

Since Equations 5-46 and 5-47 are rather unwieldy, it may be preferable to use an empirical equation that fits them quite well:

rb 0.14 2 x exp4--- 0.43x= =

rbe 0.14 xe2 ye

2+ exp E =

E4---ye

xe--------

12--- 1

xe

ye--------

ye

xe--------–

1–tan

ye

xe-------- ++=



5-48

Well Exactly at Corner of Isotropic, Square Grid

For the special case of isotropic medium and square grid, then Equations 5-46 and 5-47 become:

This value of rb, combined with Equation 5-14, should be used for the well index in five-spot calculations, if kx = ky and x =y.

rbe xe2 ye

2+ 0.3816 0.2520

xe

ye--------

0.9401 ye

xe--------

0.9401+

------------------------------------------------------------+=

rb 0.14 2 x exp 4--- 1

2---+

0.72x= =



Single Well Arbitrarily Located in Corner Block

Figure 5.11 shows a single well A located somewhere in the interior of a corner block.

Figure 5.11: Well in Corner Block

This situation is perhaps not too likely, but is included here for completeness. Images of the well and the corner block are shown. In Reference 4 it is derived that:

5-49

where E is given by Equation 5-47.

rbe

0.14 4 xe2 ye

2+

2exp 4E

reABreACreAD-----------------------------------------------------------------------=



Well at Center of Corner Block in Isotropic, Square Grid

For the special case where kx = ky , x = y, and the well is at the center of the corner block, then Equation 5-49 reduces to

or

This is not far from the rule of thumb of 0.2 x. But it does require that the well be exactly at the center of the corner block, which is not too likely.

Horizontal Well

Equation 5-35, which shows rb as a function of x, y, kx, and ky, was derived for an isolated vertical well. For a horizontal well, it appears to be sufficient to replacey by z and ky by kz, to yield:

5-50

The assumption that the well is not near any grid boundary may be hard to satisfy in the simulation of a horizontal well. The question arises: how far does the well have to be from the top or bottom boundary in order to use Equation 5-50?

Reference 5 shows that Equation 5-50 is satisfied to within 10 percent if:

5-51

where zw is the distance from the well to the nearer of the top or bottom boundary. Since kz is usually much smaller than kx, this inequality should be easy to satisfy, if x is not too much bigger than z.

rb0.14 4 2x

2 exp 2+ x 2x x

--------------------------------------------------------------=

rb 0.188x=

rb 0.28kz kx 1 2 x

2kx kz 1 2 z

2+

kz kx 1 4kx kz 1 4

+---------------------------------------------------------------------------------=

xz------

kz

kx----

1 20.9

zw

z------



However, if the inequality of Equation 5-51 is not satisfied, it is necessary to use the much more complicated general formula for rb that was derived by Babu et al6, and which is repeated in Reference 5.

Figure 5.12 displays the result of using this formula for the special case where the well is centered in a reservoir that is infinitely wide.

Figure 5.12: Effect of e and z/z on rbe for Centered Well

As a first approximation, this figure can be used even when the well is not centered and the reservoir is of finite width, provided zw is interpreted as the distance from the well to the nearer of the top or bottom boundary.



Inclined Well

The permeability-thickness for an inclined well is computed by transforming an original anisotropic diffusion equation into an equivalent isotropic equation. The equivalent isotropic equation is not discussed here; however, details are available in Muskat.9 The equivalent isotropic permeability ke is defined by

5-52

where kx, ky, and kz are rock permeabilities in x, y, and z directions, respectively. Nexus extends the theory to computation of a perforation length inclined arbitrarily in space. Projected perforation lengths Lx, Ly, and Lz onto each coordinate axis are transformed, then perforation length Lp in the transformed plane is represented by the following: 5-53

where Lx, Ly , and Lz are x, y, and z components, respectively. These, in turn, are transformed into the isotropic medium:

5-54

5-55

5-56

where l is the physical well perforation length, is the angle of the well segment with respect to vertical direction, and is the angle with respect to the x-axis as shown in Figure 5.13.

ke kxkykz 13---

=

Lp Lx2

Ly2

Lz2

+ +=

Lx lke

kx---- cossin=

Ly lke

ky---- sinsin=

Lz lke

kz---- cos=



Figure 5.13: Inclined Well Segment

Thus, the permeability-thickness for a well segment is expressed by keLp using Equation 5-52 and Equation 5-53. Note that keLp = l(kxkz)

0.5 if a well is parallel to the y-axis.

Well index WI is defined by5-57

where rb is the equivalent radius, rw is the well radius, and s is the skin factor. In computing rb, an analogy to Peaceman’s equation3 for a vertical well is adopted. However, the equivalent radius and well radius are obtained in the transformed plane similarly to the permeability-thickness. Peaceman’s equivalent radius for a well parallel to the y-axis is expressed as

5-58

where notation Rby indicates a well parallel to the y-axis. The numerator can be simplified using properties in the transformed isotropic medium.

x

y

z

l

Lx

Lz

Ly

WI 2

lnrb

rw----- s+

-------------------------=

Rby 0.14kz kx 1 2 x

2kx kz 1 2 z

2+

0.5 kz kx 1 4kx kz 1 4

+ ---------------------------------------------------------------------------------=



Transferring the denominator of Equation Equation 5-58 to Rwy , the following can be concluded.

5-59

where 5-60

where (key/kx)0.5 x and (key/kz)

0.5 z are gridblock lengths in the transformed plane and the equivalent isotropic permeabilities are equal (kx’ = kz’= key). If a well is parallel to x or z direction, the same principle can be applied to get the following:

5-61

where5-62

5-63

where5-64

For an arbitrarily inclined well, i.e., for any (,) in Figure 5.13, Nexus approximates the equivalent radius as follows.

5-65

where

Rey 0.14key

kx------- x

2 key

kz------- z

2+=

key kxkz 12---

=

Rex 0.14kex

ky------- y

2 kex

kz------- z

2+=

kex kykz 12---

=

Rez 0.14kez

kx------ x

2 kez

ky------ y

2+=

kez kxky 12---

=

Rbe rex2

rey2

rez2

+ +=



rex = Rex sin cos rey = Rey sin sin rez = Rez cos 5-66

Note that Equation 5-65 yields Equations 5-59, 5-61, and 5-63 as special cases and any equivalent radii in Equation 5-65 stay in an ellipsoid created by them.

The wellbore radius in the transformed plane is expressed in a similar manner. For example, a horizontal well that is parallel to the y-axis is approximated by 5-67

where Rw is the true pipe radius and Rw(key/kx)0.5 and Rw(key/kz)

0.5 are the well radii in major and minor axes. Similarly, 5-68

For arbitrary values of (,), the transformed wellbore radius is approximated by

5-69

whererwx = Rwx sin cos rwy = Rwy sin sin rwz = Rwz cos 5-70

Rwy

Rw

2------ key kx key kz+ =

Rwx

Rw

2------ kex ky kex kz+ =

Rwz

Rw

2------ kez kx kez ky+ =

Rwe rwx2

rwy2

rwz2

+ +=



Combining Equations 5-65 and 5-69, we obtain the generalized Peaceman’s equivalent block radius for an inclined well from

5-71

The permeability-thickness and the well index computed using Equations 5-52 through 5-71 are implemented in Nexus; but users can override them if correct values are known. Radius Rwe also forms an ellipsoid for arbitrary angles (,), as shown in Figure 5.14.

Figure 5.14: Ellipsoid

The equivalent radius as computed above may not always be accurate because of the assumptions on which Peaceman’s equation is based. Therefore, calibrate the perforation WI’s in proportion to the computed values in such a way that WI from a well test or production data is honored.

Rb

Rbe

Rwe---------Rw=

x’

y’

z’

Rwx

Rwz

Rwy



Incorporating Skin into Well Model

Equation 5-14 presented the well index in terms of rb and rw as shown below:

This equation was derived without considering the effect of skin. In this section, we show how skin may be taken into account.

Derivation of Skin Due to Altered Permeability

Figure 5.15 depicts radial flow from an outer radius, rd, into the wellbore, which has radius rw .

Figure 5.15: Radial Flow With Zone of Altered Permeability

Let the pressure at rd be pd. The reservoir has permeability k, but an inner zone of radius ra has an “altered” permeability, ka. Assuming steady state:

WI 2ln rb rw ------------------------=

pa pwf–q

2kaz----------------------ln

ra

rw-----=

pd pa–q

2kz--------------------ln

rd

ra----=

232 Well Data: Incorporating Skin into Well Model R5000.0.2


Adding and rearranging the above equations yields the following equation:

5-72

where

In this context, skin may be interpreted as causing additional pressure drop due to decreased permeability in the altered zone. However, it may be either positive or negative, depending on whether ka is less than or greater than k. The skin due to altered permeability is also referred to as “mechanical” skin.

Including Mechanical Skin In Well Index

Now suppose we consider solving the same steady-state radial problem with a simulator on a uniform square grid, with x being much smaller than rd. As mentioned above, the well block with pressure pb acts like a well with radius rb. This results in:

5-73

But the equation for the well model in terms of the well index, WI, is:5-74

Combining Equations 5-73 and 5-74 gives:

and comparison with Equation 5-72 yields:

pd pwf–q

2kz-------------------- ln

rd

rw----- sa+

=

sa

k ka–

ka-------------ln

ra

rw-----=

pd pb–q

2kz--------------------ln

rd

rb----=

q WIkz

------------- pb pwf– =

pd pwf–q

2kz--------------------ln

rd

rb---- q

kz WI -------------------------+=

lnrd

rb---- 2

WI-------+ ln

rd

rw----- sa+=

R5000.0.2 Well Data: Incorporating Skin into Well Model 233


or5-75

WI 2

lnrb

rw----- sa+

----------------------=



Skin Due to Restricted Entry

So far, we have assumed that the well fully penetrates the reservoir. When the well is not fully penetrating, there can be an additional pressure drop caused by restricted entry. This additional pressure drop can be interpreted in terms of an extra skin term, sr (also called pseudo skin). Before considering how this skin should be included in the well index, let us discuss two methods for calculating sr. .

Figure 5.16: Pseudo Skin Factor, after Brons and Marting

The first method uses the graph of Figure 5.16, due to Brons and Marting7, which shows sr as a function of two parameters, hp/ht and hte/rw , where hp is the length of the interval open to flow, ht is the total thickness of the producing zone, , and kH/kV is the ratio of horizontal to vertical permeability.

hte ht kH kV=



Figure 5.17 shows three ways in which hp and ht can be interpreted. The first is simple partial penetration; the second is a centered open interval; the third is with multiple entries. In the latter two cases, hp and ht apply to a symmetry element.

Figure 5.17: Examples of Partial Well Completion

The second method for calculating sr , due to Odeh8, allows for a single open interval anywhere in the producing zone. It is an empirical equation derived from some numerical studies:

5-76

where

and zm is the distance from the top of the sand to the middle of the open interval. (See Figure 5.18.)

The two methods are obviously not the same. However, where they can be compared (e.g., for zm = hp/2), they give comparable values for sr .

sr 1.35ht

hp----- 1– 0.825

ln hte 7+ 0.49 0.1 ln hte + ln rwc 1.95–– =

hte ht kH kV=

rwc rwe0.2126 2.753 zm ht+

=



Figure 5.18: Partial Well Completion

ht

hp

zm



Effects of Restricted Entry on Well Index

We need to distinguish two different effects of restricted entry on the well index. First, there is the “overall” restricted entry that is included in the total skin, st, determined in a well test. The skin due to this overall restricted entry, sor , must be subtracted from the total skin to get the mechanical skin:

5-77

Secondly, there is what we might call “local” restricted entry due to partial completion of the well within a layer of thickness zl. The skin due to this local restricted entry must be added to the mechanical skin. Thus, layer by layer, we have:

5-78

For example, consider the 3D situation shown in Figure 5.19. The well completely penetrates the top two layers and partially penetrates the third layer. The overall skin, st, is determined from a well test. The skin due to overall restricted entry is determined from hp1 and ht1 and subtracted from st to get sa. Because total penetration through each of the top two layers is assumed, slr = 0 there. In the third layer, slr is determined from ht2, hp2, and zm2, and added to sa, as in Equation 5-78.

sa st sor–=

WIl2

lnrb

rw----- sa slr+ +

----------------------------------=



Figure 5.19: Partial Well Completion



References

1. Peaceman, D.W., “Interpretation of Well-Block Pressures in Numerical Reservoir Simulation,” Soc. Pet. Engr. J., pp. 183-194 (June 1978).

2. Peaceman, D.W.: “Interpretation of Wellblock Pressures in Numerical Reservoir Simulation: Part 3 - Off-Center and Multiple Wells Within a Wellblock,” SPE Res. Eng., pp. 227-232 (May 1990).

3. Peaceman, D.W., “Interpretation of Well-Block Pressures in Numerical Reservoir Simulation with Non Square Grid Blocks and Anisotropic Permeability,” Soc. Pet. Eng. J., June 1983, 531-543.

4. Peaceman, D.W.: “Interpretation of Wellblock Pressures in Numerical Reservoir Simulation-Part 3: Some Additional Well Geometries,” Paper SPE 16976 presented at the SPE Annual Fall Meeting, Dallas, TX, Sept. 27-30, 1987.

5. Peaceman, D.W., “Representation of a Horizontal Well in Numerical Reservoir Simulation,” Paper SPE 21217 presented at the 11th SPE Symposium on Reservoir Simulation held in Anaheim, CA, Feb. 17-20, 1991; also SPE Adv. Tech. Ser. 1, 7-16 (1993).

6. Babu, D.K., Odeh, A.S., Al-Khalifa, A.J., and McCann, R.C., “The Relation Between Wellblock and Wellbore Pressures in Numerical Simulation of Horizontal Wells,” SPE Res. Eng., pp. 324-328 (Aug. 1991).

7. Brons, F. and Marting, V.E., “The Effect of Restricted Fluid Entry on Well Productivity,” J. Petr. Tech., pp. 172-174 (Feb. 1961); Trans. AIME, vol. 222

8. Odeh, A.S., “An Equation for Calculating Skin Factor Due to Restricted Entry,” J. Petr. Tech., pp. 964-965 (June 1980).

9. Muskat, M., The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill Book Co., 1937.

240 Well Data: References R5000.0.2


Chapter 6

Equilibrium

Introduction

This chapter discusses the methodology employed by Nexus for Model initialization. The major topics covered are:

• Mainstream path• Supercritical initialization• Gibbs sedimentation• Search for GOC

R5000.0.2 Equilibrium: Introduction 241


Mainstream Path

Consider the multiple-equilibrium-region scenario depicted in Figure 6.20 below.

Figure 6.20: Multiple Independent Equilibrium Region Scenario

The following steps are applied:

1. Independent equilibrium regions are defined by the IEQUIL array. These arrays associate blocks with equilibrium methods. Equilibrium “methods” are the user-directed information needed to generate the initial pressure, saturation and composition within these regions. These equilibrium methods include fluid contacts, initial pressures, compositions, and any special methodology that is to be applied.

2. If more than one PVT type were to exist within an equilibrium region then these user-designated regions are automatically subdivided so that only one PVT type exists within a given equilibrium region. These subdivisions will show up in some of the detailed equilibrium output reports

3. Hydrostatic phase-pressure-versus-depth tables are constructed.

Independent Equilibrium Regions

Initialization ScenarioReservoir Cross-Section

Impermeable Barriers

GOC

WOC

Independent Equilibrium Regions

Initialization ScenarioReservoir Cross-Section

Impermeable Barriers

GOC

WOC

242 Equilibrium: Mainstream Path R5000.0.2


• The depth table range of depths (dmin/dmax) closely corresponds to the actual depth span of the cells within the equilibrium region. If the DINIT is specified outside the dmin/dmax span then table dmin or dmax is adjusted to include this depth.

• If an equilibrium region depth span is less than 10 feet then the dmin/dmax will be adjusted to give a table of at least 10 feet in span.

• If a GOC is specified and is within the resulting dmin/dmax span then the PINIT will be ignored and replaced by the PSAT at the GOC.

• Any composition data is interpolated throughout the table. The composition is interpreted as the overall composition.

• Saturation pressure (PSAT) is computed throughout the table. If the type of PSAT (bubble point/dew point) does not correspond with the input data fluid type then a fatal error is reported.

• Phase pressures are calculated from a known pressure and depth datum.

- The initial datum pressure and depth, PINIT/DINIT may be specified in the water, oil or gas zones. However, these specifications are ignored in cases where a GOC is specified and within the resulting dmin/dmax span. This rule (explained above) applies no matter where the PINIT/DINIT is specified.

- A hydrostatic pressure march begins at PINIT/DINIT. Each step to an unknown position in the table, upward or downward, involves a quickly converging successive substitution iteration where the density (function of unknown composition and pressure) is computed first from the know level and then successively at the unknown level.

- During each step, a flash at the interpolated composition is performed. And the resulting phase compositions are used for density calculations.

4. The phase pressures are interpolated from the depth-tables shown in Figure 6.21 and assigned to each cell. Compositions are re-interpolated from the original compositional-depth tables (if

R5000.0.2 Equilibrium: Mainstream Path 243


entered) as is temperature. That is, they are not interpolated from the depth tables. Saturation pressures are calculated for the cells based on these interpolated compositions.

Figure 6.21: Depth Table Construction - Typical Fully Projected Pressure Profiles

PSATPSAT DEPTH Po Pg Pw PcgoPcowDenoDengDenw(ADJ)ORIG)[ft ] [psia [psia] [psia]psia[psia[lb/ft3][lb/ft3][lb/ft3][psia][psia]------------------------------------------------------------------------------------------------------5151.0 3459.5 3933.6 3000.9474.2458.637.441013.026260.58553933.6*4014.75184.0 3468.1 3936.6 3014.7468.6453.337.438913.036160.58603936.6*4014.75217.0 3476.6 3939.6 3028.6463.0448.037.436913.046060.58653939.6*4014.75250.0 3485.2 3942.6 3042.5457.4442.737.434813.055960.58713942.6*4014.75283.0 3493.8 3945.6 3056.4451.8437.437.432813.065860.58763945.6*4014.75316.0 3502.4 3948.6 3070.3446.2432.137.430713.075760.58823948.6*4014.75349.0 3511.0 3951.6 3084.2440.6426.837.428713.085660.58873951.6*4014.75382.0 3519.5 3954.6 3098.1435.1421.537.426613.095660.58933954.6*4014.75415.0 3528.1 3957.6 3111.9429.5416.237.424613.105560.58983957.6*4014.7

Reservoir Phase Pressures

Depth

Water Zone

Oil Zone

Gas Zone

Pcgo at GOC

Pcwo at WOC

Reservoir Phase Pressures

Depth

Water Zone

Oil Zone

Gas Zone

Pcgo at GOC

Pcwo at WOC



5. Determine phase saturations by performing an inverse table lookup using the input capillary pressure versus saturation tables.

• The default method (GBC for grid block center) is to use the MDEPTH or grid block center as the basis for the phase-pressure differential used in the inverse lookup. This produces grid blocks which are in hydrostatic equilibrium (well, almost, subject to some usually small effects covered below). It also has the disadvantage that small variations in the fluid contact positions can make the cell saturation change dramatically. Smaller block thicknesses are required to capture the fluid volumetrics through the transition zone with this method.

• A second option (INTSAT for integrated saturation) is to integrate the saturation versus capillary pressure relationship over the depth span of the cell. This produces potentially much more accurate fluid volumetrics. However, it does not preserve the hydrostatic equilibrium between blocks. (It is a different saturation and therefore a different capillary pressure would be associated with it and it no longer equals the phase pressure differential at the cell mid-point.)

- At this point the default option is to create a special set of additive constants for each cell which bring the system back to equilibrium. In a three-phase situation two constants are needed; one for the water-oil capillary pressure and another for the gas-oil capillary pressure. The negative possible negative impact of these constants is that while they bring the system to equilibrium at time zero, they are permanent and may not make physical sense later in the run if transition zones have changed position significantly.

- The user may also refuse to calculate non-zero capadj constants (NONEQ) and allow the system to be out of equilibrium at the start of the run. This option may obviously have negative consequences.

• In the case of thin oil zones, it is possible that there will be a gas-water transition zone above the water-oil and gas-oil transition zones. This situation is always anticipated as synthetic stand-by gas-water capillary pressure tables are routinely generated from the water-oil and gas-oil capillary pressure curves by adding the values at appropriate equivalent saturations. Inverse table lookups using the synthetic gas-water tables are used when the water-oil and gas-oil capillary pressure tables generate non-positive oil saturations.



6. Optionally, replace any of the above calculated saturations or pressure by overreading values from an external source (OVERREAD). The following rules apply:

• If overread data are present, then after performing equilibration, any pressures or saturations input as an array will replace the calculated values. If no pressure or saturation arrays are input, then this has no effect.

• Either or both Sw and Sg may be overread.

• If Sw only is overread the following rules apply where ‘*’ indicates ‘overread’ data and ‘+’ is capillary-gravity-equilibrium-based result:

- Sw = Sw*

- If(1 – Sw*) <=Sg+, then Sg = (1 – Sw*) and So = 0.

- If (1 – Sw*) > Sg+ , then Sg = Sg+ and So = (1 – Sw* - Sg+).

• Similarly, if only Sg is overread the following rules apply where ‘*’ indicates ‘overread’ data and ‘+’ is capillary-gravity-equilibrium-based result:

- Sg = Sg*

- If (1 – Sg*) <= Sw+ , then Sw = (1 – Sg*) and So = 0.

- If (1 – Sg*) > Sw+ , then Sw = Sw+ and So = (1 – Sg* - Sw+).

If both Sw and Sg are overread then:

- If Sw + Sg > 1 then Sw = 1 – Sg and So = 0

- Otherwise So = (1 – Sw* – Sg*)

- And If So < 1D-8 then So = 0

• The above rules apply for three-phase models except when there is a gas-water contact. In that situation the following applies:

- If Sw input then: Sw = Sw* and Sg = 1 – Sw*

- If Sg input then: Sg = Sg* and Sw = 1 – Sg*



• By default overreading saturations will cause the static capillary adjustment additive constant (same idea as employed for INTSAT option described above) to be calculated unless the NONEQ option is used. If pressure is overridden then no capillary pressure adjustment will be calculated regardless of the status of the NONEQ option.

• For WATEROIL models if Sw only entered:

- Sw = Sw*

- So = (1 – Sw*)

• For GASWATER models if Sw only entered:

- Sw = Sw*

- Sg = (1 – Sw*)

• For GASWATER models if Sg only entered:

- Sg = Sg*

- Sw = (1 – Sg*)

7. Assign grid block pressures

• Three-Phase

- No Capillary Pressure Adjustment (Additive constant)

• If not a GWC case and an oil saturation (See point 5 (c) above) exists then:

a. If Sw >= Swmax – tol then P = Pw + pcowmin

b. Else If Sg >= Sgmax – tol then P = Pg – pcgomax

c. Otherwise, P = Po

• If not (1) then:

a. If Sw >= Swmax – tol then P = Pw + pcowmin

b. Else If Sg >= Sgmax – tol then P = Pg – pcgomax



c. Else Po = Pw - Pcwo

• Capillary Pressure Adjustment

- If not a GWC case and an oil saturation (See point 5 (c) above) exists then:

a. If Sw >= Swmax – tol then P = Pw + pcowmin and Pcadj(wat) = 0

b. If Sw > Swmin + tol and Sw < Swmax – tol then: Po = Po and Pcadj(wat) = Pw – Po – Pcwo*. Where Pcwo* is evaluated at overread or INTSAT conditions.

c. Else if Sg > sgmax – tol then P = Pg – pcgomax and Pcadj(gas) = 0

d. Else if Sg > sgmin + tol then P = Po and Pcadj(gas) = Pg – Po – Pcgo*.

- If not (1) then:

a. If Sw >= Swmax – tol then P = Pw + pcowmin and Pcadj(wat) = 0

b. Else if Sg > sgmax – tol then P = Pg – pcgomax and Pcadj(gas) = 0

c. Else P = Pg – Pcgo* and if Sw > swmin + tol then Pcadj(wat) = Pw – Po – Pcwo*



• Water-Oil

- No Capillary Pressure Adjustment

• If Sw >= swmax – tol then P = Pw + pcowmin

• Else P = Po

- Capillary Pressure Adjustment

• If Sw >= swmax – tol then P = Po and Pcadj(gas), Pcadj(wat) = 0

• Else if Sw > swmin + tol then Pcadj(wat) = Pw – Po - Pcwo*

• Gas-Water

- No Capillary Pressure Adjustment

• If Sw > swmax – tol then P = Pw + pcgwmin

• Else P = Pg

- Capillary Pressure Adjustment

- If Sw >= swmax – tol then P = Pw + pcgwmin

- Else if Sw > swmin + tol P = Pg and Pcadj(wat) = Pw – Pg – Pcwg*

8. Reconcile Flashed Compositions with Saturations

• Black Oil

- Constant Psat

• If Sg < tol and P < Psat then Psat = P

• Else if (GWC exists) then Psat = P – (Pg – Psat) where Pg is generally what is calculated from hydrostatics. Now if P < Psat then Psat = P.

- Variable Psat

• If Sg < tol and P < Psat then Psat = P



• If So < tol then Psat = P – (Pg – Psat) where Pg is generally what is calculated from hydrostatics. Now if P < Psat then Psat = P.

• EOS

- If 1 - Sw < tol then Hydrocarbon Mass Fractions remain at interpolated values. These values will be reset when the first flash is performed outside of the equilibrium initialization routine and component masses set accordingly.

- If P < Psat then flash contents of cell.

- Look for inconsistencies between composition and saturations. Honor saturation by adjusting composition as necessary:

• Two-phase flash and two-phase lookup; Psat = P and the overall mass fraction is computed from the saturations, molar densities and flashed phase compositions.

• Two-phase flash but single-phase oil saturation lookup. Then Psat = P and the overall mass fraction is set equal to the flashed oil phase composition.

• Two-phase flash but single-phase gas saturation lookup. Then Psat = P and the overall mass fraction is set equal to the flashed gas phase composition.

• Single-phase flash (Oil)

a. Two-phase lookup; incrementally add incipient gas (from PSAT determination of single phase oil) until oil becomes saturated. This is done iteratively. And a report will automatically be generated when this occurs which details the incipient gas composition and the final oil composition. Once the oil is in a saturated state then Psat = P and the overall mass fraction is computed from the saturations, molar densities and final phase compositions.

b. Single-phase gas lookup; same procedure as above.

• Single-phase flash (GAS)



a. Two-phase lookup; incrementally add incipient oil (from PSAT determination of single phase gas) until gas becomes saturated. This is done iteratively. And a report will automatically be generated when this occurs which details the incipient oil composition and the final gas composition. Once the gas is in a saturated state then Psat = P and the overall mass fraction is computed from the saturations, molar densities and final phase compositions.

b. Single-phase oil lookup; same procedure as above.

After reconciliation of the composition and saturations then the mainstream equilibrium initialization is complete. Saturations and the overall composition is passed on to be later flashed before the first Newton is performed.



Supercritical Initialization

In some reservoirs, where the composition varies with depth, there is a smooth transition from a fluid that would be labeled a gas (exhibits a dew point) to a fluid that would be labeled oil. However, in these special situations, at all points in the hydrocarbon column the fluid pressure exceeds the Psat and is therefore undersaturated (See Figure below).

Figure 6.22: Transition from Gas to Liquid without Two-Phase GOC

There is a point where the critical temperature of the fluid equals the temperature of the reservoir. This point can be called an undersaturated GOC. Actually, there may be more than one point where the phase label flips from gas to oil or vice versa. When this option is invoked some of the above mainstream procedure is altered somewhat. The highlights are:

1. If the supercritical option is used (CRINIT) the program will verify that the fluids are not saturated anywhere within the equilibrium region domain. If this occurs it is a fatal error. This could mean that this reservoir is not supercritical or there is a data

Mole Fraction

Depth

0.0 1.0

T

P

x

T

Px

Heavy L

igh

t

Tres

Tc

UndersaturatedGOC

TemperatureMole Fraction

Depth

0.0 1.0

T

P

x

T

Px

Heavy L

igh

t

Tres

Tc

UndersaturatedGOC

Temperature

252 Equilibrium: Supercritical Initialization R5000.0.2


entry problem. The error could be with the PINIT or the compositional variation specified.

2. Flashes are not necessary during the depth table build.

3. Only an oil-water inverse table lookup is performed. Recall that there will be always only one hydrocarbon phase. The actual hydrocarbon saturation assigned is 1 – Sw and is assigned to the phase determined by the Psat test.

4. No reconciliation step is needed for the supercritical case.

R5000.0.2 Equilibrium: Supercritical Initialization 253


Gibbs Sedimentation

The Gibbs Sedimentation option (SEDIMENTATION) can be useful if the only limited reservoir composition is known and it is expected that isothermal gravity-chemical equilibrium theory should apply. Even if the applicability of the theory is in question this option provides a theoretical basis on how the composition could vary with depth.

The chemical potential – gravity equilibrium may be expressed with the following equations:

6-1

Where:

is the chemical potential of the i-th component

is the pressure a given reference depth

is the single phase composition of the fluid at the reference depth

is the pressure at depth, h

is the single phase composition of the fluid at the depth, h

is the isothermal temperature

is a given depth

is the reference depth

is the molecular weight of component i

i ref ref i i refP Z T P Z T M g h h, , , ,( ) = ( ) + −( )i Nc=1,

i

Pref

Zref

P

Z

T

h

href

Mi

254 Equilibrium: Gibbs Sedimentation R5000.0.2


This equation may also be expressed in terms of component fugacity, , as:

6-2

This represents unknowns ( and ) and . The remaining equation is that the component mole fractions must sum to 1.

The solution algorithm starts at a known composition (user input or previous step calculation) and then solves the non-linear equations making a total of five tries. These tries are:

1. Standard Newton Iteration

2. Whitson Method without what they call GDEM

3. Whitson Method with GDEM + Standard Newton

4. Whitson Method with GDEM

5. Whitson Method with GDEM with relaxed convergence constraints

The details of these iterations can be viewed if the debug option is selected as described in the “Useful Debugging Output” section below.

If a GOC is specified the algorithm will generate a compositional profile in the gas phase above the GOC and in the oil phase below the GOC.

The supercritical option (CRINIT) maybe combined with the SEDIMENTATION option.

fi

f P Z T f P Z T M g h hi ref ref i i ref, , , , exp( ) = ( ) + −( )⎡⎣ ⎤⎦

i Nc=1,

Nc +1 Z P Nc

R5000.0.2 Equilibrium: Gibbs Sedimentation 255


Search for GOC

The purpose of the Search-for-GOC option is for the situation where the user is unsure where the GOC is and suspects that it may be within the equilibrium region of interest.

To fully understand this option it is import know what the initialization algorithm will do if the user specifies a GOC above the equilibrium region effective depth table span. Under this circumstance, the composition of the oil will be modified if the fluid pressure ever drops below the saturation pressure. Effectively, a flash takes place and the resulting oil composition is used. In other words, the GOC specification is honored and the composition is adjusted.

In the Search-for-GOC option the user omits the GOC data entirely indicating to the program to look for a saturated condition. The algorithm makes two loops, first it checks to see if a saturated condition exists anywhere in the hydrocarbon column. If it does then that is labeled as the GOC. Then a second loop is performed, this time the treatment is identical to what would have happened if the user had entered the “found” GOC. So, in contrast to the paragraph above, a gas-oil transition zone will appear.

This option makes certain assumptions about the uniqueness of a GOC and can fail under certain circumstances so it is important to inspect the applicable depth-table report when using this feature.

256 Equilibrium: Search for GOC R5000.0.2


Chapter 7

Compaction

Introduction

Compaction options can affect the cell pore volume, the transmissibility between cells, and well KH factors. The user can choose between two mechanisms which can produce these changes -- pressure and change in water saturation. Effects can be reversible or irreversible (hysteretic).

Like relative permeability data and other data in Nexus, compaction input data are placed in method files -- in this case, rock property method files. (See the “Rock Properties” chapter of the Nexus Keyword Document for a description of rock property method files.)

Ordinary compressibility may be combined with these compaction options. All of the options can vary by rock property region (defined with the IROCK keyword).

R5000.0.2 Compaction: Introduction 257


Pore Volume Adjustment

The pore volume may be modified by a pressure-dependent factor, , and a water saturation dependent factor, .

7-1

where is the i-th cell's resulting pore volume and is the specified reference cell volume. The multipliers default to 1 if not applicable to a given case. When using a constant or spatial varying cr, the formula is:

7-2

Where is the current gridblock pressure and is the gridblock reference pressure.

If pressure-dependent compaction tables are in effect, the following calculations apply.

Reversible Cases

For reversible cases, the pore volume is modified as follows:7-3

where signifies an evaluation of the compaction table at a

pressure, .

The table below is an example of a compaction table.

CMTP PVMULT2000 0.99853000 0.99954000 1.00005000 1.000

M PiM Swi

PV PV M Mi i P Si wi= 0

PViPVi

0

M cr P PP i i refi i= + −( )1

Pi Prefi

M CMT PP ii= { }

CMT Pi{ }Pi

258 Compaction: Pore Volume Adjustment R5000.0.2


Irreversible Cases

For irreversible cases, the pore volume is modified as follows:7-4

when

where is a user-controllable tolerance and is the historic minimum gridblock pressure,

and when then

7-5

Figure 7.1 illustrates the difference between the irreversible compaction path and the reversible compaction path.

Figure 7.1: Reversible and Irreversible Compaction Factor Paths (Does not include tolrev -- TOLREV_P)

M CMT PP ii= { }

P P Pi tolrev mini− ≤

PtolrevPmini

P P Pi tolrev mini− >

M cr P P P CMT P PP repressure i min tolrev min tolrevi i i= + − −( )⎡

⎣⎤⎦ −1 {{ }

R5000.0.2 Compaction: Pore Volume Adjustment 259


Where is a user-controllable repressurization compressibility

and is an evaluation of the compaction table at the

pressure, .

(For a description of the repressurization method, see CR_REPRESSURE in the Nexus Keyword Document.)

Finally, if simple compressible (as expressed in Equation 7-2) and

compaction tables are used simultaneously, then the combined effect on

is multiplicative.

Water-Induced Compaction Modeling

Water-induced compaction is modeled differently for reversible cases than for irreversible cases.

For reversible cases,

7-6

where is an evaluation of the water-induced

compaction tables at, with the current gridblock water

saturation and is the initial water saturation.

For irreversible cases, when then

7-7

where is a user controllable tolerance (TOLREV_SW) and

is the historic maximum gridblock water saturation,

and when then

crrepressure

CMT P Pmin tolrevi−{ }

P Pmin tolrevi−

M Pi

M WIRTC S SS wi wiinitwi= −{ }

WIRTC S Swi wiinit−{ }S Swi wiinit− Swi

Swiinit

S S Swi wtolrev wmaxi+ ≥

M WIRTC S SS wi wiinitwi= −{ }

Swtolrev

Swmaxi

S S Swi wtolrev wmaxi+ <

260 Compaction: Pore Volume Adjustment R5000.0.2


7-8

M WIRTC S S SS wmax wtolrev wiinitwi i= − −{ }

R5000.0.2 Compaction: Pore Volume Adjustment 261


Transmissibility and Well KH Adjustment

The ij-th interblock connection transmissibility, FTRij may be modified by block pressure dependent factors TPi, TPj or a water saturation dependent factor TSwi, TSwj.

7-9

where is the input reference interblock transmissibility.

Similarly, perforation KH values in block "i" are adjusted according to

the formula

7-10

where is the unadjusted input KH value for the i-th perforation.

The multipliers default to 1 if not applicable to a given case.

The adjustment method is different for reversible cases than for irreversible cases.


7-11

where, signifies an evaluation of the compaction table at a

pressure, .

For irreversible cases,

when

7-12

FTR FTRT T T T

ij ij

P S P Si wi j wj= ⋅+ ⋅

0

2

FTRij0

KH KH T Ti i P Si wi= ⋅ ⋅0

KHi0

T CMT PP ii= { }

CMT Pi{ }Pi

P P Pi tolrev mini− ≤

T CMT PP ii= { }

262 Compaction: Transmissibility and Well KH Adjustment R5000.0.2


where, is a user-controllable tolerance and is the historic

minimum gridblock pressure,

and, when , then7-13

where, is a user-controllable repressurization compressibility

(KP_REPRESSURE) and is an evaluation of the

compaction table at pressure .

Water-Induced Compaction Table Effects

The calculation of water-induced compaction using compaction tables is different for reversible cases than for irreversible cases.


7-14

where, is an evaluation of the water-induced

compaction tables at , where is the current gridblock

water saturation and is the initial water saturation.

For irreversible cases,

7-15

when ,

is a user-controllable tolerance (TOLREV_SW) and is

the historic maximum gridblock water saturation,

and when then

PtolrevPmini

P P Pi tolrev mini− >

T kp P P P CMT P PP repressure i min tolrev min tolrevi i i= + ⋅ − −( )⎡

⎣⎤⎦ ⋅ −1 {{ }

kprepressure

CMT P Pmin tolrevi−{ }

P Pmin tolrevi−

T WIRTC S SS wi wiinitwi= −{ }

WIRTC S Swi wiinit−{ }S Swi wiinit− Swi

Swiinit

T WIRTC S SS wi wiinitwi= −{ }

S S Swi wtolrev wmaxi+ ≥

SwtolrevSwmaxi

S S Swi wtolrev wmaxi+ <

R5000.0.2 Compaction: Transmissibility and Well KH Adjustment 263


7-16

Note that Equation 7-16 becomes a constant for this part of the irreversible case.

T WIRTC S S SS wmax wtolrev wiinitwi i= − −{ }

264 Compaction: Transmissibility and Well KH Adjustment R5000.0.2


Implicitness of Calculations

Interblock transmissibility adjustment (pressure and water saturation) is done explicitly for IMPES and is fully implicit for IMPLICIT. Well KH adjustment is treated implicitly for both solution methods.

R5000.0.2 Compaction: Implicitness of Calculations 265


266 Compaction: Implicitness of Calculations R5000.0.2


Chapter 8

Relative Permeability Methods

Introduction

Like other input for Nexus (for example, equilibrium regions and PVT regions), input for relative permeability is done by methods. These methods are assigned to individual grid blocks through the IRELPM array. A method contains all of the information necessary to calculate relative permeability for all of the phases present. Therefore it contains saturation tables, three-phase relative permeability methods, hysteresis options and any other special features relevant to the calculations. Each method is independent; therefore there is total flexibility when choosing the various options in different domains.

This chapter covers in detail the following topics:

• Three-phase oil relative permeability models

• Hysteresis

• End-point scaling

• Directional relative permeability

• Relative permeability and capillary pressure adjustment near the critical point

The rules governing tabular data entry are fully covered in the “Relative Permeability and Capillary Pressure” chapter of the Nexus Keyword Document and are not repeated here.

R5000.0.2 Relative Permeability Methods 267


Three-Phase Oil Relative Permeability Models

When describing three-phase flow through porous media, the relative permeability to the intermediate-wetting phase (generally assumed to be the oil phase) normally is calculated from two sets of two-phase relative permeability data (krow and krog). Stone's Model I, Stone's Model II, and the Saturation Weighted Interpolation Model are available in Nexus to predict three-phase oil relative permeabilities. There is also an option to model three-phase water relative permeability by similar methodologies described further below.

Stone's Model I

Stone's Model I1 is invoked by entering the keyword "STONE1" in the applicable relative permeability method file.

8-1

where

8-2

8-3

8-4

8-5

kk k

k

S

S Sro

rog row

rocw

o

w g

=−( ) −( )

*

* *1 1

SS S

S Soo or

or wl

* = −− −( )1

SS

S Sgg

or wl

* =− −( )1

SS S

S Sww wl

or wl

* = −− −( )1

S SS S

S SSor orw

orw org

org wlg= −

−− −

⎛

⎝⎜⎜

⎞

⎠⎟⎟1

268 Relative Permeability Methods R5000.0.2


Or

8-6

Which equation gets used depends on the SOMOPT option that is in effect. Note that the original Stone1 methodology is equivalent to this SOMOPT option and the default is the Fayers and Matthews3 modification.

In the above equations, is the relative permeability to oil at connate water ( ), is the relative permeability to oil in the two-phase water-oil system (without gas present), is the relative permeability to oil in a gas-oil system with connate water, and is the residual oil saturation in the water-oil (j = w) or gas-oil (j = g) system.

Stone's Model II

Stone's Model II2 is invoked by entering the keyword "STONE2" in the appropriate relative permeability method file. This is the default option.

8-7

Saturation Weighted Interpolation Model

The Saturation Weighted Interpolation4 three-phase model is invoked by entering the keyword "KROINT" in the appropriate relative permeability method file.

It assumes that the oil phase is uniformly distributed in the gridblock, while the gas and water phases are completely segregated. Water saturation in the gas zone is assumed to be connate water saturation (Swl), while gas saturation in the water zone is zero. For average oil, gas, and water saturations of So, Sg, and Sw, the full breakdown of the saturation distribution is as follows.

In the gas zone, which occupies a fraction, Fg, of the pore volume:

S Sor orm=

krocw

Swlkrow

krog

Sorg

k kk

kk

k

kk k kro rocw

row

rocwrw

rog

rocwrg rg rw= +

⎛

⎝⎜

⎞

⎠⎟ +⎛

⎝⎜

⎞

⎠⎟ − −

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥



Oil saturation = So

Gas saturation = Sg + Sw - Swl

Water saturation = Swl.

In the water zone, which occupies a fraction, 1 - Fg, of the pore volume:

Oil saturation = So

Gas saturation= 0

Water saturation = Sg + Sw.

The material balance requires that the fraction of the pore volume occupied by the gas zone, Fg, be

8-8

The relative permeability to oil in the gas zone is (evaluated at

= 1 - - ), while the relative permeability to oil in the water zone

is (evaluated at = 1 - ). The average relative permeability

to oil in the gridblock, , is the volume average of the oil relative

permeabilities in the two zones:

8-9

where is gas saturation in the gas zone and is water saturation in the water zone:

8-10

FS

S S Sgg

g w wl

=+ −

krog Sg

So Swl

krowSw

So

kro

kS k S S S k S

S S Sro

g rog g w wl row w

g w wl

=( ) + −( ) ( )

+ −

* *

Sg* Sw

*

S S S Sg g w wl* = + −



8-11

S S Sw g w* = +



Guidelines for Selecting the Models

The optimal choice of a three-phase relative permeability model for any simulation study depends on the reservoir system. If laboratory three-phase relative permeability data are available, the best three-phase model can readily be determined by comparing the isoperms predicted by all three models with the data. Otherwise, a good engineering judgment may be made by comparing the isoperms predicted by all three models.

• Stone's Model II generally is regarded as too pessimistic at low oil saturations. It predicts much lower oil relative permeabilities and much higher residual oil saturations than the other models. However, use of this model should not be ruled out completely because the model has been shown to be superior to other models in one of eight systems tested by Baker.4

• Stone's Model I as modified by Fayers and Matthews3 (SOMOPT1) predicts much more favorable oil permeabilities with isoperms concave toward the 100% oil saturation apex and has been suggested to predict a value of Kro that is too high at low oil saturations for some systems.

• Saturation Weighted Interpolation Model4 may predict oil

permeabilities that are between those predicted by Stone's Model I

and Model II. The model gives apparently erroneous results in the

region of low oil isoperms if the and curves are dissimilar.

The relative permeability to oil in the three-phase region is

dominated by the two-phase curve with higher relative

permeability to oil. The predicted residual oil saturation

(corresponding to zero oil isoperm) equals the minimum

of and everywhere, except for one two-phase limit at

which a step jump in the residual oil saturation occurs (from

to at the water-oil two-phase limit if < , or from

to at the gas-oil two-phase limit if < ). The impact

of this behavior in the low oil isoperm region on simulation results

should be investigated first for any simulation study if the

Saturation Weighted Interpolation Model is to be employed.

krog krow

Sorg Sorw

Sorg

Sorw Sorg Sorw

Sorw Sorg Sorw Sorg



Hysteresis

Introduction

The hysteresis options allow the use of relative permeabilities and capillary pressures that are a function of both phase saturation and the history of the phase saturation. The option is useful for the rigorous simulation of fluid flow in reservoirs where the saturation of a fluid phase within a gridblock does not change monotonically.

Gas-phase relative permeability hysteresis is limited to the gas-phase relative permeability function (krg). Oil-phase relative permeability hysteresis is limited to the oil relative permeability function in a water-oil system (krow). Four options are available to specify gas and oil phase hysteresis: Carlson's method5linear interpolation, scaling of the drainage function, and user-entered imbibition-drainage tables. Water phase hysteresis is also available, but only the user-entered imbibition-drainage tables.

The capillary pressure hysteresis option follows a simplified version of the model proposed by Killough6. The user must enter both bounding drainage and imbibition capillary pressure curves.

Water-oil capillary pressure hysteresis may be specified independently of the relative permeability hysteresis option.

Background

The following discussion about non-wetting hysteresis applies to both gas-oil and oil-water hysteresis. Oil-water hysteresis applicability is mainly in the oil-water transition zone or areas where oil is displaced to where it previously was mainly water. Use of the gas-phase hysteresis option should be considered anytime it is anticipated that the reservoir depletion scenario will involve an increasing and then decreasing gas saturation trajectory over a significant portion or critical area of the reservoir. Two possible applications are: wag injection and gas conning.

In the discussion below it is assumed that oil is non-wetting relative to water and that gas is non-wetting relative to oil.



The relative permeability curves employed during hysteresis calculations are shown in the figure below. The primary drainage curve is used so long as the non-wetting saturation is monotonically increasing. If a saturation reversal occurs prior to reaching the bounding non-wetting saturation value, Snwbound , then an intermediate imbibition-drainage curve is constructed. The starting point for this curve is the historical maximum gas saturation, Snwmax and the ending point is the 'trapped' gas saturation, Snwtrap. There are four methods of generating the intermediate imbibition-drainage curve. In all the 'trapped' non-wetting saturation is computed from the empirical relationship of Land7.

Details of the construction methods are provided below. As long as the non-wetting saturation does not subsequently exceed, Snwmax, then the non-wetting relative permeability will continue to follow the same intermediate imbibition-drainage curve. If, however, the non-wetting saturation reaches and surpasses the historical maximum non-wetting saturation, then the primary drainage curve is rejoined. Subsequent saturation reversals follow the same pattern described except with different end points. Finally, if the non-wetting saturation reaches Snwbound, the non-wetting relative permeability stabilizes to a single bounding imbibition-drainage curve with end point at the user input value of Snwtr(Sgtr or Sotr).



Figure 8.1: Non-wetting Relative Permeability Hysteresis



Intermediate Imbibition-Drainage Construction Methods

For the first three construction methods described below, the trapped gas saturation, Sgtrap is calculated using the approach of Land.

8-12

This relationship was derived from experimental observations on the effect of initial non-wetting saturation on trapped non-wetting saturation after imbibition.

SS

SS S

nwtrapnw

nwnwtr nwbound

=⋅ −⎛

⎝⎜⎜

⎞

⎠⎟⎟ +

max

max

1 11



Linear Method

Once the Sgtrap is determined, the linear method simply creates an intermediate imbibition-drainage curve by connecting the points, Sg = Sgtrap, krg = 0 and Sg = Sgmax, krg = krg(Sgmax) with a line. An example of the resulting curves generated by Falcon's relative permeability modular tester are given below.

Figure 8.2: “Linear” Intermediate Imbibition-Drainage Curve Construction



Scaling of Primary Drainage Curve

This option generates an intermediate imbibition-drainage curve that assumes a scaled form of the primary drainage curve. That is, the portion of the primary drainage curve, , from Sgc to Sgmax, is scaled for the intermediate imbibition-drainage curve, to the range of Sgtrap to Sgmax. In equation form,

8-13

where8-14

An example of the curves generated using the 'SCALED' option is shown below.

Figure 8.3: “Scaled” Intermediate Imbibition-Drainage Curve Construction

krgPD

krgIID

k S k SrgIID

g rgPD

g( ) = ( )*

S SS S

S SS Sg gc

g gtrap

g gtrapg gc

*

maxmax= +

−−

−( )



Carlson Method

The rationale behind the Carlson approach was first devised by Land. It says that along the intermediate imbibition-drainage curve, at any given Sg, a portion of the gas saturation is isolated and non-flowing, Sgt, and a portion is connected and flowing, Sgf.

8-15

Therefore, given a way of determining Sgf as a function of Sg then the gas relative permeability could be obtained from the primary drainage curve evaluated at the 'free' or flowing gas saturation, Sgf. In equation form,

8-16

The needed relationship between Sgf, Sgt and Sg was derived by Land and is as follows:

8-17

where8-18

An example of the curves generated using the 'CARLSON' option is shown below.

S S SgIID

gtIID

gtfIID= +

k S k SrgIID

g rgPD

gf( ) = ( )

S S S SS Sgf gc gxn gxn

gtr gbound

= + + ⋅ ⋅ −⎛

⎝⎜⎜

⎞

⎠⎟⎟

1

24

1 12

S S Sgxn g gtrap= −



Figure 8.4: “Carlson” Intermediate Imbibition-Drainage Curve Construction

User-Supplied Bounding Method (Non-Wetting Hysteresis)

In this method the user inputs the bounding imbibition-drainage curve and intermediate curves are constructed from simple ratios as shown in the diagram below. Consider being on an intermediate curve; the following steps are performed:

1. A saturation, on the bounding imbibition curve which

produces the same as at of the Drainage curve is

determined.

2. is determined which is the same fraction of the bounding imbibition curve (in terms of saturation) as the saturation of interest is on the imbibition-drainage curve.

3. is looked up on the bounding imbibition curve at .

8-19

Where:

Snwinv

krnw Snwmax

Snwlookup

krnw Snwlookup

S S S S S S S Snwlookup nwinv nw nwmax nwinv nwtrapmax nwmax= + −( ) −( ) −* / nnwtrap( )



= saturation used to lookup on the Bounding Imbibition curve to produce

= historical maximum non-wetting saturation

= saturation on the bounding imbibition curve which gives an equivalent to the one from the Drainage Curve evaluated at

= non-wetting saturation for which the is needed

= non-wetting maximum trapped saturation

= current trapped non-wetting phase saturation based on Land's formula

Figure 8.5: “User” Method of Non-wetting Hysteresis

The imbibition-drainage curve constructed will be reversible until the value is exceeded by the non-wetting phase.

Snwlookup

krnw

Snwmax

Snwinv

krnw

Snwmax

Snw* krnw

Snwtrapmax

Snwtrap

Snwmax



User-Supplied Bounding Method (Wetting Hysteresis)

This method is similar in form to the non-wetting hysteresis but different in specifics. The user inputs the bounding imbibition-drainage curve and intermediate curves are constructed from simple ratios as shown in the diagram below. Consider being on an intermediate curve; the following steps are preformed:

1. A saturation, on the bounding imbibition curve which produces the same as at of the drainage curve is determined.

2. is determined which is the same fraction of the bounding imbibition curve (in terms of saturation) as the saturation of interest is on the imbibition-drainage curve.

3. is looked up on the bounding imbibition curve at .

8-20

Where:

= saturation used to lookup on the Bounding Imbibition

curve to produce

= historical maximum non-wetting saturation

= saturation on the bounding imbibition curve which gives an

equivalent to the one from the drainage curve evaluated at

= non-wetting saturation for which the is needed

= non-wetting maximum trapped saturation

The imbibition-drainage curve constructed will be reversible until the water saturation goes below the previously established value is exceeded by the non-wetting phase. Note that, like the oil-water hysteresis option described above, this option is most usual in the transition zone or where the historical minimum water

Snwinv

krw Swmax

Swlookup

krw Swlookup

S S S S S S S Swlookup winv w wmin wu winv wu wmin= + −( ) −( ) −( )* /

Swlookup

krw

Swmax

Swinv

krw

Snwmax

Sw* krnw

Swtrapmax

Swmin



saturation has room to move (i.e., is not already at connate water saturation).

Figure 8.6: “User” Method of Wetting Hysteresis



End-Point Scaling

Introduction

Nexus allows the user to model spatial differences in relative permeability and capillary pressure end points with a single set of generic relative permeability and capillary pressure curves. Both the saturation and relative permeability end points for any gridblock can be different from those of the generic rock type to which the gridblock is assigned.

The relative permeabilities and capillary pressures at a given saturation are determined from the generic curves based on the assumption that the normalized relative permeability and capillary pressure curves of the generic rock type and the gridblock are identical.

The calculation of the relative permeabilities is divided into two steps: saturation end-point scaling, then end-point relative permeability scaling. Two- and three-point saturation scaling options are available Nexus.

Just like every other relative permeability option in Nexus the end-point scaling features are applied by relative permeability region and are specified in the applicable method file.

Normalized Saturation End Points

Eight saturation end points can be defined for each gridblock, grouped by water-oil and gas-oil normalized functions. They are:

• Water-Oil Normalized Functions

• SWL - Connate water saturation (the smallest water saturation entry in a water saturation table).

• SWR - Residual water saturation (the highest water saturation for which the water is immobile).

• SWRO - Water saturation at residual oil (one minus residual oil saturation where the residual oil saturation is the highest oil saturation for which the oil is immobile in the water-oil system).



• SWU - Maximum water saturation (the highest water saturation entry in a water saturation table).

• Gas-Oil Normalized Functions

• SGL - Connate gas saturation (the smallest gas saturation entry in a gas saturation table).

• SGR - Residual gas saturation (the highest gas saturation for which the gas is immobile).

• SGRO - Gas saturation at residual oil (one minus the residual oil saturation minus the connate water saturation where the residual oil saturation is the highest oil saturation for which the oil is immobile in the gas-oil-connate water system).

• SGU - Maximum gas saturation (the highest gas saturation entry in a gas saturation table).

• Gas-Water Normalized Functions

• SGL - Connate gas saturation (the smallest gas saturation entry in a gas-water saturation table).

• SGRW - Gas saturation at residual water.

• SGU - Maximum gas saturation (the highest gas saturation entry in the gas-water table).

• Optional Capillary Pressure Only Normalized Functions

• SWLPC - Capillary pressure connate water saturation. Will be used only for water-oil capillary pressure scaling. Normally this is equal to SWL.

• SWGPC - Capillary pressure connate gas saturation. Will be used only for gas-oil and gas-water capillary pressure scaling. Normally this is equal to SGL.



Linear transformation is used to correlate gridblock saturation to the equivalent saturation in the generic rock table. This equivalent saturation is then used to compute relative permeability or capillary pressure. For example, if the saturation end points for the gridblock are SL and SR and for the rock are SL and SR , then the equivalent rock saturation, S , corresponding to the gridblock saturation, S, is calculated as:

8-21

S is then used in the table look-up for the relative permeability or capillary pressure.

Two- and three-point scaling options are available to calculate relative permeability and capillary pressure. The default option is the three-point scaling option. It uses three-point scaling for relative permeabilities and four-point scaling for capillary pressures. Whatever method is selected is used both during initialization (for inverse table lookups) and during the recurrent simulation.

SS S S S

S S

L R L

R L

’

’ ’

=−( ) −( )

−( )



Rules for Missing or Incomplete End-Point Scaling Arrays and Data Errors

It is always best to enter as many of the end-point scaling arrays as are known or can be easily calculated. Missing data requires the simulator to make some assumptions which might not be exactly what was desired. The following rules are applied to missing data or data with errors.

(In the discussion below, the signifies an array of end-point saturations.)

• In general, missing end-points assume the values from the appropriate table of their relative permeability region. Nexus attempts to fill in any missing end-point saturation data with logical choices, but these are not unique. The best practice is for the user to enter all of the end points or use the defaults only after fully comprehending the defaulting rules.

• If the tabular = (within tolerance) and has not been

entered, then that array is set equal to (input or defaulted to

table). This does not apply to two-phase gas-water systems.

• If the tabular + = 1 (within tolerance), and if was not

entered, then is set equal to 1 - . Conversely, if is not

entered and + > 1, then the is set equal to 1 - .

This only applies to three-phase systems.

• If the tabular + = 1 (within tolerance), and + >

1, then, if was not entered, is set equal to 1 - .

Conversely if is not entered then it is set equal to 1 - .

This only applies to three-phase systems.

• If the tabular = (within tolerance) and is not entered, then it is set equal to (entered or defaulted). This only applies to two-phase gas-water systems.

Data Errors

• If - (1 - ) > tolerance, then fatal error unless NOCHK is

active. This only applies to three-phase systems.

S

Swr Swl Swl

Swr

Sgu Swl Sgu

Sgu Swl Swl

Sgu Swl SwlSgu

Sgl SwuSgl Swu

Sgl Sgl Swu

SwuSgl

Swl Sgr Sgl

Sgr

Sgu Swl



• If - (1 - ) > tolerance then fatal error unless NOCHK is

active. This only applies to three-phase systems.

• If > then fatal error. This does not apply to gas-water systems.



• If > then fatal error. This does not apply to water-oil systems.

• If > then fatal error. This only applies to three-phase systems.

• If > then fatal error. This only applies to three-phase systems.

• If > then fatal error. This only applies to gas-water systems.

• If <= then fatal error. This only applies to gas-dependant water relative permeability systems (three-phase).

Two-Point Scaling Option

The keyword SCALING TWOPOINT is used to invoke the two-point scaling option. It uses two-point scaling for both capillary pressures and relative permeabilities.

Two-Point Scaling of Capillary Pressure Functions

Two-point scaling uses the connate and maximum saturations as two saturation end points in the calculation of capillary pressures:

Pcwo SWL and SWU

Pcgo SGL and SGU

Sgl Swu

Swl Swr

Swr Swro

SwroSwu

Sgl Sgr

SgrSgro

Sgro Sgu

Sgrw Sgu

Sgrw Sgl



Two-Point Scaling of Relative Permeability Functions

Two-point scaling uses the residual and maximum saturations as two saturation end points in the calculation of relative permeabilities:

krw SWR and SWU

krg SGR and SGU

krow SWL and SWRO

krog SGL and SGRO



Three-Point Scaling Option (Default Option)

The three-point scaling option uses four-point scaling for capillary pressures and three-point scaling for the relative permeabilities. When the three-point option is used, the saturation end points divide saturation into different regions. The end points used depend on the region to which the grid-point saturation belongs.

Four-Point Scaling of Capillary Pressure Functions

Four-point scaling uses connate saturation, residual saturation, water/gas saturation at residual oil, and maximum saturation as four saturation end points to calculate capillary pressure. The following end points are used in the water-oil and gas-oil capillary pressure calculation:

Pcwo SWL, SWR, SWRO, and SWU

Pcgo SGL, SGR, SGRO, and SGU

Three-Point Scaling of Relative Permeability Functions

In addition to honoring the end points used in the two-point scaling option, the three-point scaling algorithm honors the relative permeability of the mobile phase at the residual saturation of the other phase. The following end points are used in the relative permeability calculation:

krw SWR, SWRO, and SWU

krg SGR, SGRO, and SGU

krow SWL, SWR, and SWRO

krog SGL, SGR, and SGRO



Normalized End-Point Relative Permeability

Nexus allows the user to specify the rock end-point relative permeabilities for each gridblock. For a given gridblock saturation, Nexus calculates a transformed saturation for use in the rock table.

The following end-point relative permeability arrays may be entered for each gridblock:

• KRW_SWRO - The water relative permeability at residual oil saturation. (Only relevant to three-point scaling.)

• KRW_SWU - The water relative permeability saturation.

• KRG_SGRO - The gas relative permeability at residual oil saturation. (Only relevant to three-point scaling.)

• KRG_SGU - The gas relative permeability at maximum gas saturation.

• KRO_SWL - The oil relative permeability at minimum water saturation.

• KRO_SWR - The oil relative permeability at residual water saturation. (Only relevant to three-point scaling.)

• KRO_SGL - The oil relative permeability at minimum gas saturation.

• KRO_SGR - The oil relative permeability at residual gas saturation. (Only relevant to three-point scaling.)

• KRG_SGRW - The gas relative permeability at residual water saturation in a two-phase gas-water system (Only relevant to three-point scaling.)



End-Point Relative Permeability Scaling for Oil (Two-Point Scaling)

The two-phase oil relative permeabilities for a gridblock, krow and krog, are calculated from the rock curves and the gridblock to rock endpoint relative permeability ratios according to the following equations:

8-22

and

8-23

Where, , are the tabular values after any saturation scaling has

been applied, and are the tabular relative

permeability end-points, and and are the array

values.

k kKRO

KROrow row= ′ ×′

_SWL

_SWL

k kKRO

KROrog rog= ′ ×′

_SGL

_SGL

′krow′krog

KRO_SWL′ KRO_SGL′

KRO_SWL KRO_SWL



End-Point Relative Permeability Scaling for Water and Gas (Two-Point Scaling)

The water and gas relative permeabilities are computed with the following equations:

8-24

and

8-25

Where, and are the tabular values after any saturation scaling has

been applied, and are the tabular relative

permeability end-points, and and are the

array values.

k kKRW

KRWrw rw= ′ ×′

_SWU

_SWU

k kKRG

KRGrg rg= ′ ×′

_SGU

_SGU

′krw′krg

KRW_SWU′ KRG_SGU′

KRW_SWU′ KRG_SGU′



End-Point Relative Permeability Scaling for Oil (Three-Point Scaling)

For the case of the oil-water table, if then8-26

However, if , then

8-27

For the case of the gas-oil table, 8-28

However, if , then

8-29

S Sw wr tab≤ _

k KRO k KROKRO R KRO L

KRO R KROrow row= + ′ − ′( ) −( )′ −

_SWL _SWL_SW _SW

_SW __SW ′( )L

abs KRO R KRO L tol( )_SW _SW′ − ′ <

k kKRO

KROrow row= ′ ×′

_SWR

_SWR

k KRO k KROKRO R KRO L

KRO R KROrog rog= + ′ − ′( ) −( )′ −

_SGL _SGL_SG _SG

_SG __SG ′( )L

abs KRO R KRO L tol( )_SG _SG′ − ′ <

k kKRO

KROrog rog= ′ ×′

_SGL

_SGL



End-Point Relative Permeability Scaling for Water and Gas (Three-Point Scaling)

For water relative permeability if , then

8-30

Unless , then

8-31

And if , then

8-32

For gas relative permeability, if , then

8-33

Unless , then

8-34

And if , then

8-35

S Sw wro tab> _

k KRW k KRWKRW WU KRW WRO

KRW WU KRrw rw= + ′ − ′( ) −( )′ −

_SWRO _SWRO_S _S

_S WW SWRO_ ′( )

abs KRW WU KRW WRO tol( )_S _S′ − ′ <

k kKRW

KRWrw rw= ′ ×′

_SWU

_SWU

S Sw wro tab≤ _

k kKRW

KRWrw rw= ′ ×′

_SWRO

_SWRO

S Sg gro tab> _

k KRG k KRGKRG GU KRG GRO

KRG GU KRrg rg= + ′ − ′( ) −( )′ −

_SGRO _SGRO_S _S

_S GG SGRO_ ′( )

abs KRG GU KRG GRO tol( )_S _S′ − ′ <

k kKRG

KRGrg rg= ′ ×′

_SGU

_SGU

S Sg gro tab≤ _

k kKRG

KRWrg rg= ′ ×′

_SGRO

_SGRO



End-Point Relative Permeability Scaling for Gas-Water Cases (Two-Point Scaling)

For gas relative permeability, if , then

8-36

Unless , then8-37

And if , then8-38

For the case of the water phase, 8-39

However, if , then

8-40

S Sg grw tab> _

k KRG k KRGKRG GU KRG GRW

KRG GUrgw rgw= + ′ − ′( ) −( )′ −

_SGRW _SGRW_S _S

_S KKRG SGRW_ ′( )

abs KRG GU KRG GRW tol( )_S _S′ − ′ <

k kKRG

KRGrgw rgw= ′ ×′

_SGU

_SGU

S Sg grw tab≤ _

k kKRG

KRGrgw rgw= ′ ×′

_SGRW

_SGRW

k KRW k KRWKRW R KRW L

KRW R KRWrwg rwg= + ′ − ′( ) −( )′ −

_SGL _SGL_SG _SG

_SG __SG ′( )L

abs KRW R KRW L tol( )_SG _SG′ − ′ <

k kKRW

KRWrwg rwg= ′ ×′

_SGL

_SGL



Data Defaulting and Errors Applied to End-Point Relative Permeability

Nexus attempts to fill in any missing end-point relative permeability data with logical choices, but these are not unique. The best practice is for the user to enter all of the end points. When not entering all the end points, users should fully understand the defaulting rules listed below.

• For all end-point relative permeability arrays (e.g., KRW_SWL) if an array is not entered, then that array will be populated with values from the applicable table for the given cell. This default value is subject to modification. (See below).

• If a -1 value is entered for an end-point array then its value will also be taken from the applicable table, but this case will be treated as if the value had been entered in the array. (See below for the possible distinction.) This is different from the case where no array is entered and a default is taken from a table — in that situation the array is considered to have not been entered.

Three-Point Scaling Defaulting Rules

Note: The following defaulting rules also apply to water-oil two phase systems, except for references to gas quantities.

• If the table values for and differ by less

than a small tolerance AND was not entered, then

is set equal to .



is set equal to .

• If the table values for and differ by

greater than a small tolerance AND entered but not

then preserve the ratio of the table by setting

8-41

KRO_SGR′ KRO_SGL′KRO_SGL

KRO_SGL KRO_SGR

KRO_SGL′ KRO_SGR′

KRO_SGR

KRO_SGR KRO_SGL

KRO_SGR′ KRO_SGL′

KRO_SGR

KRO_SGL

KRO_SGL=KRO_SGRKRO_SGL

KRO_SGR×

′′




greater than a small tolerance AND was entered but

not , then preserve the ratio of the table by setting

8-42

• If after the above, > ,then

is set equal to .



is set equal to .



is set equal to .




8-43




8-44

KRO_SGR′ KRO_SGL′

KRO_SGL

KRO_SGR

KRO_SGR=KRO_SGLKRO_SGR

KRO_SGL×

′′

KRO_SGL KRO_SGR KRO_SGL

KRO_SGR

KRO_SWL′ KRO_SWR′

KRW_SGL

KRW_SGL KRO_SWR


KRW_SGR

KRO_SWR KRW_SGL


KRO_SGR

KRO_SGL

KRO_SWL=KRO_SWRKRO_SWL

KRO_SWR×

′′

KRO_SWR′ KRO_SWL′KRO_SWL

KRO_SWR

KRO_SWR=KRO_SWLKRO_SWR

KRO_SWL×

′′



• If after the above, > ,

then is set equal to .


less than a small tolerance AND was not entered,



less than a small tolerance, AND was not entered,





8-45




8-46

• If after the above, < then

is set equal to .



is set equal to .

KRO_SWL KRO_SWR

KRO_SWL KRO_SWR

KRW_SWU′ KRW_SWRO′

KRW_SWU

KRW_SWU KRW_SWRO


KRW_SWRO

KRW_SWRO KRW_SWU


KRW_SWU

KRW_SWRO

KRW_SWRO=KRW_SWUKRW_SWRO

KRW_SWU×

′′


KRW_SWRO

KRW_SWU

KRW_SWU=KRW_SWROKRW_SWU

KRW_SWRO×

′′

KRW_SWU KRW_SWRO

KRW_SWU KRW_SWRO

KRG_SGU′ KRG_SGRO′

KRG_SGRO

KRG_SGRO KRG_SGU






8-47


greater than a small tolerance AND was entered, but


8-48

• If after the above, < , then

is set equal to .

Three-Point Scaling Defaulting Rules (Gas-Water)


less than by a small tolerance, AND was not

entered, then is set equal to .


greater than by a small tolerance, AND was entered

but not , then preserve the ratio of the table by

setting

8-49


KRG_SGU

KRG_SGRO

KRG_SGRO=KRG_SGUKRG_SGRO

KRG_SGU×

′′


KRG_SGRO

KRG_SGU

KRG_SGU=KRG_SGROKRG_SGU

KRG_SGRO×

′′

KRG_SGU KRG_SGRO

KRG_SGU KRG_SGRO

KRG_SGU′ KRG_SGRW′

KRG_SGRW

KRG_SGRW KRG_SGU

KRG_SGU′ KRG_SGRW′KRG_SGU

KRG_SGRW

KRG_SGRW=KRG_SGUKRG_SGRW

KRG_SGU×

′′




greater than by a small tolerance AND was entered

but not , then preserve the ratio of the table by setting

8-50

If, after the above, < , then is

set equal to .

KRG_SGU′ KRG_SGRW′KRG_SGRW

KRG_SGU

KRG_SGU=KRG_SGRWKRG_SGU

KRG_SGRW×

′′

KRG_SGU KRG_SGRW KRG_SGU

KRG_SGRW



Directional Relative Permeability

The directional relative permeability option in Nexus allows each gridblock to have different relative permeability values in each flow direction, rather than simply one computation per gridblock. For vertical equilibrium problems (when VE option is invoked), the program internally generates the relative permeability functions. Use of the directional relative permeability option results in a different set of functions generated for the areal and vertical directions. For non-VE (vertical equilibrium) problems, the user may assign multiple saturation function tables to each gridblock.



IFT Adjustment of Capillary Pressure and Gas/Oil Relative Permeability

EOS type fluid grids may go through a compositional path that leads to the critical point of the hydrocarbon mixture. In a gridblock near the critical point, when a hydrocarbon mixture splits into two phases, the compositions of the phases are similar. For this reason, interfacial tension and capillary pressure between the phases become small and approach zero, the oil and gas residual saturations decrease, and the gas and oil relative permeability curves approach straight lines near the critical point.

Nexus has a special option that automatically adjusts relative permeabilities and gas-oil capillary pressure in near-critical blocks. The adjustment correlates these rock properties to gas-oil interfacial tension as follows:

8-51

8-52

8-53

k f k f k SS S f

S S fro ror

m wo org

w org

= ( ) + − ( )⎡⎣ ⎤⎦ ( ) − ( )− − ( )

11

k f k f k SS S f

S S frg rgr

m wg gc

w gc

= ( ) + − ( )⎡⎣ ⎤⎦ ( ) − ( )− − ( )

11

P Pcgo cgor

ref

=



where , , , are the adjusted relative permeabilities to oil,

gas, and gas-oil capillary pressure, respectively; , , are

the nonadjusted respective values as calculated by Nexus (by the active

three-phase method) from the rock curves; and is the relative

permeability to miscible hydrocarbon fluid near the critical point. This

value is a function of water saturation only. Also, is the residual

oil saturation to gas, is the gas-oil interfacial tension, is the

reference interfacial tension of the gas-oil system on which the

capillary pressure rock curve has been measured, and is a

function of the interfacial tension defined as:

8-54

Here, is the threshold interfacial tension below which the above relative permeability adjustment is used. For interfacial tensions greater than the threshold value, is assigned a value of one. Exponent e is a positive number generally in the range from 0.1 to 0.25.

krokrg Pcgo

kror krg

rPcgo

r

k Sm w( )

Sorg

ref

Pcgor f ( )

fe

( ) = ⎛

⎝⎜

⎞

⎠⎟*

*

f ( )



The relative permeability to miscible hydrocarbon fluid, is defined as the following arithmetical average:

8-55

The gas-oil interfacial tension depends on the oil and gas compositions and is calculated from the correlation:

8-56

where is the parachor of component i; , are molar

densities; and and and the mole fractions in the liquid and gas

phases, respectively.

k Sm w( )

k S k S S S k S S Sm w ror

o w g rgr

g w o( ) = = − =( ) + = − =( )⎡⎣ ⎤⎦1

21 0 1 0, ,

14

1

21

= −( )⎡⎣ ⎤⎦=∑ P x ych o i g ii

nc

i

Pchio g

xi yi



References

1. Stone, H.L.: "Probability Model for Estimating Three-Phase Relative Permeability," JPT (Feb 1970) 214; Trans., AIME (1970) 249.

2. Stone, H.L.:"Estimation of Three-Phase Relative Permeability and Residual Oil Data," J. Cdn. Pet. Tech. (April 1973) 12, N. 4, 53.

3. Fayers, F.J. and Matthews, J.D., "Evaluation of Normalized Stone’s Methods for Estimating Three-Phase Relative Permeabilities," Soc. Pet. Eng. J., pp. 225-232 (Apr. 1984)

4. Baker, L.E., "Three-Phase Relative Permeability Correlations," paper SPE/DOE 17369 presented at the 1988 SPE/DOE Enhanced Oil Recovery Symposium, Tulsa, OK, April 17-20.

5. Carlson, F. E., "Simulation of Relative Permeability Hysteresis to the Nonwetting Phase," SPE Paper 10157 presented at the SPE-AIME 56th Ann. Fall Mtg., San Antonio, TX, Oct.1981.

6. Killough, J. E., "Reservoir Simulation with History-Dependent Saturation Functions," SPE Paper 5106 presented at the SPE-AIME 49th Ann. Fall Mtg., Houston, TX, Oct. 1974; also Soc. Pet. Eng. J., Feb. 1976, 37-48.

7. Land, C. S., "Calculation of Imbibition Relative Permeability for Two- and Three-Phase Flow From Rock Properties," Soc. Pet. Eng. J., Trans. AIME, 243, 149-156, June 1968.



Chapter 9

Tracers

Tracer Calculations

Tracer calculations consist of the explicit timestepping of component molar or mass conservation equations at the end of a timestep.

At the end of a timestep, the pressure and fluxes between gridblocks have been set.

The molar conservation equation for component i and tracer m is

9-1

where

cj,i,m is the concentration of tracer m in component i of phase j.

It is assumed that

9-2

This is the default assumption used in VIP also, but VIP also allows the user to alter the proportion of the tracer that will split between phases. This could come in useful for many reasons. For example, the tracer may represent a fraction of a pseudo-component that has a different K-value than the pseudo-component as a whole. Alternatively, it may be used to reduce dispersion effects in large gridblocks.

cimMi n 1+cimMi n

–

t------------------------------------------------------------- Fc im

n

c 1=

nconn

Qimn

+=

Fc im Tkrj

j--------jxj i cj i m

n

j 1=

nphases

pon Pc oj

n jMjg nD–+

c

=

i 1 nc 1 m+ 1 ntracers = =

cj i m ci m=

R5000.0.2 Tracers: Tracer Calculations 307


Qim is the source/sink term for the tracer. For example, it could be the stream tracer concentration multiplied by the molar flow rate of component i.

The tracer solution does not necessarily use the same timestep size as the general reservoir simulation. Because the calculation is being performed explicitly, using the simulation timestep may cause instability, especially if the grid is being solved implicitly and large timesteps are being taken. Therefore, the tracer calculation may be divided into many timesteps where the reservoir variables are held constant and the concentrations are updated for every sub-timestep. The sub-timestep is essentially calculated so that the molar (or mass) flux into or out of the gridblock does not exceed the amount of moles (or mass) assigned to the tracer of a component. However, a minimum timestep is imposed, so it is still possible that the tracer calculations are performed at a timestep exceeding the stability limit.

308 Tracers: Tracer Calculations R5000.0.2


Chapter 10

Grid

Introduction

Nexus has two forms for reading the grid array property values. In the basic form, no grid structure is necessary and array data, for active cells only, are supplied in separate files by property. This basic form of array input is referred to as the “unstructured” form. Generating array data in an unstructured form usually requires a separate program. The second form of array data input uses a structured grid format and is therefore referred to as the “structured grid” form. The input for the structured grid form is like that of the VIP-CORE ARRAYS data section.

R5000.0.2 Grid: Introduction 309


Unstructured Grid Form

When the unstructured form of array data input is used, the individual property array files are specified in the casename.fcs file. Data in the casename.fcs file are organized by section. Array data can be specified in the PROPERTY_FILES, INITIALIZATION_FILES and ROCK_ARRAY_FILES sections. For details on file formats, contents and requirements, see the "Array Data" chapter of the Nexus Keyword Document.

Additionally, connection transmissibility data are required input. The connection data files are specified with the CONNECTIONS keyword in the PROPERTY_FILES section of the casename.fcs file. For file format and content details see the "Connection Data " chapter of the Nexus Keyword Document.

Map Output

With the unstructured form of array data input, the default form of map output to a flat map file follows that of the input data and assumes a grid of shape NB*1*1, where NB is the total number simulation of cells.

However, if GLOBALCELL array data are available, and contain values that are consistent with the natural cell order of a structured grid (including inactive cells), then the simulator can be instructed to write flat map data that corresponds to the structure of the GLOBALCELL data. This requires additional data about the shape of the structured grid (or LGR grids) to be supplied in the OPTIONS file of the PROPERTY_FILES section. The grid shape data are introduced by the STRUCTURED_MAPS keyword in the OPTIONS file. For details and restrictions see the "Structured Grid Data" section in the "Options Data" chapter of the Nexus Keyword Document.

The simulator can also write map data directly to a VDB, provided that the VDB contains INIT class data with the physical grid description (corner point CORP data) for the specified case name, and the total number of cells with PV greater than zero matches the total number of cells in the simulator. When writing directly to a VDB, the grid shape data are extracted from the VDB; therefore, STRUCTURED_MAPS data in the OPTIONS file if present will be ignored.

310 Grid: Unstructured Grid Form R5000.0.2


Structured Grid Form

The structured form of array data input uses the GRID_FILES section instead of the PROPERTY_FILES section in the casename.fcs file. All the required array data will be specified in the STRUCTURED_GRID file instead of the PROPERTY_FILES, INITIALIZATION_FILES and ROCK_FILES sections, as required by the unstructured grid form. In addition to array property data, grid geometry data are also required and specified in the STRUCTURED_GRID file. For details on the contents of the STRUCTURED_GRID file see "Appendix A: Structured Grid Data" of the Nexus Keyword Document.

Grid Geometry Data

Nexus always uses the corner point geometry for all the grid related computations. Each grid cell is independently specified by the coordinates of its eight corners. Corner point data can be directly input using the CORP array option. Alternatively the grid geometry can be defined either with the DX, DY, DZ, etc. arrays, or with the DXB, DYB, DZB, etc. arrays, and Nexus will automatically compute the coordinates of each cell's eight corners. All calculations that require cell geometry (for example - cell volumes, transmissibilities, etc.) are computed from the cell corner points.

Fault Specification

Faults and pinchouts are automatically detected and transmissibility connections created when the grid geometry is defined with the CORP data option. The DX or DXB grid specification options do not have any facility for defining faults or pinchouts.

R5000.0.2 Grid: Structured Grid Form 311


Calculations of Cell Bulk Volume and Center Depth

The bulk volume Vb, and center depth Dc, of a cell are defined as follows:

10-1

10-2

where z is the Z-coordinate. A numerical integration technique is used to estimate the three-dimensional integrals over the cell volume V in the above expressions. Each cell is mapped into a unit cube using a local coordinate transformation function. A quadrature formula with one, two, or three quadrature points is used to approximate the three-dimensional integral in the unit cube. The CORNER card is used to specify the numbers of the quadrature points. See the "Corner-Point Geometry" appendix of this manual for details.

Calculations of Cell Dimensions

The cell dimensions DXC, DYC and DZC that are output to the map file and "vdb" are computed from the average lengths of the four lines that join the relevant corner points along each direction. The cell thickness is set equal to the value of DZC.

Vb dVV=

Dc1

Vb------ z dV

V=

312 Grid: Structured Grid Form R5000.0.2


Calculations of Intercell Transmissibility

Two methods of calculating inter-cell transmissibility are available - the standard option, which uses "harmonic integration,” and the NEWTRAN option.

See the "Corner-Point Geometry" appendix of this manual for details on both options.

Standard Transmissibility Option

The following expression is used for calculations of inter-block transmissibilities in the X-direction in the standard Nexus option (see Figure 10.1):

10-3

Figure 10.1: Fault Block Connection

where

Aijarea of mutual intersection of Blocks I and J

Tij

CDARCY TMLTXi

Airight

AijTXiright

------------------------Aj

left

AijTXjleft

---------------------+

----------------------------------------------------=

I

J

Ai right

Aj left

Aij



, areas of the right face of the i’th block and the left face of the j’th block (see Figure 10.1)

, right and left transmissibilities of Blocks I and J in the X-direction,

NEWTRAN Transmissibility Option

The NEWTRAN method, which is similar to the corresponding ECLIPSE technique, has been incorporated in VIP-EXECUTIVE as an option to improve the compatibility of both simulators. This option can be invoked using the NEWTRAN keyword in the CORNER card. The following expression is used to calculate the inter-block transmissibility in this option:

10-4

where

CDARCY Darcy‘s constant in the appropriate units (0.008527 in metric units and 0.001127 in field units)

TMLTXi transmissibility multiplier in the X-direction for the i’th block

TXi =

RNTGi net-to-gross ratio, which appears in the X- and Y-transmissibility, but not in the Z-transmissibility

PERMXi permeability in the X-direction in the i’th block

Ax, Ay, Az X-, Y-, and Z-projections of a mutual intersection of Blocks i and j; i.e., projection on the y-z, x-z, and x-y planes, respectively.

DIx, DIy, DIz X-, Y-, and Z-components of the distance between the center of the i’th gridblock and the center of relevant face of Block i.

Airight

Ajleft

TXiright

TXjleft

Tij

CDARCY TMLTXi1

TXi-------- 1

TXj--------+

-----------------------------------------------=

PERMXiRNTGi

AxDIx AyDIy AzDIz+ +

DIx2

DIy2

DIz2

+ +---------------------------------------------------------



The expressions for transmissibilities in the Y- and Z-directions are similar, but the net-to-gross ratio is not included in the expression for the Z-transmissibility.



Local Grid Refinement (LGR)

Cartesian grid refinements are supported in Nexus. A Cartesian grid refinement consists of any rectangular area comprising one or more grid cells that have been refined into a finer grid structure. Any grid cell within the rectangular area may subsequently be omitted from the refinement. The following illustration shows a refined area REF2 within a refined area REF1, within a coarse base grid.

Figure 10.2: Example of Cartesian Refinement

When creating a Cartesian refinement, the following properties must be defined:

• Grid name

• Range of coarse grid cells to be refined.

• Number of x, y, z fine grid divisions per parent grid division.

Optionally, the range of grid cells to be included or omitted within the specified area may also be defined.



LGR Keyword Parameters

Local grid refinements are defined through a nested set of keyword parameters and data, describing the type, characteristics, and location of the refinement relative to its parent grid. The nesting of the data sets defines the levels of refinements within refinements. This represents the minimum amount of data required for the LGR option.

The keyword parameters below are used to describe the refinement structure shown above in Figure 10.2.

NX NY NZ NCOMP 5 5 1 2 LGR BASEGRID CARTREF REF1 2 4 2 4 1 1 2 3 2 2 2 2 1 CARTREF REF2 2 3 3 4 1 1 3 2 4 2 1 ENDREF ENDREF ENDLGR

In this sequence of keywords and parameters, each keyword pertains to a specific group of parameters, which are entered to the side or below the keyword. For instance, the NX/NY/NZ keywords are used to describe the number of grid cells in the coarse grid area. The CARTREF keywords and following data specify the names and corresponding structure of the Cartesian refinement grids, including the i, j, k range of the refinements, the frequency of refinements in the refined area, and any parts of the refined area to be omitted.

For more information about LGR keyword parameters, refer to "Appendix A: Structured Grid Data" of the Nexus Keyword Document.



Propagation of Reservoir Properties to LGR Cells

When LGR grids are defined, the user has the option to specify all of the properties for the coarse and refined grid cells or only to specify properties for the coarse grid cells. Where properties are specified only for the coarse grid cells, nexus will calculate unique depths for each refined grid cell, based on interpolation of the corner points of the coarse grid, but it will not calculate unique initial reservoir properties for each refined grid cell. Instead, each refined grid cell will have the same properties as its parent coarse grid cell.

Reservoir properties for specific grids may be entered using the keywords ARRAY (grid name).

Example:

ARRAYS REF1 POR VALUE INCLUDE POR_REF1.data

Grid Coarsening

The Grid Coarsening option is a feature that enables the user to combine adjacent grid cells into coarser grid cells, thereby reducing the number of cells used for simulation and reduce run time.

Grid coarsening data is entered with the COARSEN keyword. Detailed information on the COARSEN keyword can be found in the "Appendix A: Structured Grid Data" of the Nexus Keyword Document.

Coarse Cell Properties

The properties of coarse blocks are calculated from the properties of the constituent fine blocks using a pore volume weighed average:

A

Ai PVi

i

PVii

--------------------------------=



The following properties are exceptions:For DX,DY,DZ,HNET a bulk volume weighted average is used, and the result multiplied by the number of fine blocks in the appropriate direction.

Porosity is calculated from PV,BV,DZ and HNET as follows:

DX nx

DXi BVi

i

BVii

-------------------------------------=

DY ny

DYi BVi

i

BVii

-------------------------------------=

DZ nz

DZi BVi

i

BVii

-------------------------------------=

HNET nz

HNETi BVi

i

BVii

---------------------------------------------=

PORPVBV------- DZ

HNET----------------=



When table indices ISAT,ISATI,ICMT,OILTRF ,GASTRF, IEQUIL, IREGION, IPVT, IPVTW, ITRAN, IWIRC are different for fine blocks within a coarse block, a WARNING message is printed on the standard output file, and they are assigned to the indices of the fine blocks whose sum represents the largest pore volume within the coarse block.

The half transmissibilities of the coarse block are calculated from those of the constituent fine blocks using the tubes in parallel method.

The fine block transmissibility values will include changes applied with the TOVER cards, but will not include changes applied with the OVER, VOVER cards. Transmissibility values at coarse block boundaries may be modified through the non-standard transmissibility multipliers.

Average coarse block permeabilities KX, KY and KZ that are used for calculating well, influx and polymer properties, and for other uses by VIP-EXEC are calculated from the average coarse block properties as follows:

TX+

TX- 2

1

TX+

i

------------ 1

TX-i

-----------+

i--------------------------------------------

j k= =

TY+

TY- 2

1

TY+

j

------------ 1

TY-j

----------+

j-------------------------------------------

i k= =

TZ+

TZ- 2

1

TZ+

k

------------- 1

TZ-k

-----------+

k---------------------------------------------

i j= =

KXTX

-

2 HNET----------------------- DX

DY--------=



The face areas of the coarse block are calculated from the sum of the face areas of the fine blocks that make a coarse block face.

KYTY

-

2 HNET----------------------- DY

DX--------=

KZTZ

-

2 DX--------------- DZ

DY--------=



Chapter 11

Procedures

Introduction

Procedures are sets of instructions which direct Nexus to perform various (user-specified) actions depending on whether various (user specified) conditions have been satisfied. A FORTRAN-like syntax is used, allowing a great deal of flexibility in specifying constraints and actions. (See the “Procedures” chapter of the keyword document for details of the syntax). The procedures are input as part of the surface network file.

During a timestep, the network is first solved using the most recent recurrent data input, with fixed reservoir pressures and fluid mobilities. On the first Newton iteration of a timestep, and also on the second Newton iteration if connections have opened (started producing or injecting), the procedures are executed. (See the network chapter for a detailed description of the network calculations performed during a timestep). The procedures can retrieve information from this network solution, such as connection rates, node pressures etc, and take actions, such as changing rate constraints, making changes to network connections etc, based on that information. Note that the network solution is not updated after each action (because that is computationally expensive) unless the user directs the program to resolve the network, using the solve_network function. However, after all procedures have been executed, the program will then resolve the network if any actions would result in a change to the network solution.

Nexus provides functions to retrieve the flow rates from the network potential solution (the qp function). The network potential (obtained by solving the network without targets) is normally calculated after the procedures have been executed, so the qp function will actually return flow rates calculated during the previous Newton iteration. The user can force the potential solution to be resolved by using the solve_network_potential function. When this function is used, then if the procedure has already taken actions which would change the network solution, such as changing constraints, the network solution is resolved, and then the network potential is resolved. The network solution is resolved because the network potential calculation uses

R5000.0.2 Procedures: Introduction 323


information from the network solution if connections have reopened or shut in. Subsequent functions will then reflect the fact that the network has been resolved.

A set of procedures is enclosed between the PROCS and ENDPROCS keywords, and there can be multiple sets of procedures. Procedures remain active from the time that they are first input, until the end of the run, unless subsequently cleared. A set of procedures may be given a name, which can then be used to clear the procedure. A set of procedures may also be given a priority. Procedures are executed in the order of increasing priority, with procedures having the same priority executing in the order in which they were input.

324 Procedures: Introduction R5000.0.2


Procedure Functions

Nexus provides many functions which can be used in Procedures. These are listed in the Procedures section of the keyword document. Functions usually require one or more arguments and return a result which may be any of the basic variable types, such as integer or real. The syntax used for a function is as follows:

result = function(arg1, arg2, arg3, …)

where

result is the result returned by the function

function is the name of the function

arg1, arg2, arg3 etc are arguments required by the function

Function arguments can be keywords, strings, constants, procedure variables, or network names such as connection or node names. The keyword document lists what type each argument of a function should be. There are basic functions, such as abs (absolute value), max (maximum of two real or integer values) etc, functions to retrieve network data, such as p (pressure at a node or well), gor (gas oil ratio of a connection or well) etc, functions to change the network, such as constraint (set a network constraint), activate (activate a connection) etc, and functions to manipulate lists, such as intersection (the intersection of two lists) etc. If the return type is an array or list, the function will allocate the space for the array or list.

Many functions use logical arrays as “masks” to determine when to perform the specified action. For example:

PROCSLOGICAL_1D mask1, resultmask1 = gor(conlist,all1d) > 3.0result = deactivate(conlist,mask1)ENDPROCS

Here mask1 is a logical array, the value of which will be true for any connection in the connection list named conlist, if the GOR of that connection is greater than 3. The function deactivate will then deactivate all connections in the connection list conlist, for which mask1 is true, so all connections with a GOR greater than 3 will be deactivated. Note that the argument all1d is a special variable which takes the value of true for all elements of an array.

R5000.0.2 Procedures: Procedure Functions 325


Network names, such as connection, node, well and target names, which are used as function arguments must have been previously defined in the surface network file.

Although most functions return results, it is not necessary to set a variable equal to the function unless you want to store the result for some purpose.

Most functions are “Overloaded”. This means that the result of the function or result returned by the function depend on the number and type of arguments used when invoking the function. For example, the WOR function can return the Water/Oil ratio for a well, a connection, a well list, a connection list, the completions in a well, or all the completions for all wells in a well list, depending on the arguments used.

Because the procedures are executed every timestep, the user must specify logic in the procedures that prevents unwanted actions. For example

PROCSLOGICAL pmin_reduced = falseREAL X! get the oil rate in the FIELD connection at surface

conditionsX = Q("OS",FIELD)IF(X < 400.0)THEN IF(not pmin_reduced)THEN ! Set the miniumum pressure constraint at the FIELD node

to 100 psi CONSTRAINT("PMIN",FIELD,100.0)pmin_reduced = true

ELSE! If the field oil rate falls below 200, stop the run ABANDON(FALSE) ENDIF ENDIFENDPROCS

In this example, when the oil rate for connection FIELD falls below 400, the minimum pressure at the FIELD node is reduced to 100 psi (which presumably the user expects to result in an increase in oil rate. A logical variable is then set to true, indicating that this action has occurred. When the oil rate again falls below 400, this program will then execute the abandon function, which stops the run, instead of trying to reset the minimum pressure to 100 psi (which would accomplish nothing).

326 Procedures: Procedure Functions R5000.0.2


The value of each variable is preserved from the previous time each set of procedures was executed. In the above example, this value of pmin_reduced is initially set to FALSE the first time the procedure set is executed (due to the declaration LOGICAL pmin_reduced = false), then when it is set to true, this value is then saved and available to all subsequent executions of the procedure set. Variables are only able to be used within the procedure set in which they are declared, unless they are declared as STATIC, in which case all procedure sets can use that variable. The names of STATIC variables must be unique across all procedure sets. I.e if an integer is declared as INTEGER, STATIC i, in one set of procedures, then i can not be declared as a variable in any other procedure set.

R5000.0.2 Procedures: Procedure Functions 327


Debugging Procedures

Debug output can be turned on using the set_debug function. The program then outputs the arguments to, and result of, every function call and operation that is executed to the .dbg file. In addition, the printout function can be used to print out the current contents of any procedure variable.

328 Procedures: Debugging Procedures R5000.0.2


Chapter 12

Aquifers

Introduction

When modeling aquifers, Nexus always treats outer boundaries of a reservoir model as sealing barriers to flow. The influx/efflux of fluids from outside the grid is treated by source/sink terms.

There are two methods available in Nexus to model aquifer influx/efflux resulting from changes in reservoir conditions. The first approach uses one of two analytical models which predict the idealized aquifer response. The second approach extends the grid system to cover the aquifer with the reservoir. In the latter case, interaction between the reservoir and the aquifer automatically is taken into account. Each of these methods has positive and negative aspects. The analytical methods are easy to use. They are computationally efficient and do not require a significant amount of additional computer memory. However, the analytical models may be too restrictive because of assumptions and simplifications. The grid extension approach has the possible disadvantage of requiring significantly more memory and computations.

R5000.0.2 Aquifers: Introduction 329


Analytic Model — The Carter-Tracy Aquifer

The Carter-Tracy1 aquifer analytic model may be applied to an aquifer of radial or linear geometry. The aquifer and the reservoir communicate through the boundary AB as shown in in Figure 12.1.

Figure 12.1: Aquifer Geometry

The water influx from the aquifer into the reservoir is the response to pressure changes at this boundary. Water flow in the aquifer is described by Equation 12-1 with the initial and boundary conditions:

12-1

12-2

12-3

12-4

Here, is aquifer porosity, w is water viscosity, ct is aquifer total compressibility, and k is aquifer permeability.

Van Everdingen and Hurst2 derived the following superposition form of the solution of the above problem:

AquiferA

B

Reservoirre

ra

wct

k--------------p

t------ 1

r--- r----- r

pr------=

p t 0= pi=

pr re=

pe t =

pr------

r ra=

0=

330 Aquifers: Analytic Model — The Carter-Tracy Aquifer R5000.0.2


12-5

where

12-6

12-7

12-8

12-9

12-10

In Equation 12-5, W(t) is the cumulative water influx from the aquifer and Q(td) is the standard function derived by Van Everdingen and Hurst, which gives the cumulative influx in the case of the unit pressure change at the boundary r = re. Parameter t0, which is given by Equations 12-7 and 12-8, is the aquifer time constant. It is used to define the dimensionless time td according to Equation 12-6. Parameter B, as defined by Equations 12-9 and 12-10, is the aquifer capacity parameter. Equation 12-9 contains two constants: h is the aquifer thickness and is the angle (in radians) subtended by the aquifer (Figure 12.1). The and parameters are unit-dependent constants. The terms w and L represent the width and length for the linear aquifer case.

W t Bdpe td1

dtd1--------------------Q td td1– dtd1

0

td

–=

tdtt0----=

t0

wct re2

2k-----------------------= RADIAL

t0 wct L

2

2k------------------------= LINEAR

Bre

2h ct 1

-------------------------= RADIAL

BwL h ct

1------------------------= LINEAR

R5000.0.2 Aquifers: Analytic Model — The Carter-Tracy Aquifer 331


Carter and Tracy derived an approximation of the solution (Equation 12-5). This approximate solution of the problem (Equations 12-1 through 12-4) is given by:

12-11

Here, t is the timestep, td is the dimensionless timestep, and Pd(td) is a function derived by Van Everdingen and Hurst.

The Carter and Tracy solution does not require integration and therefore is more efficient. In the Nexus implementation of the Carter and Tracy solution, the user may provide the dimensionless pressure function Pd(td) in a tabular form through input data. One can find tabulated pressure functions Pd(td) for finite aquifers in the published literature.4 The dimensionless pressure function for infinite radial aquifer4 is given in the table below. This is the default table that will be used if the user does not provide an input table.

W t t+ W t B pi p t t+ – W t P'd td td+ –

Pd td td+ tdP'd td td+ –-------------------------------------------------------------------------------------------td+=

332 Aquifers: Analytic Model — The Carter-Tracy Aquifer R5000.0.2


tD pD tD pD

.01 0.112 10 1.651

.05 0.229 15 1.829

.1 0.315 20 1.960

.15 0.376 25 2.067

.2 0.424 30 2.147

.25 0.469 40 2.282

.3 0.503 50 2.388

.4 0.564 60 2.476

.5 0.616 70 2.550

.6 0.659 80 2.615

.7 0.702 90 2.672

.8 0.735 100 2.723

.9 0.772 150 2.921

1.0 0.802 200 3.064

1.5 0.927 250 3.173

2.0 1.020 300 3.262

2.5 1.101 400 3.406

3.0 1.169 500 3.516

4.0 1.275 600 3.608

5.0 1.362 700 3.684

6.0 1.436 800 3.750

7.0 1.500 900 3.809

8.0 1.556 1000 3.860

9.0 1.604

Dimensionless Pressure Function for Infinite Radial Aquifer4

R5000.0.2 Aquifers: Analytic Model — The Carter-Tracy Aquifer 333


Analytical Model - The Fetkovich Aquifer

The Fetkovich aquifer model3 is a more direct computational method of performing water influx calculations. In this approach the flow of aquifer water into a hydrocarbon reservoir is modeled in precisely the same way as the flow of oil from a reservoir into a well. An inflow equation of the form

12-12

is used where,

= aquifer productivity index,

= pressure at the oil or gas water contact,

= average pressure in the aquifer.

The water influx is evaluated using the simple aquifer material balance

12-13

in which,

= the initial pressure in the aquifer and reservoir,

= the total compressibility,

= initial aquifer water in place.

This balance can be alternatively expressed as

12-14

where

12-15

dWe

dt---------- J pa p– =

J

p

pa

We ctWi pi pa– =

pi

ct

Wi

pa pi 1We

Wei--------–

=

Wei ctWi pi=

334 Aquifers: Analytical Model - The Fetkovich Aquifer R5000.0.2


is defined as the initial amount of encroachable water and represents the maximum possible expansion of the aquifer. Substituting the average aquifer pressure expression into the inflow equation results

12-16

After manipulation, it can be shown that 12-17

Pa is evaluated from Equation 12-14 and 12-18

dWe

dt---------- J pi 1

We

Wei--------–

p–=

W t Wei

Pi-------- Pa Pn– 1 e

Jpit

Wei

----------- –

–

=

Pn p t p t t+ +2

--------------------------------------=

R5000.0.2 Aquifers: Analytical Model - The Fetkovich Aquifer 335


Numerical Aquifer

In the numerical aquifer treatment, the grid system is extended to cover the aquifer with the reservoir. This can be done in several ways. One way is to extend the reservoir grid system as shown in Figure 12.2 to cover both the reservoir and aquifer.

Figure 12.2: Natural Extension of Reservoir Grid

A model with the naturally extended grid system may require a significant number of additional gridblocks in the model. This significantly increases memory and CPU requirements.

Another method is to represent the aquifer by several blocks with large pore volumes which are connected to reservoir blocks around the edges of the grid as shown in Figure 12.3. This is done by defining nonstandard connections between gridblocks.

Figure 12.3: Aquifer with Arbitrary Connected Blocks

Improvements made to the Nexus unstructured grid solver make solver considerations for the choice between these methods less important than they were to VIP, where the first method was better for the solver.

Aquifer

Reservoir

Aquifer

Reservoir

336 Aquifers: Numerical Aquifer R5000.0.2


References

1. Carter, R. D. and Tracy, G. W., “An Improved Method for Calculating Water Influx,” Trans. AIME, 219, 415-417.

2. Van Everdingen, A. F. and Hurst, W., “The Application of the Laplace Transformation to Flow Problems in Reservoirs,” Trans. AIME, 186, 305.

3. Fetkovitch, M.J., “A Simplified Approach to Water Influx Calculations-Finite Aquifer Systems”, J.Pet. Tech., July,1971, 814-828

4. Bradley, H.B., Petroleum Engineering Handbook, First Printing, Chap. 38, Society of Petroleum Engineers, 1987

R5000.0.2 Aquifers: References 337


338 Aquifers: References R5000.0.2


Appendix A: Corner-Point Geometry

Mapping of Gridblock to Unit Cube

When the user chooses the corner-point geometry option, the location of the eight corner points of each gridblock is quite arbitrary. In order to calculate bulk volumes, block centers, and transmissibilities, Nexus maps each gridblock into a unit cube. That mapping is described here.

Two-Dimensional Mapping

To introduce the mapping, let us first consider the simpler two-dimensional problem of mapping an arbitrary quadrilateral in the x-y plane into a unit square in the u-v plane, as shown in Figure A-A.1.

Figure A.1: Mapping of Quadrilateral to Unit Square

Let

A-1

A-2

It is easy to see that u = 0 corresponds to side 1-4, u = 1 to side 2-3, v = 0 to side 1-2, and v = 1 to side 4-3.

The differential area dx dy is given by:

dx dy = J(u,v)du dv

x x1 x2 x1– u x4 x1– v x1 x3 x2– x4–+ uv+ + +=

y y1 y2 y1– u y4 y1– v y1 y3 y2– y4–+ uv+ + +=

R5000.0.2 Corner-Point Geometry: Mapping of Gridblock to Unit Cube 339


where J(u,v) is the so-called Jacobian of the mapping of the unit square back onto the quadrilateral. It may be considered to be a scaling, or stretching, factor of the mapping. It is given by the determinant of a certain matrix of partial derivatives, as follows:

A-3

where, from Equations A-A-1 and A-A-2,

Thus, the area of the quadrilateral may be obtained by integrating the Jacobian over the unit square:

A-4

The centroid, or center of gravity, may be obtained by the integrals:

A-5

J u v x y u v ----------------

xu------ x

v-----

yu------ y

v-----

= =

xu------ x2 x1– x1 x3 x2– x4–+ v+=

xv----- x4 x1– x1 x3 x2– x4–+ u+=

yu------ y2 y1– y1 y3 y2– y4–+ v+=

yv----- y4 y1– y1 y3 y2– y4–+ u+=

A J u v du dv

v=0

1

u=0

1

=

xc1A--- x u v J u v du dv

v=0

1

u=0

1

=

340 Corner-Point Geometry: Mapping of Gridblock to Unit Cube R5000.0.2


A-6

where x(u,v), and y(u,v), correspond, respectively, to Equations A-A-1 and A-A-2.

Three-Dimensional Mapping

Figure A.2: The Eight Corners of a Gridblock

In Nexus’s corner-point option, each gridblock is a solid defined by the location of its eight corners, as shown in Figure A-A.2. Analogously to Equations A-A-1 and A-A-2, the block can be mapped to a unit cube by the following equations:

A-7

A-8

where A-9

yc1A--- y u v J u v du dv

v=0

1

u=0

1

=

x p1 x u p2 x v p3 x w p4 x uv p5 x vw p6 x uw p7 x uvw p8 x+ + + + + + +=

y p1 y u p2 y v p3 y w p4 y uv p5 y vw p6 y uw p7 y uvw p8 y+ + + + + + +=

z p1 z u p2 z v p3 z w p4 z uv p5 z vw p6 z uw p7 z uvw p8 z+ + + + + + +=

p1 x x2 x1–= p2 x x4 x1–= p3 x x5 x1–=



This works similarly for p1,y , ..., p8,y in terms of the y‘s, and p1,z , ..., p8,z in terms of the z‘s. It can be seen that u = 0 corresponds to face 1-4-8-5, u = 1 to face 2-3-7-6, v = 0 to face 1-2-6-5, v = 1 to face 4-3-7-8, etc.

The differential volume dx dy dz in “real” space is given by:

dx dy dz = J(u,v,w) du dv dw

where the Jacobian J(u,v,w) is given byA-10

The partial derivatives are given by:

A-11

A-12

A-13

Similarly fory/u, z/u, and so forth.

p4 x x1 x3 x2– x4–+= p5 x x1 x8 x4– x5–+= p6 x x1 x6 x2– x5–+=

p7 x x2 x4 x5 x7 x1– x3 x6 x8–––+ + += p8 x x1=

J u v w x y z u v w -----------------------

xu------ x

v----- x

w-------

yu------ y

v----- y

w-------

zu------ z

v----- z

w-------

= =

xu------ p1 x p4 x v p6 x w p7 x vw+ + +=

xv----- p2 x p4 x u p5 x w p7 x uw+ + +=

xw------- p3 x p5 x v p6 x u p7 x uv+ + +=



Volumetric Calculations

The bulk volume of the gridblock is obtained by integrating the Jacobian of Equation A-A-10 over the unit cube:

A-14

The centroid, or block center, is obtained by:

A-15

A-16

A-17

V J u v w du dv dw

w=0

1

v=0

1

u=0

1

=

xc1V--- x u v w J u v w du dv dw

w=0

1

v=0

1

u=0

1

=

yc1V--- y u v w J u v w du dv dw

w=0

1

v=0

1

u=0

1

=

zc1V--- z u v w J u v w du dv dw

w=0

1

v=0

1

u=0

1

=



Finally, let us consider how we might obtain the “average” thickness of the block in the vertical direction.

Figure A.3: Elemental Tube for Determining Thickness

In Figure A-A.3, consider the elemental tube (shown by dashed lines) bounded by u, u+du, v, and v+dv. The thickness of the tube, i.e., its vertical extent, is z(u,v,1) - z(u,v,0).

The volume of the tube is:

Thus the volume-averaged thickness of the entire block is

which may be rearranged to the final form:

A-18

J u v w dw

w=0

1

du dv

DZ1V--- z u v 1 z u v 0 – J u v w dw

w=0

1

du dv

v=0

1

u=0

1

=

DZ1V--- z u v 1 z u v 0 – J u v w du dv dw

w=0

1

v=0

1

u=0

1

=



Figure A.4: Elemental Tube for Determining DX

Now we consider how we might obtain the average distance from the left face to the right face of the block. In Figure A-A.4, consider the element tube bounded by v, v+dv, w, and w+dw. The length of that tube from the left face to the right face is

A-19

Define the distance between two points by

Then the expression in Equation A-A-19 for the length of the tube can be written

x 1 v w x 0 v w – 2 y 1 v w y 0 v w – 2 z 1 v w z 0 v 2 – 2+ +

D u2 v2 w2 u1 v1 w1 x u2 v2 w2 x u1 v1 w1 – 2=

y u2 v2 w2 y u1 v1 w1 – 2 z u2 v2 w2 z u1 v1 w1 – 2

1 2

+ +

D 1 v w 0 v w



Thus the volume averaged distance from the left face to the right face is

which may be rearranged to the final formA-20

Similarly,A-21

Integration by Gaussian Quadrature

A single integral may be evaluated quite accurately by the N-point summation:

A-22

for properly chosen values of wk and xk. The wk are weights and the xk are Gauss points, or quadrature points, at which the function f(x) is evaluated. For N=1, 2, or 3, the weights and Gauss points are given by:

DX1V--- D 1 v w 0 v w J u v w dw

w=0

1

dudv

w=0

1

v=0

1

=

DX1V--- D 1 v w 0 v w J u v w du dv dw

w=0

1

v=0

1

u=0

1

=

DX1V--- D u 1 w u 0 w J u v w du dv dw

w=0

1

v=0

1

u=0

1

=

f x dx

x=0

1

wk f xk

k 1=

N

N 1: w1 1= = x112---=

N 2: w1 w212---= = = x1

12--- 1 1

3-------–

x2 12--- 1 1

3-------+

= =

N 3: w1 w3518------ w2 4

9---= = = = x1

12--- 1 3

5---–

x2 12---= = x3 1

2--- 1 3

5---+

=



If f(x) is a polynomial of degree d, then Gaussian quadrature is exact if d 2N-1. Thus 2-point quadrature is exact for a cubic, and 3-point quadrature is exact for a fifth-degree polynomial.

The triple integrals of the previous section may be approximated by the triple summations:

A-23

where Ni, Nj, and Nk are, respectively, the number of quadrature points in the x-, y-, and z-directions. These numbers are entered on the CORNER card as iquads, jquads, and kquads.

It turns out that the integrands within all the triple integrals of the preceding section for volumetric calculations are composite polynomials of low enough degree that iquads = jquads = kquads = 2 provides exact integration. This is true no matter how arbitrarily the corner points are located.

Of course, if the gridblocks are simple rectangular parallelepipeds, iquads = jquads = kquads = 1 would be sufficient, but then there would be little point in using the corner-point option.

However, the story is different for evaluating transmissibilities. We shall see later that if the harmonic integration option is chosen, the integrands are no longer polynomials. In that case, the user should specify iquads = jquads = kquads = 3 to get the most accurate integration.

f u v w du dv dw

w=0

1

v=0

1

u=0

1

wiwjwk f ui vj wk

k 1=

Nk

j 1=

Nj

i 1=

Ni



Transmissibility Calculations

For corner-point geometry, Nexus provides two options for calculating the transmissibilities between adjacent gridblocks: the standard Nexus option that uses “harmonic integration” (HARINT), and the NEWTRAN option, which is similar to the corresponding ECLIPSE technique. Both of these are described below, first for the simpler 2-D case and then for the full 3-D case.

Calculation of Transmissibility in 2D

Simplest Case: Orthogonal Grid

Figure A.5: Orthogonal Gridblocks

Consider the two rectangular blocks in Figure A-A.5, where the points A and B are the centroids of their respective blocks. Each block has its own permeability, kA or kB. As this may be considered to be flow through two resistances in series, the transmissibility from A to B is given by:

A-24

where

and

TAB1

1TA------ 1

TB------+

-------------------=

TA kAz W LA=



In this simple 2D case, z, which is perpendicular to the plane of Figure A-A.5, is considered constant. Note that TA may be considered to be the “half-block transmissibility” from the point A to the right face of block A.

Non-Orthogonal Grid with Parallel Sides

Figure A.6: Nonorthogonal Gridblocks with Parallel Sides

In Figure A-A.6, we consider the somewhat more general case where the gridblocks are not orthogonal, but the layer boundaries are parallel. Equation A-A-24 still holds, with TA again being given by:

We note that LA is now specifically the distance from the centroid A to the point C, where C is the center of the interface between blocks A and B. If LAB is the length of that interface, and if is the angle between the interface and the parallel sides, then clearly:

Alternatively, let n be the unit vector normal to the interface at point C, and let be the angle between line AC and the normal n. As and are complementary angles, then:

and

TB kBz W LB=

LAB

TA kAz W LA=

W LAB sin=

W LAB cos=



A-25

We shall see that it will be also useful to express this as the following resistance from the point A to the right face:

A-26

Non-Orthogonal Grid with Non-Parallel Sides

Figure A.7: Gridblockwith Non-Parallel Sides

In Figure A-A.7, we show the more general case of a single gridblock in which none of the sides are parallel. Point A is still the centroid, and again we wish to calculate the half-block transmissibility from point A to the right face. For this purpose, recognize that, in mapping the gridblock to a unit square, the variable u represents a fractional distance from the left face to the right face. Let point 5 be a point moving along the upper edge, such that

and let point 6 be a similar point moving along the lower edge, such that:

TA kAz LAB cos LA=

1TA------

LA

kAz LAB cos-------------------------------------=

x5 u x1 u x2 x1– += y5 u y1 u y2 y1– +=

x6 u x3 u x4 x3– += y6 u y3 u y4 y3– +=



Then the line L56(u), connecting points 5 and 6, is a moving line that slices the block as shown in Figure A-A.8.

Figure A.8: Slice Parametrized by u

In particular, u = 0 corresponds to the left edge, u = 1 corresponds to the right edge, and uc is the value of u corresponding to the line L56 that passes through the centroid. (Note that uc is not necessarily equal to 0.5!)

We may conceptualize the half-block transmissibility from the slice going through the centroid to the right face as being the sum of transmissibilities of a collection of tubes formed by lines of constant v, as shown in Figure A-A.9.

Figure A.9: Collection of Tubes and Slices



In Figure A-A.9, we also show a sequence of slices formed by lines of constant u. Let us call the intersection of a tube and a slice a chunk. A typical chunk is shown in Figure A-A.10.

Figure A.10: Tube, Slice, and Chunk

By analogy with Equation A-A-26, the resistance of that typical chunk is given by:

wheres is the distance along the tube lying within the slice, L is the distance along the slice within the chunk, n(u,v) is the normal to the slice, and (u,v) is the angle between the tube and that normal. But the area of the chunk is given by A = z L, so the resistance of the typical chunk can be expressed somewhat more succinctly as:

skAz L u v cos---------------------------------------------------

skAA u v cos------------------------------------------



In the limit, then, the resistance of the tube from the slice going through the centroid to the right face is given by the integral:

Because of the appearance of the area A in the denominator, this integral is referred to as a “harmonic integral”, by analogy to the harmonic series , where the term n appears in the denominator.

Finally, the transmissibility of all the tubes in parallel is given, in the limit, by:

A-27

1TA------

dsdu------ u v

kAA u v cos------------------------------------------du

u=uc

u=1

=

1n---

TA1

dsdu------ u v

kAdAdv------- u v cos

------------------------------------------du

u = uc

u = 1

--------------------------------------------------------------- dv

v = 0

1

=



Alternate Approach for Gridblock with Non-Parallel Sides

Figure A.11: Gridblockwith Nonparallel Sides

Consider the gridblock with nonparallel sides that is shown in Figure A-A.11. We wish to calculate the half-block transmissibility between the centroid and the right face of that block. Again, point A is the centroid; point C is the center of the right face. The right face is the interface between block A and block B (which is not shown). Let AC be the vector from A to C. Let CD be a vector normal to the right edge, whose length is equal to the length of the right edge, that is:

Again, let be the angle between AC and CD .

Now, while Equation A-A-25 was derived for a gridblock with parallel sides, as in Figure A-A.6, it can also be applied to this more general situation. Thus we can write

CD LAB=

TA kAzLAB cos

LA---------------------- kAz

LALAB cos

LA 2-----------------------------= =



where . Then we can write the half-block transmissibility as:

Now, the scalar (also known as inner or dot) product of two vectors a and b having angle between them can be expressed in the following two ways:

where ax and ay are the x- and y-components of the vector a, and bx and by are the x- and y-components of the vector b, respectively. Thus we see that the half-block transmissibility can also be expressed as:

But, since vectors EF and CD are perpendicular,

Then

A-28

This is the 2-D version of NEWTRAN.

The quantity Lx is to be interpreted as the x-projection of the interface between blocks A and B on the y-axis. The quantity Ly can be interpreted as the y-projection of that interface on the x-axis. It is important that the correct signs of Lx and Ly be used. Thus, in Figure A-A.11, Lx is positive while Ly is negative.

LA AC=

TA kAzAC CD cos

AC2

------------------------------------=

a b a b cos=

a b axbx ayby+=

TA kAzxC xA– xC xD– yC yA– yC yD– +

xC xA– 2 yC yA– 2+-------------------------------------------------------------------------------------------------=

xC xD– yE yF– Lx= =

yC yD– xF xE– Ly= =

TA kAzLx xC xA– Ly yC yA– +

xC xA– 2yC yA– 2+

---------------------------------------------------------------=



Calculation of Transmissibility in 3D (HARINT)

Equation A-A-27, which was derived as the 2D version of HARINT, may be extended to the general 3D case, where the eight corners of the gridblock are completely arbitrary. However, considerable care must be taken in interpreting and calculating the various terms in it. In doing so, we will be making use of the mapping of the gridblock into the unit cube, as described in the "Three-Dimensional Mapping"section.

That extended equation is, then:A-29

Tubes and Slices in 3D

In Figure A.2 on page -341, let Equations A-A-7, A-A-8, and A-A-9 be the functions that map the coordinates of real space (x,y,z) into the coordinates of the unit cube (u,v,w). Tubes are formed by surfaces of constant v intersecting surfaces of constant w. On the other hand, any surface of constant u over the range

defines a slice of the gridblock (in real space) that corresponds to the slice shown in Figure A-A.8. In general, the slice will not be a planar surface but, rather, a bilinear surface.

TA1

dsdu------ u v w

kAdA

dv dw--------------- u v w cos

--------------------------------------------------------du

u = uc

u = 1

----------------------------------------------------------------------------dv dw

w = 0

1

v = 0

1

=

0 v 1 0 w 1



Differential Area of the Slice

The differential area of this nonplanar slice is computed by1:

A-30

whereA-31

withA-32

The partial derivatives are obtained from Equations A-A-7, A-A-8, and A-A-9 by:

Similarly for ¹y / v, z v, and so forth.

Location of Centroid, in Real Space, and in the Unit Cube

The coordinates of the centroid in real space, xc , yc , zc , are found by application of Equations A-A-15, A-A-16, and A-A-17. Finding the coordinates of that centroid in the unit cube, uc, vc, wc, involves inverting the mapping functions in Equations A-A-7, A-A-8, and A-A-9. This nontrivial task is accomplished by a Newtonian iteration, which converges very rapidly. Prior to the first iteration, the first guess is uc = vc = wc = 0.5. After convergence, the following is satisfied:

dA D u v w dv dw=

D u v w Jxy2

Jyz2

Jzx2+ +=

Jxy

xv----- y

v-----

xw------- y

w-------

Jyz

yv----- z

v-----

yw------- z

w-------

Jzx

zv----- x

v-----

zw------- x

w-------

=;=;=

xv----- p2 x p4 x u p5 x w p7 x uw+ + +=

xw------- p3 x p5 x v p6 x u p7 x uv+ + +=

xc x uc vc wc =

yc y uc vc wc =

zc z uc vc wc =



where x(u,v,w), y(u,v,w), and z(u,v,w) are the polynomials defined in Equations A-A-7, A-A-8, and A-A-9.

Evaluation of Arc Length Derivative

Along each tube of constant v and w, the derivative of arc length is given by:

A-33

The partial derivatives in Equation A-A-33 are given above in Equation A-A-11.

Evaluation of cos (u,v,w)

The only quantity in the integrand of Equation A-A-29 now left to be evaluated is the cosine of , the angle between the tube and the normal to the slice. Let p be a vector along the tube, and n be a vector normal to the slice. Then, from the two ways of expressing the scalar (or dot) product of the two vectors in 3D:

we have

orA-34

The components of p are clearly equal to the derivatives of x, y, and z with respect to u. That is:

dsdu------ u v w

dx

du-------- 2

dy

du-------- 2

dz

du-------- 2

+ +=

p n p n cos=

p n pxnx pyny pznz+ +=

cospxnx pyny pznz+ +

p n--------------------------------------------=

cospxnx pyny pznz+ +

px2

py2

pz2

+ + nx2

ny2

nz2

+ + ------------------------------------------------------------------------=



The normal to the slice is obtained by first finding the equation for the plane in real space tangent to the slice where the tube intersects it. To get this equation, we find two lines (or vectors) in the slice through the point u, v, w (in the unit cube) that correspond to the intersection point x, y, z (in real space). See Figure A-A.12.

Figure A.12: Lines a and b Defining Plane Tangent to Slice

Let line a be at constant u and v; let line b be at constant u and w. Then the components of a and b are:

The equation of a plane through the arbitrary point x1, y1, z1, parallel to a and b is2:

pxxu------= py; y

u------= pz; z

u------=

u=constant

axxw------- ;= ay

yw------- ;= az

zw-------=

bxxv----- ;= by

yv----- ;= bz

zv-----=

x x1– y y1– z z1–

ax ay az

bx by bz

0=



If the equation for that plane is expressed as:

then A is the cofactor of (x-x1), B is the cofactor of (y-y1), and C is the cofactor of (z-z1). Thus:

Note that A, B, and C are equal, respectively, to -Jyz, -Jzx, and -Jxy of Equation A-A-32. But the quantities A, B, and C are also equal to the components of the normal:

Now we have all the quantities necessary to compute cos u,v,wbyEquation A-A-34. We also have all the quantities necessary to compute the integrals of Equation A-A-29:

A-35

Ax By Cz+ + D=

Aay az

by bz

yw-------z

v----- z

w-------y

v-----–= =

Bax az

bx bz

–zw-------x

v----- x

w-------z

v-----–= =

Cax ay

bx by

xw-------y

v----- y

w-------x

v-----–= =

nx A ;= ny B ;= nz C=

TA1

dsdu------ u v w

kAD u v w u v w cos-----------------------------------------------------------------du

u = uc

u = 1

--------------------------------------------------------------------------------------dv dw

w = 0

1

v = 0

1

=



Half-Block Transmissibility to Other Faces

All the above discussion of HARINT has dealt with the calculation of half-block transmissibility from the slice through the centroid to the right face. To get the half-block transmissibility from the slice through the centroid to the left face, it suffices to change the limits of integration In Equation A-A-35 as follows:

A-36

To get half-block transmissibilities from the slice through the centroid to the front and back faces, it suffices to rotate the gridblock about the z-axis, and use the program that calculates the integrals of Equations A-A-35 and A-A-36. Finally, to get half-block transmissibilities from the slice through the centroid to the top and bottom faces, it suffices to rotate the gridblock about the y-axis, and again use that same program.

Calculation of Transmissibility in 3D by the NEWTRAN Option

Equation A-A-19 for the half-block transmissibility is easily generalized to the 3D case as follows:

A-37

Point A is again the centroid of the gridblock, while point C is to be interpreted now as the center of the right face. The four corners of the right face are projected onto each of the three coordinate planes, as shown in Figure A-A.13. Ax is the area of the projection onto the y-z plane, Ay is the area of the projection onto the x-z plane, while Az is the area of the projection onto the x-y plane. It is even more important in this 3-D case that considerable care be taken to get the correct sign of these area terms.

TA1

dsdu------ u v w

kAD u v w u v w cos-----------------------------------------------------------------du

u = 0

u = uc

------------------------------------------------------------------------------------dv dw

w = 0

1

v = 0

1

=

TA kA

Ax xC xA– Ay yC yA– Az zC zA– + +

xC xA– 2 yC yA– 2zC zA– 2+ +

-------------------------------------------------------------------------------------------------=



Figure A.13: Projections of Right Face onto the Three Coordinate Planes



Half-Block Transmissibility to All Six Faces

While Equation A-A-37 was described as the equation for the half-block transmissibility from the centroid to the right face, actually it applies to all six faces. Point C is now interpreted as the center of the relevant face, while Ax, Ay , and Az are the areas of the projections of that face. The sign of TA may turn out to be negative, so its absolute value is used.

Calculation of Full Transmissibility Between Gridblocks

Unfaulted Case

As pointed out in the previous sections, the half-block transmissibility from the centroid to all six faces is computed for each gridblock. These must be combined to yield the full transmissibility between adjacent gridblocks. For example, if gridblock B is to the right of gridblock A, then the transmissibility between their centroids is:

A-38

where TA is the half-block transmissibility from the centroid of block A to the right face of block A, and TB is the half-block transmissibility from the centroid of block B to the left face of block B. Equation A-A-38 is, of course, the same as Equation A-A-24. Similar combinations are made to calculate TY between two adjacent blocks in the same y-row, and to calculate TZ between two adjacent blocks in the same vertical column.

TXAB1

1TA------ 1

TB------+

-------------------=



Faulted Case

Figure A.14: Fault Block Connection

In the case of the FAULT option, the adjacent blocks may share only part of the common face, as illustrated in Figure A-A.14. In this figure, AA is the total area of the right face of block A, and is used in the calculation of TA. Similarly, AB is the total area of the left face of block B, and is used in the calculation of TB. The area of their mutual intersection is AAB. Then Equation A-A-38 is modified as follows:

A-39

TXAB

AAB

AA

TA------

AB

TB------+

--------------------=



Choice Between HARINT and NEWTRAN Options

The following 2D radial problem provides a test for comparing the accuracy of the above two options for transmissibility. It involves a four-block (2 x 2) model in which the corner points are given by

Figure A-A.15 shows this model for the particular case of = 45°.

Figure A.15: Radial problem, with = 45°

xij i j 1– cos=

yij 3 i j 1– sin+=



It can be shown that if the grid were truly radial, rather than approximated by this corner-point geometry, then the angular transmissibility between Blocks 1 and 3 in Figure A-A.15 would be:

while that between Blocks 2 and 4 would be:

In addition, the radial transmissibility, between Blocks 1 and 2, as well as between Blocks 3 and 4, would be:

NOTE: in these equations is in radians.

These equations should hold in the limit as is reduced to zero.

Results for the two options are shown in Figures A.16, A.17, and A.18. In Figure A-A.16, the angular transmissibility between Blocks 1 and 3, multiplied by /z, is plotted against ,and we see that HARINT converges to the correct value for small , and is only a few percent low for as large as 20-30 degrees. NEWTRAN is considerably poorer, and converges to an incorrect value. Similar results are shown in Figure A-A.17 for the angular transmissibility between Blocks 3 and 4.

In Figure A-A.18, the radial transmissibility, divided by x z, is plotted against , and here we see that HARINT gives the correct answer for all , while NEWTRAN does not.

It may be concluded that HARINT is clearly superior to NEWTRAN and is therefore the option of choice. NEWTRAN is offered only as an alternative option, to provide compatibility with the corresponding technique in the ECLIPSE simulator. When that compatibility is not required, HARINT is the recommended, and hence the default, option.

T1 3–z ln 2 1

----------------------------=

T2 4–z ln 3 2

----------------------------=

TR z

ln33 23–33 22–----------------- ln

23 13–22 12–-----------------–

------------------------------------------------------------- z

ln19 57 3-------------

-----------------------= =



Figure A.16: Comparison of HARINT and NEWTRAN for Angular Transmissibility Between Blocks 1 and 2



Figure A.17: Comparison of HARINT and NEWTRAN for Angular Transmissibility Between Blocks 3 and 4



Figure A.18: Comparison of HARINT and NEWTRAN for Radial Tranmissibility



References

1. Franklin, P., A Treatise on Advanced Calculus, John Wiley & Sons, New York, 1940, Page 375, Equation 110.

2. Recktorys, K., Survey of Applicable Mathematics, M.I.T. Press, Cambridge, Mass., 1969, Page 240, Theorem 11.

370 Corner-Point Geometry: References R5000.0.2

nexus® technical reference guide - halliburton

Documents