Chapter 48

Reservoir Simulation K.H. Coats, Scientific Software-Intercom

Introduction Webster’s dictionary defines simulate as to assume the appearance ofwithout the reality. Simulation of petroleum reservoir performance refers to the construction and operation of a model whose behavior assumes the ap- pearance of actual reservoir behavior. The model itself is either physical (for example, a laboratory sandpack) or mathematical. A mathematical model is simply a set of equations that, subject to certain assumptions, describes the physical processes active in the reservoir. Although the model itself obviously lacks the reality of the oil or gas field, the behavior of a valid model simulates (assumes the appearance of) that of the field.

The purpose of simulation is estimation of field per- formance (e.g., oil recovery) under one or more produc- ing schemes. Whereas the field can be produced only once, at considerable expense, a model can be produced or run many times at low expense over a short period of time. Observation of model performance under different producing conditions aids selection of an optimal set of producing conditions for the reservoir.

The tools of reservoir simulation range from the in- tuition and judgment of the engineer to complex mathematical models requiring use of digital computers. The question is not whether to simulate but, rather, which tool or method to use. This chapter attempts to summarize the evolution and current status of reservoir simulation practice involving usage of the mathematical, computer- ized models. The relatively modern nature of this prac- tice is indicated by the first edition of this handbook (1962) not including a chapter on reservoir simulation.

The nearly exponential growth in annual rate of simulation-related publications from the mid-1960’s to the present indicates the industry’s widespread acceptance of mathematical simulation as an engineering tool. This ac- ceptance has been and remains qualified by questioning and improvement of accuracy in simulation model results. Thus a significant portion of the extensive literature deals

with model (1) evaluation or validation through com- parison of field (laboratory) and model results and (2) im- provement by use of new techniques related to model mathematics and representation of reservoir fluid and rock description parameters.

The volume and increasing complexity of publications related to the latter item preclude a detailed mathematical description of current simulation technology in this chapter. Rather, emphasis is given to a general descrip- tion of reservoir simulation models, how and why they are used, choice of different types of models for different reservoir problems, and reliability of simulation results in the face of model assumptions and uncertainty in reser- voir fluid and rock description parameters. The chapter concludes with an abbreviated description of simulation model technology consisting of comments on a number of highly technical publications. Various texts l-4 give detailed descriptions of simulation technology through me late 1970’s, including finite-difference approximations, model formulations, iterative solution techniques, and stability analyses.

A Brief History In a broad sense, reservoir simulation has been practiced since the beginning of petroleum engineering in the 1930’s. Before 1960, engineering calculations consisted largely of analytical methods, ‘$ zero-dimensional material balances, 7,8 and one-dimensional (1 D) Buckley- Leverett9,10 calculations.

The term simulation became common in the early 1960’s, as predictive methods evolved into relatively sophisticated computer programs. These programs represented a major advancement because they allowed solution of large sets of finite-difference equations describ- ing two- and three-dimensional (2D and 3D), transient, multiphase flow in heterogeneous porous media. This ad- vancement was made possible by the rapid evolution of

PETROLEUM ENGINEERING HANDBOOK

I - 0 IMENSIONAL

fL3Gm ii y c- x

2-DIMENSIONAL

CROSS-SECTION

(*g

Fig. 48.1-l-, 2-, and 3D grids.

large-scale, high-speed digital computer. and development of numerical mathematical methods for solving large systems of finite-difference equations.

During the 1960’s, reservoir simulation efforts were devoted largely to two-phase gas/water and three-phase blackail reservoir problems. Recovery methods simulated were limited essentially to depletion or pressure maintenance. It was possible to develop a single simula- tion model capable of addressing most reservoir problems encountered. This concept of a single, general model always has appealed to operating companies because it significantly reduces the cost of training and usage and, potentially, the cost of model development and maintenance.

During tbe 1970’s, the picture changed markedly. The sharp rise in oil prices and governmental trends toward deregulation and partial funding of field projects led to a proliferation of enhanced-recovery processes. This led to simulation of processes that extended beyond conven- tional depletion and pressure maintenance to miscible flooding, chemical flooding, CO2 injection, steam or hot- water stimulation/flooding, and in-situ combustion. A relatively comfortable understanding of two-component (gas and oil) hydrocarbon behavior in simple immiscible flow was replaced by a struggle to unravel and characterize the physics of oil displacement under the in- fluence of temperature, chemical agents, and complex multicomponent phase behavior. In addition to simple multiphase flow in porous media, simulators had to reflect chemical absorption and degradation, emulsifying and in- terfacial tension (IFT) reduction effects, reaction kinetics, and other thermal effects and complex equilibrium phase behavior. This proliferation of recovery methods in the

1970’s caused a departure from the single-model concept as individual models were developed to represent each of these new recovery schemes.

Research during the 1970’s resulted in many signifi- cant advances in simulation model formulations and numerical solution methods. These advances allowed simulation of more complex recovery processes and/or reduced computing costs through increased stability of the formulations and efficiency of the numerical solution methods.

General Description of Simulation Models A number of papers”-l4 present general, largely non- mathematical discussions of reservoir simulation. Odeh ’ ’ gives an excellent description of the conceptual simplici- ty of a simulation model. He illustrates the subdivision of a reservoir into a 2- or 3D network of gridblocks and then shows that the simulation model equations are basically the familiar volumetric material balance equation7y8 written for each phase for each gridblock. The phase flow rates between each gridblock and its two, four, or six (in lD, 2D, or 3D cases, respectively) adja- cent blocks are represented by Darcy’s law modified by the relative permeability concept. Fig. 48.1 illustrates l-, 2-, and 3D grids representing a portion of a reservoir. The block and its two or four neighbors are denoted by B and N in the 1D and 2D grids. One can visualize an interior block of the 3D grid with its six neighbors, two on either side of the block in the n, y, and z directions.

The subsea depths to the top surface of each grid in Fig. 48.1 vary with areal position, reflecting reservoir forma- tion dip. Reservoir properties such as permeability and porosity, and fluid properties such as pressure, temperature, and composition, are assumed uniform throughout a given gridblock. However, reservoir and fluid properties vary from one block to another; fluid properties for each gridblock also vary with time during the simulation period.

A simulation model is a set of partial-difference equa- tions requiring numerical solution as opposed to a set of partial differential equations amenable to analytical solu- tion. Tbe reasons for this are (1) reservoir heterogeneity- variable permeability and porosity and irregular geometry, (2) nonlinearity of relative permeability and capillary pressure vs. saturation relationships, and (3) nonlinearity of fluid PVT properties as functions of pressure, composi- tion, and temperature. The models require high-speed digital computers because of the large amount of arithmetic associated with the solutions.

The large amount of arithmetic performed by a simula- tion model stems from the large number of gridblocks representing the reservoir and from the number and com- plexity of equations describing the oil-recovery process. Total arithmetic or computing expense for a given model run is at least linearly proportional to the total number of gridblocks, N,N,N,, where N,, NY, and N, are the numbers of gridblocks specified in the X, y and z direc- tions , respectively.

The individual gridblocks are customarily identified by subscripts i, j, k, where blocks are numbered i = 1,2.. .N, in the x direction, j=1,2...N, in the y direction, and k= 1,2.. .N, in the z direction. Most simulators use no- flow or closed boundary conditions at the exterior bound- aries [x=(O,LX), y=(O,L,) and z=(O,L,)] with provision

RESERVOIR SIMULATION 48-3

for aquifer influx along the areally exterior boundary. ‘I he nonrectangular, areal (x-y) shapes of most reservoirs are represented by zero gridblock porosity and permeability in the appropriate area1 portions of the x-y grid.

Preceding statements described the simulation model as a set of equations expressing conservation of mass for each phase for each gridblock. More precisely, the model equations express conservation of mass of each reservoir fluid component for each block, The number and identi- ty of these components depend on the nature of the original reservoir fluid and the particular oil-recovery process, as discussed in the following. The total number of mass con- servation equations is then N,N,N,N, where N is the number of components necessary to describe the reser- voir fluids.

Each conservation equation states that the mass rate of flow into a gridblock minus the mass rate of flow out must equal the rate of change or accumulation of mass within the block. These N mass-balance equations (one for each component) apply to each gridblock. The block is an open system, in the thermodynamic sense, because of fluid flow between the block and its six neighbors and fluid injec- tion or production if a well is perforated in the block.

The center of gridblock (i,j,k) is located at (xi.yj,zk). This block has six neighboring blocks (i-tl,j,k),(i,j-t 1,k) and (i,j,kk 1). For brevity and clarity, the interblock flow rates are written here in terms of only r-direction flow between blocks (i- 1 ,j,k) and (i,j,k), the indices j and k are suppressed, and the general symbol C, denotes con- centration (mass/volume) of component I in the various phases. The three immiscible phases (water, oil, and gas) are denoted by subscripts w, o, and g, respectively.

The interblock flow rate of component I, according to Darcy’s law modified by relative permeability, is

+%l,(ApO -y,AZ)+&@, -ysAZ) , 1 P8 J

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where 41 =

k= A=

L=

k rP =

p’p = C IP =

APP =

YP =

Ax = z=

component I interblock flow rate, mass/time,

absolute permeability, AyiAZk =cross-sectional area normal to

flow, distance between adjacent block centers,

(AX-1 fAri)/ relative permeability to phase P

(P=w,o,g). viscosity of phase P concentration of component I in phase P,

mass/volume, pressure of phase P specific weight of phase P X,-l -x,, where x is p or Z, and subsea depth, measured positively

downward.

If subscript J=1,2,3 is used to denote phases w,o,g respectively, then Eq. 1 simplifies to

The first term in parentheses is the interblock trunsmissi- b&y, TIJ, for flow of component I in phase J, requir- ing evaluation here at (i- %,j,k)-i.e., between blocks i - 1 and i. The M/L portion of T is normally calculated as the harmonic or series-resistance mean value using block i- 1 and block i properties. The remaining portion of T normally is evaluated at the upstream gridblock- i.e., the block from which the phase is flowing. Thus Eq. 2 becomes simply

representing interblock flow of component I from gridblock i- 1 to gridblock i.

The right-hand or accumulation terms of the mass balances are

;6 [

3

d c (SJCIJ) , . . . . . . . . . . . . . . , . . J=l 1 (4)

where V = grid block volume, hxi AYj AZk 6 = time difference operator, 6X=X,,+, -X,, n = time level, t,+l =t,+At,

At = timestep, $I = porosity, fraction, and

SJ = saturation of phase J, fraction of pore space.

Eqs. 3 and 4 give the final form of the component I mass-balance equation for gridblock (i,j,k) as

3

C A[TIJ(APj-r./Az)l-4p~ J=l

(5)

where q,,I is the mass rate of production of component I from the block resulting from any well perforated in the block and the Laplacian term of type A(TAp) is defined as

and

48-4 PETROLEUM ENGINEERING HANDBOOK

)

For the general case where each component is present (soluble) in all three phases, Eq. 5 is Nequations in 3N+6 unknowns. The unknowns are 3N CIJ values, 3 phase saturations, and 3 phase pressures. Thus an additional 2N-t 6 equations are required for a determinate or solvable model having equal numbers of equations and unknowns. The N Eqs. 5 are referred to as primary equations while the additional 2N+6 equations are denoted constraint equations. The constraint equations are manipulated in the model programming to eliminate 2N+6 variables (unknowns) in terms of the remaining N (primary) unknowns. The result is then the set of N primary Eqs. 5 in N primary unknowns. The constraint equations are relations between unknowns pertaining only to the par- ticular gridblock (iJ,k) to which Eqs. 5 apply. The N primary Eqs. 5, however, involve unknowns (e.g., pi) at the gridblock (i,j,k) and its six neighboring blocks, ow- ing to the nature of the interblock flow terms on the left- hand side.

The 2N+6 constraint equations are illustrated here for the case of an isothermal, compositional model where the N components are HZ0 and N- 1 hydrocarbon com- ponents (e.g., methane, ethane.. .C ,). The first three con- straints are

S,+S,+S,=l.O, . . . . . . . . . ..(6)

po -pw =Poy,(S,), . . . . . . . . . . . . (7)

and

pg -p. =P,,(S,), . . . . . . . . . . . . . . (8)

where P,, =water/oil capillary pressure and P,, =gasJ oil capillary pressure.

These constraints express the requirement that the phase saturations sum to unity and also eliminate the water and gas phase pressures in terms of the unknown oil pressure phase using capillary pressure curves. For this composi- tional case, concentration C,J =p Jx/J where pi is the mo- lar density of phase J (moMvolume) and xIJ is mol fraction of component I in phase J. The next three con- straints require that the mol fractions of all components sum to unity in each of the three phases,

N

c x[J=l.o . . . . . . . . . . . . , , . . . . . I=1

where J=w,o,g or 1,2,3. The remaining 2N constraints express equilibrium of each component among the three phases,

j-r, =fr, . . . . . . . . . . . . . (104

and

j-i,=fr, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (lob

where fIJ is the fugacity of component I in phase J. These fugacities can be expressed in terms of mol fractions and pressure by use of an equation of state (EOS). Altemative- ly, they can be replaced by equilibrium K-value relation-

a

ships (e.g., y=Kx) with K-values given as functions of pressure or of pressure and composition.

The 2N + 6 constraint Eqs .6 through 10 are manipulated to eliminate one-phase saturation, two-phase pressures and 2N-3 mole fractions (x,J) from the N primary Eqs. 5. The final result is a model consisting of the N Eq. 5 in N unknowns consisting of two saturations, one pressure, and N-3 mole fractions. Each coeffkient or term remain- ing in the N primary equations is then either one of the N primary unknowns or a function of one or more of the primary unknowns.

Types of Models Different types of simulation models are used to describe different mechanisms associated with different oil- recovery processes. The most widely used types are black oil, compositional, thermal, and chemical flood. The four basic recovery mechanisms for recovering oil from reser- voirs are (1) fluid expansion, (2) displacement, (3) gravity drainage, and (4) capillary imbibition. Simple fluid ex- pansion with pressure decline results in oil expulsion from and subsequent flow through the porous matrix. Oil is displaced by gas and injected or naturally encroaching water. Gravity drainage, caused by positive (water/oil and oil/gas) density differences, aids oil recovery by causing upward drainage of oil from below an advancing bottom- water drive and downward drainage from above a declin- ing gas/oil contact. Finally, imbibition, generally normal to the flow direction, can be an important recovery mechanism in lateral waterfloods in heterogeneous sands with large vertical variation of permeability.

Accommodation of compositional and the enhanced- recovery processes in this discussion requires the addi- tion of a fifth mechanism, oil mobilization. This loosely defined term includes widely differing phenomena that create or mobilize recoverable oil. Some of these phenomena are not really distinct from the first four.

The black-oil model accounts for the four basic mechanisms in simulation of oil recovery by natural deple- tion or pressure maintenance (e.g., waterflooding). This isothermal model applies to reservoirs containing immisci- ble water, oil, and gas phases with a simple pressure- dependent solubility of the gas component in the oil phase. The two-component representation of the hydrocarbon content I5 presumes constant (pressure-independent) com- positions of the oil component and the gas component, no volatility of the oil component in the gas phase, no solubility of the oil and gas components in the water phase, and no volatility of water (H20) in the oil and gas phases. The oil component is stock-tank oil and its unit of mass is 1 STB (1 bbl at stock-tank pressure and temperature). The gas component is surface system gas and its unit of mass is 1 standard cubic foot (scf). The water component unit of mass is 1 STB. For water and gas, components and phases are identical while the oil phase is a mixture of the oil component and the gas component.

The number of components (N) and therefore the number of Eqs. 5 per gridblock is three for the black-oil model. Table 48.1 gives the definitions of component con- centrations, CIJ, for this model. The water phase, gas phase, and saturated oil phase, reciprocal formation volume factors, b, (STB/RB), bR (scf/RB), and b, (STB/RB), respectively, are given smgle-valued functions of pressure. For undersaturated oil, b, is dependent on

RESERVOIR SIMULATION 48-5

pressure and solution gas (R, , scf/STB). As discussed later original work in formulations’6m’8 led to a number of papers ‘9m23 describing black-oil models during the 1960’s.

The remaining model types discussed here account for some mobilization mechanisms in addition to the four basic recovery mechanisms. The isothermal compositional model represents reservoir fluids by N components, in- cluding water and N- 1 hydrocarbon components. Generally, but not necessarily, solubilities of water in the oil and gas phases and of hydrocarbon components in the water phase are considered negligible. For water, then, the concentration in Eqs. 5 is as given in Table 48.1. The hydrocarbon component I concentration CIJ is P JX IJ, as mentioned earlier, for J=o,g or 2,3. Gas/oil phase equilibrium and phase densities within each gridblock are calculated using equilibrium K-values from pressure- and composition-dependent correlations or, more recently, from EOS’s. 25-28 Unlike the black-oil model, the com- positional model can represent the mobilization of oil by outright (single-contact) or dynamic (multicontact) miscibility, oil swelling and viscosity reduction by solu- tion of an injected nonequilibrium gas (e.g., CO,), and stripping or vaporization of an oil’s lighter ends by in- jection of a dry gas. With one exception,29 recent papers 29-33 describing compositional models are based on equilibrium K-values obtained from EOS’s.

A thermal simulation model is a set of N conservation equations, similar to Eq. 5, which expresses conserva- tion of mass of H2 0 and N-2 hydrocarbon components and conservation of energy. With energy designated as “component” N, the last (I=N) of Eqs. 5 becomes the energy balance upon addition of terms representing heat conduction and overburden heat loss. An additional re- quirement is the use of pJHJ for cNJ in the well and in- terblock flow terms and p J UJ for CNJ in the right side accumulation term. HJ and UJ are enthalpy and inter- nal energy, respectively, energy/mole. If the in-situ com- bustion capability is included then the mass conservation equations include source (sink) terms represented by Ar- rhenius reaction rate expressions for cracking and oxida- tion of hydrocarbon components and the energy balance includes heat of reaction terms. For the same number of fluid components, a thermal model has one more (energy) conservation equation than the compositional model and one additional unknown, temperature T.

For steam-injection processes, thermal model com- ponents are typically H 2 0, heavy (nonvolatile) and light (solution gas or distillable) hydrocarbon components and energy. For in-situ combustion studies, typical com- ponents are HzO, heavy-oil component, a lighter (distillable) oil component, solid coke, 02, CO*, N2, and energy. Frequently CO2 and N2 are lumped as one component to reduce computing expense. The steam tables and/or an EOS are used to calculate liquid Hz0 (water phase) properties and the Hz0 gas/water phase K-value as functions of pressure and temperature. In most applica- tions, Hz0 is assumed insoluble in the oil phase. In most

current models, the distribution of other (non-H20) com- ponents among all phases is represented by user-provided K-values dependent on only pressure and temperature.

Thermal simulators are applied to steam-injection or in- situ combustion processes in heavy-oil reservoirs where oil is mobilized primarily by (1) reduction of oil viscosi-

TABLE 48.1-DEFINITIONS OF CONCENTRATIONS C,, FOR THE

BLACK-OIL MODEL

Phase J=l J=2 J=3

I Component Water Oil Gas - __ - 1 water bw 0 0 2 oil 0 bo 0

3 gas 0 b,Rs b,

ty with increased temperature, (2) distillation of in- termediate hydrocarbon components from the oil phase to the more mobile gas phase, and (3) cracking of the oil phase [usually above 500°F (26O”C)l with subsequent distillation. Thermal models developed from 1965 to 198234-40 generally exhibit a trend toward inclusion of more dimensions, more components and dual capability of steamflood and in-situ combustion.

Chemical flood models include polymer, micellar (sur- factant), and alkaline (caustic). Polymer waterflooding im- proves oil recovery by lowering the oil/water mobility ratio, by reducing the effective permeability to water, and/or by increasing water viscosity. In micellar flooding, surfactants greatly reduce oil/water IFT, thereby solubiliz- ing oil into the micelles and forming an oil bank. 4’ The surfactant slug and mobilized oil normally are propelled toward the production well by a graded bank of polymer- thickened water. The mechanisms responsible for im- proved oil recovery in alkaline flooding are thought to include low IFT, wettability alteration, and emulsifica- tion. 42 Chemical flooding processes involve complicated fluid/fluid and rock/fluid interactions such as adsorption, ion exchange, viscous shear, and three- (or more) phase flow. Several recent papers 43-45 describe implementation of these complex chemical flood mechanisms in numerical simulators.

The four types of models described above are defined or distinguished by the recovery process and the nature of the original reservoir fluid. Considering the nature of the reservoir formation leads to a fifth, fractured-matrix type of simulation model. While in theory any recovery process can be implemented in a fractured-matrix reser- voir, most simulation work reported to date is concerned with black oil fracture&matrix models. Three-dimensional models are described by Thomas et a1.“6 for the three- phase case and by Gilman and Kazemi4’ for two-phase water-oil flow. Their models consider a discontinuous ar- ray of matrix blocks in a continuous 3D fracture network. Flow throughout the reservoir and to the wells occurs in the fracture system and the matrix blocks are treated as sink/source terms in that system. Their model equations include the set of N conservation Eqs. 6 written for each gridblock in the fracture system. Each gridblock may con- tain a number of similarly behaving matrix blocks. However, additional terms are added to Eqs. 6, represent- ing matrix-fracture flow. Also, for each gridblock addi- tional equations are required to express mass conservation of each component in the matrix blocks included in the gridblock. These additional equations can be eliminated or combined with the basic N (fracture system) flow equations4”v4’ so that the final model includes only N equations (per block) possessing interblock flow terms. Blaskovich et a1.48 describe a fractured-matrix model

48-6 PETROLEUM ENGINEERING HANDBOOK

which allows for reservoir-wide flow through the matrix as well as the fracture system. This extension leads to a model including 2N equations (per block) possessing in- terblock flow terms.

Model Input Data and Calculated Results A simulation model requires three types of input data. First, reservoir description data include (1) overall geometry, (2) grid size specification, (3) permeability, porosity, and elevation for each gridblock, and (4) relative permeability and capillary pressure vs. saturation func- tions or tables. Geological and petrophysical work, which involves logs and core analyses, is necessary for Items 1 and 3. Laboratory tests on core samples yield estimates of relative permeability and capillary pressure relation- ships. Second, fluid PVT properties, such as formation volume factors, solution gas or component equilibrium K-values, and viscosities are obtained by laboratory tests. Finally, well locations, perforated intervals, and produc- tivity indices (PI’s) must be specified. Each well must be assigned a production (injection) rate schedule and/or a limiting producing (injecting) pressure for use in calculating well deliverability (injectivity).

Model output or calculated results include spatial distributions of fluid pressure, saturations and composi- tions, and producing GOR and WOR and injection/pro- duction rate (for wells on injectivitylproductivity) for each well at the end of each timestep of the computations. In- ternal manipulation of these results gives average reser- voir pressure and instantaneous rates and cumulative injection/production of oil, gas, and water by well and total field vs. time.

Current models offer various levels of visual output display features that ease the engineer’s assimilation and interpretation of simulator results. Example features are contour maps of pressure, saturations, compositions and temperature, concise tabular summaries of individual well or well-group performance, and field or well timeplots of quantities such as production rates and WOR’s and GOR’s.

Purpose of Reservoir Simulation Reservoir simulation is used to estimate recovery for a given existing producing scheme (forecasting) to evaluate the effects on recovery of altered operating conditions, and to compare economics of different recovery methods. Black-oil models have been widely applied to forecast oil recovery and to estimate the effects on oil recovery of (1) well pattern and spacing, (2) well completion inter- vals, (3) gas and/or water coning as a function of rate, (4) producing rate, (5) augmenting a natural water drive by water injection and desirability of flank or peripheral as opposed to pattern waterflooding, (6) inlill drilling, and (7) gas vs. water vs. water-alternating-gas (WAG) in- jection.

A few of many reported studies are briefly mentioned here. Henderson et al.49 applied a single- (gas) phase model to optimize the locations and numbers of wells necessary to meet peak deliverability requirements in a gas storage field. Mann and Johnson” showed good agreement between model-predicted and actual field per- formance. Thomas and Driscol15’ applied a black-oil model in estimating locations of bypassed oil for the

P ur-

pose of designing an infdl drilling plan. Two studies 2,53

report extensive results related to rate-sensitivity of various Alberta reservoirs subjected to water-oil displacements. Thakur ef al. 54 applied a black-oil model to characterize (through history matching) offshore Nigerian reservoirs and estimate incremental recovery of waterflooding over natural depletion, with infill drilling and removal of allowable rates.

Compositional models also are used for most of Pur- poses 1 through 7 listed previously, but only in cases where the black-oil assumption of constant composition oil and gas components is invalid. Example compositional model applications include (1) depletion of a volatile oil or gas condensate reservoir where hydrocarbon phase compositions and properties vary significantly with pressure below bubble- or dewpoint, (2) injection of non- equilibrium gas (dry or enriched) into an oil reservoir to mobilize oil by vaporization into the more mobile gas phase or by attainment of outright (single-contact) or dynamic (multicontact) miscibility, and (3) injection of CO2 into an oil reservoir to mobilize oil by stripping of light ends, oil viscosity reduction, and oil swelling.

Compositional simulation has been performed to estimate (1) loss of recovery caused by liquid dropout dur- ing depletion of retrograde gas condensate reservoirs and the reduction of this loss by full or partial cycling (rein- jection of gas from surface facilities) and (2) effects of pressure level, injected gas composition, and CO2 or N2 injection on oil recovery by vaporization or miscibility. Graue and Zana55 describe application of a compositional model in estimating Rangely (CO) field oil recovery by CO2 injection as a function of injected composition and pressure level. Results of compositional simulation of a CO2 project include CO;! breakthrough time and rate and composition of produced fluids. These are required to design production facilities and CO2 recycling strate- gies. 56 Modeling is also useful to optimize pattern size and CO2 /water-injection rates to overcome the effects of reservoir heterogeneity. 57

Thermal models are applied in reservoir studies of in- situ combustion and are used to simulate performance of cyclic steam simulation and steamflooding. In steam in- jection, questions addressed by simulation relate to effects of injected steam quality and injection rate, operating pressure level, and inclusion of gas with the injected steam. One question in cyclic stimulation concerns the optimal time periods per cycle for steam injection, soak, and production. The flooding case introduces the issues of well pattern and spacing. A number of steam-injection field studies using models have been published. Herrera and HanzlikS8 compare field data and model results for a cyclic stimulation operation. Williams59 discusses field performance and model results for stimulation and flooding, and Meldaum discusses field and model results related to addition of gas to the in’ected steam. Gomaa et al. 6’ and Moughamian et al. 61 applied steamflood simulation in identifying and optimizing operating parameters in pilot and field drive operations.

Numerical simulation has been used to estimate chemical flood performance in a reservoir environment where the processes are very complex and many reser- voir parameters affect the results. Chemical flood simula- tion has been used to construct a screening algorithm for the selection of reservoirs suitable for micellar/polymer flooding63 and to examine competing EOR strategies-

RESERVOIR SIMULATION 48-7

e.g., CO* vs. surfactant flooding.@’ For caustic65 and polymeP applications, as well as for the micellar proc- ess, chemical flood modeling is useful to discern control- ling process mechanisms and to identify laboratory data required for process description.

In recent years, simulation has been used increasingly to estimate and compare recoveries from a given reser- voir under alternative enhanced-recovery processes, such as CO2 injection, thermal methods (steam injection and in-situ combustion), and several types of chemical flooding.

Considerations in Practical Application of Simulation Models This section describes the procedure followed and cer- tain questions faced by the engineer conducting a reser- voir simulation study. The engineer must select the appropriate type of simulation model, select the grid net- work, and specify rock and fluid description data. Then the engineer must attempt to reduce or at least estimate inaccuracies in simulation results which stem from uncer- tain rock/fluid description data and from spatial trunca- tion error.

Selection of Model Type As mentioned earlier, selection of model type may de- pend on both the nature of the original reservoir fluid and upon the recovery process(es) to be studied. As a rough guide, an original reservoir oil solution gas or GOR value, R,, below 2,000 scf/STB indicates a black oil, whereas a higher value indicates a volatile oil or retrograde gas condensate requiring compositional treatment.

For a black-oil reservoir, a black-oil model may be used to study natural depletion, water injection and/or equilibrium gas injection operations. However, a com- positional model is generally necessary to estimate recovery by injection of dry or enriched gas, solvent, or CO*. An exception here is the applicability of a modified black-oil model67 in simulation of CO2 or solvent injec- tion where outright (single-contact) miscibility occurs.

Compositional simulation is generally employed for volatile oil reservoirs. However, a less expensive black- oil model study is adequate for simulating above- bubblepoint waterflooding performance. Compositional models have generally been employed to study retrograde gas condensate reservoirs. The compositional model is necessary in the case of below-dewpoint cycling. However, in some cases natural depletion or above- dewpoint cycling can be simulated at less expense using a black-oil model modified to account for volatility of the oil (condensate) component in the gas phase. 6sm70

The alkaline and surfactant-flood processes generally require use of the complex chemical flood simulation model. However, some (augmented) black-oil models of- fer the capability to simulate polymer and alkaline flooding.

Selection of Model Grid Selection of the x-y-z gridblock network involves many factors, including available budget and the engineer’s judgment and experience. For any type of model, the arithmetic or computing expense per timestep is at least linearly proportional to the total number of gridblocks

employed. The computing expense of a single model run is proportional to the product of the number of gridblocks and the number of timesteps required by the model to cover the total time period of interest. In many cases, the timestep size is controlled by the maximum rate of change (overall gridblocks) in one or more calculated quantities such as pressure and saturations. This maximum rate of change generally occurs at or near a well or in the vicini- ty of a flood front. A doubling of the number of gridblocks can result approximately in a doubling of this maximum rate of change since each gridblock is (on the average) one-half as large. The average timestep size, then, might decrease by a factor of two if the number of blocks were doubled. The final result is a computing expense per model run which can approach a proportionality to the square of the total number of gridblocks. This indicates the importance of selecting the smallest number of gridblocks consistent with reservoir/well description, recovery process characteristics, and the questions asked regarding reservoir performance.

The number of gridblocks and resultant study computing expense are the lowest in cases where the engineer can justify use of a representative element of the total field as the basis for the model study. This may be possible in reservoirs developed with repeated well patterns, for any recovery process-waterflooding, CO* injection, steamflooding, etc. In such cases, the representative ele- ment ideally should be a symmetrical element of the reser- voir. In strict terms, this requires (1) a repeated, regular pattern of identically completed and operated wells, (2) a horizontal, areally homogeneous reservoir formation of uniform thickness, and (3) areally uniform initial fluid saturation distributions. If these conditions were met, then questions regarding total field optimization, forecasting and comparative evaluation of recovery processes could be addressed inexpensively by simulation of the single pat- tern (element).

While actual reservoirs never satisfy these conditions exactly, representative-element simulation studies are fre- quently performed for repeated pattern processes. In some cases, a substantial portion of the reservoir may exhibit only moderate areal heterogeneity and thickness variation. Resultant variation in performance from one pattern to another may be sufficiently small for engineering purposes to justify scale-up of single-pattern results to total field performance.

Representative-element simulation is often performed where the study purpose is comparative evaluation of alternative recovery processes as opposed to forecasting of total field performance for a specific process and operating scheme. The justification of single-element simulation implied in such cases is that the resultant rank- ing of alternative processes is unaffected by the variations in pattern (element) properties over the field. This justification can be and frequently is checked by repeating the various process simulations for two or more patterns of different properties representative of different portions of the reservoir.

Finally, the relatively inexpensive single-element simulation applies to design or optimization studies of a specific recovery process operated in a repeated pattern mode. For a repeated-pattern steamflood, single-pattern model runs have been performed to “optimize” pattern type (e.g., five-, seven- or nine-spot) and size, injected

48-8 PETROLEUM ENGINEERING HANDBOOK

steam quality and rate, well completions, etc. Occasional publications describe single-pattern simulation studies us- ing a one-quarter five-spot or one-quarter nine-spot as the symmetrical element of the respective pattern. Actually, a one-eighth five-spot or nine-spot (and xZ seven-spot) are the smallest symmetrical elements and should be used to minimize computing expense. 71

Currently, a major portion of the industry-wide effort and computing expense in simulation studies is associated with total-field forecasting of black-oil reservoir perfor- mance under a sequence of recovery processes. Typical- ly, the engineer must select a 3D grid for a large reservoir with significant heterogeneity, large areal variation in dip and thickness, irregular well locations and increasing numbers of wells with successive development stages. The engineer may face a several- to many-year period of historical performance under natural depletion, frequently with some natural water encroachment. Study objectives may include history matching, followed by matching and forecasting for a waterflood period, in turn followed by forecasting for some tertiary scheme such as CO2 in- jection.

The total number of gridblocks is the product of the number of areal blocks, N,N,, and the number of grid layers, N,. Different considerations enter into selection of these two numbers of spacings. Factors indicating a need for fine area1 grid spacing are high well density and sharp or rapid changes (areally) in permeability, porosi- ty, thickness, and dip. Since these factors frequently vary over the field, thex- and y-direction grid spacings are often nonuniform. Grid spacings generally increase toward the downdip reservoir boundaries and increase greatly with distance into the aquifer if the latter is present and included in the grid.

In general, of course, the number of area1 gridblocks required increases with the size of the reservoir and the number of wells. However, grid spacings ranging from very fine to very coarse may be appropriate for different reservoirs of comparable size. The smallest numbers of areal blocks (coarsest areal spacings) are associated with reservoir studies limited to natural depletion and crestal or flank gas and/or water injection. In such a case, a coarse grid may result in a number of area1 blocks that include two or more similar type (e.g., production) wells, with little loss in engineering significance of the simulator results. Large numbers of area1 blocks may be required in cases of pattern waterfloods or enhanced recovery proc- esses. A rough guide in this case is the need for at least two, preferably three or more, gridblocks separating each injection-production well pair. However, recent studies describe estimation of pseudorelative-permeability curves, which allow adjacent-block placement of an injectoripro- ducer well pair. 72,73

The major factors affecting the number of grid layers (vertical gridblocks) required are the formation stratifica- tion, vertical communication, and total thickness. Many reservoirs possess a number of formation layers, which correlate from well to well over much of or all the field. Variations of layer thickness, permeability, and porosity may be significant areally and even greater from one layer to another. The vertical communication (vertical permeability) between adjacent layer-pairs may vary from zero to very high, both areally and from one layer-pair to another. In general, at least one grid layer should be

used for each correlatable formation layer. However, common sense and budget constraints argue against detini- tion of a large number of very thin grid layers. Three- dimensional reservoir studies typically employ 4 to 12 grid layers, and one or more of these grid layers may be a lumped representation of several thin formation layers.

The need for subdivision of one formation layer into two or more grid layers depends on the layer thickness and fluid-segregation characteristics of the recovery proc- ess and operating rates. Most recovery processes result in moderate to severe gravity segregation of oil and in- jected fluids; injected water or gas tend to underrun or override oil, respectively; many steamflood projects ex- hibit severe override of oil by the steam. A formation layer that has significant thickness and zero to poor vertical communication with layers above and below may exhibit a pronounced phase segregation and require two or more grid layers. In the idealized example of a fieldwide, pro- nounced gravity override in a vertically homogeneous reservoir, a variable grid spacing increasing from top to bottom might be specified. That is, four layers of thicknesses 5, 10,20, and 25 ft might give more accurate results than four layers of equal 15ft thickness.

A customary approach to determining NZ involves use of the simulation model itself in 2D cross-sectional (X-Z slice) mode. For the particular recovery process of in- terest, X-Z model runs are performed by using different numbers of grid layers. Pseudorelative-permeability curves reflecting phase segregation are calculated from model runs performed with fine vertical grid spac- ing. 74-76 These pseudocurves are then used in equivalent x-z model runs using fewer grid layers to obtain coarse (vertical) definition results similar to the fine-spacing “correct” results. The fewer grid layers of the coarse definition are then employed in the 3D reservoir study grid. This concept of generating pseudocurves for coarse vertical grids that reproduce vertical fine-grid results (us- ing rock or laboratory relative permeabilities) has been extended to the areal spacing problem, 72.75 as mentioned earlier.

Obviously, a minimum computing expense follows from use of a single grid layer representing the entire forma- tion thickness. This results in a 2D X-Y area1 grid as op- posed to a 3D grid and occasionally is justified in the two extremes of a very high vertical permeability and a layered formation with zero vertical permeability. Pseudorelative permeability and capillary pressure curves are discussed for the former case in papers describing the vertical equilibrium (VE) concept21,77 and for the latter case by Hearn. 78

Specification of Reservoir Rock and Fluid Description Data

Geological and petrophysical work based on logs and core analyses yields maps of structure, net dh, and w1 products for each of the several reservoir layers. The kh and +h data often are augmented or modified by results of drillstem, pressure buildup, and pulse tests. For each layer, the engineer can overlay his area1 x-y grid spacing network on these maps and read off the values of subsea depth, I#& and kh at the center of each gridblock. These values along with gross thickness of each block are then transposed to a data file in a format compatible with that

RESERVOIR SIMULATION 48-9

required by the simulation model. Current research ef- fort is directed toward developing computer programs that accept digitized core analysis, log and geological data, the selected grid network, and, through mapping and in- terpolation techniques, automatically prepare the simula- tion input data file.

Laboratory core analysis work includes measurement of relative-permeability, k,., and capillary-pressure, P,, curves for a number of field cores. Variations in rock lithology may result in different sets of k, and P, curves for different layers and/or different areal portions of the reservoir. Most simulation models allow multiple sets of such data in tabular form with assignment of each set to a user-specified layer/portion of the reservoir. If the rock water/oil (gas/oil) capillary pressure values are small, the water/oil (gas/oil) transition zone in the reservoir may be a very small fraction of total formation thickness. In such cases, pseudocapillary-pressure curve(s) should be used. l7

For black-oil studies, laboratory tests are performed to determine gas compressibility factor and saturated oil and gas viscosities vs. pressure. Differential and/or constant- composition expansion tests on oil samples yield the saturated oil pressure-dependent formation volume fac- tor, B, (RB/STB), and solution gas, R, (scf/STB). The resulting oil and associated gas properties vs. pressure are entered in the data file in tabular form compatible with simulator input requirements. For gas condensate deple- tion studies, constant-volume and constantcomposition expansion tests yield the required pressure-dependent liq- uid content, CL (STB/scf), and condensate density values.

A wide variety of laboratory tests are performed for compositional model studies that involve injection of a nonequilibrium fluid (dry or enriched gas, CO?, N2, etc.). Swelling tests yield relative volumes, saturation pressures, and equilibrium phase compositions for each of a sequence of mixtures of, say, 1 mole of original reser- voir oil and injected fluid. 79 Various single- and multicontact tests may be augmented by ID corefloods and/or slim-tube displacements. Orr et al. 8o*81 discuss a variety of C02-oil laboratory tests. Much of the laboratory PVT test data must be processed to yield cor- relations or a calibrated EOSs2-85 for simulator input re- quirements.

History Matching In most simulation studies, reservoirs have some period of historical performance data that include WOR, GOR, individual phase rates and cumulatives, and pressure measurements by well. Ideally, periodic (e.g., monthly), accurate measurements of all these data would be recorded and available for all wells. In the typical case, many of these data are unrecorded or unavailable and some of the reported values may be of questionable accuracy.

The reservoir description based on log and core analysis data reflects a very small (volumetric) sampling of the reservoir. The historical reservoir-performance data reflect the reservoir description, and its impact on pressure/fluid movement behavior, on a much larger scale. The previously mentioned geological and petrophysical work yields an initial reservoir description. History matching yields a refinement of that description, which improves agreement between model results and observed reservoir behavior. The history-match phase of

the simulation study entails a sequence of model runs in which input reservoir description parameters are altered to improve this agreement. This is a trial-and-error pro- cedure frequently requiring considerable engineering judgment and experience. The description parameters ob- tained from the geological/petrophysical work often are used to establish legitimate ranges of parameter variation in the history-match model runs.

This history-match phase can consume half or more of the total simulation study (match plus prediction) com- puting effort and expense, depending on the length of the history period, complexity of the reservoir, and amount of available performance data.

Refs. 86 through 90 describe methods and applications of inverse simulation or automatic history matching. This concept requires user-specification of a finite set of reser- voir description parameters to be determined (e.g., zonal permeability, porosity values), a finite set of observed reservoir performance data to be matched, and a regres- sion procedure coded interactively with the simulator. A single computer submittal is then performed, which in turn executes many model history runs. The regression pro- cedure automatically varies description-parameter values from run to run to determine that set of description- parameter values that maximize agreement between model results and the set of observed data. This concept is especially appealing to the engineers who have experi- enced the frequently high frustration levels associated with trial-and-error matching of complex reservoir behavior. However, to date the trial-and-error procedure still predominates with isolated successes reported with automatic history matching. Two factors complicating the latter approach are (1) the expense of the required single computer submittal can be very large, (2) the a priori choice of description parameters (or zonation) can be dif- ficult, subjective, and lead to a questionable reservoir description.

Validity of Simulation Results Uncertainties or errors in simulation model results may arise from (1) questionable assumptions or mechanisms not represented in the differential form of the model, (2) spatial and time truncation error introduced by replace- ment of the model differential equations by finite- difference approximations, and (3) inadequately known reservoir rock and/or fluid description data. In addition, the exact solution of the difference equations is not at- tained because of round-off error introduced by the finite word length of the computer. Round-off error is general- ly negligible compared with errors from the other three sources. With some exceptions, the above sources of er- ror are listed in order of increasing importance. However, successful history matching can reverse the importance of the second and third sources.

Comparisons of model and laboratory experiment results can indicate model validity in the absence of the Uncertainty 3 above. Several such comparisons show good model-experiment agreement for gas/oil sys- tems,91,92 water/oil coning,93 and fractured-matrix im- bibition. 94

Model Assumptions

An assumption common to many black-oil models is com- plete re-solution of free gas in accordance with the

48-I 0 PETROLEUM ENGINEERING HANDBOOK

saturated R,(p) curve during repressurization. This may be a poor assumption in a case where gridblock thickness is large and gas/oil gravity (vertical) segregation is pro- nounced. Prior to repressurization in a given block, the free gas may exist as a high gas saturation in only the up- per portion of the block. This contradicts its representa- tion in the model as a lower saturation distributed throughout the entire block volume. In the segregated state, the gas will redissolve only in the lower or residual oil saturation in the upper, gas-occupied portion of the block volume. However, the model will allow re-solution in the entire block’s oil volume. Pressure hysteresis in the R,(p) curve has been used to cope with this problem; an alternative remedy where the computing budget per- mits is the use of more grid layers.

An assumption common in early black-oil models was that the reservoir oil obeyed a single pair of B,(p) and R,(p) curves. Some black-oil reservoirs exhibit a signifi- cant variation of oil API gravity and PVT behavior with depth or with depth and areal location. In some such cases, this variation can be represented in a black-oil model by simply allowing initial solution gas R, to vary with depth in the undersaturated oil column, retaining a single set of B,(p) and R,(p) curves. In other cases, multiple sets of these curves and two oil components are necessary and the single oil-type assumption in a black-oil model can lead to appreciable error.

Mechanisms or phenomena that are significant in some reservoirs and may not be represented in the model in- clude compaction, hysteresis in wetting and nonwetting relative permeabilities, and interlayer wellbore crossflow. The latter is a particularly difficult modeling problem and the subject of continuing research. A production well com- pleted in a number of layers may exhibit production from some layers and, simultaneously, injection (backflow or recirculation) into others. Factors that promote this possibility are low-pressure drawdown (high PI and/or low rate) and poor vertical communication between the reservoir layers in the vicinity of the well. A rigorous treatment of this problem requires modeling of wellbore multiphase hydraulics and phase segregation combined with calculation of correct phase mixtures for the layers undergoing injection.

Spatial Truncation Error Spatial and time truncation error theoretically can be reduced to any desired low level by sufficiently reducing gridblock dimensions and timestep size. However, the resultant increased number of blocks and timesteps fre- quently lead to prohibitive computer expense and memory storage requirements.

Time truncation error is generally insignificant. In most applications timestep size is restricted by considerations other than time truncation error, such as model stability, frequencies of printout, and frequencies of changes in well data (rates, completions, new wells, etc.). In any given case, the level of time truncation error can be estimated by repeating a run or portion of a run with a smaller (or larger) timestep. Insensitivity of results to timestep size indicates low-time truncation error.

Spatial truncation error appears in the forms of numerical dispersion, grid-orientation effects, and error in calculated well WOR and GOR values. Spatial trunca- tion error can be expressed in mathematical terms through

complex manipulation of the model differential equations and Taylor series expansions. In simpler terms, this er- ror can be viewed as a consequence of replacing the physical continuum (reservoir formation) by a 3D network of mixing cells (gridblocks). This consequence is the con- tradictory requirements that any variable value (pressure, saturation, temperature, concentration) simultaneously represents the value at the grid point (e.g. block center) and the entire block’s volumetric average value. This re- quirement is not met (1) during a frontal displacement as a sharp front enters the gridblock, (2) when gravity forces result in phase segregation within the block’s thickness, and/or (3) when area1 cusping or coning causes sharp localized saturation gradients within the block volume.

Numerical dispersion generally appears as falsely smeared spatial gradients of water saturation in waterflooding, temperature in steamflooding, solvent in miscible flooding, and chemical agent in chemical flooding. This excessive smearing occurs primarily in the areal (X or y) directions and, if uncontrolled, results in too early calculated breakthrough times of water (heat, solvent, etc.) at production wells. This numerical disper- sion generally increases with increasing areal gridblock size (AX and AJJ). Lantz95 quantitatively related the dif- ference equation truncation error term to an artificial, second-order diffusion term in the differential equation.

The engineer can anticipate possibly significant numerical dispersion effects in simulating two types of miscible displacement. The first type is slug or bank, as opposed to continuous, injection of solvent or CO?. Numerical dispersion erodes the calculated solvent con- centration within the bank. If miscibility requires maintenance of solvent bank integrity or a certain solvent peak concentration, then this numerical dispersion can result in a calculated (false) loss of miscibility. The sec- ond type is multicontact miscibility For continuous sol- vent injection in 1D simulations, several studies 3’X97 report the need for 100 to 300 gridblocks to reduce the effect of numerical dispersion on miscible front velocity.

Kyte and Berry 75 describe control of numerical disper- sion in simulation of waterflooding through large area1 gridblocks. They use pseudorelative-permeability curves obtained from detailed (fine-grid) cross-sectional simula- tions. Harpole and Hearn98 used their method in a 3D black-oil study. To date, steamflood simulation general- ly has been confined to pattern studies for which a suffi- cient number of gridblocks between unlike wells is used to minimize numerical dispersion effects. Killough et al. 73 describe their reduction of numerical dispersion in a stratified, heterogeneous, repeated pattern black-oil reservoir study. They performed fine-grid, 3D single- pattern simulations and then used regression to determine pseudorelative permeabilities for a 2 x 2 four-block area1 grid representation of the pattern. Agreement between the 3D fine-grid results and four-block pattern results was good enough to allow fieldwide simulation by use of the latter coarse, areal definition. Several recent papers99-‘01 describe local grid refinement, the method of character- istics, and other methods to reduce numerical dispersion effects.

Pronounced grid-orientation effects have been noted in simulation of adverse mobility ratio floods with models incorporating the commonly used five-point difference scheme and single-point upstream weighting. The value

RESERVOIR SIMULATION 48-l 1

5-POINT

--------- g-POINT

Fig. 48.2-Five-point and nine-point difference schemes

of the coefficient krJCIJIpJ in Eq. 2 obviously affects the interblock Darcy flow rate from gridblock i-Z to block i. Intuition might dictate evaluation of this coefficient at some average of variable values (pressure, saturations, etc.) in the two blocks. However, considerations of stabili- ty and numerical dispersion frequently have led to its evaluation at conditions existing in the block from which the flow occurs-i.e., the upstream block. This is referred to as single-point upstream weighting. The five-point dif- ference scheme is reflected in the form of terms of type A(7Ap) in Eq. 5 for the case of 2D flow. These terms represent the interblock Darcy flow rates in the mass balance equation for each gridblock. The solid arrows of Fig. 48.2 illustrate these flow rates between the gridblock and each of its four neighbors.

A strong rid-orientation effect was first reported by Todd et al. ” for highly adverse mobility waterfloods and later observed for pattern steamfloods. lo3 An area1 grid with the usual perpendicular x and y axes may be placed over a five-spot pattern with the x axis either parallel to or at a 45” angle to the line connecting the in- jector to a producer (Fig. 48.3). These parallel and diagonal grids lo2 can result in markedly different calculated shapes of the water or steam front and the breakthrough times. This difference was reduced by the nine-point finite difference formulation described by Yanosik and McCracken, to4 illustrated by the four ex- tra dashed-line diagonal flow terms in Fig. 48.2. The nine- point scheme has been programmed into many simulators treating adverse mobility ratio waterfloods, steamfloods, and CO2 solvent displacements.

As an example of this grid orientation effect, Fig. 48.3 shows a 3-acre nine-spot steamflood pattern with the diagonal grid and 45”-shifted parallel grid. This pattern

. A / \ / \ i’ 2 ‘3

PARALLEL-\ /

GRID \ r-7 -DIAGONAL

/ GRID /

.

3 INJECTION WELL . PRODUCTION WELL

Fig. 48.3-Nine-spot grids

has three types of wells-labeled 1 (injector), 2 (near pro- ducer), and 3 (far producer). Reservoir formation and fluid properties and well rates used in the simulation model are reported elsewhere. l4 The calculated results in Table 48.2 show the pronounced effect of grid orientation on steam breakthrough times calculated by use of the five- point difference scheme. Obviously, steam should arrive at the near producer, Well 2, before it reaches the far pro- ducer, Well 3. The parallel grid with the five-point scheme actually gives breakthrough at Well 3 at 117 days, before breakthrough at Well 2 (204 days). Table 48.2 shows that the nine-point difference scheme virtually eliminates the effect of grid orientation for this problem. Figure 48.4 uses parallel and diagonal grids to show calculated steam- front shapes at 80 days for the two different schemes. The difference between the nine-point fronts for two grids is small and about equal to the error of manual interpolation.

A two-point, upstream weighting method lo2 was pro- posed to reduce both numerical dispersion and grid- orientation effects. Abou-Kassem and Aziz lot discuss this and other methods 1oe-108 for reducing the orientation ef- fect. They conclude that the nine-point scheme is the most effective in reducing steamflood grid-orientation effects. Two studies105-‘09 show very significant reduction of grid-orientation effects in pattern steamflood simulation results when areally homogeneous, square grids (AX= Ay=constant) are used with the Yanosik and McCracken nine-point scheme. However, the effects persist for a non- square, uniform grid (Ax=2Ay)‘09 and the latter scheme yields physically unreasonable results for the cases of heterogeneity and nonuniform grids where AX (or Ay) varies with x (y). The latter shortcoming is addressed by several recent papers110‘112 that propose new or altered nine-point schemes. Frauenthal et al. ‘I3 describe a modified five-point difference scheme and Pruess et al. It4 present a seven-point, hexagonal gridblock scheme for reducing grid-orientation effects.

The engineer can anticipate possibly significant grid- orientation effects in simulating single- or repeated- pattern, adverse mobility ratio displacements. Preliminary areal, single-pattern model runs allow estimation of the level of such effects and the need for use of a nine-point scheme or other remedy.

The discussion and references cited obviously indicate the current concern regarding effects of numerical- dispersion and grid-orientation effects on the validity of

PETROLEUM ENGINEERING HANDBOOK

48-12

GRID DIFFERENCE-SCHEME

PARALLEL 5-POINT --- DIAGONAL 5-POINT ---- EITHER g-POINT

TIME-80 DAYS

0 INJECTOR

l PRODUCER

Fig. 48.4-Calculated shape of steamflood front in a nine-spot pattern.

simulation results. However, these numerical effects are not serious in many simulation studies. Numerical- dispersion effects are generally demonstrated as smear- ing of theoretically sharp fronts in l- or 2D horizontal displacements in homogeneous formations. Actual reser- voir behavior frequently reflects strong gravity effects such as a gas or solvent override or a water underrun. These gravity effects combined with reservoir structure (areal variation in dip angle) can have an influence on fluid movement patterns, which dominates the numerical ef- fects just discussed. In addition, reservoir heterogeneity can play the same relatively dominant role as gravity forces. In highly stratified or layered reservoirs, the dif- ferent rates of travel of injected fluid through different layers can dominate the numerical dispersion effect at the leading edges of the individual layer displacement fronts. Finally, a given level of numerical-dispersion or grid- orientation effect is acceptable if its impact on calculated reservoir performance is inconsequential in an engineer- ing sense-i.e., in light of the questions being asked.

The model itself often can be used to estimate the level of these numerical errors and the degree of their accepta- bility. Before selecting the full study grid, preliminary model runs using grids of varying coarseness can be per- formed for a representative cross-section or 3D portion

TABLE 48.2-CALCULATED STEAM BREAKTHROUGH TIMES (DAYS) FOR A NINE-SPOT PATTERN

Well 2 Well 3

Diagonal Parallel Diagonal Parallel

Five-point 47.0 204 1,400 117 Nine-point 87.7 75.5 900 1,000

of the reservoir. The results can be helpful in selecting the coarsest grid spacing compatible with acceptably low numerical dispersion.

In fieldwide simulation, spatial truncation error may af- fect calculated values of well productivity, wellbore pro- ducing pressure, and WOR and GOR. Without special measures, the model calculates the well behavior with only the gridblock’s average values of pressure, saturations, etc. However, the actual well behavior may reflect near- well coning, liquid dropout, or gas evolution effects. The dimensions of this near-well region may be two orders of magnitude smaller than the areal block dimensions (b, Ay). Thus the block’s average conditions may provide a poor basis for calculating well behavior. This problem can be significant for a well completed throughout for- mation thickness and even more significant for a partial- ly penetrating well.

The simplest remedy to this problem is applicable in some cases where good vertical communication results in a high degree of vertical phase segregation. In this case, well pseudorelative-permeability curves have been used. 1’5,1’6 These curves reflect the location of the com- pletion interval and relate well behavior to average block conditions. A more complicated approach requires use of multivariable correlations relating well WOR and GOR to average block conditions. ’ “g118 These correlations are developed from a number of single-well r-z (radial-depth) model runs with fine grid spacing near the wellbore. The most rigorous treatment of this problem incorporates in- dividual 1D radial or 2D ~-2 simulations for each well simultaneously within the fieldwide 3D simulation. ’ 19,120 Again, in any given case, the model itself can be used to estimate the severity of this problem through com- parison of single-well, r-z and representative 3D (portion of reservoir) model results.

Uncertain Reservoir Description Data Errors in reservoir description data clearly contribute to

errors in simulation model results. Since the description data are never exactly known, one might infer that model results are necessarily erroneous and unreliable. A number of considerations contribute, in contradiction of this in- ference, to model results being widely used to select and to design oil-recovery processes and to forecast oil recovery.

Accurate determination of all reservoir description data is not necessary for reliability of model results. The re- quired accuracy of any description parameter is propor- tional to its influence on computed results (reservoir performance). The simulation model should be used to perform preliminary sensitivity runs to determine which description data are important. Expense and effort should then be concentrated on obtaining or refining only those “sensitive” description data. The particular parameters found to be important will vary from study to study, depending on the nature of the reservoir, the recovery process(es) of interest, and study objectives or questions. For example, if computed oil recovery is insensitive to wide variations in the gas relative-permeability curve, then the accuracy of this curve might deserve little attention. In a case where the gravity drainage mechanism is domi- nant, the oil relative-permeability curve at low and midrange oil saturations has a large effect on oil recovery and deserves effort of definition. Gas viscosity, relative

RESERVOIR SIMULATION 48-13

.

permeability, and capillary pressure may play virtually no role and their accuracies are irrelevant. All phase relative permeabilities may be unimportant in a natural or flank waterflood of a relatively clean, thick high-relief sand where gravity forces are dominant with pronounced phase segregation. Only relative-permeability curve end- points may be important in such cases. However, thinner sand or stratification, lower permeability and/or higher rates can increase the importance of water and oil relative- permeability curve shapes. While capillary pressure is unimportant in many reservoir studies, it can provide the dominant, cross-imbibition, mechanism in waterflooding thin, heterogeneous water-wet sands.

In some studies, the engineer is less concerned with the absolute accuracies of both model results and description data than with the sensitivity of calculated results to varia- tions in those data. An example is a study performed to compare oil recoveries under alternative recovery proc- esses. Model runs performed for each process, with reser- voir description data varied over estimated ranges of uncertainty, may yield substantially invariant process rankings and incremental oil recovery differences. If so, any significant history-matching effort may be un- necessary and the only concern regarding accuracy of description data should be the estimated ranges of uncer- tainty. Another example is a design study of a given recovery process performed to optimize pattern type and size, well completions, and rates. Model runs, as just described, may show that minimal history-match and/or laboratory efforts for reservoir description are necessary to meet the study objective.

As previously mentioned, reservoir description data are altered through history matching to improve agreement between model results and reservoir performance data. Frequently, the study objectives involve estimation of reservoir performance under displacement conditions not present or recovery processes not active during the his- tory period. In such cases, some description parameters that significantly influence future performance may not be reflected in the historical performance. An example is a heavy-oil reservoir that was produced for nearly 40 years under natural depletion with no water drive. Solu- tion gas was very low and interstitial water saturation was immobile. A 50% water cut developed in time as a large pressure decline caused water mobility through water ex- pansion and porosity reduction. Performance data includ- ed WOR, GOR, and pressure data for a number of wells. The only description parameters influencing these data were formation permeability, compressibility, critical gas saturation, water relative permeability at saturations slight- ly above SwC, and gas relative permeability at saturations slightly above S,,. The history-match effort gave a good match of performance with a unique set of these parameter values. However, it provided no information regarding the full-range relative-permeability curves necessary to estimate oil recovery under waterflood and steam stimu- lation or flooding. Laboratory relative-permeability meas- urements and waterflood and thermal pilots were conducted in this case.

Laboratory work and well pressure testing can be per- formed to estimate values of some reservoir description parameters that are not reflected in performance data. These parameters, together with others determined by history matching, can be used in model runs to estimate

oil recovery under the various alternative recovery schemes within the study scope. Field pilot tests then may be planned for one or more of the recovery processes, subject to the model results and engineering judgment.

Argument has persisted for years regarding uniqueness of the reservoir description obtained by history matching. A thorough treatment of this question requires length and mathematical complexity beyond the scope of this chapter. Any such treatment requires careful definition of terms. For example, define a reservoir description as a bounded set of m numbers {Xi} representing selected zonal permeabilities and porosities and parameters characteriz- ing relative permeability curves. Let the sets of N numbers {dj*}, {dj} represent observed and model calculated per- formance data where dj=dj(x, ,x2.,.x,). If N>m, each xi affects one or more dj, and the d. are independent functions of {Xi} (in a mathematics i sense undefined here), then with rare exceptions a unique set of parameter values {Xi} will minimize the difference between the observed and calculated data. An altered zonation gives a physically different parameter set {ii}. Again, a unique set of values of {ai} generally will minimize the dif- ference between observed and calculated data. However, for this two-parameter set “experiment,” comparable matches of observed data would allow a claim of non- uniqueness.

As a practical matter, study budget and time constraints prevent exhaustive trials of different parameter sets and even limit the number of model runs with different com- binations of parameter values within a given set. General- ly, difficulty encountered in a history-match effort is that of finding any reasonable description that gives good agreement with history. The effort rarely ends with dif- ficulty in selecting among significantly different reservoir descriptions that give comparably good matches. In any event, the pertinent question regarding reservoir- description data is not related to correctness or uniqueness in an absolute sense. The pertinent question concerns the engineering significance of variations in parameter values within ranges of uncertainty. As discussed previously, the model itself is useful in estimating this significance.

Simulation Technology Simulation technology can be divided roughly into the categories of model definition, model formulation, solu- tion techniques, and special techniques related to numerical dispersion control, viscous fingering, and grid- orientation effects. Model definition includes specifica- tion of the problem (process) addressed, component iden- tities, mass transport laws or expressions, fluid PVT and rock property relationships and, finally, the set of finite- difference equations expressing conservation of mass for each gridblock. These equations are generally nonlinear. Before they can be solved for pressures, saturations, etc., they must be linearized and manipulated into a set of simultaneous linear algebraic equations. The term for- mulation refers to these manipulations and the final form of this set of linearized equations. In a general sense, this set of equations can be expressed in the matrix form Ap = b where A is a very sparse, banded Nb xN~ matrix and the known t, and unknown e are column vectors of dimen- sion Nb . A rapidly expanding portion of the simulation literature describes increasingly efficient, iterative solu- tion techniques for this problem.

48-14 PETROLEUM ENGINEERING HANDBOOK

Model Formulations

In 1959, Douglas et al. I6 proposed leap-ffog and simultaneous formulations for incompressible 2D two- phase flow. During 1960-69 a number of authors21-24 described two- and three-phase, 2D and 3D black-oil models based on this simultaneous formulation. In 1960, Stone and Garder I8 and Sheldon et al. I7 introduced the concept of eliminating saturation derivatives among the black-oil model equations to obtain a sin le difference equation in pressure. Fagin and Stewart 1$ in 1966 and Breitenbach et al. *O in 1968 described three-phase black- oil models based on this implicit-pressureiexplicit- saturation (IMPES) formulation. The IMPES formulation is explicit in saturation and composition in that relative permeabilities and concentrations are expressed explicit- ly in the interblock flow terms. Solution of the pressure equations over the grid is followed by an explicit updating of phase saturations and compositions in each gridblock.

In 1969, Blair and Weinaug12’ published a fully im- plicit formulation which expresses all terms in the inter- block flow and well production expressions implicitly. This requires simultaneous solution of all N model equa- tions. A number of later papers describe implementation of the implicit formulation in black-oil, “* composi- tional 3’ and thermal 39 models.

In 1970, MacDonald’23 improved the stability of the IMPES method for the two-phase water/oil case by following the pressure equation solution with solution of a water-saturation equation over the grid using implicit (new-time-level or end-of-timestep) values of relative permeabilities in the interblock flow terms. Spillette et al. 124 extended this concept to the three-phase case and called the formulation sequential.

The IMPES formulation can become unstable if the volumetric flow through a gridblock in a timestep exceeds a small fraction of the block PV. The more stable sequen- tial formulation remains stable to much larger ratios of gridblock volumetric throughput/PV. The tolerable throughput ratio for the implicit formulation is significant- ly larger than that of the sequential method. Arithmetic (or computing cost) per timestep and timestep size both increase from IMPES to sequential to implicit formula- tions. Since the total cost of simulating a given time period is proportional to the product of arithmetic per timestep and timestep size, all three formulations are used widely today.

The sequential formulation can fail to preserve material balances in some problems where adjacent

Q ridblock com-

positions differ greatly. ‘25 Meijerink ‘* described a stabilized IMPES formulation, which improves the stabili- ty of IMPES to a lesser extent than the sequential method but reduces material balance error in regions of steep com- position gradients.

Thomas and Thurnau 127 describe an adaptive implicit formulation, which allows different levels of implicitness in different gridblocks. These various levels may change with timestep number and with iteration number within a given timestep. As previously mentioned, the implicit formulation 12’ requires simultaneous solution of N equa- tions for each block over the entire grid. The correspond- ing arithmetic effort of solution is proportional to N3. Since N= 1 for IMPES, the implicit formulation obviously requires considerably more computing time per timestep

than does IMPES. For each gridblock, the adaptive im- plicit method internally senses (without user intervention) which dependent variables (e.g., saturations, pressure, mole fractions) require implicit dating for stability. For most practical reservoir problems this results in one equa- tion per gridblock for a major fraction of the grid and an overall average number of equations per block con- siderably less than N. Thus the method can attain the stability of the implicit formulation with considerably less computing expense. Also, computer storage requirements are reduced significantly. Future implementations of this formulation may contribute to increased model reliabili- ty (stability) and efficiency in simulations of all types of recovery processes.

Single-well coning studies generally involve radial grid spacings, resulting in very small gridblocks near the well and large throughput ratios. For these studies, the IMPES formulation is unsuitable, and the implicit formulation is generally the most efficient. ‘** For field-scale, 3D black- oil studies, the overall computing time is frequently less with the sequential than with the IMPES or implicit for- mulation. The typical black-oil simulator applied today in 1 ,OOO- or more gridblock, field-scale studies is an IM- PES model with a user-specified option of sequential solu- tion. Smaller black-oil studies and preliminary cross- sectional, coning, and sensitivity studies associated with the large problems frequently employ the implicit formula- tion. Recent thermal models involve implicit formulations. With one exception 3’ recent compositional models29-33 are based on the IMPES formulation.

The popular IMPES and more recent implicit formula- tions are illustrated here for the case of 3D two-phase flow of water and undersaturated oil. This illustration is in the form of the Newton-Raphson procedure, which Blair and Weinaug l2 ’ used in describing their implicit formulation. For clarity, rock compressibility, gravity, and capillary pressure are neglected, and phase (component) produc- tion rates are fixed, independent of pressure and satura- tions. The terms explicit and implicit refer to the time level of evaluation for variables or terms in the left side, in- terblock flow terms of Eqs. 5. Explicit dating denotes evaluation at the beginning of the timestep, t, (level n), while implicit dating denotes evaluation at the end of the timestep, t,+i (level n+ 1). The implicit formulation of Eq. 5, then, appears for each gridblock as

=f(S,,p)=O . . . . ., . . . . . . . . . . . . (lla)

and

A(T,Ap)-qpo -+,S, -@Jo),1

=g(S,,p)=O, . . . . . . . . . . . . . . . . . . . (llb)

where T = interblock transmissibility, p = pressure,

qpp = production rate of phase P, oil or water VP = PV of gridblock,

RESERVOIR SIMULATION 48-I 5

4

Atp = timestep bp = reciprocal formation volume factor of

phase P, ST vol/res vol, Sp = saturation of phase P, and

f, g = function of.

Gridblock indices ij, and k on all terms are suppressed for clarity. All terms at time level n are known from the previous timestep’s calculations. The absence of time level subscript denotes the implicit level, n + 1. Thus all terms T,,T,,b,,b,,S,,S,,p in Eq. 11 represent unknown values at time level n + 1, For all Nb gridblocks, Eqs. 11 are ~Nz. eouations in the 2Nh unknowns ~s,ikniYltPi,‘kn+I 11 a;k ‘two e&rations

}. For a particular gridblock, Eqs. in the 14 unknowns consisting of

(S,,p) pairs in the block and its six neighbors. Thesesix neighbor pairs are introduced by the Laplacian interblock flow terms. Oil saturations are not additional unknowns since S, =I-S,; the transmissibilities T and reciprocal formation volume factors (b) are functions of S ,,, and p.

Application of the well-known Newton-Raphson itera- tive procedure to Eq. 11 gives

and

f6p=o . . . . . . . . . . . . . . . . . . . . . . . . (12

b? p g(s,,p)=g(sppf)+ - W, ( 1 ls 6S, w Wb)

where P is iteration number, superscript P denotes evalua- tion at (S&‘), hp=p’+’ -pp, and S,f,,pp approach the desired S, ,p values as 4 increases. The terms containing derivatives are actually sums of seven terms because of the previously mentioned functional dependence on neighboring block unknowns. Substituting from Eqs. 11 into Eqs. 12, performing the differentiation and rearrang- ing the result gives

A(7-p11A&S,)+A(TP,2A8p)- VP -b&E, AI

“P - pt(S,b”c,)Y6P+f(S~,PY)=O .

and

A(TpZ, A6S,)+A(Tp*2A6p)+ VP -b,ySS, At

“P -- At (S,b,c,)pGp+g(S~,pe)=O, .

or in condensed matrix form,

A(Tc)-Cc+B=O, . . . . . . .

(134

(13b)

. (14)

where T and C are 2 X 2 matrices and P and R are 2 X 1 - column vectors:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)

The coefficients Tll , TzI arise from saturation deriva- tives of relative permeabilities in the transmissibilities. The C matrix elements are obvious upon inspection of Eqs. 13 (e.g., c,* = VplA,(S,b,c,). The compressibil- ities (c, ,c,) appear through the definition c = (1 ib)dbidp.

Eq. 14 written for all gridblocks is a set of Nb linear parabolic difference equations in the Nt, unknowns {Pi,.,k}. Following solution by a direct or iterative tech- nique, the new iterates are calculated as S,$+’ =S$+ 6Sw ,P p+ ’ =pp+Gp. Coefficients are recalculated at

1+1 (SK, .p “I ) for all blocks and Eq. 14 is solved again. These outer or Newton iterations are continued until the maximum values over the grid of { I AS,,, I , I Sp I } are less than some prescribed tolerances. The term inner itera- tions refers to iterations performed by an iterative solu- tion technique in solving Eq. 14 for a given Newton iteration.

The IMPES formulation treats transmissibilities ex- plicitly SO that the first terms in Eqs. 11 are A(T, Ap) and A(T,,Ap), where T,, and T,, are calculated from known saturations at time level n. This results, upon ap- plication of Eq. 12, in zero values for transmissibility derivatives with respect to neighboring block saturations and Eq. 16 replaces Eq. 13:

-z(S b ,+ ,c,)‘6p+f(S$,,pp)=0.. .(16a)

and

A(T,,Asp) + %b$S,

-~(S,b,c,)‘6p+g(S&pp)=0. . . (16b)

The single saturation unknown, 6S,, can be eliminated by multiplying Eqs. 16a and 16b by B,fz and BJ, respec- tively, and adding to obtain

B,$ A(T,.,A&p) + B,PA(T,,,A&p)

-J$S ,L,cM.+SSUcO)E~p+B~~fY+BljigY=O,. (17)

where B is the formation volume factor, l/b. This is a set of Nb single or scalar, linear parabolic dif-

ference equations in the Nb pressure unknowns {Gp+ } As described before, a number of outer or Newton itera- tions are performed with pressure updated and coefficients

48-16 PETROLEUM ENGINEERING HANDBOOK

recalculated after each iteration. After convergence, saturation S, is explicitly calculated, block by block, from Eq. 1 la (with T,, in the first term). Eq. 17 can be written more simply for the section to follow as

A(TAGp)-c++r=O, . . . . . . (18)

where T, 6p,c, and r are scalars.

Solution Techniques Eq. 18 written for all Nb gridblocks can be expressed in matrix form as

AP=b, . . _ . . . . . . . . . . . . . . . . . . . . (19)

where A is a nonsymmetric, sparse Nb xN~ matrix and &’ is the Nb x 1 column vector (Sp Uk ) . Direct solution (Gaussian elimination) or iterative methods can be used to solve Eq. 19. The arithmetic effort required in direct solution strongly depends on the pattern of nonzero elements in the A matrix. This pattern in turn depends upon the particular linear ordering or numbering of the Nb gridblocks. An ordering is simply a one-to-one cor- respondence between a linear index m = 1,2 . Nh and the gridblock indices {ij,k}, i= 1,2. . .N,, j=1,2,. . .N,, k = 1,2. . NZ . Here the term natural ordering denotes numbering the blocks consecutively first in the shortest direction, then in the next shortest direction, and finally in the longest direction. For example, if N, >N, > N,, then

m=k+(j-l)N,+(i-l)NYN,. . . . . . . (20)

Breitenbach et al., ‘29 Peaceman, ’ and others illustrate the diagonal-band form of the A matrix and minimum direct solution effort which result from this natural ordering.

The half bandwidth of the A matrix is N,N, and the arithmetic effort (number of multiplications) of direct solu- tion is roughly proportional to Nb(NyN,)*. Iterative methods require an arithmetic effort roughly proportional to Nb. Thus increasing problem size, Nh, renders iterative solution increasingly preferable to direct solu- tion. For large problems, computer storage requirement is also significantly less for iterative than direct solution.

Price and Coats I30 described reduced bandwidth direct solution methods based on diagonal (D2) and alternate- diagonal (D4) gridblock orderings. For certain test prob- lems and iterative methods, they showed a D4 direct/iterative work ratio less than unity for half band- widths up to about 30. Compared with natural ordering, D4 ordering can reduce direct solution computational ef- fort by factors up to four and six for the 2D and 3D cases, respectively. Woo et al. 13’ described other techniques that take advantage of matrix sparsity to reduce direct solution effort. In spite of these advancements in direct solution, iterative methods remain preferable for large reservoir studies.

Successive-overreluxation (SOR) iterative methods described bv Young 132,‘33 have been used in simulators from the early 1960’s. Block SOR (BSOR) methods, in- cluding line (LSOR), two-line, and planar SOR, have proved especially popular. The BSOR methods require direct solution within each block, which means that direct

solution must be performed for tridiagonal matrices in LSOR and pentadiagonal matrices in planar SOR. As the block size is increased the arithmetic work per iteration increases because of the increased arithmetic associated with this direct solution within each block. However, con- vergence rate generally increases and the total number of iterations correspondingly decreases with increasing block size. To some extent, optimal block size can be determined by mathematical analysis ‘33,134 of this tradeoff between work per iteration and number of itera- tions. The SOR methods remain popular because of ease of coding, low computer storage requirement and automatic determination of the optimum value of the single iteration parameter. Varga ‘~4 describes the power method for this parameter determination and Breitenbach et al. 129 illustrate its application.

In 1971, Watts135 presented an additive correction method which improves LSOR convergence rate in highly anisotropic problems. A highly anisotropic problem is one where, throughout the grid, transmissibibties in one direc- tion are much

B reater than those in other direction(s). Set-

tari and Aziz ’ 6 extended Watts’ method to other iterative solution techniques.

Alternating-direction iterative methods (ADI) were developed for 2D by Peaceman and Rachford 137 in 1955 and for 3D by Douglas and Rachford ‘38 in 1956. These methods were widely used in simulation throughout the 1960’s and into the 1970’s. The AD1 methods require a sequence or set of iteration parameters. While mathe- matical analysis yields an optimal parameter set for cer- tain cases, 1,137 actual reservoir cases frequently require some trial-and-error effort.

In 1968, Stone 139 described the strongly-implicit pro- cedure (SIP); Weinstein et al. t4’ described SIP in 3D. Again, a set of iteration parameters is required. Parameter estimation methods associated with AD1 have proved useful for SIP * but, again, some trial-and-error effort is required or beneficial in man reservoir studies.

A number of studiest29~136~15~141 compare direct solu- tion, LSOR, ADI, and SIP methods for a variety of test and reservoir problems. There is no simple answer to which method is best. The ranking of the methods is problem-dependent in that it depends on the range of varia- tion in coefficient (transmissibility) values in the A matrix and the pattern (e.g., highly anisotropic), if any, of their variation. In general, the more difficult reservoir problems have a very large range from the smaller to larger transmissibilities and this large ratio is not uniformly associated with a particular direction throughout the grid. SIP became widely used throughout the 1970’s and re- mains in use today because it frequently outperforms the other methods in these difficult cases.

A new class or type of iterative methods is the subject of a number of papers’4’-‘50 published since the mid 1970’s. Basically, the methods involve approximate fac- torization of the A matrix into an LU product, followed by an iterative sequence

wpk+’ -pk)=rk, , . . . . . . . . . (21)

where L = a lower triangular matrix (all entries Pii =0

for j>i),

RESERVOIR SIMULATION 48-17

V = an upper triangular matrix (all entries uii = 0 forj<i),

k = iterate number, and rk = b-Apk.

Convergence is accelerated by the conjugate gradi- ent 3 14’ orthomin, 143 or other techniques. Because of the sparse, banded nature of the lower and upper triangular matrices L and V, the arithmetic work per iteration of solving Eq. 21 is a very small fraction of that required in direct solution of the original problem, Eq. 19.

These new methods seem very attractive in that they require no iteration parameter and are more robust than previous methods. That is, they generally exhibit fast or reasonable convergence rates, even for difficult reservoir problems where older methods fail or converge slowly. Two studies 148,‘50 showed convergence for the difficult thermal (steamflood) problem where negative transmis- sibilities can occur. 15’ The interested reader should con- sult the reference sections of Refs. 141 through 150 for a number of equally good papers dealing with these new iterative methods.

Code Vectorization The computational speed, storage, and vectorization capabilities of computer hardware have increased sharp- ly in the past few years. As an example, the Cray-I S com- puter provides up to 4,000,OOO decimal-words of storage, compared with a typically available 100,000 words on most machines used until 1975. Recently introduced com- puters offer sharp increases in computational speeds and speed/cost ratios. In addition, vector processing capabilities of Control Data Corp. and Cray supercom- puters significantly increase the efficiency of simulators coded to use this vectorization.

This increased machine size, speed, and vector process- ing contribute to the feasibility of larger reservoir studies. Until the middle 1970’s, most black-oil studies involved grids of 3,000 or fewer blocks. In 1980, Mrosovsky et al. 15* described a Prudhoe Bay field study with more than 16,000 active gridblocks. Studies on the supercom- puters with grids of more than 30,000 blocks have been performed recently.

The vector-processing capability has had a strong im- pact on simulation technology. The resultant increase in feasible study size has spurred the development of new, faster iterative solution techniques. Several papers ‘52-‘56 describe the contribution of code vectorization to reduced computing expense and the need to develop or redesign model code to take advantage of the vectorization.

Nomenclature A = a nonsymmetric, sparse Nb x Nh matrix A = Ay,AZk =cross-sectional area normal to

flow bp = reciprocal formation volume factor of

phase P, ST vol/res vol C = 2x2 matrix

C,J = concentration of component I in phase J C,p = concentration of component I in phase P CNJ = concentration of component N in phase .I f,J = fugacity of component I in phase J

HJ = enthalpy, energy/mole k rP = relative permeability to phase P (P=w,o,g)

L= a lower triangular matrix (all entries e,=O

L= forj>i), see Eq. 21

distance between adjacent block centers,

m= Nb = N, = N, = N, =

n= P=

P ego = P two =

9I =

(AT~-~ +Ax;)/2, see Eq. 1 1, 2. _ .Nb, linear index total number of gridblocks, N,N,N, number of gridblocks in x direction number of gridblocks in y direction number of gridblocks in z direction time level, t n+, =t, +A\t Nbxl column vector {6pijk}, see Eq. 19 gas/oil capillary pressure water/oil capillary pressure component I interblock flow rate,

mass/time qpi = mass rate of production of component I

qpp = production rate of phase P (water or oil) R= 2 X 1 column vector

s,, = critical gas saturation SJ = saturation of phase J

s WC = critical water saturation T= interblock transmissibility and T=2 x 2

matrix TN = interblock transmissibility for flow of

component I in phase .I T,, = oil transmissibility at time level 12 T wn = water transmissibility at time level n

V = an upper triangular matrix (all entries uij=O forj<i)

UJ = internal energy, energy/mole I/ = grid block volume, hi A!J kzk

Ax = xi-1 -Xi, where x is p or Z x/J = mol fraction of component I in phase J yp = specific weight of phase P

6 = time difference operator, 6X=X,+, -X, pp = viscosity of phase P

Subscripts i, j, k = gridblock indices

J = w,o,g or 1,2,3, n = step number

Superscripts k = iteration number P = iteration number

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48-18 PETROLEUM ENGINEERING HANDBOOK

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46. Thomas, L.K., Dixon, T.N., and Pierson, R.G.: “Fractured Reser- voir Simulation,” Sot. Pet. Eng. J. (Feb. 1983) 42-54.

47. Gilman, J.R. and Kazemi, H.: “Improvements in Simulation of Naturally Fractured Reservoirs,” Sot. Pet. Eng. J. (Aug. 1983) 659-707.

48. Blaskovich, F.T. er al.: “A Multicomponent Isothermal System for Efficient Reservoir Simulation,” paper SPE 11480 presented at the 1983 SPE Middle East Oil Show, Manama, March 14-17.

49. Henderson, J.H.. Dempsey, J.R. and Tyler, J.C.: “Use of Numerical Models to Develop and Operate Gas Storage Reser- voirs,” J. Pet. Tech. (Nov. 1969) 1239-46.

50. Mann, L.D. and Johnson, G.A.: “Predicted Results of Numeric Grid Models Compared With Actual Field Performance.” J. Pet. Tech. (Nov. 1970) 1390-98.

51. Thomas, J.E. and Driscoll, V.J.: “A Modeling Approach for Op- timizing Waterflood Performance, Slaughter Field Chickenwire Pattern,” J. Pet. Tech. (July 1973) 757-63.

52. “A Study of the Sensitivity of Oil Recovery to Production Rate.” Proc., Alberta Energy Resources Conservation Board, No. 75 11 (Feb. 1974) Schedule 1, Shell Canada Ltd.

53. Stright, D.H. Jr.. Bennion, D. W., and Aziz, K.: “Influence of Production Rate on the Recovery of Oil From Horizontal Waterfloods,” J. Pet. Tech. (May 1975) 555-63.

54. Thakur, G.C., et al. : “G-2 and G-3 Reservoux, Delta South Field, Nigeria: Part 2-Simulation of Water Injection,” J. Pet. Tech. (Jan. 1982) 148-58.

55. Graue, D.J. and Zana, E.T.: “Studv of a Possible CO? Flood in Rangely Field, Colorado,” J. Pet.‘Tech. (July 1981) 1~12-18.

56. Bloomquist, C.W., Fuller, K.L., and Moranville, M.B.: “Miscible Gas Enhanced Oil Recovery Economics and the Effects of the Windfall Profit Tax,” paper SPE 10274 presented at the 1981 SPE Annual Technical Coifeience and Exhibition, San Antonio, Oct. 5-7.

57. Todd, M.R., Cobb, W.M., and McCarter, E.D. : “CO2 Flood Performance Evaluation for the Cornell Unit, Wasson San Andres Field,” J. Pet. Tech. (Oct. 1982) 1583-90.

58. Herrera, J.Q. and Hanzlik, E.J.: “Steam Stimulation History Match ofMultiwel1 Pattern in the Sl-B Zone, Cat Canyon Field,” paper SPE 7969 presented at the 1979 California Regional Meeting, Ven- tura, April 18-20.

59. Williams, R.L.: “Steamflood Pilot Design for a Massive, Steep- ly Dipping Reservoir,” paper SPE 10321 presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, Oct. 5-7.

60. Meldau. R. F., Shipley, R.G., and Coats, K.H .: “Cyclic Gas/Steam Stimulation of Heavy-Oil Wells,” J. Pet. Tech. (Oct. 1981) 1990-98.

61. Gomaa, E.E., Duerksen, J.H., and Woo, P.T.: “Designing a Steamflood Pilot in the Thick Monarch Sand of the Midway-Sunset Field,” J. Per. Tech. (Dec. 1977) 1559-68.

62. Moughamian, J.M., et al. : “Simulation and Design of Steam Drive in a Vertical Reservoir,” J. Per. Tech. (July 1982) 1546-54.

63. “Selection of Reservoirs Amenable to Micellar Flooding,” First Annual Report, Dept. of Energy (Dec. 1980) BC/OOO48-20.

64 Fayers, F.J., Hawes, R.I., and Mathews, J.D.: “Some Aspects

RESERVOIR SIMULATION 48-19

of the Potential Application of Surfactants or CO2 as EOR Proc- essesinNorth Sea Reservoirs,” J. Per. Tech. (Sept. 1981) 1617-27.

65. dezabala, E.F., er al.: “A Chemical Theory for Linear Alkaline Flooding,” Sot. Pet. Eng. J. (April 1982) 245-58.

66. Patton, J.T., Coats, K.H., and Colegrove, G.T.: “Prediction of Polymer Flood Performance,” Sot. Per. Eng. J. (March 1971) 72-84; Trans., AIME, 251.

67. Todd, M.R. and Longstaff, W.J.: “The Development. Testing, and Application of a Numerical Simulator for Predicting Misci- ble Flood Performance,” J. Per. Tech. (July 1972) 874-82; Trans., AIME, 253.

68. Cook, R.E., Jacoby, R.H., and Ramesh, A.B.: “A Beta-Type Reservoir Simulator for Approximating Compositional Effects Dur- ing Gas Injection,” Sot. Per. Eng. J. (Oct. 1974) 471-81.

69. Patton, J.T., Coats, K.H., and Spence, K.: “Carbon Dioxide Well Simulation: Part I-A Parametric Study,” J. Per. Tech. (Aug. 1982) 1798-1804.

70. Coats, K.H.: “Simulation of Gas Condensate Reservoir Perfor- mance,” paper SPE 10512 presented at the 1982 SPE Reservoir Simulation Symposium, New Orleans, Feb. 1-3.

71. Coats, K.H.: “Simulation of l/8 Five-/Nine-Spot Patterns,” Sot. Pet. Eng. J. (Dec. 1982) 902.

72. Killough, I.E., et al.: “The Kuparek River Field: A Regression Approach to Pseudorelative Permeabilities,” paper SPE 1053 1, presented at the 1982 SPE Symposium on Reservoir Simulation, New Orleans, Feb. 1-3.

73. Killough, I.E., er al.: “The Prudhoe Bay Field: Simulation of a COIIIpleX ReSeNOir,” Proc., Intl. Petroleum Exhibition and Technical Symposium, Beijing (1982) 777-94.

74. Jacks, H.H., Smith, O.E., and Mattax, C.C. : “The Modeling of a Three-Dimensional Reservoir With a Two-Dimensional Reser- voir Simulator-The Use of Dynamic Pseudo Functions,” Sot. Pef. Eng. J. (June 1973) 175-85.

75. Kyte, J.R. and Berry, D.W.: “New Pseudo Functions to Control Numerical Dispersion,” Sot. Pet. Eng. J. (Aug. 1975) 269-76.

76. Killough, J.E. and Foster. H.P. Jr.: “Reservoir Simulation of the Empire Abe Field: The Use of Pseudos in a Multilayered System.” Sot. Per. Ert~. J. (Oct. 1979) 279-88.

77. Coats, K.H., Dempsey. J.R.. and Henderson, J.H.: “The Use of Vertical Equihbrrum in Two-Dimensional Simulation of Three- Dimensional Reservoir Performance,” Sot. Per. Enx. J. (March 1971) 63-71; Trans.. AIME, 251.

78. Hear”, C.L.: “Simulation of Stratified Waterflooding by Pseudo Relative Permeability Curves,” J. Per. Trch. (July 1971) 80-13.

79. Simon. R., Rosman. A.. and Zana. ET.: “Phase-Behavior Proper- ties of CO*-Reservoir Oil System,” Sot. Pet. Eng. J. (Feb. 1978) 20-26.

80. Orr, F.M. and Silva, M.K.: “Equilibrium Phase Compositions of CO 2 /Hydrocarbon Mixtures-Part 1: Measurement by a Con- tinuous Multiple-Contact Experiment,” Sot. Per. Eng. J. (April 1983) 272-80.

81. Orr, F.M., Silva, M.K., and Lien, C.: “Equilibrium Phase Com- positions of CO2iCrude Oil Mixtures-Part 2: Comparison of Con- tinuous Multiple-Contact and Slim-Tube Displacement Tests,” Sot. Pet. Eng. J. (April 1983) 281-91.

82. Katz, D.L. and Firoozabadi, A.. “Predicting Phase Behavior of Condensate/Crude-Oil Systems Using Methane Interaction Coef- ficients,” J. Pet. Tech. (Nov. 1978) 1649-55; Trans., AIME, 265.

83. Yarborough, L.: “Application of a Generalized Equation of State to Petroleum Reservoir Fluids,” Equations of St&e in Engineer- ing, Advances in Chemisrgv Series, K.C. Chao and R.L. Robn- son (eds.), American Chemical Society, Washington, D.C. (1979). 182, 385-435.

84. Whitson, C.H. and Tarp, S.B.: “Evaluating Constant-Volume Depletion Data,” J. Per. Tech. (March 1983) 610-20.

85. Coats, K.H. and Smart. G.T.: “Application of a Regression-Based EOS PVT Program to Laboratory Data,” paper SPE 11197, presented at the 1982 SPE Annual Technical Conference and EX- hibition, New Orleans, Sept. 26-29.

86. Coats, K.H., Dempsey, J.R., and Henderson, J.H.: “A New Technique for Determining Reservoir Description from Field Per- formance Data,” Sot. Pet. Eng. J. (March 1970) 66-74; Trans., AIME, 249.

87. Thomas, L.K. and Hellurns, L.J.: “A Nonlinear Automatic History Matching Technique for Reservoir Simulation Models,” Sot. Pet. Eng. J. (Dec. 1972) 508-14; Trans.. AIME, 253.

88. Wasserman, M.L., Emanuel, A.S., and Seinfeld, J.H.: “Prac- tical Application of Optimal-Control Theory to History-Matching

Multiphase Simulator Models,” Sot. Pet. Eng. J. (Aug. 1975) 347-55; Trans., AIME, 259.

89. Bobcrg, T.C., et al.: “Application of Inverse Simulation to a Com- plex Multireservoir System,” J. Pet. Tech. (July 1974) 801-08; Trans., AIME, 257.

90. Watson, A.T., et al. : “History Matching Two-Phase Petroleum Reservoirs,” Sot. Per. Eng. J. (Dec. 1980) 521-32.

91. Blair, P.M. and Peaceman, D.W.: “An Experimental Verifica- tion of a Two-Dimensional Technique for Computing Performance of Gas-Drive Reservoirs,” Sot. Pet. Eng. J. (March 1963) 19-27; Trans., AIME, 228.

92. Ridings, R.L., et al. : “Experimental and Calculated Behavior of Dissolved-Gas-Drive Systems,” Sot. Pet. Eng. J. (March 1963) 41-48; Trans., AIME, 228.

93. Mungan, N.: “A Theoretical and Experimental Coning Study,” Sot. Pet. Eng. J. (June 1975) 247-54: Trans., AIME, 259.

94. Kazemi, H. and Merrill, L.S.: “Numerical Simulation of Water Imbibition in Fractured Cores,” Sot. Per. Eng. J. (June 1979) 175-82.

95. Lantz, R.B.: “Quantitative Evaluation of Numerical Diffusion (Truncation Error),” Sm. Pet. Eng. J. (Sept. 1971) 315-20: Trans., AJME. 251.

96. Van-Quy, N., Simandoux, P., and Corteville. I.: “A Numerical Study of Diphasic Multicomponent Flow,” Sot. Pet. Eflg. J. (April 1972) 171-84; Trans., AIME, 253.

97. Fussell, D.D., Shelton, J.L., and Griffith, J.D. : “Effect of ‘Rich’ Gas Composition on Multiple-Contact Miscible Displacement- A Cell-to-Cell Flash Model Study,” Sot. Per. Eng. J. (Dec. 1976) 310-16; Trans., AIME, 261.

98. Harpole, K.J. and Heam, C.L.: “The Role of Numerical Simula- tion in Reservoir Management of a West Texas Carbonate Reser- voir,” froc., Intl. Exhibition and Technical Symposium. Beijing (1982) 759-76.

99. Heinemann, Z.E., et al.: “Using Local Grid Refinement in a Multiple-Application Reservoir Simulator,” Proc., SPE Sym- posium on Reservoir Simulation, San Francisco (1983) 205-18.

100. Ewing, R.E., Russell, T.F., and Wheeler, M.F.: “Simulation of Miscible Displacement Using Mixed Methods and a Modified Method of Characteristics,” Proc., SPE Symposium on Reser- voir Simulation, San Francisco (1983) 71-82.

101. Carr, A.H. and Christie, M.A.: “Controlling Numerical Diffu- sion in Reservoir Simulation Using Flux-Corrected Transport.” Proc., SPE Symposium on Reservoir Simulation, San Francisco (1983) 25-32.

102. Todd, M.R., O’Dell, P.M., and Hirasaki, G.J.: “Methods for In- creased Accuracy in Numerical Reservoir Simulators,” Sot. Pet. Eng. J. (Dec. 1972) 515-30; Trans.. AIME, 253.

103. Coats, K.H., et al.: “Three-Dimensional Simulation of Steamflooding,” Sot. Pet. Eng. J. (Dec. 1974) 573-92; Trans., AIME, 257.

104 Yanosik, J.L. and McCracken, T.A.: “A Nine-Point, Finite- Difference Reservoir Simulator for Realistic Prediction of Adverse Mobility Ratio Displacements,” Sot. Pet. Eng. J. (Aug. 1979) 253-62; Trans., AIME, 267.

105. Abou-Kassem, J.H. and Aziz. K.: “Grid Orientation During Steam Displacement,” paper SPE 10497 presented at the 1982 SPE Sym- posium on Reservoir Simulation, New Orleans, Feb. 1-3.

106 Holloway, C.C., Thomas, L.K., and Pierson, R.G.: “Reduction of Grid Orientation Effects in Reservoir Simulation,” paper SPE 5522 presented at the 1975 SPE Annual Technical Conference and Exhibition, Dallas, Sept. 28-Oct. 1.

107. Robertson, G.E. and Woo, P.T.: “Grid-Orientation Effects and the Use of Orthogonal Curvilinear Coordinates tn Reservon Simula- tion,” Sot. Pet. Eng. J. (Feb. 1978) 13-19.

108. Vinsome, P.K.W. and Au, A.D.K.: “One Approach to the Grid Orientation Problem in Reservoir Simulation,” paper SPE 8247 presented at the 1979 SPE Annual Technical Conference and Ex- hibition, Las Vegas, Sept. 23-26.

109. Coats, K.H. and Ramesh, A.B.: “Effects of Grid Type and Dif- ference Scheme on Pattern Steamflood Simulation Results,” paper SPE 11079 presented at the 1982 SPE Annual Technical Conference and Exhibition, New Orleans, Sept. 26-29.

110. Bet-tiger, W.I. and Padmanabhan. L.: “Finite-Difference Soh- tions to Grid Orientation Problems Using IMPES,” paper SPE 12250 presented at the 1983 SPE Symposium on Reservoir Simula- tion, San Francisco, Nov. 16-18.

111. Shah, P.C.: “A Nine-Point Finite Difference Gperator for Reduc- tion of the Grid Orientation Effect,” paper SPE 12251 presented

48-20 PETROLEUM ENGINEERING HANDBOOK

at the 1983 SPE Symposium on Reservoir Simulation, San Fran- cisco, Nov. 16-18.

112. Coats, K.H. and Modine, A.D.: “A Consistent Method for Calculating Transmissibilities in Nine-Point Difference Equations,” paper SPE 12248 presented at the 1983 SPE Symposium on Reser- voir Simulation, San Francisco, Nov. 16-18.

113. Frauenthal, J.C., Towler. B.F., and diFranco, R.: “Reduction of Grid-Orientation Effects in Reservoir Simulation By Generalized Upstream Weighting,” paper SPE 11593 presented at the 1983 SPE Symposium on Reservoir Simulation, San Francisco, Nov. 16-18.

114. Pruess, K. and Bodvarsson, G.S.: “A Seven-Point Finite- Difference Method for Improved Grid Orientation Performance in Pattern Steamfloods,” paper SPE 12252 presented at the 1983 SPE Symposium on Reservoir Simulation, San Francisco, Nov. 16-18.

115. Emmanuel, A.S. and Cook, G.W.: “Pseudo-Relative Permeability for Well Modeling,” Sm. Pet. Eng. J. (Feb. 1974) 7-9.

116. Chappelear, J.E. and Hirasaki, G.J.: “A Model of Oil-Water Con- ing for Two-Dimensional, Area1 Reservoir Simulation,” Sot. Per. Eng. J. (April 1976) 65-72; Trans., AIME, 261.

117. Woods, E.G. and Khurana, A.K.: “Pseudofunctions for Water Coning in a Three-Dimensional Reservoir Simulator,” Sot. Pet. Enn. J. (Aun. 1977) 251-62.

118. Ad&gton, D.V.: “An Approach to Gas-Coning Correlations for a Large Grid Cell Reservoir Simulator,” J. Pet. Tech. (Nov. 1981) 2267-74.

119. Akbar, A.M., Arnold, M.D., and Harvey, A.H.: “Numerical Simulation of Individual Wells in a Field Simulation Model,” Sot. Pet. Eng. J. (Aug. 1974) 315-20.

120. Mrosovsky. I. and Ridings, R.L.: “Two-Dimensional Radial Treat- ment of Wells Within a Three-Dimensional Reservoir Model,” Sot. Pet. Eng J. (April 1974) 127-31.

121. Blair, P.M. and Weinaug, C.F.: “Solution of Two-Phase Flow Problems Using lmphcit Difference Equations,” Sot. Per. Eng. J. (Dec. 1969) 417-24; Trans., AIME, 246.

122. Bansal, P.P. et a[.: “A Strongly Coupled, Fully Implicit. Three- Dimensional, Three-Phase Reservoir Simulator,” paper SPE 8329 presented at the 1979 SPE Annual Technical Conference and Ex- hibition, Las Vegas, Sept. 23-26.

123. MacDonald, R.C. and Coats, K.H.: “Methods for Numerical Simulation of Water and Gas Coning,” Ser. Pet. Eng. J. (Dec. 1970) 425-36; Trans., AIME, 249.

124. Spillette. A.G.. Hill&ad. J.G., and Stone, H.L.: “A High-Stabilitv Sequential-Solution Approach to Reservoir Simulation,“-paper SPE 4542 presented at the SPE 1973 Annual Meeting, Las Vegas, Sept. 30-Oct. 3.

125. Coats, K.H.: “A Highly Implicit Steamflood Model,” Sot. Pet. Eng. J. (Oct. 1978) 369-83.

126. Meijerink, J.A.: “A New Stabilized Method for Use in IMPES- Type Numerical Reservoir Simulators,” paper SPE 5247 presented at the 1974 SPE Annual Meeting, Houston, Oct. 6-9.

127. Thomas, G.W. and Thumau, D.H.: “Reservoir Simulation Us- ing an Adaptive implicit Method,” Sot. Pet. Eng. J. (Oct. 1983) 759-68.

128. Trimble, R.H. and McDonald, A.E.: “A Strongly Coupled, Ful- ly implicit, Three-Dimensional, Three-Phase Well Coning Model,” Sot. Pet. Eng. J. (Aug. 1981) 454-58.

129. Breitenbach, E.A., Thurnau, D.H., and Van Poollen, H.K.: “Solu- tion of the Immiscible Fluid Flow Simulation Equations,” Sot. Per. Eng. J. (June 1969) 155-69.

130. Price H.S. and Coats, K.H.: “Direct Methods in Reservoir Simula- tion,” Sot. Pet. Eng. J. (June 1974) 295-308; Trans., AIME, 257.

13 I, Woo, P.T., Roberts, S.J., and Gustavson, F.G.: ‘ ‘Apphcation of Sparse Matrix Techniques in Reservoir Simulation,” Sparse Matrix Computations, J.R Bunch and D.E. Rose (eds.), Academic Press Inc.; Washington, D.C. (1976) 427-38.

132. Young, D.M.: “The Numerical Solution of Elliptic and Parabolic Partial Differential Equations,” Survey ofNumerical Analysis, J. Todd (ed.), McGraw-Hill Book Co. Inc., New York City (1963) 380-438.

133. Young, D.M : Iterative Solution of Large Linear Systems. Academic Press Inc., Washington, D.C. (1971).

134. Varga, R.S.: Matrix I&rat& Analy‘sis, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1962) 322.

135. Watts, J.W : “An Iterative Matrix Solution Method Suitable for Anisotropic Problems,” Sot. Per. Eng. J. (March 1971) 47-5 I; Trans., AIME, 251.

136. Settari, A. and Aziz, K.: “A Generalization of the Additive Cor- rection Methods for the lterative Solution of Matrix Equations,” Sm. Ind. Appl. Math. J. Number Analysis (1973) 10, 506-21.

137. Peaceman, D.W. and Rachford, H.H.: “The Numerical Solution of Parabolic and Elliptic Differential Equations,” Sot. Ind. Appi. Math. J. (1955) 3, 28-41.

138. Douglas, J. and Rachford, H.H.: “On the Numerical Solution of Heat Conduction Problems in Two and Three Space Variables,” Trans., American Math. Sot. (1956) 82, 421-39.

139. Stone, H.L.: “Iterative Solution of lmpliclt Approximation of Multidimensional Partial Differential Equations,” Sot. Ind. Appl. Math. J. Number Analysis (1968) 5, 530-58.

140. Weinstein, H.G.. Stone, H.L., and Kwan. T.V.: “Iterative Pro- cedure for Solution of Systems of Parabolic and Elliptic Equations in Three Dimensions,” IEC Fundamentals (1969) 8, 281-87.

141. Watts, J.W. III: “A Conjugate Gradient-Truncated Direct Method for the Iterative Solution of the Reservoir Simulation Pressure Equa- tion,” Sot. Per. Eng. J. (June 1981) 345-53.

142. Hestenes, M.R. and Stiefel, E.: “Methods of Conjugate Gradients for Solving Linear Systems,” J. of Research (1952) 49, 509-36.

143. Concus, P. and Golub, G.H.: “A Generalized Conjugate Gradient Method for Nonsymmetric Systems of Linear Equations,” Report STAN-CS-76-535, Stanford U.. Stanford, CA (Jan. 1976).

144. Vinsome, P.K.W.: “Orthomin, an Iterative Method for Solving Sparse Banded Sets of Simultaneous Linear Equations.” paper SPE 5729 presented at the 1976 SPE Symposium on Numerical Simula- tion of Reservoir Performance, Los Angeles, Feb. 19-20.

145. Meijerink, J.A. and Van Der Worst, H.A.: “An Iterative Solu- tion Method for Linear Systems of Which the Coefficient Matrix is a Symmetric M-Matrix,” Math. of Camp. (Jan. 1977) 148-62.

146. Kershaw, D.S.: “The Incomplete Cholesky-Conjugate Gradient Method for the Iterative Solution of Systems of Linear Equations,” J. Compt. Physics (1978) 26, 43-65.

147. Young, D.M. and Jea. N.C.: “Generalized Conjugate Gradient Acceleration of Nonsymmetric Iterative Methods,” Linear Algebraic Applicarions (1980) 34, 159-94.

148. Tan, T.B.S. and Letkeman, J.P.: “Application of D4 Ordering and Minimization in an Effective Partial Matrix Inverse Iterative Method,” paper SPE 10493 presented at the 1982 SPE Symposium on Reservoir Simulation, New Orleans, Feb. 1-3.

149. Behie, A. and Forsyth, P.A.: “Practical Considerations for In- complete Factorization Methods in Reservoir Simulation.” paper SPE 12263 presented at the 1983 SPE Symposium on Reservoir Simulation, San Francisco, Nov. 16-18.

150. Wallis, J.R.: “Incomplete Gaussian Elimination as a Precondi- tioning for Generalized Conjugate Gradient Acceleration,” paper SPE 12265 presented at the 1983 SPE Symposium on Reservoir Simulation, San Francisco, Nov. 16-18.

151. Coats, K.H.: “Reservoir Simulation: A General Model Formula- tion and Associated Physical/Numerical Sources of Instability,” Bounaivy and Interior Layers-Computational and Asymptotic Methods, J.J. Miller (ed.), Boole Press, Dublin (1980) 62-76.

152. Mrosovsky, I., Wong, J.Y., and Lampe. H.W.: “Construction of a Large Field Simulator on a Vector Computer,” J. Pet. Tech. (Dec. 1980) 2253-64.

153. Woo, P.T.: “Application of Array Processor to Sparse Elimina- tion,” Proc., paper SPE 7674 presented at the 1979 SPE Sym- posium on Reservoir Simulation, Denver, Jan. 31-Feb. 2.

154. Nolen, J.S., Kuba, D.W., and Kasic, M.J. Jr.: “Application of Vector Processors to Solve Finite Difference Equations,” Sot. Pet. Eng. J. (Aug. 1981) 447-53.

155. Calahan, D.A.: “Performance of Linear Algebra Codes on the CRAY-I,” Sot. Pet. Eng. J. (Oct. 1981) 558-64.

156. Killough, J.E. and Levesque, J.M.: “Reservoir Simulation and the In-House Vector Processor: Experience for the First Year,” paper SPE 10521 presented at the 1982 SPE Symposium on Reser- voir Simulation, New Orleans, Feb. 1-3.

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