development and application of a 3d equation-of-state

Copyright

by

Rohollah Abdollah Pour

2011

Development and Application of a 3D Equation-of-State

Compositional Fluid-Flow Simulator in Cylindrical

Coordinates for Near-Wellbore Phenomena

by

Rohollah Abdollah Pour, B.S.; M.S.; M.S.E.

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy


December 2011

The Dissertation Committee for Rohollah Abdollah Pour

certifies that this is the approved version of the following dissertation :




Committee:

Carlos Torres-Verdın, Supervisor

Kamy Sepehrnoori, Supervisor

Mojdeh Delshad

Leszek Demkowicz

Russell T. Johns

To my parents

Abolfazl and Fatemeh

Acknowledgments

I had the chance to work with two outstanding professors at the University of Texas at

Austin. They helped me directly or indirectly to succeed in my academic research, pro-

fessional attitude, and personal life. I am deeply indebted to my supervisor, Dr. Carlos

Torres-Verdın for setting the bar high. I would like to thank him for his valuable guid-

ance, endless patience, and sharing his expertise. I also extend my thanks to Dr. Kamy

Sepehrnoori for guiding me during all steps of my research. He encouraged and helped me

to find solutions for many of the research problems.

I would like to extend my sincere appreciation to the committee members of my

dissertation Dr. Mojdeh Delshad, Dr. Leszek Demkowicz, and Dr. Russell T. Johns for

taking the time to review dissertation and delivering their instructive guidance and feedback.

I would like to thank the faculty member Dr. Larry Lake and Dr. Ekwere J. Peters, who

were my reference and help to answer several of my questions. I was very lucky to have a

distinguished petrophysicist, David Kennedy, as my mentor and friend; thank you Dave for

all your help and feedback related to my dissertation. I also want to appreciate all work

and help, I had from Dr. Roger Terzian, Cheryl Kruzie, Frankie L. Hart, and Jana Cox. I

would like to send a gratitude note to Reynaldo Casanova for his help in handling university

paperworks.

In the development of this simulator, I performed several benchmark comparisons

and tests with simulators from Computer Modeling Group. I would like to thank Kan-

haiyalal Patel for his assistance and support. I would like to say thanks to David Pardo

Zubiaur for mentoring me on the first year; his instructions and method of development were

helped me to perform this research. I am also grateful for detailed discussions with my col-

v

leagues Farhad Tarahhom, Abdoljalil Varavei, Mayank Malik, Kaveh Ahmadi, Renzo Ange-

les Boza, Meghdad Roshanfekr, Gholamreza Garmeh, Mehdi Haghshenas, Javad Behseresht,

Amir Forooqnia, and Abdolhamid Hadibeik.

I am also thankful of my friends Seyed Reza Yousefi, Javad Behseresht, Mehdy

Haghshenas, Amir Forooqnia, Vahid Shabro, Andrew Popielski, Philippe Marouby, Olabode

Ijasan, Waleed Fazelipour, Jorge Sanchez, Ankur Ghandi, Robert K Mallan, David Wolf,

Alberto Mendoza Chavez, Amirreza Rahmani, Ali Moinfar, Mahdy Shirdel, Kyati Rai,

Ryosuke Okuno, Farshad Lalehrokh, Tatyana Torskaya, Kanay Jerath, Abhishek Bansal,

Haryanto Adiguna, Siddharth Mishra, Antoine Montaut, Edwin Ortega, and Chicheng Xu,

for their good company and their assistance during research.

This research was made possible with funding by The University of Texas at Austin’s

Research Consortium on Formation Evaluation, jointly sponsored by Aramco, Anadarko

Petroleum Corporation, Baker-Hughes, British Gas, BHP Billiton, BP, Chevron, Cono-

coPhillips, ENI, ExxonMobil, Halliburton Energy Services, Marathon Oil Corporation, Mex-

ican Institute for Petroleum, Petrobras, Schlumberger, Shell International E&P, StatoilHy-

dro, TOTAL, and Weatherford.

Last but not least, I want to thank my family: mother, father, brothers and my

beautiful sister. To them, I dedicate my dissertation.

Rohollah Abdollah Pour


December 2011

vi




Publication No.

Rohollah Abdollah Pour, Ph.D.

The University of Texas at Austin, 2011

Supervisors: Carlos Torres-Verdın and Kamy Sepehrnoori

Well logs and formation testers are routinely used for detection and quantification of hy-

drocarbon reserves. Overbalanced drilling causes invasion of mud filtrate into permeable

rocks, hence radial displacement of in-situ saturating fluids away from the wellbore. The

spatial distribution of fluids in the near-wellbore region remains affected by a multitude of

petrophysical and fluid factors originating from the process of mud-filtrate invasion. Con-

sequently, depending on the type of drilling mud (e.g. water- and oil-base muds) and the

influence of mud filtrate, well logs and formation-tester measurements are sensitive to a

combination of in-situ (original) fluids and mud filtrate in addition to petrophysical prop-

erties of the invaded formations. This behavior can often impair the reliable assessment

of hydrocarbon saturation and formation storage/mobility. The effect of mud-filtrate in-

vasion on well logs and formation-tester measurements acquired in vertical wells has been

vii

extensively documented in the past. Much work is still needed to understand and quantify

the influence of mud-filtrate invasion on well logs acquired in horizontal and deviated wells,

where the spatial distribution of fluids in the near-wellbore region is not axial-symmetric

in general, and can be appreciably affected by gravity segregation, permeability anisotropy,

capillary pressure, and flow barriers.

This dissertation develops a general algorithm to simulate the process of mud-filtrate

invasion in vertical and deviated wells for drilling conditions that involve water- and oil-

base mud. The algorithm is formulated in cylindrical coordinates to take advantage of the

geometrical embedding imposed by the wellbore in the spatial distribution of fluids within

invaded formations. In addition, the algorithm reproduces the formation of mudcake due to

invasion in permeable formations and allows the simulation of pressure and fractional flow-

rate measurements acquired with dual-packer and point-probe formation testers after the

onset of invasion. An equation-of-state (EOS) formulation is invoked to simulate invasion

with both water- and oil-base muds into rock formations saturated with water, oil, gas, or

stable combinations of the three fluids. The algorithm also allows the simulation of physical

dispersion, fluid miscibility, and wettability alteration.

Discretized fluid flow equations are solved with an implicit pressure and explicit

concentration (IMPEC) scheme. Thermodynamic equilibrium and mass balance, together

with volume constraint equations govern the time-space evolution of molar and fluid-phase

concentrations. Calculations of pressure-volume-temperature (PVT) properties of the hy-

drocarbon phase are performed with Peng-Robinson’s equation of state. A full-tensor per-

meability formulation is implemented with mass balance equations to accurately model fluid

flow behavior in horizontal and deviated wells. The simulator is rigorously and successfully

verified with both analytical solutions and commercial simulators.

Numerical simulations performed over a wide range of fluid and petrophysical con-

ditions confirm the strong influence that well deviation angle can have on the spatial distri-

bution of fluid saturation resulting from invasion, especially in the vicinity of flow barriers.

Analysis on the effect of physical dispersion on the radial distribution of salt concentration

shows that electrical resistivity logs could be greatly affected by salt dispersivity when the

invading fluid has lower salinity than in-situ water. The effect of emulsifiers and oil-wetting

agents present in oil-base mud was studied to quantify wettability alteration and changes

viii

in residual water saturation. It was found that wettability alteration releases a fraction

of otherwise irreducible water during invasion and this causes electrical resistivity logs to

exhibit an abnormal trend from shallow- to deep-sensing apparent resistivity. Simulation

of formation-tester measurements acquired in deviated wells indicates that (i) invasion in-

creases the pressure drop during both drawdown and buildup regimes, (ii) bed-boundary

effects increase as the wellbore deviation angle increases, and (iii) a probe facing upward

around the perimeter of the wellbore achieves the fastest fluid clean-up when the density of

invading fluid is larger than that of in-situ fluid.

ix

Contents

Acknowledgments v

Abstract vii

List of Tables xvii

List of Figures xx

Chapter 1 Introduction 1

1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Review of Relevant Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Compositional Simulator . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Mud-Filtrate Invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Physical Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 Wettability Alteration . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.5 Formation-Tester Measurements Acquired in Horizontal and Deviated

Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Review of Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 2 Mathematical Models 14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

x

2.2 Mass Conservation and Constitutive Equations . . . . . . . . . . . . . . . . . 15

2.3 Auxiliary Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Saturation Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 Porosity Dependency on Pressure . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Phase Molar Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.4 Phase Composition Constraint . . . . . . . . . . . . . . . . . . . . . . 20

2.3.5 Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.6 Phase Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.7 Phase Mass Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.8 Phase Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.9 Relative Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.10 Phase Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Pressure Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Physical Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Moles of Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Modeling Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7.1 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.1.1 Lohrenz et al.’s correlation . . . . . . . . . . . . . . . . . . . 29

2.7.1.2 Quarter-Power Mixing rule . . . . . . . . . . . . . . . . . . . 31

2.7.1.3 Linear Mixing rule . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7.2 Relative Permeability Models . . . . . . . . . . . . . . . . . . . . . . . 31

2.7.2.1 Stone’s Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.7.2.2 Table Lookup for Relative Permeability . . . . . . . . . . . . 33

2.7.3 Capillary Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.7.4 Interfacial Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.8 Phase Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.8.1 Peng-Robinsion’s Equation of State . . . . . . . . . . . . . . . . . . . 36

2.8.1.1 Fugacity of Components . . . . . . . . . . . . . . . . . . . . . 38

xi

2.8.2 Molar and Mass Density . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.8.3 Derivatives of the Pressure Equation . . . . . . . . . . . . . . . . . . . 39

2.8.3.1 Derivative of Total Volume of Fluid with Respect to Moles

of Components . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.8.3.2 Derivative of Total Volume of Fluid with Respect to Pressure 41

2.9 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 3 Computational Approach 44

3.1 Reservoir Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Discretization of the Pressure Equation . . . . . . . . . . . . . . . . . . . . . 48

3.3 Calculation of Transmissibilities . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3.1 Upstream Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.2 Fluid-Phase Transmissibility . . . . . . . . . . . . . . . . . . . . . . . 57

3.3.3 Capillary-Pressure Term . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.4 Gravity Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 Discretization of the Molar Mass Equation . . . . . . . . . . . . . . . . . . . . 59

3.5 Phase Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5.1 Phase Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5.1.1 Tangent-Plane Distance Approach . . . . . . . . . . . . . . . 61

3.5.1.2 Flash Calculation . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.1.2.1 Successive Substitution Method . . . . . . . . . . . 63

3.5.1.2.2 Newton’s Method . . . . . . . . . . . . . . . . . . . 64

3.5.1.3 Combination of the Successive Substitution Method and New-

ton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5.2 Phase Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6 Boundary and Well Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6.2 Well Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

xii

3.6.2.1 Injection with Constant Volume Rate . . . . . . . . . . . . . 67

3.6.2.2 Injection with Constant Bottomhole Pressure . . . . . . . . . 68

3.6.2.3 Production with Constant Volumetric Rate . . . . . . . . . . 69

3.6.2.4 Production with Constant Bottomhole Pressure . . . . . . . 70

3.7 Computation of Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.8 Material Balance Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.9 Automatic Time-Step Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.10 Structure and Solution of the Pressure Equation . . . . . . . . . . . . . . . . 74

Chapter 4 Verification of the Simulator 76

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Description of Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.1 Rock Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.2 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 One-Dimensional Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.1 Two-Phase Flow Simulations . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.1.1 Gas-Water Simulation . . . . . . . . . . . . . . . . . . . . . . 78

4.3.1.2 Oil-Water Simulation . . . . . . . . . . . . . . . . . . . . . . 81

4.3.2 Three-Phase Flow Simulations . . . . . . . . . . . . . . . . . . . . . . 83

4.3.3 Variable Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.4 Dispersion of Salt Concentration . . . . . . . . . . . . . . . . . . . . . 86

4.3.4.1 Case 1: Rock Type I . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.4.2 Case 2: Rock Type II . . . . . . . . . . . . . . . . . . . . . . 91

4.4 Two-Dimensional Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.1 Two-Dimensional Axisymmetric Simulations . . . . . . . . . . . . . . 94

4.4.2 Two-Dimensional Radial Simulation . . . . . . . . . . . . . . . . . . . 95

4.4.3 Two-Dimensional Horizontal-Well Simulations . . . . . . . . . . . . . 100

4.4.3.1 Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xiii

4.4.3.2 Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.4.3.3 Case III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5 Three-Dimensional Cylindrical Simulation . . . . . . . . . . . . . . . . . . . . 105

4.5.1 Sampling after WBMF Invasion into an Oil-Bearing Formation . . . . 109

4.5.2 Sampling after OBMF Invasion into a Gas-Bearing Formation . . . . . 109

4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Chapter 5 Simulation of Mud-Filtrate Invasion 117

5.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Validation of the Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3 Simulations of the Process of Mud-Filtrate Invasion . . . . . . . . . . . . . . . 124

5.3.1 Case Study of Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . 124

5.3.2 Case Study of Three-Phase Flow . . . . . . . . . . . . . . . . . . . . . 126

5.3.3 Comparison of Oil- and Water-Base Mud-Filtrate . . . . . . . . . . . . 129

5.3.4 Mud-Filtrate Invasion In Vertical Wells . . . . . . . . . . . . . . . . . 130

5.3.5 Mud-Filtrate Invasion in Deviated Wells . . . . . . . . . . . . . . . . . 133

5.3.6 Physical Dispersion During Mud-Filtrate Invasion . . . . . . . . . . . 134

5.3.7 Injection of the Fresh Water . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3.8 Injection of Salty Water . . . . . . . . . . . . . . . . . . . . . . . . . . 137


Chapter 6 Simulation of Wettability Alteration 145

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.2 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.3 Mudcake Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4 Wettability Alteration Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.5 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.6 Flow Rate of Mud-Filtrate Invasion . . . . . . . . . . . . . . . . . . . . . . . . 150

xiv

6.6.1 Effect of Formation Permeability on the Flow Rate of Mud-Filtrate

Invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.6.2 Effect of Mudcake Permeability on the Flow Rate of Mud-Filtrate

Invasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.6.3 Effect of Wettability Alteration on the Rate of Mud-Filtrate Invasion 155

6.7 Wettability Alteration Effects on Saturation and Resistivity . . . . . . . . . . 155

6.7.1 Effect of Different OBM Emulsifiers on Wettability Alteration . . . . . 157

6.7.2 Effect of Mudcake Reference Permeability . . . . . . . . . . . . . . . . 158

6.8 Wettability Alteration in Oil- and Gas-Bearing Formations and Correspond-

ing Effect on Water Saturation and Electrical Resistivity . . . . . . . . . . . . 160

6.8.1 Oil-Base Mud-Filtrate Invasion Into an Oil-Saturated Formation . . . 162

6.8.2 Oil-Base Mud-Filtrate Invasion into a Gas-Bearing Formation . . . . . 165


Chapter 7 Simulation of Formation-Tester Measurements Acquired in De-

viated Wells 171

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3 Simulations of Dual-Packer Formation-Tester Measurements . . . . . . . . . . 176

7.4 Petrophysical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.5 Mud-Filtrate Invasion in Deviated Wells . . . . . . . . . . . . . . . . . . . . . 179

7.6 Simulation of Probe-Type FTMs Acquired in Thinly-Bedded Formations . . . 184

7.6.1 Drawdown-Buildup Test in Deviated Wells . . . . . . . . . . . . . . . 184

7.6.2 Cleanup Time and Fluid Sampling . . . . . . . . . . . . . . . . . . . . 186

7.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Chapter 8 Summary, Conclusions, and Recommendations 194

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

8.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

xv

8.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . 200

Nomenclature 204

Appendix A Discretization of Physical Dispersion Terms 216

Appendix B Permeability Tensor Transformation 219

Bibliography 223

xvi

List of Tables

2.1 List of variables included in equation (2.9). . . . . . . . . . . . . . . . . . . . 17

2.2 List of auxiliary relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 List of parameters in Lohrenz et al.’s (1964) viscosity correlation, eq. (2.60). . 31

4.1 Absolute permeability, porosity, residual water saturation, and residual oil

saturation for three synthetic rock types assumed in this chapter. . . . . . . . 78

4.2 Properties of hydrocarbon components assumed in the simulator verification;

Pcrit, Tcrit, ω, Mw, Vcrit, and Ψi are critical pressure, critical temperature,

acentric factor, molecular weight, critical molar volume, and parachor of the

components, respectively. IC4, IC5, and FC6 through FC18 are pseudo com-

ponents (Source: CMG-WinProp). . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 Assumed properties for the water component. . . . . . . . . . . . . . . . . . . 80

4.4 Properties assumed in the description of the reservoir. . . . . . . . . . . . . . 80

4.5 Assumed initial reservoir properties for gas and water. . . . . . . . . . . . . 80

4.6 Summary of initial conditions assumed for the reservoir containing oil and

water fluid phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.7 Summary of geometrical, fluid, petrophysical, and Brooks-Corey’s properties

assumed in the simulations described in Section 4.3.4. . . . . . . . . . . . . . 92

4.8 Summary of petrophysical and fluid properties for different rock types as-

sumed in Section 4.3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

xvii

4.9 Summary of parameters assumed in the Lohrenz et al.’s (1964) viscosity cor-

relation 2.60 for the simulations described in Section 4.3.4. . . . . . . . . . . . 92

4.10 Summary of geometrical and numerical parameters assumed for the numerical

simulation described in Section 4.4.1. . . . . . . . . . . . . . . . . . . . . . . 97

4.11 Summary of formation rock, rock fluid properties, initial conditions, and

boundary conditions assumed in sections 4.4.1 and 4.4.2. . . . . . . . . . . . . 97

4.12 Summary of geometrical and numetrical parameters assumed in the simula-

tions described in Section 4.4.2. . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.13 Summary of geometrical, fluid, and petrophysical properties assumed in the

simulations described in Section 4.4.3. . . . . . . . . . . . . . . . . . . . . . . 101

4.14 Summary of PVT properties of in-situ hydrocarbon components assumed in

the EOS calculations described in Section 4.4.3. . . . . . . . . . . . . . . . . 102

4.15 Summary of geometrical, petrophysical, and numerical properties/parameters

assumed in the simulations described in Section 4.5.1. . . . . . . . . . . . . . 110

5.1 Summary of assumed mudcake parameters used in the numerical simulation

of mud-filtrate invasion (field Mud 97074) (Dewan and Chenevert, 2001; Wu,

2004). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Summary of assumed mudcake properties in the numerical simulations of

mud-filtrate invasion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.3 List of parameters assumed in the description of the reservoir. . . . . . . . . . 124

5.4 List of parameters assumed in this chapter for Archie’s (1942) equation to

calculate rock electrical resistivity. . . . . . . . . . . . . . . . . . . . . . . . . 127

5.5 Absolute permeability, porosity, residual water saturation, and residual oil

saturation for three synthetic rock types assumed in Sections 5.3.4 and 5.3.5.

Figure 5.12 shows the relative permeability and capillary pressure curves

corresponding to these rock types. . . . . . . . . . . . . . . . . . . . . . . . . 132

xviii


assumed in the simulations described in this chapter. . . . . . . . . . . . . . 150

6.2 Summary of PVT properties of in-situ hydrocarbon and mud-filtrate com-

ponents assumed in equation-of-state calculations described in this chapter.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.3 Summary of Brooks-Corey’s properties assumed in the simulations described

in this chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.4 Summary of petrophysical and fluid properties for different rock types as-

sumed in the simulations of the process of mud-filtrate invasion described in

this chapter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.5 Summary of PVT properties for in-situ hydrocarbon and mud-filtrate com-

ponents assumed in equation-of-state calculations described in this chapter.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.6 Summary of mudcake and mud filtrate properties assumed in the simulations

of the process of mud-filtrate invasion. . . . . . . . . . . . . . . . . . . . . . 167

7.1 Summary of petrophysical properties assumed for different rock types in the

numerical simulations described in this chapter. . . . . . . . . . . . . . . . . 179


assumed in the simulations described in this chapter. . . . . . . . . . . . . . 181

7.3 Summary of PVT properties and in-situ hydrocarbon components assumed

in the equation-of-state calculations described in this chapter (Source: CMG-

WinProp). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

xix

List of Figures

3.1 Discreption of a point in a discretized grid block in cylindrical coordinates. . 45

3.2 Description of discretization of a grid block with neighboring blocks in (a)

horizontal plane and (b) vertical direction. Indices r,Θ, and Z identify radial,

azimuthal, and vertical locations, respectively. Subscripts r, θ, and z identify

element numbers in radial, azimuthal, and vertical directions. . . . . . . . . 46

3.3 Discreption initialization of grid blocks when a petrophysical bed boundary

does not conform with the gridding system. Based on the location of block

center with respect to the bed boundary, petrophysical properties are initial-

ized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Structure of the matrix for pressure equation (3.35) when constructed for a

model with 4x2x3 grids in radial, azimuthal, and vertical directions. . . . . . 75

4.1 Water-oil (a) capillary pressure and (b) relative permeability curves of rock

types studied in this dissertation. Variables kro and krw are relative perme-

ability of oil and water, respectively. . . . . . . . . . . . . . . . . . . . . . . . 77

xx

4.2 Rock Type 1: Comparison of calculated (a) pressure and (b) water satura-

tion with CMG-GEM and UTFEC along the radial direction at three different

times after the onset of injection. Initial pressure = 1500 [psi], initial water

saturation = 0.25, and initial composition (0.3, 0.6, and 0.1) for components

(C1, C2, and C3). The maximum time of water injection is 1 day with a

constant flow rate of 10 [STW/day]. . . . . . . . . . . . . . . . . . . . . . . . 82

4.3 Rock-type 1: Comparison of results for (a) salt concentration and (b) elec-

trical resistivity calculated with CMG-STARS and UTFEC along the radial

direction at three different times after the onset of injection. Initial pressure

= 1500 [psi], initial water saturation = 0.25, and initial composition (0.3,

0.6, and 0.1) of components (C1, C2, and C3). The maximum time of water

injection is 1 day with a constant flow rate of 10 [STW/day]. . . . . . . . . . 82

4.4 Rock Type 2: Comparison of calculated (a) pressure and (a) water satu-

ration with CMG-GEM and UTFEC along the radial direction at different

times. The boundary condition is 1 day injection of oil with a composition

(0.1, 0.9) of FC10 and FC18, and with a bottomhole pressure constraint of

3800 [psi]. After injection, fluid withdrawal takes place for 1 day with a

constant flow rate of 5 [bbl/day]. . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Rock Type 2: Comparison of results for resistivity calculated with CMG-

GEM and UTFEC along the radial direction at different times. The boundary

condition is 1 day injection of oil with a composition (0.1, 0.9) of FC10 and

FC18, and with a bottomhole pressure constraint of 3800 [psi]. After injection,

fluid withdrawal takes place for 1 day with a constant flow rate of 5 [bbl/day]. 84

4.6 Rock Type 3: Comparison of calculated (a) pressure and (b) water satura-

tion with CMG-GEM and UTFEC along the radial direction at three different

times after the onset of injection. The boundary condition is 1 day of injec-

tion of oil with a composition (0.1, 0.3, 0.6) of components (C1, C3, and FC7)

imposed by a constraining bottomhole pressure of 1300 [psi]. . . . . . . . . . 85

xxi

4.7 Rock Type 3: Comparison of calculated (a) oil and (b) gas saturations with

CMG-GEM and UTFEC along the radial direction at three different times

after the onset of injection. The boundary condition is 1 day injection of oil

with a composition (0.1, 0.3, 0.6) of components (C1, C3, and FC7) imposed

by a constraining bottomhole pressure of 1300 [psi]. . . . . . . . . . . . . . . 85

4.8 Rock Type 2: Comparison of (a) pressure and (b) water saturation cal-

culated with CMG-GEM and UTFEC along the radial direction at three

different times after the onset of injection. The boundary condition is 1 day

injection of oil with a composition (0.15, 0.15, 0.35, 0.35) of components (C1,

C2, FC6, and FC7), and with a constraint of bottomhole pressure equal to

1800 [psi]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.9 Rock type 2: Comparison of (a) oil and (b) gas saturations calculated with

CMG-GEM and UTFEC along the radial direction at three different times

after the onset of injection. The boundary condition is 1 day injection of oil

with a composition (0.15, 0.15, 0.35, 0.35) of components (C1, C2, FC6, and

FC7), and with a constraint of bottomhole pressure equal to 1800 [psi]. . . . 87

4.10 Phase envelope for four hydrocarbon components C1, C2, FC6, and FC7 with

a composition of (0.55, 0.35, 0.05, and 0.05). . . . . . . . . . . . . . . . . . . . 87

4.11 Assumed time-variation of injection flow rate. . . . . . . . . . . . . . . . . . . 88

4.12 Variable Flow Rate: Comparison of radial profiles of pressure at different

times after the onset of injection. Panel (a) shows that pressures increase at

the beginning of injection and panel (b) shows that pressures decrease with

time after 0.002 day of injection. Dynamic flow rate corresponding to those

simulations is shown in Figure 4.11. Invasion times are (a) = [0.011, 0.108,

1.088, 10.877, 108.771] seconds and (b) = [0.012, 0.126, 1.259] days. . . . . . 88

xxii

4.13 Rock Type 2: Comparison of radial profiles of (a) water saturation and (b)

salt concentration at different times after the onset of injection. Figure (4.11)

shows the imposed flow rate at different invasion times. Radial profiles are

shown at invasion times = [0.011, 0.109, 1.088, 10.877, 108.771, 1087.715,

10877.1552, 108771.552] seconds after the onset of injection. . . . . . . . . . 89

4.14 Dispersivity data measured for different rock types and different scales (Plot

adapted from John (2008)). This figure compares dispersivities measured

from laboratory echo tests, field scale echo tests (single well transmission

test), and with the traditional forward flow method. . . . . . . . . . . . . . . 89

4.15 Rock Fluid Properties: (a) water-oil relative permeability and (b) cap-

illary pressure curves assumed for Rock Type I (solid lines) and Rock Type

II (dotted lines); krw and kro are relative premeabilities of water and oil,

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.16 Radial distributions of (a) water saturation and (b) salt concentration calcu-

lated after 1 day from the onset of water injection with a constant flow rate of

0.5 [bbl/day]. The dashed blue and solid red curves identify water saturation

calculated with UTFEC and CMG-STARS, respectively. Initial water satu-

ration is equal to 0.20 and residual water saturation is equal to 0.07. Connate

water salinity equals 168 [kppm NaCl] and invading-water salinity equals 3

[kppm NaCl]. The formation exhibits the petrophysical properties of Rock

Type I (described in Table 4.8 and Figure 4.15). . . . . . . . . . . . . . . . . 93

xxiii

4.17 Radial distributions of (a) water saturation (b) salt concentration calculated

1 day after the onset of water injection with a constant rate of 0.5 [bbl/day].

The dashed blue and solid red curves identify water saturation calculated with

UTFEC and CMG-STARS, respectively. Initial water saturation is equal to

0.20 which is equal to residual water. Connate water salinity equals 3 [kppm

NaCl] and invading-water salinity equals 168 [kppm NaCl]. The formation

exhibits the petrophysical properties of Rock Type II (described in Table 4.8

and Figure 4.15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.18 2D Axisymmetric Model: Spatial distributions of (a) water saturation

calculated with UTFEC and (b) difference between water saturations cal-

culated with UTFEC and CMG-GEM. Initially, it was assumed that the

formation was invaded to a radial depth of 2.5 [ft] before the onset of fluid

sampling. Sampling takes place between the depths of 2139 to 2140 [ft] at

a constant rate of 10 [bbl/day] for 12 [hrs]. In the above figures, radial and

vertical distances are displayed in logarithmic and linear scales, respectively.

The formation exhibits petrophysical properties of Rock Type I described

in Table 4.1 and Figure 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.19 2D Axisymmetric Model: Spatial distributions of (a) pressure calculated

with UTFEC and (b) relative difference between pressures calcualted with

UTFEC and CMG-GEM. Initially, it was assumed that the formation was

invaded to a radial depth of 2.5 [ft] before the onset of fluid sampling. Sam-

pling takes place between the depths of 2139 to 2140 [ft] at a constant rate

of 10 [bbl/day] for 12 [hrs]. In the above figures, radial and vertical distances

are displayed in logarithmic and linear scales, respectively. The formation ex-

hibits petrophysical properties of Rock Type I described in Table 4.1 and Fig-

ure 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

xxiv

4.20 2D Axisymmetric Model: Time evolution of fractional flow of water, Fw,

for fluid sampled at the sand face during fluid withdrawal. The maximum dif-

ference between calculated fractional-flow curves obtained with UTFEC and

CMG-GEM is less than 4×10−3. The formation exhibits petrophysical prop-

erties of Rock Type I described in Table 4.1 and Figure 4.1. Figures 4.18(a)

and 4.19(a) show spatial distributions of water saturation and pressure cor-

responding to this sampling process. . . . . . . . . . . . . . . . . . . . . . . . 96

4.21 2D Radial Model: Spatial distributions (planar view) of (a) water satura-

tion obtained with UTFEC and (b) the difference between water saturations

calculated with UTFEC and CMG-GEM. It was assumed that the formation

was invaded to a radial length of 2.5 [ft] before the onset of sampling. Fluid

sampling takes place within an azimuthal angle from 0 to 18 at a constant

flow rate of 10 [bbl/day] for 12 [hrs]. The formation exhibits a permeabil-

ity of 100 [md] and a porosity of 0.25 [fraction]. Remaining petrophysical

properties are those of Rock Type 3, described in Table 4.1 and Figure 4.1. . 99

4.22 2D Radial Model: Spatial distributions (planar view) of (a) pressure ob-

tained with UTFEC and (b) the relative difference between pressures calcu-

lated with UTFEC and CMG-GEM. It was assumed that the formation was

invaded to a radial length of 2.5 [ft] before the onset of sampling. Fluid sam-

pling takes place within an azimuthal angle from 0 to 18 at a constant flow

rate of 10 [bbl/day] for 12 [hrs]. The Formation exhibits a permeability of 100

[md] and a porosity of 0.25 [fraction]. Remaining petrophysical properties are

those of Rock Type 3, described in Table 4.1 and Figure 4.1. . . . . . . . . . 99

xxv

4.23 2D Radial: Time evolution for water fractional flow, Fw, of the fluid sampled

at the sand face during fluid pumpout. Maximum difference between simu-

lation results calculated with UTFEC and CMG-GEM is less than 2× 10−3.

Formation exhibits a permeability of 100 [md] and a porosity of 0.25 [frac-

tion]. Remaining petrophysical properties are those of Rock Type 3 described

in Table 4.1 and Figure 4.1. Figures 4.21(a) and 4.22(a) show spatial dis-

tributions of water saturation and pressure corresponding to this sampling

process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.24 Water-oil relative permeability curves assumed for the base case correspond-

ing to simulations described in Section 4.4.3; krw and kro are relative perme-

abilities of water and oil fluid phases, respectively. . . . . . . . . . . . . . . . 102

4.25 2D Radial Horizontal Well: 2D spatial (cross section of a plane perpen-

dicular to the well axis) distributions of (a) water saturation obtained with

UTFEC and (b) the difference between water saturations calculated using

CMG-GEM and UTFEC after 10 days from the onset of water injection with

a constant flow rate of 0.00475 [bbl/day]. Initially, water saturation is equal

to residual water saturation, Swi = Swirr = 0.38 [fraction]. Saturating oil

exhibits a specific density of 0.87 and formation permeability is equal to 1000

[md]. Remaining properties of the formation are those of the base case de-

scribed in Table 4.13 and Figure 4.24. . . . . . . . . . . . . . . . . . . . . . . 104



UTFEC and (b) the difference between water saturations calculated with


a constant flow rate of 0.00475 [bbl/day]. Saturating oil exhibits specific

density of 0.76 and formation permeability is equal to 500 [md]. Remaining

properties of the formation are those of the base case. This case study is

described in Section 4.4.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

xxvi



UTFEC and (b) the difference between water saturations calculated with


a flow rate of 0.095 [bbl/day]. Horizontal permeability is equal to 100 [md]

and Raniso = 10. Remaining petrophysical properties of the formation are

those of base case described in Table 4.13 and Figure 4.24. Section 4.4.3.3

describes this case study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.28 3D Cylindrical Model: Geometrical description of a deviated well (for

vertical wells, θw = 0) in cylindrical coordinates used in the formulation of

fluid-flow equations described in this dissertation. In this graph, r, θj , and z

designate the radial location, azimuthal angle, and vertical location, respec-

tively; n is the unit normal vector to the bedding plane, h is bed thickness, zp

is the vertical distance from probe to a bed boundary, θw is wellbore deviation

angle measured from the bedding normal vector, n, and qmf is mud-filtrate

flow rate; krr, kθθ, and kzz are diagonal terms of the permeability tensor after

transformation to cylindrical coordinates. . . . . . . . . . . . . . . . . . . . . 107

4.29 Description of the probe-type formation tester assumed in the numerical sim-

ulations of fluid withdrawal performed with the developed algorithm. . . . . . 108

4.30 3D Cylindrical Vertical Well: 3D spatial distributions of (a) water satura-

tion and (b) the difference between water saturations calculated with CMG-

GEM and UTFEC after 12 hours from the onset of fluid sampling. Fluid

withdrawal takes place through azimuthal angles 252 to 288 [degrees]. For-

mation petrophysical properties are those of Rock Type 1 described in Ta-

ble 4.1. The formation was invaded with WBM to a radial length of 2.5 [ft]

prior to fluid withdrawal. Fluid sampling takes place with a constant flow

rate of 10 [bbl/day]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xxvii

4.31 3D Cylindrical Vertical Well: 3D spatial distributions of (a) pressure

obtained with UTFEC and (b) the relative difference between pressures cal-

culated with CMG-GEM and UTFEC after 12 hours from the onset of fluid

sampling. Fluid withdrawal takes place through azimuthal angles 252 to 288

[degrees]. Formation petrophysical properties are those of Rock Rype 1 de-

scribed in Table 4.1. The formation was invaded with WBM to a radial

length of 2.5 [ft] prior to fluid withdrawal. Fluid sampling takes place with a

constant flow rate of 10 [bbl/day]. . . . . . . . . . . . . . . . . . . . . . . . . 112

4.32 3D Cylindrical Vertical Well: Time evolution of the fractional flow of

water, Fw, for fluid sampled at the sand face during fluid withdrawal. . . . . 114

4.33 3D Cylindrical Vertical Well: Time evolution of GOR for the fluid sam-

pled at the sand face during pumpout. . . . . . . . . . . . . . . . . . . . . . 114

4.34 3D Cylindrical Vertical Well: 3D spatial distribution of hydrocarbon

components (a) C1 and (b) FC18 obtained with UTFEC after 0.5 days from

the onset of fluid sampling through azimuthal angles 0 to 18 and at a depth

of 2133 − 2135 [ft]. The formation was previously invaded with OBMF to a

radial length of 2.5 [ft]. Fluid sampling takes place with a constant flow

rate of 10 [bbl/day]. Due to symmetry, a half-cylinder model is used in the

numerical simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.35 3D Cylindrical Vertical Well: 3D spatial distribution (a) pressure ob-

tained with UTFEC and (b) the relative difference between pressures calcu-

lated using using UTFEC and CMG-GEM after 12 [hrs] from the onset of fluid

sampling through azimuthal angles 0 to 18 and at a depth of 2133 − 2135

[ft]. The formation was previously invaded with OBMF to a radial length of

2.5 [ft]. Fluid sampling takes place with a constant flow rate of 10 [bbl/day].

Due to symmetry, a half-cylinder model is used in the numerical simulation. . 116

xxviii

5.1 Comparison of volume of filtrate obtained with numerical simulations using

UTFECS against that measured in the laboratory with field Mud 97074 (De-

wan and Chenevert, 2001). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 Time variation of mudcake overbalance pressure. The formation exhibits the

properties of Rock Type 3 described in Table 4.1 and Figure 4.1. . . . . . . . 122

5.3 Time variation of mud-filtrate flow rate after the onset of invasion into a

formation with (a) Rock Type I and (b) Rock Type III (rock types are de-

scribed in Table 4.1 and Figure 4.1). For each rock type, the following cases

are considered: presence of mudcake and no mudcake at the well boundary. . 123

5.4 Time variation of (a) mudcake thickness and (b) flow rate after the onset of

invasion for different values of reference mudcake permeability. The invaded

formation exhibits the petrophysical properties of Rock Type 1 described

in Table 4.1 and Figure 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.5 Two-Phase Flow: Comparison of radial distributions of (a) pressure and (b)

water saturation at different times after the onset of invasion for two cases:

(i) without presence of mudcake and (ii) with presence of mudcake. Initial

P=3500 [psi], Sw = 0.25 [fraction]. Well constraint is 1 day of WBMF inva-

sion with BHP=3800 [psi]. Mudcake reference permeability, Kmc0=0.3 [md],

mudcake reference porosity, φmc0=0.3 [fraction], and solid fraction, fs=0.06

[fraction]. The invaded formation exhibits the petrophysical properties of

Rock Type 1 described in Table 4.1 and Figure 4.1. . . . . . . . . . . . . . . . 125

5.6 Three-Phase Flow: Time variations of (a) mudcake thickness and (b) flow

rate after the onset of invasion for different values of reference mudcake per-

meability. The invaded formation exhibits the petrophysical properties of

Rock Type 3 described in Table 4.1 and Figure 4.1. . . . . . . . . . . . . . . . 127

xxix

5.7 Three-Phase Flow: Comparison of radial profiles of (a) pressure and (b)

water saturation calculated at different times after the onset of invasion for

two cases: without and with presence of mudcake. Initial P=500 [psi], Sw =

0.25 [fraction], temperature, T=200 [F], and composition (0.4, 0.3, and 0.3)

for pseudo components (C1, C3, and FC7). Well constraint is 1 day of water-

base mud invasion with BHP=1300 [psi]. Mudcake reference permeability,

Kmc0=0.3 [md], mudcake reference porosity, φmc0=0.3 [fraction], and solid

fraction, fs=0.06 [fraction]. The invaded formation exhibits the petrophysical


5.8 Three-Phase Flow: Comparison of radial profiles of (a) oil and (b) gas

saturation calculated at different times after the onset of invasion for two

cases: without and with presence of mudcake. Initial P=500 [psi], Sw = 0.25

[fraction], temperature, T=200 [F], and composition (0.4, 0.3, and 0.3) for

pseudo components (C1, C3, and FC7). Well constraint is 1 day of water-

base mud invasion with BHP=1300 [psi]. Mudcake reference permeability,

Kmc0=0.3 [md], mudcake reference porosity, φmc0=0.3 [fraction], and solid

fraction, fs=0.06 [fraction]. The invaded formation exhibits the petrophysical


5.9 Three-Phase Flow: Comparison of radila profiles of (a) salt concentration

and (b) electrical resistivity calculated at different times after the onset of

invasion for two cases: without and with presence of mudcake. Initial P=500

[psi], Sw = 0.25 [fraction], temperature, T=200 [F], and composition (0.4,

0.3, and 0.3) for pseudo components (C1, C3, and FC7). Well constraint is 1

day of water-base mud invasion with BHP=1300 [psi]. Mudcake reference per-

meability, Kmc0=0.3 [md], mudcake reference porosity, φmc0=0.3 [fraction],

and solid fraction, fs=0.06 [fraction]. The invaded formation exhibits the

petrophysical properties of Rock Type 3 described in Table 4.1 and Figure 4.1. 129

xxx


invasion of water-base and oil-base mud (µo = 0.5 [cp]). The invaded forma-

tion has the petrophysical properties of Rock Type 3 described in Table 4.1

and Figure 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130


invasion of water-base and oil-base mud (µo = 2.0 [cp]). The invaded forma-

tion has the petrophysical properties of Rock Type 3 described in Table 4.1

and Figure 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.12 Water-oil (a) capillary pressure and (b) relative permeability curves of rock

types studied in Sections 5.3.4 and 5.3.5. Variables kro and krw are relative

permeability of oil and water, respectively. Rock types 5-I, 5-II, and 5-III are

identified with square, circle, and star markers, respectively. . . . . . . . . . 131

5.13 Gas-oil (a) capillary pressure and (b) relative permeability curves of rock

types studied in Sections 5.3.4 and 5.3.5. Variables kro and krg are relative

permeability of oil and gas, respectively. Rock types 5-I, 5-II, and 5-III are

identified with square, circle, and star markers, respectively. . . . . . . . . . 132

5.14 Spatial (radial and vertical directions) distributions of (a) water saturation

and (a) gas saturation after three days from the onset of oil-base mud-filtrate

invasion into a formation with petrophysical properties of Rock-Type 5-III.

Overbalance pressure is assumed equal to 300 [psi], and mudcake reference

permeability is 0.03 [md] (described in Table 5.5 and Figure 5.12). . . . . . . 133

xxxi

5.15 Spatial distributions of water saturation after 10 days from the onset of water-

base mud-filtrate invasion into a formation with three petrophysical layers

(vertical axis is the true vertical depth). Wellbore deviation angle, θw, is

equal to 45 [degrees]. Overbalance pressure is assumed to be 300 [psi]. The

petrophysical properties of top, middle, and bottom layers are those of rock

5-I, 5-II, and 5-III, respectively (described in Table 5.5 and Figure 5.12).

Prior to WBM invasion, water saturations in all layers were assumed equal

to residual saturation. Saturating oil has an API of 55°. . . . . . . . . . . . . 135

5.16 Spatial distributions of salt concentration after 10 days from the onset of

water-base mud-filtrate invasion into a formation with three petrophysical

layers (vertical axis is the true vertical depth). Wellbore deviation angle, θw,

is equal to 45 [degrees]. The petrophysical properties of top, middle, and

bottom layers are those of rock 5-I, 5-II, and 5-III (described in Table 5.5

and Figure 5.12), respectively. Prior to WBM invasion, water saturations

in all layers were assumed equal to residual saturation. Connate water has

a salinity equal to 160 [kppm NaCl], whereas invading water has a salinity

equal to 3 [kppm NaCl]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.17 Radial distributions of water saturation calculated after 1 day from the onset

of water injection with a constant rate of 0.5 [bbl/day]. The dashed blue curve

identifies water saturation calculated with UTFEC, and the solid red curve

identifies water saturation obtained with CMG-STARS. Initially, the invaded

formation exhibits water saturation equal to 0.20 [fraction] and residual water

saturations equal to (a) 0.07 [fraction] (Sw,movable = 0.13 [fraction]) and (a) 20

[fraction] (Sw,movable = 0). Petrophysical properties of the invaded formation

are those of (a) Rock Type I and (b) Rock Type II described in Table 4.8. . 138

xxxii

5.18 Radial distributions of (a) salt concentration and (b) electrical resistivity

calculated after 1 day from the onset of water injection with a constant rate

of 0.5 [bbl/day]. The dashed, dotted, and dashed-dotted curves correspond

to radial profiles for dispersivity values (in equations (2.38) through (2.43)):

αl1 = α = 0, α = 0.2, and α = 1 [ft], respectively. Salt concentration in

the invaded formation is 168 [kppm NaCl], whereas salt concentration in the

invading water is 3 [kppm NaCl]. Figure 5.17(a) shows the radial distribution

of water saturation corresponding to this case. . . . . . . . . . . . . . . . . . 139

5.19 Radial distributions of (a) salt concentration and (b) electrical resistivity cal-

culated after 1 day from the onset of water injection with a constant flow rate



αl1 = α = 0, α = 0.2, and α = 1 [ft], respectively. Salt concentration in

the invaded formation is 168 [kppm NaCl], whereas salt concentration in the

invading water is 3 [kppm NaCl]. Figure 5.17(b) shows the radial distribution

of water saturation corresponding to this case. . . . . . . . . . . . . . . . . . 140


calculated after 1 [day] from the onset of water injection with a constant rate



αl1 = α = 0, α = 0.2, and α = 1 [ft], respectively. Initially, formation

is assumed to have water saturation equal to 0.20 [fraction] and residual

water saturation is equal to 0.07 [fraction] (Sw,movable = 0.13 [fraction]).

Salt concentration in the invaded formation is 3 [kppm NaCl], whereas salt

concentration in the invading water is 168 [kppm NaCl]. Figure 5.17(a) shows

the radial distribution of water saturation corresponding to this case. . . . . . 143

xxxiii


calculated after 1 [day] from the onset of water injection with a constant rate



αl1 = α = 0, α = 0.2, and α = 1 [ft], respectively. Initially, the formation is

assumed to have a water saturation equal to residual saturation (Swi = 0.20

[fraction] and Sw,movable = 0). Salt concentration in the invaded formation

is 3 [kppm NaCl], whereas salt concentration in the invading water is 168

[kppm NaCl]. Figure 5.17(b) shows the radial distribution of water saturation

corresponding to this case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.1 Water-oil relative permeability curves assumed for water-wet and oil-wet con-

ditions. Variables kro and krw are relative permeability of oil and water,

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.2 Water-oil capillary pressure, Pcow, curves assumed for water-wet and an oil-

wet rock surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.3 Calculated time variations of (a) mudcake thickness and (b) mud-filtrate flow

rate after the onset of invasion for different values of mudcake reference per-

meability. Formation permeability is assumed equal to 300 [md]; remaining

petrophysical properties are those of the base case. . . . . . . . . . . . . . . . 153

6.4 Calculated time variations of (a) mudcake thickness and (b) mud-filtrate flow

rate after the onset of invasion for different values of mudcake reference per-

meability. Formation permeability is assumed equal to 1 [md]; remaining

petrophysical properties are those of the base case. . . . . . . . . . . . . . . . 154

xxxiv

6.5 Radial distributions of (a) water saturation and (b) rock electrical resistivity

calculated at different times after the onset of invasion with OBMF con-

taining surfactant. Initially, the formation is assumed to be water-wet with

water saturation equal to residual saturation (0.16 [fraction]). After wettabil-

ity alteration, residual water saturation decreases to 0.12 [fraction]. Mudcake

reference permeability is assumed to be equal to 0.003 [md] and initial over-

balance pressure is 300 [psi]. Table 6.1 lists the parameters used in Archie’s

equation to calculate rock resistivities. Mud-filtrate viscosity is equal to 10

[cp]. Formation petrophysical properties are those of the base case. . . . . . . 156


calculated after 3 [days] of invasion with OBMF containing surfactant for

different values of reference mudcake permeability. Initially, the formation

is assumed to be water-wet with water saturation equal to residual satura-

tion (0.16 [fraction]). After wettability alteration, residual water saturation

decrease to 0.14, 0.12, 0.10, and 0.08 [fraction]. Overbalance pressure is 300

[psi]. Table 6.1 lists the parameters used in Archie’s equation to calculate

electrical resistivity values. Formation petrophysical properties are those of

the base case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158


calculated after 3 days from the onset of invasion with OBMF containing

surfactant for different values of reference mudcake permeability. Initially,

the formation is assumed to be water-wet with a water saturation equal to

residual saturation (16%). After wettability alteration, residual water satu-

ration decreases to 12%. Overbalance pressure is equal to 300 [psi]. Table 6.1

lists the parameters used in Archie’s equation to calculate resistivity values.

Petrophysical properties of the formation are those of the base case. . . . . . 159

xxxv

6.8 Array-induction (AIT) apparent resistivity curves simulated for the case of

OBMF invasion with mudcake reference permeability values equal to (a) 0.3

and (b) 0.003 md. Figures 6.7(a) and 6.7(b) show the corresponding radial

distributions of water saturation and rock electrical resistivity. . . . . . . . . 160

6.9 Water-oil relative permeability curves assumed for Rock Type I (dashed

lines), Rock Type II (solid lines), and Rock Type III (dotted lines) for two

different wettability conditions. Blue and red curves identify water- and oil-

wet conditions, respectively. residual water-saturation for oil-wet conditions

is smaller than that of water-wet conditions. . . . . . . . . . . . . . . . . . . . 161

6.10 Water-oil capillary pressure curves assumed for Rock Type I (dashed lines),

Rock Type II (solid lines), and Rock Type III (dotted lines) for two different

wettability conditions. Blue and red curves identify water- and oil-wet condi-

tions, respectively. In the case of oil-wet conditions, oil is the wetting phase

and capillary pressure becomes negative. . . . . . . . . . . . . . . . . . . . . . 161

6.11 Spatial (radial and vertical directions) distribution of (a) water saturation, (b)

electrical resistivity, and (c) array-induction apparent resistivitys log calcu-

lated after invasion of OMBF containing surfactant into an oil-saturated for-

mation. The formation exhibits petrophysical properties of Rock Type I

described in Table 6.4 and Figures 6.9 and 6.10. Archie’s properties for the

calculation of electrical resistivity are those listed in Table 6.1. . . . . . . . . 164


electrical resistivity, and (c) array-induction apparent resistivity logs calcu-


mation. The formation exhibits petrophysical properties of Rock Type II



xxxvi




mation. The formation exhibits petrophysical properties of Rock Type III





lated after invasion of OMBF containing surfactant into a gas-saturated for-

mation. The formation exhibits petrophysical properties of Rock Type I






mation. The formation exhibits petrophysical properties of Rock Type II






mation. The formation exhibits petrophysical properties of Rock Type III



xxxvii

7.1 Description of a dual packer-probe WFT deployed in a deviated well. The

observation probe does not withdraw fluid; it only measures fluid pressure

variations. In this diagram, n is the unit normal vector to bedding plane, θw

is wellbore deviation from the bedding normal vector, h is formation thickness

(100 [ft]), 2lw is the length of the dual packer (6 [in]), zo is distance of the

observation probe from the packer center (6 [in]), and zw is the distance of

the packer center from the lower bed boundary (200 [ft]). . . . . . . . . . . . 175

7.2 Comparison of pressure time variation at the packer center and an observa-

tion probe calculated using UTFEC and those calculated with an analytical

expression (Onur et al., 2004). Figure 7.1 describes the configuration of a

dual packer used to conduct the pressure test. The formation is assumed to

be saturated with water. It is assumed that packer pumps out fluid with a

constant flow rate of 10 [cc/sec] with the pressure test consisting of 1 [min]

fluid withdrawal followed by 9 [min] of pressure buildup. The observation

probe is located at 6 [in] above the packer center. Formation properties are

as follows: (a) Rock Type II: kh = 5 [md], kv = 0.5 [md], and porosity =

0.12 [fraction], (b) Rock Type II: kh = 5 [md], kv = 0.5 [md], and porosity =

0.12 [fraction], (c) Rock Type I: kh = 500 [md], kv = 50, and porosity = 0.32

[fraction], and (d) Rock Type I: kh = 500 [md], kv = 100 [md], and porosity

= 0.32 [fraction]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.3 Water-oil (a) relative permeability and (b) capillary pressure curves assumed

for (i) Rock Type I (solid lines) and (ii) Rock Type II (dot-dashed lines).

The symbols krw and kro designate relative permeabilities of water and oil,

respectively, fluid phases. Petrophysical properties of the rock types are given

in Table 7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

xxxviii

7.4 Spatial distribution of water saturation after 5 days from the onset of WBMF

invasion under an overbalance pressure of 200 [psi]. Wellbore deviations

are (a) 45, (b) 60, and (c) 80 degrees. It is assumed that mudcake refer-

ence permeability is 0.003 [md]. Petrophysical properties of the formation

are those of Rock Type I (described in Table 7.1 and Figure 7.3). . . . . . . . 182

7.5 Spatial distribution of water saturation after 5 days from the onset of WBMF

invasion under an overbalance pressure of 200 [psi]. Wellbore deviations

are (a) 45, (b) 60, and (c) 80 degrees. It is assumed that mudcake refer-

ence permeability is 0.003 [md]. Petrophysical properties of the formation

are those of Rock Type II (described in Table 7.1 and Figure 7.3). . . . . . . 183

7.6 Geometrical description of a deviated well model in cylindrical coordinates.

The variables r, θj , and z designate the radial location, azimuthal angle, and

vertical location, respectively; n is the unit normal vector to the bedding

plane, h is the bed thickness, zp is the probe vertical distance from the lower

horizontal boundary, and θw is wellbore deviation from the bedding normal

vector, n. Formation thickness is assume to be 10 [ft]; probes 1 and 2 are

located at vertical distances of 0.5 and 5 [ft], respectively, measured from the

lower shale boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

7.7 Time variations of pressure simulated during drawdown-buildup tests with

probe-type FTs conducted at points 1 and 2 (described in Figure 7.6) within a

thinly-bedded formation. In this graph, dashed lines and solid lines indentify

pressure variations at locations 1 and 2, respectively, in Figure 7.6. Prior to

the pressure test the formation has undergone (a) no invasion, and (b) WBM

invasion. Petrophysical properties of the formation are those of Rock Type

II (described in Table 7.1 and Figure 7.3). It is assumed that the formation

exhibits an isotropic permeability, i.e., Raniso= 1. The wellbore inclination

angle from the normal to bedding plane is assumed equal to 80 [degrees]. . . 186

xxxix

7.8 Comparison of pressure time variations recorded at probes 1 (dashed lines)

and 2 (solid lines) in formations with two different initial conditions: (i) not

invaded, and (ii) WBM invaded. Synthetic pressure responses are calculated

in formations penetrated with wells with deviation angles (a) 80 and (b) 30

degrees. Petrophysical properties of the formation are those of Rock Type

II (described in Table 7.1 and Figure 7.3). It is assumed that the formation

exhibits an isotropic permeability, i.e., Raniso= 1. Figure 7.6 describes the

geometrical configuration of the synthetic model. . . . . . . . . . . . . . . . . 187

7.9 Comparison of pressure time variations recorded at probes 1 (dashed lines)

and 2 (solid lines) in formations with two different initial conditions: (i) not

invaded, and (ii) WBM invaded. Synthetic pressure responses are calculated

in a formation with wellbore deviations of 30 degrees. It is assumed that

the formation exhibits an anisotropic permeability of 10 (Raniso= 10). Fig-

ure 7.8(b) shows pressure time variations when Raniso= 1. The petrophysical

properties of the formation are those of Rock Type II (described in Table 7.1

and Figure 7.3). Figure 7.6 describes the geometrical configuration of the

synthetic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

xl

7.10 Comparison of pressure time variations recorded with a probe-type FT de-

ployed in deviated wells. Synthetic pressure tests are conducted in three

wellbore deviation angles of 45, 60, and 80 [degrees]. The pressure tests are

conducted in deviated wells penetrated into formations with petrophysical

properties of (a) Rock Type I, and (b) Rock Type II. In (a) probe is located

at point 2, and in (b) probe is located in point 1; Figure 7.6 describes the

geometrical properties associated with this simulation. It is assumed that

the formation has been previously invaded with WBM before the onset of

pressure test. Figures 7.4 and 7.5 show the distribution of water saturation

after WBMF invasion. It is assumed that formation exhibits an anisotropic

permeability of Raniso= 10. Table 7.1 and Figure 7.3 describe petrophysical

properties of each rock type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

7.11 Spatial distribution of water saturation after 2.4 hours from the onset of fluid

sampling with a probe-type FT. Sampling takes place when the probe (a)

faces downward, (c) faces to the side, (b) faces upward, and (d) faces to the

side. Wellbore deviation is 80 degrees. Sampling takes place after WBMF

invasion for 5 days; Figure 7.4(c) shows the spatial distribution of water

saturation before the onset of fluid sampling. Petrophysical properties of the

formation are those of Rock Type I (as described in Table 7.1 and Figure 7.3).190

7.12 Spatial distribution of water saturation after 2.4 hours from the onset of fluid

sampling with a probe-type FT. Sampling takes place when the probe (a)

faces downward, (c) faces to the side, (b) faces upward, and (d) faces to the

side. Wellbore deviation is 45 degrees. Sampling takes place after WBMF

invasion for 5 days; Figure 7.4(a) shows the spatial distribution of water

saturation before the onset of fluid sampling. Petrophysical properties of the

formation are those of Rock Type I (as described in Table 7.1 and Figure 7.3).191

xli

7.13 Simulated time evolution of fractional flow of water during fluid withdrawal

at different azimuthal angles plotted in a (a) linear-linear and (b) log-log

scale. Wellbore deviation is equal to 60 degrees. Petrophysical properties of

the formation are those of Rock Type I (described in Table 7.1 and Figure 7.3).192

7.14 Simulated time evolution of fractional flow of water during fluid withdrawal

at different azimuthal angles plotted in a log-log scale. Wellbore deviations

are (a) 80 and (b) 45 degrees. The simulations were performed for a simple

probe FT. Petrophysical properties of the formation are those of Rock Type

I (as described in Table 7.1 and Figure 7.3). . . . . . . . . . . . . . . . . . . . 192

xlii

Chapter 1

Introduction

This chapter outlines the research objectives of the dissertation, reviews relevant literature

in the development of near-wellbore fluid-flow simulators, and previews all the subsequent

chapters.

1.1 Problem Statement

Oil-base mud (OBM) is increasingly being used to enhance the speed and efficiency of

drilling, decrease washouts, achieve better hole control, and minimize the swelling of shales (Ma-

lik, 2008; Theys, 1999). The composition of hydrocarbon components in the injected mud

filtrate is usually different from that of formation fluid. The miscibility of hydrocarbon

components in different fluid phases causes variations in properties such as relative perme-

ability, viscosity, compressibility, molar mass density, gas-oil ratio, etc. The spatial fluid

distribution after mud-filtrate invasion in the vicinity of a vertical well tends to be axisym-

metric. However, in deviated wells gravity segregation, capillary pressure, heterogeneity,

and permeability anisotropy bring about non-symmetric distributions of invading fluid in

the near-wellbore region. An eccentric distribution of mud-filtrate around the wellbore has

been observed in previous studies (Alpak et al., 2003; Angeles et al., 2011; Moinfar et al.,

1

2010). On the other hand, oil-base mud filtrate (OBMF) contains emulsifier and oil-wetting

agents. Experimental studies (Sharma and Wunderlich, 1985; Yan et al., 1993) show that

surfactants within OBMF can change the wettability of the rock surface. It was found that

residual water saturation decreases in those regions in contact with OBMF emulsifiers. Field

measurements (Salazar and Martin, 2010; Salazar and Torres-Verdın, 2009) also indicate

that changes in wettability and, consequently, water saturation take place in the vicinity of

the wellbore after OBMF invasion. A comprehensive study of mud-filtrate invasion requires

the development of a method which considers prominent alterations in mudcake, fluid, and

rock-fluid (relative permeability and capillary pressure) properties.

Invasion affects several borehole measurements including electrical resistivity, sonic

slowness, neutron porosity, density, and pressure. Measurements acquired with formation

testers (FT) enable petrophysicists to estimate formation properties such as horizontal and

vertical permeabilities, skin, and initial pressure. Traditionally, pressure transient anal-

yses were interpreted using single-phase analytical expressions by neglecting the effect of

mud-filtrate invasion. Pressure-transient well tests with probe-type FTs normally require

a few minutes for fluid withdrawal and a few minutes for pressure buildup; therefore, the

assumption of single-phase flow can introduce errors in estimations performed from pressure-

transient measurements. Studies by Malik et al. (2007) and Angeles et al. (2007b) show that

OBM and water-base mud (WBM) invasion affect the estimation of permeability performed

with formation-tester measurements (FTM). In high-angle wells, invasion and its resulting

asymmetric spatial distribution of mud filtrate around the wellbore affect formation-tester

measurements (Angeles et al., 2011, 2009). Preceeding analyses on FTMs (Abbaszadeh and

Hegeman, 1990; Angeles et al., 2005; Dubost et al., 2004; Onur et al., 2004) were performed

with the assumption that the FT tool was located at the center of the formation. However,

field experiments reveal that FTMs in deviated wells are affected by nearby bed boundaries.

Wireline formation testers (WFT) are used to collect fluid samples from multiple

zones in a well. WFTs often calculate compositional and pressure-volume-temperature

(PVT) properties of the collected fluid sample at reservoir condition. However, fluid samples

2

acquired within a formation drilled with OBM are often contaminated with mud filtrate. A

high contamination level (above 10-15% for oil and 1-3% for gas condensate) renders the

hydrocarbon sample inadequate for fluid characterization (Dong et al., 2006). To circumvent

this problem, service companies in the oil industry have developed new sampling tools such

as simple, focused, and oval-focused probes (Hadibeik et al., 2009). Due to expense and risks

associated with a long cleanup time, early estimation of cleanup time has been of interest

to many researchers (Angeles et al., 2009; Malik et al., 2009a; Sherwood, 2005). Several of

the previous published works that approximated the time for contamination cleanup with

a probe-type FT assumed either a vertical well or immiscible flow (Gok et al., 2006; Malik

et al., 2009c; Mullins and Schroer, 2000; Sheng, 2006; Sherwood, 2005). Previous analysis

methods of FTMs in high-angle wells (Angeles et al., 2011, 2009) were accurate only within

a limited range of wellbore deviation angles. Moreover, due to the non-conformality of

simulation grids with wellbore geometry, such simulations required complicated grid design,

grid refinements, and adjustments to formation transmissibilities.

To the best of the author’s knowledge, thus far there has been no attempt to develop

a method which fully integrates the processes of mud-filtrate invasion and formation-tester

sampling in horizontal and deviated wells. The simulator should enforce a phase-behavior

algorithm to calculate fluid properties in the presence of spatial variations of fluid composi-

tion and pressure. Moreover, a three-dimensional (3D) configuration is needed to simulate

invasion in high-angle wells as well as probe-type FTMs in vertical, horizontal, and deviated

wells.

The central objective of this dissertation is to develop a 3D cylindrical fluid-flow

simulator to numerically model invasion and formation-tester measurements acquired at

any wellbore deviation. The developed simulator integrates several near-wellbore promi-

nent fluid-flow phenomena such as multi-phase fluid flow, physical dispersion, miscibility,

mudcake growth, and wettability alteration.

Cylindrical coordinates were chosen for the numerical algorithm for several following

reasons: (i) the method’s applications (simulating mud-filtrate invasion and FTMs) concern

3

a single-well model, (ii) it benefits from the axial symmetry to solve for invasion taking

place in vertical wells, (iii) numerical grids are conformal with the wellbore geometry, (iv)

when simulating FTMs, there are smooth variations in the spatial (radial, azimuthal, and

vertical directions) distributions of pressure and fluid phases in the vicinity of the wellbore.

1.2 Review of Relevant Literature

I have surveyed the literature describing topics in fluid-flow simulations, including: for-

mulation of compositional simulators, mud-filtrate invasion, wettability alteration, physical

dispersion, and formation-tester measurements.

1.2.1 Compositional Simulator

In a compositional fluid-flow simulator, mass conservation equations together with ther-

modynamic equilibrium equations are solved to calculate pressure, concentration of com-

ponents, composition of phases, and subsequently, fluid saturations. There are numerous

published works describing the development of compositional simulators for different appli-

cations. For example, Thele (1984) and Thele et al. (1983) include a thorough literature

review of compositional simulators that use the K-value approach. This dissertation imple-

ments an equation-of-state (EOS) approach to develop a mathematical model for simulating

fluid flow through porous media. The formulation of EOS compositional simulators is di-

vided into the following categories:

Fully-implicit formulations:

Coats (1980) developed a fully-implicit, 3D, three-phase simulator, which included rel-

ative permeability and capillary pressure in his fluid-flow mathematical formulations;

however, his formulations neglected physical dispersion. Coats used a modified ver-

sion of the Redlich-Kwong EOS (1949) to compute properties of hydrocarbon phases.

He applied the Newton-Raphson’s method to simultaneously compute pressure, fluid

saturation, and phase compositions.

4

Chien et al. (1985) also developed a fully-implicit formulation, but selected a different

set of primary variables from those of Coats’ formulation. In Chien et al.’s formulation,

aqueous and hydrocarbon fluid phases were immiscible. They implemented the Peng-

Robinson’s EOS (1976) to model hydrocarbon phase behavior.

IMPES-type formulations:

Nghiem et al. (1981) and Nghiem (1983) proposed an IMPES-type formulations for the

development of a compositional fluid-flow simulator. In their mathematical formula-

tion, hydrocarbon components were allowed to enter the aqueous phase. However, wa-

ter was assumed to remain only in the aqueous phase. They used the Peng-Robinson’s

EOS (1976) to calculate properties of hydrocarbon phases and Henry’s law to calculate

the concentration of hydrocarbon components in the aqueous phase.

Acs et al. (1985) developed an IMPES-type formulation, similar to Nghiem et al.’s (1981)

formulation. In Acs et al.’s algorithm, fluid-flow equations included composition-

dependent terms. Their pressure equation was derived based on the premise that pore

volume should be filled with the total volume of the fluid.

Chang et al. (1990) and Chang (1990) used an approach similar to Acs et al.’s. They

implemented a more efficient formulation for constructing the pressure matrix and

included the case of four-phase flow.

Malik et al. (2007) developed a compositional simulator for near-wellbore problems

using a similar approach to that of Chang et al.’s (1990). They designed their simulator

assuming azimuthal symmetry in formation properties with respect to a vertical axis.

Recently, Pour (2008) used Acs’s formulation to develop a new one-dimensional (1D)

radial compositional simulator for near-wellbore applications.

Sequential implicit compositional formulation:

Watts (1986) expanded Acs et al.’s formulation to solve the pressure equation implic-

itly and developed a sequential semi-implicit algorithm solving for time dependent fluid

saturation. In Watts’ formulation, both capillary pressure and relative permeability

5

were calculated implicitly.

Adaptive-implicit formulation:

Collins et al. (1992) developed a compositional simulator with the idea that only a few

grid blocks required an implicit solution while the remaining blocks did not. Based on

changes of fluid saturation and overall fluid composition, they exchanged grid blocks

into implicit or explicit groups; the Newton-Raphson’s method is implemented to

calculate pressure and composition. Subsequently, phase composition was computed

using flash calculations.

In summary, considering the stability, numerical efficiency, accuracy, and imple-

mentation simplicity for different near-wellbore physical models, I chose an IMPEC-type

discretization scheme to develop the simulator discribed in this dissertation.

1.2.2 Mud-Filtrate Invasion

Mud-filtrate invasion affects borehole logging measurements such as electrical resistivity,

neutron porosity log, sonic slowness, and formation testing. The radial length of inves-

tigation of most of well-logging instruments is limited to a few inches. For this reason,

understanding mud-filtrate invasion is important in well-log analysis. Mud-filtrate invasion

takes place because of the overbalance pressure originated from the difference between hy-

drostatic pressure at the wellbore and reservoir fluid pressure. In the last five decades,

extensive studies have been conducted concerning formation damage and especially mud-

filtrate invasion (Bennion, 1999; Bennion et al., 1996; Civan and Engler, 1994; Ding, 2011;

Donaldson and Chernoglazov, 1987; Hawkins Jr., 1956; Iscan et al., 2007; Suryanarayana

et al., 2007; Windarto et al., 2011; Wu et al., 2004; Yan et al., 1993). A number of pub-

lished works (Civan, 2007; Ding and Renard, 2005; Moghadasi et al., 2006; Parn-anurak and

Engler, 2005; Wu et al., 2009) have proposed standalone invasion models.

Laboratory experiments have been performed to quantify the invasion flow rate.

For instance, Ferguson and Klotz (1954), Bezemerand and Havenaar (1966), and Fordham

6

et al. (1988) obtained relations for invasion flow rates; however, these investigations were

limited to specific types of mud. Dewan and Chenevert (2001) proposed relations for the

time variation of mudcake properties such as mudcake permeability and porosity. In their

model, variations of mudcake properties were related to dynamic alterations of mudcake

overbalance pressure. In addition, they found that porosity and permeability of mudcake

decreased with time as the overbalance pressure on mudcake decreased (Dewan and Ch-

enevert, 2001). Dewan and Chenevert’s experiment provided a dynamic mudcake model

for which Wu (2004) later developed a coupled invasion simulator. However, Wu’s method

was limited to the case of immiscible mud-filtrate invasion. Lee (2008), Malik (2008), and

Salazar (2008) also used an extended version of Wu’s algorithm to simulate WBM- and

OBM-filtrate invasion. Recently, Pour (2008) and Pour et al. (2011b) coupled Dewan and

Chenevert’s experimentally validated mudcake growth model to a 1D compositional fluid-

flow simulator. This algorithm was reliable for case of immiscible and partially to fully

miscible mud-filtrate invasion. Similar to Pour (2008), Ding (2011) used a productivity

index to couple a formation damage model for the near-wellbore zone with a two-phase

flow simulator. All previous studies modeled the process of mudcake growth as a 1D radial

process. This assumption is accurate for the case of a vertical well in low-permeablility for-

mations. However, in high-angle and horizontal wells, mudcake thickness is not symmetric

around the wellbore.

In this dissertation, the 1D mudcake growth model is extended to simulate invasion

around horizontal and deviated wellbores where mudcake thickness and flow rate are not

constant around the circumference of wellbore. In addition to coupling of a dynamic mud-

cake growth model to the reservoir, a wettability alteration model is also implemented in

the simulation algorithm (described in Section 1.2.4).

1.2.3 Physical Dispersion

Dispersion is the net effect of molecular diffusion and hydrodynamic dispersion (Lake, 1989).

Molecular diffusion is the net mass transport in space of a component due to chemical

7

potential. Fick (1855) formulated molecular diffusion as a diffusive flux. Hydrodynamic

dispersion refers to local velocity gradients within the pore network, length and velocity of

locally heterogeneous streamlines, and pore mechanical mixing.

Bear (1972) expressed dispersion as a tensor, where the geometry of porous me-

dia, molecular diffusion, fluid saturation, and fluid phase velocity determined each entry

of the diffusion tensor (Equation 2.37). Lake and Hirasaki (1981) and Sternberg and

Greenkorn (1994) showed that macroscopic heterogeneities, layering, and cross-flow can

lead to large dispersivities in field-scale studies. Taylora and Howard (1987) showed that

dispersion incrementally increases with distance traveled by a tracer. Moreover, it has been

shown that dispersion affects reservoir recovery. Chang (1990) used Bear’s model for phys-

ical dispersion and implemented it in a compositional fluid-flow simulator (UTCOMP). In

this dissertation, I implement a mathematical model for dispersion to study physical dis-

persion of aqueous phase salt concentration during fluid flow in porous media. The study

quantifies physical dispersion effects on the spatial distribution of salt concentration, and

consequently, on rock electrical resistivity.

1.2.4 Wettability Alteration

OBMs often contain some surfactants – cationic and anionic – to suspend fluid components

in the drilling-fluid mixture (Schramm, 2000). Surfactants in OBM wet the surface of

cuttings and facilitate their removal.

Sharma and Wunderlich (1985) studied wettability alteration due to water-base

mud-filtrate (WBMF) invasion. Their experiments show that oil-wet surfaces become less

strongly oil-wet after contact with filtrate. Later, Menezes et al. (1989) experimentally

investigated several mechanisms that change sandstone wettability after their interaction

with OBM. Their experiments showed that contact angle and capillary pressure can change

drastically under the influence of OBMF surfactants. Ballard and Dawe (1988) studied the

influence of OBMF surfactants on the wettability of glass surfaces. They showed that even

small concentrations of surfactants in mud filtrate can make the rock surface more oil-wet;

8

the effect was a reduction of water saturation below the original residual water saturation.

Yan et al. (1993) used a combined Amott/USBM method to study variations of

wettability by calculating variations in contact angle after OBMF interacted with the rock

surface; they showed that several wetting agents such as EZ Mul and DV-33 significantly

changed rock wettability. More recently, Gambino et al. (2001) performed a series of ex-

periments to study the damage associated with invasion of mud filtrate. They showed

that different mechanisms during drilling and cementing led to formation damage such as

wettability alteration, kaolinite migration, and insoluble salt precipitation. Salazar and

Torres-Verdın (2009) compared the radial distribution of fluid saturation associated with

WBM and OBM and showed that a water bank could develop in the radial profile of water

saturation as a consequence of wettability alteration due to OBMF invasion. Some of the

resistivity logs acquired after OBMF invasion into oil-bearing formations indicate abnor-

mally high values of water saturations near the wellbore (Pour et al., 2011b; Salazar and

Martin, 2010).

1.2.5 Formation-Tester Measurements Acquired in Horizontal and

Deviated Wells

Formation testers are conventionally used to perform mini-drawdown-buildup tests and fluid

sampling. Pressure transient measurements can be used to estimate important parameters in

formation production such as horizontal and vertical permeability and also initial pressure.

Oil companies are interested in FTs because of their low cost and minimal environmental

impact compared to production tests. Being small in size (a diameter of 0.6 [in]), a FT can

perform pressure-transient tests as well as fluid sampling in thin strips of permeable layers.

Formation-tester measurements are affected by (i) productivity of the formation (An-

geles et al., 2007a; Dussan V. and Sharma, 1992; Haddad et al., 2000; Onur et al., 2004),

including mobility of saturating fluids, permeability, anisotropy, porosity, thickness, and

drainage radius; (ii) mud-filtrate invasion (Gok et al., 2006; Hooper et al., 1999; Malik

et al., 2006, 2009b; Pham et al., 2005; Phelps et al., 1984; Proett et al., 2002; Waid et al.,

9

1992), leading to variations in fluid saturations and properties around the wellbore (e.g., in

the case of OBMF invasion into a gas-bearing formation, miscibility between OBMF and

in-situ gas changes properties of both oil and gas fluid phases); (iii) location with respect to

bed boundaries (Abbaszadeh and Hegeman, 1990; Angeles et al., 2005; Dubost et al., 2004;

Onur et al., 2004); and (iv) wellbore deviation angle (Angeles et al., 2011, 2009; Onur et al.,

2004, 2009; Wu et al., 2004), where the asymmetric distribution of mud-filtrate around the

wellbore, permeability anisotropy, and bed-boundaries affect FTMs acquired in deviated

wells. Because of the aforementioned effects, the application of single-phase analytical ex-

pressions (Abbaszadeh and Hegeman, 1990; Cinco-Ley et al., 1975; Kuchuk and Wilkinson,

1991; Onur et al., 2004), and black-oil near-wellbore simulators (Wei et al., 2004; Wu, 2004)

are limited to few particular situations.

Numerous previous studies were either performed for single-phase flow without con-

sideration of mud-filtrate invasion, or the measurements were acquired in vertical wells.

Doll (1956) was the first to introduce a method to estimate permeability anisotropy in ad-

dition to formation permeability. Proett et al. (2001a) developed a time-domain analytical

solution for sampling with multiple WFT probes. In their modeling, Proett et al. considered

the effect of mud-filtrate invasion prior to the pressure-test. They also studied the effect of

permeability anisotropy on pressure-test measurements. Onur et al. (2004) proposed an an-

alytical expression for single-phase spherical flow into a dual packer-probe WFT deployed in

a slanted well. However, they neglected the effect of mud-filtrate invasion in their modeling

of pressure-transient measurements. In highly deviated wells in the presence of mud-filtrate

invasion, Angeles et al. (2011) advanced one of the first studies of formation-tester mea-

surements. They used a commercial fluid-flow simulator and constructed a corner-point

geometry model in Cartesian coordinates to simulate both mud-filtrate invasion and fluid

sampling. The algorithm was only reliable for a limited range of wellbore deviation angles

because Angeles et al.’s model did not use orthogonal grids and did not account for full

tensor permeability. There is a need for the development of a fluid-flow simulator that

can reliably account for complex variations of formation properties in the wellbore vicinity,

10

simulate multiphase flow including gas and oil, with no limit on wellbore deviation angles.

1.3 Research Objectives

Development of a 3D compositional equation-of-state fluid-flow simulator is the main objec-

tive of this dissertation. The simulator is specifically targeted for near-wellbore problems,

and is based on the following assumptions:

Isothermal condition in a reservoir,

No chemical reaction or precipitation,

Darcy’s law for multiphase fluid flow,

A generalized Fick’s law for physical dispersion of salt concentration,

A slightly compressible formation, and

Multiple components in different fluid phases.

The method should be capable of:

Simulating oil-water, gas-water, gas-oil and gas-oil-water fluid flow,

Tracking of spatial-time variations in salt concentration,

Accounting for physical dispersion of salt concentration in the aqueous phase,

Applying different boundary conditions at the wellbore during injection and fluid

production,

Simulating formation-tester measurements,

Taking into account the effect of mud-filtrate invasion,

Modeling wettability alteration induced by OBMF invasion, and

11

Applying all the aforementioned capabilities to any wellbore deviation from 0 to 90

degrees.

Having verified the developed simulator, I conduct the following studies:

Simulation of the effect of wettability alteration on the radial distribution of fluid

saturation after OBMF invasion and consequently, apparent resistivity curves,

Simulation of the effect of dispersion of aqueous salt concentration in the radial dis-

tribution of salinity and electrical resistivity, and

Simulation of formation-tester measurements acquired in high-angle wells.

1.4 Review of Chapters

This dissertation describes the development of a 3D EOS compositional fluid-flow simulator

(called UTFEC) in cylindrical coordinates for general near-wellbore applications.

In Chapter 2, I describe the mathematical formulation of the physical models used

in the development of the simulator. This chapter explains the assumptions made concern-

ing mass conservation and constitutive equations, a pressure equation, models for physical

properties, and phase behavior. Chapter 3 discusses the discretization of the pressure equa-

tion, molar mass equations, phase behavior algorithms, definition of well and boundary

conditions such as those of injection and fluid withdrawal, phase behavior, computation of

phase saturations, and automatic time-step control.

Chapter 4 discusses the verification of the developed simulator by comparing re-

sults calculated with UTFEC against those obtained with commercial reservoir simulators

and also analytical expressions. Verification tests include several cases for 1D radial, two-

dimensional (2D) axis-symmetric, 2D radial, 3D vertical wells, 2D horizontal wells, and

3D deviated wells. Case studies include multiphase flow conditions such as gas-water, oil-

water, and gas-oil-water. Chapter 5 discusses the effect of mudcake growth on mud-filtrate

12

flow rate and, consequently, the radial distributions of fluid saturations around the well-

bore. In this chapter, I also study the effect of physical dispersion for aqueous salt on the

radial distribution of salinity and electrical resistivity after mud-filtrate invasion. OBMF

invasion, wettability alteration, and the subsequent effect on the radial distribution of elec-

trical resistivity are discussed in Chapter 6. Chapter 7 studies pressure-transient analysis

and fluid sampling in thinly-bedded laminations penetrated by highly-deviated wells. Fi-

nally, Chapter 8 summarizes the conclusions stemming from this dissertation and provides

recommendations for future work.

13

Chapter 2

Mathematical Models

2.1 Introduction

This chapter describes the mathematical formulations adopted in the dissertation for multi-

phase, multi-component, and multi-dimensional fluid flow through porous media and specif-

ically for near-wellbore applications. I apply volume constraint, material balance, and

thermodynamic equilibrium to derive fluid-flow partial differential equations along with

boundary and initial conditions. The following assumptions are made in the mathematical

formulation:

1. Reservoir is isothermal,

2. Reservoir is impermeable at an infinite radial distance unless in the presence of a

constant pressure aquifer,

3. There is no chemical reactions or precipitation between fluid and rock,

4. Formation is slightly compressible,

5. Darcy’s law for multi-phase flow is valid,

6. The aqueous phase consists of only water and salt components,

14

7. Physical dispersion in hydrocarbon fluid phases is negligible, and

8. Injection, invasion, and production are considered as source and sink terms.

The developed simulator assumes variations of properties in radial, azimuthal, and vertical

directions and models three-phase flow in a porous and permeable media, including water,

gas, and oil. Physical dispersion of the salt component in the aqueous phase is also included

in the formulation.

2.2 Mass Conservation and Constitutive Equations

This section describes the mathematical equations for multi-phase, multi-component systems

in an isothermal porous medium. For component i, the general mass balance equation can

be written as (Lake, 1989)

∂Wi

∂t+∇ ·

−→Fi −Ri = 0, i = 1, . . . , nc, nc + 1, (2.1)

where Wi,−→Fi, and Ri are the accumulation, flux, and source terms, respectively; nc is the

number of hydrocarbon components, and nc + 1 is the water component. In the above

equation, the accumulation term for the pore space, Wi, can be expressed as

Wi = φ

np∑j=1

ξjSjxij , i = 1, . . . , nc, nc + 1, (2.2)

where φ is total porosity, np is the number of fluid phases, xij is the mole fraction of

component i in phase j, ξj is the molar density of phase j, and Sj is the saturation of fluid

phase j.

I assume that there is no mass transfer between the hydrocarbon components and

the aqueous phase, and that the aqueous phase consists of only the water component. More-

over, the water component does not affect phase behavior. Based on this last assumption,

15

equation (2.2) can be modified to

Wi = φ

np∑j=2

ξjSjxij for i = 1, . . . , nc, (2.3)

for hydrocarbon components, and

Wnc+1 = φξ1S1, (2.4)

for water. In the modeling described in this dissertation, j = 1 is the choice for the aqueous

phase, j = 2 is for the oil phase, and j = 3 is for the gas phase. In equation (2.1), the flux

term,−→Fi, can be expressed as

−→Fi =

np∑j=1

ξjxij−→uj − φξjSjKij · ∇xij , (2.5)

where −→uj is the superficial velocity of phase j, and Kij is the dispersion tensor. By applying

Darcy’s multi-phase relation for superficial velocity, I obtain

−→u j = k · λrj(∇Pj − γj∇D), (2.6)

where k is permeability tensor, γj is specific weight of phase j, D is depth, Pj is the pressure

of phase j, and λrj is mobility with respect to the reference phase (in this dissertation, the

oil phase), which can be expressed in terms of relative permeability, krj , and viscosity, µj ,

as

λrj =krjµj. (2.7)

The developed simulator, UTFEC, includes the dispersion tensor only for salt concentration.

The form of this tensor and constituting terms are explained in Section 2.5. In equation (2.1),

16

Table 2.1: List of variables included in equation (2.9).

Variables Unit Number of Variables

φ fraction 1

Sj fraction np

ξj lb/ft3 np

xij fraction nc(np − 1)

krj fraction np

µj cp np

Pj psi np

γj fraction np

qi lbm/day nc + 1

total number of variables: ncnp + 6np + 2

the source term is correlated with the well condition as follows:

Ri =qiVb

∀ i = 1, . . . , nc, nc + 1, (2.8)

where Vb is the bulk volume and qi is the molar flow rate of each component, here considered

positive for grid blocks with an injection boundary condition and negative in grid blocks

with a production boundary condition; only for the first grid block qi may be nonzero. After

substituting equations (2.2),(2.5), and (2.8) into equation (2.1), I obtain

∂

∂t

φ np∑j=1

ξjSjxij

+−→∇ ·

np∑j=1

ξjxij−→u j − φξjSjKij∇xij

− qiVb

= 0,

∀ i = 1, . . . , nc, nc + 1. (2.9)

Equation (2.9) is a set of coupled partial differential equations, which are nonlinear with

respect to concentratoin of components and pressure; there are ncnp + 6np + 2 variables

which are listed in Table 2.1.

17

2.3 Auxiliary Relations

In equation (2.9), there are ncnp + 6np + 2 variables. In order to determine the unknown

parameters, I use the equations described below.

2.3.1 Saturation Constraint

The sum of saturations in each grid block must be equal to one, i.e.,

np∑j=1

Sj = 1. (2.10)

where Sj is the saturation of fluid phase j.

2.3.2 Porosity Dependency on Pressure

Formation porosity is a function of pressure, namely,

φ = φ(P ), (2.11)

where P is pressure of the reference fluid phase (fluid phase oil in the dissertation).

2.3.3 Phase Molar Density

Molar density of each hydrocarbon phase at a given temperature is a function of the pressure

and composition of each phase, to wit,

ξj = ξj(P,−→X j) ∀ j = 2 . . . , np, (2.12)

where P is pressure of the reference fluid phase and−→X j is the composition of fluid phase j,

i.e.,−→X j = [x1j , x2j , . . . , xncj ]. Molar density of the aqueous phase is a function of pressure

18

Tab

le2.2

:L

ist

of

au

xil

iary

rela

tion

s.

Equ

atio

ns

Nam

eU

nit

Nu

mb

erof

Equati

on

s

Equ

atio

n2.

9m

ass

con

serv

ati

on

lbm

/d

ay/ft

3nc

+1

f ij

=f ir

ph

ase

equ

ilib

riu

mp

sinc(np−

2)

Pj

=Pj(−→ S

,−→ xj)

pre

ssu

repsi

np−

1∑ n p j=

1Sj

=1

satu

rati

on

con

stra

int

fract

ion

1

krj

=krj(P,−→ S

)re

lati

vep

erm

eab

ilit

yfr

act

ion

np

∑ n p j=1xij

=1

ph

ase

com

posi

tion

const

rain

tfr

act

ion

np−

1

q i=q i

(P,−→ S

,−→ x)

wel

lm

od

ellb

m/d

aync

+1

γ1

=γ

1(P

),w

ate

rsp

ecifi

cd

ensi

tyfr

act

ion

1

γj

=γj(P,−→ xj)

hyd

roca

rbon

flu

idp

hase

spec

ific

den

sity

fract

ion

np−

1

j=

2,...,np−

1ca

lcu

late

dw

ith

equati

on

of

state

ξ 1=ξ 1

(P),

wat

erm

ola

rd

ensi

tylb

m/ft

31

ξ j=ξ j

(P,−→ xj)

hyd

roca

rbon

flu

idp

hase

mola

rd

ensi

tylb

m/ft

3np−

1

j=

2,...,np−

1ca

lcu

late

dw

ith

equati

on

of

state

µ1

=co

nst

ant

wate

rvis

cosi

tycp

1

µj

=µj(P,−→ xj)

hyd

roca

rbon

flu

idp

hase

vis

cosi

tycp

np

j=

2,...,np−

1

φ=φ

(P)

form

ati

on

poro

sity

fract

ion

1

tota

lnu

mb

erof

equ

ati

on

s:ncnp

+6np

+2

19

and salt concentration of the aqueous phase, i.e.,

ξ1 = ξ1(P, xsalt,1). (2.13)

2.3.4 Phase Composition Constraint

The sum of mole fractions of the components in each fluid phase should be equal to one,

namely,nc∑i=1

xij = 1. (2.14)

2.3.5 Flow Rate

Flow rate of each component is a function of the pressure difference between sand face and

reservoir, saturation of phases, and composition of fluid phases, i.e.,

qi = qi(P,−→S ,−→X ), (2.15)

where−→S is saturation of all fluid phases, i.e.,

−→S = [S1, S2, S3]; and

−→X is the molar fraction

of components of the upstream fluid (described in Section 3.6).

2.3.6 Phase Pressure

Pressure of each fluid phase can be related to the reference pressure with its corresponding

capillary pressure between those phases, to wit,

Pj = Pr + Pcrj , ∀ j = 1, . . . , np (j 6= r), (2.16)

where Pj is the pressure of fluid phase j, Pr is the pressure of the reference fluid phase (oil

phase is the reference pressure in this dissertation), and Pcrj is capillary pressure between

fluid phase j and pressure of the reference fluid phase. In the above equation, capillary

20

pressure is a function of phase saturation and composition, i.e.,

Pcrj = Pcrj(−→S ,−→X ), ∀ j = 1, . . . , np (j 6= r), (2.17)

where−→S is saturation of all fluid phases and

−→X is the molar fraction of components.

2.3.7 Phase Mass Density

Mass density of each hydrocarbon phase at a given temperature is a function of the pressure

and composition of that phase, i.e.,

γj = γj(P,−→X j), ∀ j = 2, . . . , np. (2.18)

where P is the pressure of the reference fluid phase and−→X j is the composition of fluid phase

j, i.e.,−→X j = [x1j , x2j , . . . , xncj ]. The mass density of the aqueous phase is a function of

pressure and salt concentration of the aqueous phase, namely,

γ1 = γ1(P, xsalt,1). (2.19)

2.3.8 Phase Viscosity

Viscosity of each hydrocarbon phase at a given temperature is a function of pressure and

composition of that phase, namely,

µj = µj(P,−→X j), ∀ j = 2, . . . , np. (2.20)

For the aqueous phase, viscosity depends on temperature, pressure, and salt concentration

of the aqueous phase, i.e.,

µ1 = µ1(T, P, xsalt,1). (2.21)

In equations (2.20) and (2.21), P is the pressure of reference fluid phase.

21

2.3.9 Relative Permeability

Relative permeability can be expressed as a function of saturation, to wit,

krj = krj(−→S ). (2.22)

2.3.10 Phase Equilibrium

The assumption of thermodynamic equilibrium in each grid block yields the following rela-

tions for hydrocarbon phases (Chapter 10, Smith and Van Ness, 2004):

fij = fir, ∀ i = 1, . . . , nc & j = 2, . . . , np (j 6= r), (2.23)

in which, the fugacity of component i in phase j, fij , is obtained from a cubic equation of

state.

Table 2.2 summarizes the auxiliary relations used in the formulations assumed in

this dissertation. There are a total of ncnp+6np+2 equations, which is equal to the number

of variables.

2.4 Pressure Equation

The numerical scheme implemented in the development of this compositional simulator is

referred to as implicit-pressure explicit concentration (IMPEC), which is similar to that of

UTCOMP (Chang, 1990). In the IMPEC method, the pressure equation is solved implicitly,

whereas moles of components are calculated explicitly. Saturations of fluid phases are also

explicitly related to moles of components. At most three co-existing fluid phases, are consid-

ered in the simulations. The fundamental equation for pressure is based on the assumption

that pore volume contains the total volume of the fluid, that is,

Vt

[P,

nc+1∑i=1

Ni

]= Vp(P ), (2.24)

22

where Ni is the number of moles of component i and Vt is the total fluid volume, which is

a function of pressure and total number of moles of hydrocarbons,∑nc+1i=1 Ni, in the pore

volume, Vp. Differentiating both sides of equation (2.24) with respect to time, and using

the chain rule yields

(∂Vt∂P

)Ni

(∂P

∂t) +

nc+1∑i=1

( ∂Vt∂Ni

)P,Nk,(k 6=i)

(∂Ni∂t

) = (dVpdP

)(∂P

∂t). (2.25)

In what follows, I describe various components of equation (2.25) as functions of primary

known parameters:

1. Relation fordVpdP

:

From the assumption of slight and constant compressibility for the formation, one can

write the following equation for porosity:

φ = φ0[1 + cf (P − P 0)], (2.26)

where φ0 is porosity at reference reservoir pressure, P 0, and cf is rock compressibility.

The definition of pore volume indicates that

Vp = Vbφ. (2.27)

Differentiating both sides of equation (2.27) with respect to pressure gives

dVpdP

= V 0p cf , (2.28)

where V 0p is the pore volume at the reference pressure.

2. Relation for∂Ni∂t

:

The total moles for each component, Ni, at each grid block is equal to the accumulation

23

in that grid, VbWi. Using equation (2.2) yields

Ni = Vp

np∑j=1

ξjSjxij . (2.29)

From equations (2.29) and (2.9), I obtain a differential equation for moles of each

component, namely,

∂Ni∂t

= −Vb∇ ·np∑j=1

[ξjxij

−→u j − φξjSjKij · ∇xij]

+ qi ∀ i = 1, . . . , nc, nc + 1,

(2.30)

where np is the number of coexisting fluid phases, qi is the molar rate of component

i, ξj is the molar density of fluid phase j, −→u j is the velocity of fluid phase j, and xij

is the molar fraction of component i in fluid phase j.

3. Relation for( ∂Vt∂Ni

)P,Nk(k 6=i)

:

A partial molar volume can be defined for each component as suggested Chang (1990),

namely,

Vti =( ∂Vt∂Ni

)P,Nk(k 6=i)

. (2.31)

Substituting equations (2.28), (2.30), and (2.31) into equation (2.25) yields:

(∂Vt∂P

)Ni

(∂P

∂t)− Vb

nc+1∑i=1

Vti∇ ·np∑j=1

[ξjxij

−→u j − φξjSjKij∇ · xij]

+

nc+1∑i=1

Vtiqi = V 0p cf (

∂P

∂t). (2.32)

In the above equation, I have used oil pressure as the reference equation; all other fluid

pressures are related to oil pressure plus the corresponding capillary pressure equation1, to

wit,

Pj = P + Pc2j , (2.33)

1 Capillary pressure can have a negative value compared to the conventional water-oil capillary pressure.

24

where P is the pressure of oil phase, Pc2j is the capillary pressure of fluid phase j with

respect to oil phase, and Pj is the pressure of fluid phase j. Substituting equations (2.33)

and (2.6) into equation (2.32) yields

[V 0p cf −

(∂Vt∂P

)Ni

]∂P

∂t− Vb

nc+1∑i=1

Vti∇ ·np∑j=1

kλrjξjxij∇P

= Vb

nc+1∑i=1

Vti∇ ·np∑j=1

[kλrjξjxij · (∇Pc2j − γj∇D)

]+

Vb

nc+1∑i=1

Vti∇ ·np∑j=1

φξjSjKij∇xij +

nc+1∑i=1

Vtiqi. (2.34)

2.5 Physical Dispersion

In Section 2.2, I included the effect of dispersion in the flux term of aqueous salt, i.e.,

−→Fi =

np∑j=1

[ξjxij

−→uj − φξjSjKij · ∇xij], (2.35)

where −→uj is the darcy velocity or superficial velocity of fluid phase j, and Kij is the dispersion

tensor. Subsequently, I obtain the material balance equation as

∂

∂t

φ np∑j=1

ξjSjxij

+∇ ·np∑j=1

[ξjxij

−→u j − φξjSjKij · ∇xij]− qiVb

= 0,

for i = nc + 1, (2.36)

where ξj is the molar density of phase j, Sj is the saturation of fluid phase j, xij is molar

fraction of component i in fluid phase j, qi is the molar rate of component i, and Vb is bulk

volume. Bear (1972) showed that dispersion has a tensorial form; the value of each entry of

the dispersion tensor depends on geometry, fluid-phase velocity, and molecular diffusion. In

this dissertation, I describe physical dispersion with a full tensor in cylindrical coordinates,

25

namely,

Kij =

Krr,ij Krθ,ij Krz,ij

Kθr,ij Kθθ,ij Kθz,ij

Kzr,ij Kzθ,ij Kzz,ij

, (2.37)

where each entry of Kij is the net contribution of molecular diffusion and mechanical dis-

persion. Bear (1972) defines each element of the tensor for multi-phase, multi-component

flow in a homogeneous, isotropic media with the following equations:

Krr,ij =Dij

τ+αljφSj

u2rj

|uj |+αtjφSj

u2θj

|uj |+αtjφSj

u2zj

|uj |, (2.38)

Kθθ,ij =Dij

τ+αljφSj

u2θj

|uj |+αtjφSj

u2rj

|uj |+αtjφSj

u2zj

|uj |, (2.39)

Kzz,ij =Dij

τ+αljφSj

u2zj

|uj |+αtjφSj

u2rj

|uj |+αtjφSj

u2θj

|uj |, (2.40)

Krθ,ij = Kθr,ij =(αlj − αtj)

φSj

urjuθj|uj |

, (2.41)

Krz,ij = Kzr,ij =(αlj − αtj)

φSj

urjuzj|uj |

, and (2.42)

Ktz,ij = Kzt,ij =(αlj − αtj)

φSj

uθjuzj|uj |

, (2.43)

where αlj is the longitudinal dispersivity of phase j, αtj is the transverse dispersivity of phase

j, Dij is the molecular diffusion coefficient of component i in phase j, τ is the tortuosity of

the porous media, and urj , uθj = rθ, and uzj are the velocities of fluid phase j in the r, θ,

and z directions, respectively. In equations (2.38) through (2.43) |uj | is the magnitude of

the velocity vector, that is,

|uj | =√u2rj + u2

θj + u2zj . (2.44)

26

In this dissertation, I only include the dispersion of aqueous salt concentration in the fluid-

flow equations. The effect of dispersion in other fluid phases is assumed to be negligible.

2.6 Moles of Components

After solving the volume constraint equation, eq. (2.34), I find moles of each component at

the current pressure using equation (2.30), namely,

∂Ni∂t

= Vb∇ ·np∑j=1

[ξjxijkλj(∇Pj − γj∇D) + φξjSjKij∇xij

]+ qi,

∀ i = 1, . . . , nc, nc + 1, (2.45)

where Ni is the total moles of component i, Vb is the bulk volume, np is the number of co-

existing fluid phases, ξj is molar density of fluid phase j, xij is mole fraction of component

i in fluid phase j, k is permeability tensor, λj is mobility of fluid phase j (λj = krj/µj), Pj

the pressure of fluid phase j, γj is specific density of fluid phase j, D is depth, φ is porosity,

Kij is dispersion tensor for component i in fluid phase j, qi is molar rate of component i,

and nc is the number of components. The calculation of salt concentration in the aqueous

phase is also based on equation (2.45). Similar to xij , I define salt mole fraction in the

aqueous phase as

xsalt,1 =NsaltNwater

, (2.46)

where Nsalt and Nwater are number of the moles of salt and water components, respectively.

2.7 Modeling Physical Properties

This section briefly explains the models implemented for calculation of viscosity, relative

permeability, capillary pressure, and interfacial tension.

27

2.7.1 Viscosity

A comprehensive study to estimate viscosity of the aqueous phase was conducted by Kestin

et al. (1981). They proposed an experimentally-tested correlation consisting of 32 parame-

ters to calculate water viscosity as a function of temperature, pressure, and concentration

of NaCl. Later, McCain (1991) suggested a correlation for the calculation of water viscosity

at atmospheric pressure. Comparison of water viscosities calculated from Kestin et al. and

those obtained using McCain’s relation indicates a very good agreement. I use the correla-

tion suggested by McCain for the calculation of water viscosity. In McCain’s model, water

viscosity at 14.7 [psi] is given by

µw,14.7 = AmcTBmc , (2.47)

where

Amc = 109.574− 8.40561× xsalt,1 + 0.313314× x2salt,1 + 8.72213× x3

salt,1, (2.48)

and

Bmc = 1.12166− 2.63951× 10−2 × xsalt,1 + 6.79461× 10−4 × x2salt,1

+ 5.47119× 10−5 × x3salt,1 − 1.55586× 10−6 × x4

salt,1, (2.49)

where xsalt,1 is the concentration of salt in the aqueous phase (up to 26 wt%) and T is the

temperature in the range of 100 to 400 [F ]. Subsequent to the calculation of water viscosity

at 14.7 [psi], µw,14.7, I calculate the viscosity of water at formation pressure, µw, using

µwµw,14.7

= 0.9994 + 4.0295× 10−5 × pr1 + 3.1062× 10−9 × p2r1, (2.50)

where

pr1 =P

14.7. (2.51)

28

In equation (2.51), P is formation pressure [psi].

For the calculation of hydrocarbon-phase viscosity, I implement the following op-

tional correlations: (1) Lohrenz et al. (1964), (2) quarter-power mixing rule (Chang, 1990),

and (3) linear mixing rule (Chang, 1990).

2.7.1.1 Lohrenz et al.’s correlation

Lohrenz et al. (1964) combined several correlations to calculate hydrocarbon-phase viscosity.

The procedure for the calculation of hydrocarbon-phase viscosity is as follows:

Step 1: Using the Stiel and Thodos’s correlation (1961), I compute the low pressure, pure

component viscosity as

µi = 3.4× 10−4T0.94ri

ζifor Tri ≤ 1.5, (2.52)

or

µi = 1.776× 10−4 (4.58× Tri − 1.67)5/8

ζifor Tri > 1.5, (2.53)

where µi is the low-pressure viscosity of component i, Tri is the reduced temperature of

component i ( TTci

), and ζi is the viscosity parameter of component i and is calculated

using the equation

ζi = 5.44Tci

1/6

W1/2ti P

2/3ci

, (2.54)

where Wti is molecular weight of component i, Tci is critical temperature of component

i, and Pci is the critical pressure of component i.

Step 2: Using Herning and Zipperer’s equation (Herning and Zipperer, 1936), I calculate the

low-pressure viscosity, namely,

µ∗j =

nc∑i=1

xij µi√Wti

nc∑i=1

xij√Wti

, (2.55)

29

where µ∗j is the low-pressure viscosity of hydrocarbon phase j, xij is the mole fraction

of component i in phase j, and nc is the number of hydrocarbon components in

hydrocarbon phase j.

Step 3: I compute the reduced phase molar density, ξjr, as

ξjr = ξj

nc∑i=1

xijVci, (2.56)

where Vci is critical volume of component i and ξj is molar density of hydrocarbon

phase j. Moreover, I calculate the mixture viscosity parameter, ηj , from

ηj = 5.44

(

nc∑i=1

xijTci)1/6

(

nc∑i=1

xijWti)1/2(

nc∑i=1

xijPci)2/3

. (2.57)

Step 4: Using Jossi et al.’s correlation (Jossi et al., 1962), I calculate hydrocarbon-phase vis-

cosity at phase pressure with

µj = µ∗j + 2.05× ξjrηj

if ξjr ≤ 0.18, (2.58)

or

µj =µ∗j + χ4

j − 1

104 × ηjif ξjr > 0.18, (2.59)

where χj is a viscosity parameter defined by

χj = a0 ξjr + a1 ξ2jr + a2 ξ

3jr + a3 ξ

3jr + a4 ξ

4jr, (2.60)

where parameters a0 through a4 are listed in Table 2.3. Lohrenz et al.’s correlation

is the preferred choice in compositional reservoir simulators. However, this model

is not accurate when predicting hydrocarbon viscosities (Pederson and Christensen,

2007). In the developed simulator, parameters Vci, and a0 through a4 are input by the

30

Table 2.3: List of parameters in Lohrenz et al.’s (1964) viscosity correlation, eq. (2.60).

Parameter Unit Value

a0 dimensionless 1.0230a1 dimensionless 0.23364a2 dimensionless 0.58533a3 dimensionless 0.40758a4 dimensionless 0.093324

user. Adjusting this set of parameters enables one to match experimental data when

calculating viscosity.

2.7.1.2 Quarter-Power Mixing rule

The quarter-power mixing rule (Chang, 1990) for the viscosity of each hydrocarbon phase

is given by

µj = (

nc∑i=1

xij µ1/4i )−4, for j = 2, . . . , np, (2.61)

where µi is the viscosity of a pure component.

2.7.1.3 Linear Mixing rule

The linear mixing rule (Chang, 1990) for hydrocarbon-phase viscosity is given by

µj =

nc∑i=1

xij µi for j = 2, . . . , np, (2.62)

where µi is the viscosity of a pure component.

2.7.2 Relative Permeability Models

There are several relative permeability models which are currently used in simulators.

Stone’s model 2 (Stone, 1973), Baker’s model (Delshad and Pope, 1989), Pope’s model (Delshad

and Pope, 1989), and Corey’s model (Corey, 1986) are a few examples. These models are

31

described in Chang (1990). In the simulator developed here, relative permeability of the oil

phase in three-phase flow is calculated using Stone’s model 2.

2.7.2.1 Stone’s Model 2

In this dissertation, oil-water, two-phase relative permeability is calculated using Corey’s

model (Corey, 1986), namely,

kr21 = k0r2

( 1− S1 − S2r1

1− S1r − S2r1

)e21, (2.63)

whereas oil-gas two-phase relative permeability is calculated with Corey’s model, given by

kr23 = k0r2

( 1− S3 − S2r3

1− S1r − S2r3 − S3r

)e23, (2.64)

where

kr21 is relative permeability of the oil phase, 2, flowing with water phase, 1,

kr23 is relative permeability of gas phase, 3, flowing with oil phase, 2,

k0r2 is endpoint relative permeability of the oil phase, 2,

S1 is saturation of the aqueous phase, 1,

S2r1 is residual saturation of the oil phase, 2, flowing with the aqueous phase, 1,

S2r3 is residual saturation of the oil phase, 2, flowing with gas phase, 3,

S1r is residual saturation of the aqueous phase,

S3r is residual saturation of the gas phase,

e21 is exponent of relative permeability of the oil phase, 2, flowing with the aqueous

phase, 1, and

32

e23 is exponent of relative permeability of the oil phase, 2, flowing with the gas phase,

3.

The relative permeability of water and gas are calculated similarly with Corey’s

model, given by

kr1 = k0r1

( S1 − S1r

1− S1r − S2r1

)e1, (2.65)

and

kr3 = k0r3

( S3 − S3r

1− S1r − S2r3 − S3r

)e3, (2.66)

where

kr1 is relative permeability of the aqueous phase, 1,

kr3 is relative permeability of the gas phase, 3,

k0r1 is endpoint relative permeability of the aqueous phase, 1,

k0r3 is endpoint relative permeability of the gas phase, 3,

e1 is exponent of relative permeability function of the aqueous phase, 1, and

e3 is exponent of relative permeability function of the gas phase, 3.

Finally, relative permeability of the oil phase for three-phase flow is calculated

as (Stone, 1973)

kr2 = k0r2

[(kr21

k0r2

+ kr1)(kr23

k0r2

+ kr3)− kr1 − kr3]. (2.67)

2.7.2.2 Table Lookup for Relative Permeability

Relative permeabilities of water and gas are found from linear interpolation. For two-

phase gas-oil and oil-water flow, the relative permeability of oil is calculated from linear

interpolation of the input table. Relative permeability of oil in three-phase flow is computed

from Stone’s model 2, equation (2.67).

33

2.7.3 Capillary Pressure

Capillary pressure is a function of parameters, such as interfacial tension, permeability,

porosity, and saturation (Behseresht et al., 2009, 2008; Leverett, 1941). Assuming water as

wetting fluid phase, capillary pressure for three-phase water-oil-gas flow is given by (Chang,

1990)

Pc21 = −Cpcσ12

√φ

k(1− S1)Epc , (2.68)

and

Pc23 = Cpcσ23

√φ

k(

S1

S2 + S3

)Epc , (2.69)

where:

Pc21 is capillary pressure between oil phase, 2, and aqueous phase, 1,

Pc23 is capillary pressure between oil phase, 2, and gas phase, 3,

Cpc is a constant determined from matching a water/oil experimental capillary pres-

sure,

σ12 is interfacial tension between the aqueous phase, 1, and the oil phase, 2; this

parameter is an input from user,

σ32 is interfacial tension between gas phase, 3, and the oil phase, 2; this parameter is

calculated using equation (2.75),

φ is total porosity,

k is permeability,

Epc is the exponent of capillary pressure which is determined from matching a water/oil

experimental capillary pressure, and

Sj is normalized saturation, defined as

S1 =S1 − S1r

1− S1r − S2r − S3r, (2.70)

34

S2 =1− S1 − S2r − S3

1− S1r − S2r − S3r, (2.71)

and

S3 =S3 − S3r

1− S1r − S2r − S3r, (2.72)

where the equivalent residual saturation of oil phase, S2r, is defined by (Chang, 1990)

S2r = bS2r1 + (1− b)S2r3, (2.73)

with

b = 1− S3

1− S1r − S2r3. (2.74)

The above parameters are determined from curve matching of laboratory experiments of

water-oil capillary pressure.

2.7.4 Interfacial Tension

The interfacial tension between water and hydrocarbon phases is assumed to be constant.

Macleod-Sudgen (Reid et al., 1987) introduced an equation to describe the interfacial tension

between hydrocarbon phases which relates interfacial tension to molar density, ξj , phase

composition, xij , and parachor of a component, Ψi, given by

σjr =

[0.016018

nc∑i=1

Ψi(ξjxij − ξrxir)

]4

, (2.75)

where subscript r identifies the reference fluid phase.

2.8 Phase Behavior

In this dissertation, hydrocarbon phase behavior is modeled using a modified version of

Peng-Robinson’s equation of state (PR-EOS) (Peng and Robinsion, 1976).

35

2.8.1 Peng-Robinsion’s Equation of State

Peng and Robinson modified Van der Waals’s equation of state to read as

[P +

α(T )

v(v + b) + b(v − b)

](v − b) = RT, (2.76)

where v is molar volume, P is pressure of the reference fluid phase, T is temperature, R is

the universal gas constant, and α and b are constants calculated as

α(T ) = Ωaα (RTc)

2

Pc, (2.77)

and

b = ΩbRTcPc

, (2.78)

where

Ωa = 0.45724, Ωb = 0.0778, (2.79)

and

α = [1 +m(1−√T

Tc)]2, (2.80)

where

m = 0.37464 + 1.54226 ω − 0.26992 ω2. (2.81)

In the above equations, Tc is temperature at the critical point, Pc is pressure at the critical

point, and ω is the acentric factor of the hydrocarbon component. Maintaining simplicity,

this equation is more reliable than any other EOS because the prediction of liquid-phase den-

sity is closer to experimental data (Firoozabadi, 1999; Pederson and Christensen, 2007). The

universal critical compressibility factor for pure components obtained with equation (2.76)

is 0.307, whereas that obtained with Soave-Redlich-Kwong equation of state (SRK) is 0.333,

but both are larger than predictions by experimental data (generally of the order of 0.25

to 0.29) (Pederson and Christensen, 2007). Later Peng and Robinson (1978) corrected

36

equation (2.81) for ω > 0.49 as

m = 0.379642 + 1.48503 ω − 0.164423 ω2 + 0.016666 ω3. (2.82)

The compressibility factor of the hydrocarbon phase, Z, is defined as

Z =Pv

RT. (2.83)

I express PR-EOS in the form of a cubic equation of fluid phase compressibility, Z, as

Z3 − (1−B)Z2 + (A− 3B2 − 2B)Z − (AB −B2 −B3) = 0, (2.84)

where

A =aP

(RT )2, (2.85)

and

B =bP

RT, (2.86)

where a and b are constants of the EOS for hydrocarbon fluid phase and are calculated as

a =

nc∑i=1

nc∑k=1

xixkaik, (2.87)

where aik is calculated from

aik = (1− δik)√aiak, (2.88)

and

b =

nc∑i=1

xibi. (2.89)

In the above equations, ai and ak are constant a of the EOS for components i and k,

respectively, bi is constant b of the EOS for component i, Kronecker delta, δik is the binary

interaction between components i and k, and xi is the mole fraction of component i in the

37

fluid phase.

2.8.1.1 Fugacity of Components

The fugacity of component i in a mixture can be computed from the following equa-

tion (Firoozabadi, 1999):

lnϕi = lnfixiP

=bib

(Z − 1)− ln(Z −B)

− A

2√

2B

[2

a

nc∑k=1

xkaik −bib

]× ln

(Z + (1 +√

2)B

Z + (1−√

2)B

), (2.90)

where:

ϕi is fugacity coefficient of component i,

fi is fugacity of component i,

xi is mole fraction of component i in the fluid phase,

P is pressure of the reference fluid phase,

bi and aik are constants of the EOS,

A, B, a, and b are constants of the EOS for the fluid phase, and

Z is the compressibility of the fluid phase.

2.8.2 Molar and Mass Density

From the EOS, I compute the compressibility factor, Z, of each hydrocarbon phase. The

molar density of hydrocarbon phase j, ξj , is given by

ξj =1

vj=

P

ZjRT, (2.91)

where P is pressure of reference fluid phase, Zj is the compressibility of the fluid phase, R

is the universal gas constant, and T is temperature [R]. The mass density of hydrocarbon

38

fluid phase j is given by

ρj = ξj

nc∑i=1

xijMwi, (2.92)

where Mwi is the molecular weight of component i. The water density at standard conditions

is given by McCain’s correlation (1991)

ρ0w = ρ0

1 = 62.368 + 0.438603× xsalt,1 + 1.60074× 10−3 × x2salt,1, (2.93)

where density of water is in lbm/ft3, and xsalt,1 is salinity in weight percent. In this research,

I assume that water is slightly compressible. Consequently, water mass density is written

as

ρw = ρ1 = ρ01[1 + c1(P − P 0

1 )], (2.94)

where ρ01 is the water mass density at reference pressure, c1 is compressibility of water at

reference pressure, P 01 , and ρ1 is mass density of water at pressure P . Molar density of

water is calculated by

ξ1 =ρ1

Mw,water, (2.95)

where ξ1 is molar density of water and Mw,water is molecular weight of water.

2.8.3 Derivatives of the Pressure Equation

Analytical computation of partial derivatives of total fluid volume is needed to solve the

volume-constraint equation (eq. (2.34)). Accordingly, I briefly formulate the derivatives in

the following section, and refer to Chang (1990) for additional details.

39

2.8.3.1 Derivative of Total Volume of Fluid with Respect to Moles of Compo-

nents

The partial derivative of total fluid volume with respect to moles of component was intro-

duced in equation (2.31), defined as

Vti =( ∂Vt∂Ni

)P,Nr(r 6=i)

=∂

∂Ni(

np∑j=1

njvj), ∀ i = 1, . . . , nc, nc + 1, (2.96)

where Vti is partial molar volume of component i, Vt is total fluid volume, Ni is total moles

of component i, nj is total moles of all components in hydrocarbon phase j, vj is molar

volume of fluid phase j, nc is number of components, and np is number of fluid phases. To

begin with, the partial molar volume of water component becomes

Vt(nc+1) = v1, (2.97)

where v1 is the molar volume of water at pressure P . For the case of hydrocarbon compo-

nents, the partial molar volume, Vti, is given by

Vti =

np∑j=2

nc∑i=1

[vj + nj∂vj∂nkj

](∂nkj∂Ni

)P,Nr(r 6=i)

, (2.98)

where nkj is moles of component k in hydrocarbon phase j. Using the EOS, I compute the

partial molar volume derivative as

∂vj∂nkj

=RT

P

(∂Zj∂nkj

), for j = 2, . . . , np and k = 1, . . . , nc, (2.99)

where R is the universal gas constant, T is temperature, P is pressure, and Zj is compress-

ibility of hydrocarbon phase j.

For the case of two hydrocarbon phases,∂nk2

∂Niis computed by solving the following system

40

of equations:

nc∑k=1

[∂ln fs2∂nk2

+∂ln fs3∂nk3

](∂nk2

∂Ni

)=(∂ ln fs3∂ni3

),

for s = 1, . . . , nc and i = 1, . . . , nc, (2.100)

where fsj is the fugacity of component s in hydrocarbon phase j, and nij is the moles of

component i in hydrocarbon phase j.

2.8.3.2 Derivative of Total Volume of Fluid with Respect to Pressure

The derivative of total fluid volume with respect to pressure is given by

(∂Vt∂P

)Ni

=

(∂V1

∂P

)Ni

+

np∑j=2

(∂Vj∂P

)Ni

, (2.101)

where Vt is total fluid volume, P is pressure, V1 is volume of the aqueous phase, and Vj is

volume of the hydrocarbon phase j.

For the aqueous phase, and having assumed slightly compressiblity, the derivative can be

expressed as (∂V1

∂P

)Ni

= −n1ξ01c1ξ21

, (2.102)

where ξ01 is molar density of water at reference pressure, ξ1 is molar density of water at

pressure P , n1 is moles of water component, and c1 is the compressibility of water.

For hydrocarbon phases, the corresponding derivative is calculated from the EOS with

(∂Vj∂P

)Ni

=

nc∑k=1

[vj + nj(∂vj∂nkj

)](∂nkj∂P

)Ni

+ nj

(∂vj∂P

)nrj, for j = 2, . . . , np. (2.103)

In the above equation, vj is molar volume of phase j, nj is the moles of all components in

fluid phase j, nkj is the moles of component k in fluid phase j, and Ni is the total moles of

component i.

41

In equation (2.99), I analytically calculated∂vj∂nkj

, where vj is molar volume of fluid phase j

and nkj is moles of component k in fluid phase j. Similarly, I expressed the partial derivative

of molar volume with respect to pressure,∂vj∂P

, as

∂vj∂P

=RT

P 2[P (

∂Zj∂P

)− Zj ] for j = 2, . . . , np, (2.104)

where vj is molar volume of fluid phase j, P is pressure, T is the temperature, R is the

universal gas constant, Zj is compressibility of fluid phase j, and np is number of phases.

For the case of two hydrocarbon phases, I found the partial derivative of moles of component

k in fluid phase 2 with respect to pressure,(∂nk2

∂P

)by solving the following system of

equations:

nc∑k=1

[∂ln fs2∂nk2

+∂ln fs3∂nk3

](∂nk2

∂P

)=(∂ ln fs3

∂P

)nr3−(∂ ln fs2

∂P

)nr2,

for s = 1, . . . , nc and i = 1, . . . , nc, (2.105)

where fsj is fugacity of component s in fluid phase j, nkj is moles of component k in fluid

phase j, P is pressure, and nc is number of components.

2.9 Initial and Boundary Conditions

I initialize the reservoir at the start of the simulation. Initial conditions determine com-

position of hydrocarbon components, salinity of the aqueous phase, and specific choices for

initialization of pressure and water saturation. For example, if the depth of the water-oil

contact is given, pressure and water saturation are calculated by enforcing gravity-capillary

pressure equillibrium. Saturation of hydrocarbon phases (for gas-oil) are calculated with

the assumption of thermodynamic equilibrium inside an isothermal reservoir.

In this dissertation, the external radial boundary of the reservoir can either be

an impermeable rock or an infinite acting aquifer. When the external radial boundary is

42

assumed impermeable, the normal flux of components is zero, i.e.,

−→n ·−→F ij = 0, (2.106)

where −→n is the unit normal vector to the boundary, and

−→F ij = ξjxij

−→u j , (2.107)

where ξj is molar density of fluid phase j, xij is molar fraction of component i in fluid

phase j, and −→u j is velocity of fluid phase j. The assumption of an aquifer at the terminal

boundary enables water influx into the reservoir. I use the steady-state aquifer model for

the calculation of influx. Accordingly, the pressure at the external boundary of the aquifer

is constant and water influx flow rate is given by

Qw = CtA (PAq − Pave), (2.108)

where CtA is total compressibility of the aquifer (the sum of water compressibility and

rock compressibility), PAq is aquifer pressure at the external boundary, and Pave is average

formation pressure at the aquifer-reservoir boundary. At the wellbore, there are several

boundary conditions such as injection, invasion, and production with either a controlled

flow rate or a controlled wellbore pressure.

43

Chapter 3

Computational Approach

This chapter describes the discretization of the pressure and mass conservation equations

using the finite-difference method. I discretize the fluid-flow equations in three-dimensional

(3D) cylindrical coordinates to take advantage of the geometrical embedding imposed by

the wellbore in the spatial distribution of fluids within invaded formations. The simulation

algorithm is based on solving the pressure equation in an implicit form and updating the

hydrocarbon fluid-phase compositions explicitly. Following the calculation of overall com-

position, the composition of each fluid phase is determined with a phase-behavior scheme.

I describe different boundary conditions, saturation calculations for different fluid phases,

and the algorithm to update the time step during each simulation.

3.1 Reservoir Discretization

In this dissertation, I develop a fluid-flow simulator for near-wellbore applications, including

simulations of mud-filtrate invasion and formation-tester measurements. In formation eval-

uation applications, a single-well model is selected because most of the variations occur in

the vicinity of the wellbore. Cylindrical coordinates are preferred for developing a numerical

algorithm in a single well because (i) it can benefit from the axial symmetry to simulate

44

z

ΔZZz+1/2

ΔZz Zz‐1/2(rr,Θθ,Zz)

y

Θ

Θθ‐1/2 y

xΘθ+1/2

Θθ

D

Δrr ΔΘθr 1/2

rr‐1/2xrr

rr+1/2

Figure 3.1: Discreption of a point in a discretized grid block in cylindrical coordinates.

invasion taking place in vertical wells, (ii) the well axis coincides with the longitudinal axis

of the cylinder, (iii) when simulating formation-tester measurements, there are smooth vari-

ations in the spatial (radial, azimuthal, and vertical directions) distributions of pressure and

fluid-phase saturations near the wellbore.

Reservoir discretization involves dividing the cylinder into nr, nθ, and nz in the

radial, azimuthal, and vertical directions, respectively. Figure 3.1 shows a point in space

described in cylindrical coordinates. The center of the block is identified by (rr, Θθ, Zz).

Figures 3.2(a) and 3.2(b) show that the block with indices (r, θ, z) is surrounded by blocks

(r − 1, θ, z) and (r + 1, θ, z) in the radial direction, (r, θ − 1, z) and (r, θ + 1, z) in

the azimuthal direction, and (r, θ, z − 1) and (r, θ, z + 1) in the vertical direction. Block

boundaries have indices (r−1/2, θ, z+1) and (r+1/2, θ, z) in the radial direction, (r, θ−1/2,

z) and (r, θ + 1/2, z) in the azimuthal direction, and (r, θ, z − 1/2) and (r, θ, z + 1/2)

in the vertical direction. Variables on block boundaries (for example fluid phase specific

density and molar density) are calculated using an upstream weighting method as described

in Section 3.3.1. Moreover, fluid phase velocities are calculated at block boundaries. Primary

variables including pressure and component concentrations are calculated at the center of

the block.

45

Δr

y

Δrrrr‐1

rr‐1/2 (rr,Θθ,Zz)

(rr‐1,Θθ,Zz)(rr,Θθ+1,Zz)

rrrr+1/2

rr+1

(rr+1, Θθ,zz)ΔΘθ+1

Θθ

Θθ+1/2Θθ+1

rr+1ΔΘθ

ΔΘθ‐1

Θθ

Θθ‐1/2

x

Θθ‐1

(a)

z

(r Θθ Z )

(rr, Θθ,Zz+1)

ΔZz

Zz+1/2

(rr, Θθ,Zz)

(rr, Θθ,Zz‐1)

Zz‐1/2

r

rD

(b)

Figure 3.2: Description of discretization of a grid block with neighboring blocks in (a)horizontal plane and (b) vertical direction. Indices r,Θ, and Z identify radial, azimuthal,and vertical locations, respectively. Subscripts r, θ, and z identify element numbers in radial,azimuthal, and vertical directions.

Permeability at block boundaries are calculated using harmonic averaging, i.e.,

krr,r+1/2 =ln(rr+1/rr)

ln(rr+1/rr+1/2)

kr+1+

ln(rr+1/2/rr)

kr

, (3.1)

kzz,z+1/2 =∆Zz+1 + ∆Zz∆Zz+1

kzz,z+1+ ∆Zz

kzz,z

, (3.2)

and

kθθ,θ+1/2 =∆Θθ+1 + ∆Θθ

∆Θθ+1

kθθ,θ+1+ ∆Θθ

kθθ,θ

. (3.3)

In the numerical model applied in this dissertation, radial grid blocks are spaced

logarithmically in r2 sizes (Aziz and Settari, 1979), namely,

r2r+1/2 =

r2r+1 − r2

r

ln( rr+1

rr), (3.4)

46

y

P t h i l

Wellbore

Petrophysical Boundary

RT

RT2

RT

RT2RT2

RT1

RT1

RT1RT2

xRT1

RT1

RT: Rock Type

Figure 3.3: Discreption initialization of grid blocks when a petrophysical bed boundarydoes not conform with the gridding system. Based on the location of block center withrespect to the bed boundary, petrophysical properties are initialized.

and

rr+1 = αlogrr, (3.5)

where

αnrlog =ReRw

, (3.6)

and Re and Rw are drainage and wellbore radii, respectively. Grid sizes in the azimuthal

and vertical directions are arbitrarily defined by the user.

Grid blocks in models for vertical wells are conformal to the geometry of the well

and bed boundaries. However, grid blocks in cylindrical models for deviated and horizontal

wells do not coincide with boundary lines. In this dissertation, petrophysical properties and

saturating fluid are initialized based on the location of the block center from the boundary

line. For instance, in Figure 3.3, grid blocks with their center above the petrophysical

boundary line have properties of Rock Type 2 and those below have properties of Rock

Type 1.

47

3.2 Discretization of the Pressure Equation

From Chapter 2, Section 2.4, the pressure equation is given by

[V 0p cf −

(∂Vt∂P

)Ni

]∂P

∂t− Vb

nc+1∑i=1

Vti∇ ·

np∑j=1

kλrjξjxij · ∇P

= Vb

nc+1∑i=1

Vti∇ ·

np∑j=1

kλrjξjxij · (∇Pc2j − γj∇D)

+

Vb

nc+1∑i=1

Vti∇ ·np∑j=1

φξjSjKij · ∇xij +

nc+1∑i=1

Vtiqi, (3.7)

where:

nc is number of components,

np is number of co-existing fluid phases,

V 0p is pore volume at reference pressure,

Vb is bulk volume,

cf is rock compressibility,

P is pressure,

Ni is total moles of component i,

Vti is partial molar volume defined in equation (2.96),

k is permeability tensor,

λrj is relative mobility of fluid phase j,

ξj is molar density of fluid phase j,

φ is rock porosity,

Sj is saturation of fluid phase j,

48

xij is molar fraction of component i in fluid phase j,

Kij is dispersion tensor,

Pc2j is capillary pressure of fluid phase j and oil phase, and

qi is flow rate of component i.

Using the finite-difference method, the time derivative becomes

[V 0p cf −

(∂Vt∂P

)Ni

](∂P

∂t

)u

1

∆t

(V 0p cf −

∂Vt∂P

)nrθz

(Pn+1rθz − P

nrθz

), (3.8)

where the subscript rθz identifies the spatial coordinates (cylindrical coordinates) and super-

scripts n and n+ 1 indicate discretized time levels. Variables such as relative permeability,

molar density, porosity, etc are calculated at the corresponding time level. The dot product

of a tensor, D, and a vector, −→a , is a vector, namely,

−→c = D · −→a , (3.9)

with the entries of the vector defined as

ci = Dijaj . (3.10)

Using the definition of divergence and gradient in cylindrical coordinates (Chapter 2, Lake,

1989) and equation (3.10), spatial derivatives (divergence of the pressure gradient) in equa-

tion (3.7) can be expressed as

∇ ·[kλrjξjxij · ∇P

]= PR + PΘ + PZ, (3.11)

where

PR =1

r

∂

∂r

[λrjξjxij

(krrr

∂P

∂r+ krθ

∂P

∂θ+ krzr

∂P

∂z

)], (3.12)

49

PΘ =1

r

∂

∂θ

[λrjξjxij

(krθ

∂P

∂r+ kθθ

1

r

∂P

∂θ+ kθz

∂P

∂z

)], (3.13)

and

PZ =∂

∂z

[λrjξjxij

(krz

∂P

∂r+ kθz

1

r

∂P

∂θ+ kzz

∂P

∂z

)]. (3.14)

Likewise, the gravity term in equation (3.7) is given by

∇ ·[kλrjξjxijγj · ∇D

]= GR + GΘ + GZ, (3.15)

where

GR =1

r

∂

∂r

[λrjξjxijγj

(krrr

∂D

∂r+ krθ

∂D

∂θ+ krzr

∂D

∂z

)], (3.16)

GΘ =1

r

∂

∂θ

[λrjξjxijγj

(krθ

∂D

∂r+ kθθ

1

r

∂D

∂θ+ kθz

∂D

∂z

)], (3.17)

and

GZ =∂

∂z

[λrjξjxijγj

(krz

∂D

∂r+ kθz

1

r

∂D

∂θ+ kzz

∂D

∂z

)]. (3.18)

Finally, the dispersion term in equation (3.7) can be discretized as

∇ ·[φξjSjKij · ∇xij

]= JRij + JΘij + JZij , (3.19)

where

JRij =1

r

∂

∂r

[φξjSj

(Krr,ijr

∂xij∂r

+Krθ,ij∂xij∂θ

+Krz,ijr∂xij∂z

)], (3.20)

JΘij =1

r

∂

∂θ

[φξjSj

(Krθ,ij

∂xij∂r

+Kθθ,ij

r

∂xij∂θ

+Kθz,ij∂xij∂z

)], (3.21)

and

JZij =∂

∂z

[φξjSj

(Krz,ij

∂xij∂r

+Kθz,ij

r

∂xij∂θ

+Kzz,ij∂xij∂z

)]. (3.22)

50

3.3 Calculation of Transmissibilities

In this section, I find expressions for the transmissibilies in equations (3.12) through (3.22).

By substituting Υ = r2 in the first term on the right-hand side of equation (3.12) and

rearranging the derivatives, I obtain

Prr =1

r

∂

∂r(kλrjξjxijr

∂P

∂r) = 4

∂

∂Υ(Υkλrjξjxij

∂P

∂Υ). (3.23)

Similarly, Prθ and Prz are expressed as

Prθ =1

r

∂

∂r(kλrjξjxijr

1

r

∂P

∂θ) = 2

∂

∂Υ(kλrjξjxij

∂P

∂θ), (3.24)

and

Prz =1

r

∂

∂r(kλrjξjxijr

∂P

∂z) = 2

∂

∂Υ(kλrjξjxijr

∂P

∂z). (3.25)

Subsequently, using the central-difference scheme to discretize equations (3.23) through (3.25),

I obtain

Prr =4

Υr+1/2 −Υr−1/2

[(Υkrrλrjξjxij)(r+1/2)

Pr+1 − PrΥr+1 −Υr

−

(Υkrrλrjξjxij)(r−1/2)

Pr − Pr−1

Υr −Υr−1

], (3.26)

Prθ =2

Υr+1 −Υr−1

[(krθλrjξjxij)(r+1)

Pr+1,θ+1 − Pr+1,θ−1

Θθ+1 −Θθ−1−

(krθλrjξjxij)(r−1)

Pr−1,θ+1 − Pr−1,θ−1

Θθ+1 −Θθ−1

], (3.27)

51

and

Prz =2

Υr+1 −Υr−1

[(krzλrjξjxijr)(r+1)

Pr+1,z+1 − Pr+1,z−1

Zz+1 − Zz−1−

(krzλrjξjxijr)(r−1)

Pr−1,z+1 − Pr−1,z−1

Zz+1 − Zz−1

]. (3.28)

Following the above discretization scheme, equation (3.13) can be expanded as

Pθr =1

rr

1

Θθ+1 −Θθ−1

[(krθλrjξjxij

)(θ+1)

Pr+1,θ+1,z − Pr−1,θ+1,z

rr+1 − rr−1−

(krθλrjξjxij

)(θ−1)

Pr+1,θ−1,z − Pr−1,θ−1,z

rr+1 − rr−1

], (3.29)

Pθθ =1

r2r

1

Θθ+1 −Θθ−1

[(kθθλrjξjxij

)(θ+1)

Pr,θ+1,z − Pr,θ,zΘθ+1 −Θθ

−

(kθθλrjξjxij

)(θ−1)

Pr,θ,z − Pr,θ−1,z

Θθ −Θθ−1

], (3.30)

and

Pθz =1

rr

1

Θθ+1 −Θθ−1

[(kθzλrjξjxij

)(θ+1)

Pr,θ+1,z+1 − Pr,θ,z−1

Zz+1 − Zz−1−

(kθzλrjξjxij

)(θ−1)

Pr,θ−1,z+1 − Pr,θ−1/2,z−1

Zz+1 − Zz−1

]. (3.31)

Analogously, equation (3.14) is elaborated as

Pzr =1

Zz+1 − Zz−1

[(krzλrjξjxij

)(z+1)

Pr+1,θ,z+1 − Pr−1,θ,z+1

rr+1 − rr−1−

(krzλrjξjxij

)(z−1)

Pr+1,θ,z−1 − Pr−1,θ,z−1

rr+1 − rr−1

], (3.32)

52

Pzθ =1

rr

1

Zz+1

[(kθzλrjξjxij

)(z+1)

Pr,θ+1,z+1 − Pr,θ−1,z+1

Θθ+1 −Θθ−1−

(kθzλrjξjxij

)(z−1)

Pr,θ+1,z−1 − Pr,θ−1,z−1

Θθ+1 −Θθ−1

], (3.33)

Pzz =1

Zz+1/2 − Zz−1/2

[(kzzλrjξjxij

)(z+1/2)

Pr,θ,z+1 − Pr,θ,zZz+1 − Zz

−

(kzzλrjξjxij

)(z−1/2)

Pr,θ,z − Pr,θ,z−1

Zz − Zz−1

], (3.34)

where subscripts l − 1/2, l, and l + 1/2 (l = r, θ, z) indicate, respectively, left, center, and

right side of the discretization element identified with index l. Discretization of the spatial

derivatives of physical dispersion terms is explained in Appendix A.

After substituting the finite-difference expressions, I express equation (3.7) as

(V 0p cf −

∂Vt∂P

)nxyz

Pn+1rθz − (∆rT∆rP

n+1 + ∆θT∆θPn+1 + ∆zT∆zP

n+1)

= (Vt − Vp)nrθz +(V 0p cf −

∂Vt∂P

)nrθzPnrθz+

∆t

nc+1∑i=1

(Vti)nrθz

qi + ∆t(Fcap − Fgrav − Fdisp

)nrθz, (3.35)

where Fcap, Fgrav, and Fdisp are capillary pressure, gravity, and dispersion terms, respec-

tively, in the pressure equation. In equation (3.35), (Vt − V p)nrθz is the difference between

fluid volume and pore volume at the previous time step. This difference arises because

of numerical errors in the calculation of pressure and saturation from the previous time

step (Acs et al., 1985; Spillette et al., 1973). In equation (3.35), ∆rT∆rPn+1, ∆θT∆θP

n+1,

and ∆zT∆zPn+1 are given by

∆rT∆rPn+1 = Arr+1/2(Pr+1 − Pr)−Arr−1/2(Pr − Pr−1)+

Ar+1,θ(Pr+1,θ+1 − Pr+1,θ−1)−Ar−1,θ(Pr−1,θ+1 − Pr−1,θ−1)+

Ar+1,z(Pr+1,z+1 − Pr+1,z)−Ar−1,z(Pr−1,z − Pr−1,z−1), (3.36)

53

∆θT∆θPn+1 = Ar,θ+1(Pr+1,θ+1 − Pr−1.θ+1)−Ar,θ−1(Pr+1,θ−1 − Pr−1,θ−1)+

Aθθ+1/2(Pθ+1 − Pθ)−Aθθ−1/2(Pθ − Pθ−1)+

Aθ+1,z(Pθ+1,z+1 − Pθ+1,z−1)−Aθ−1,z(Pθ−1,z+1 − Pθ−1,z−1), (3.37)

and

∆zT∆zPn+1 = Ar,z+1(Pr+1,z+1 − Pr+1,z+1)−Ar,z−1(Pr+1,z−1 − Pr−1,z−1)+

Aθ,z+1(Pθ+1,z+1 − Pθ−1,z+1)−Aθ,z−1(Pθ+1,z−1 − Pθ−1,z−1)+

Azz+1/2(Pz+1 − Pz)−Azz−1/2(Pz − Pz−1), (3.38)

where Arr±1/2, Aθθ±1/2, and Azz+1/2 are calculated by

Amm±1/2 = ∆t

nc+1∑i=1

(Vti)nm

np∑j=1

(xijTj

)nm±1/2

for m=r, θ, and z. (3.39)

In equation (3.39), Tj is transmissibility of fluid phase j which is defined in Section 3.3.2.

Likewise, in equations (3.36) through (3.38), Ar±1,θ, Ar±1,z, Ar,θ±1, Aθ±1,z, Ar,z±1, and

Aθ,z±1 are related to non-diagonal terms of the permeability tensor, given by

Ar±1,θ =2Vb∆t

(r2r+1 − r2

r−1)(Θθ+1 −Θθ−1)

nc+1∑i=1

(Vti)nrθz

np∑j=1

(krθxijλjξj)nr±1, (3.40)

Ar±1,z =2Vb∆t

(r2r+1 − r2

r−1)(Zz+1 − Zz−1)

nc+1∑i=1

(Vti)nrθz

np∑j=1

(rkrzxijλjξj)nr±1, (3.41)

Ar,θ±1 =Vb∆t

rr(rr+1 − rr−1)(Θθ+1 −Θθ−1)

nc+1∑i=1

(Vti)nrθz

np∑j=1

(kθrxijλjξj)nθ±1, (3.42)

Aθ±1,z =Vb∆t

rr(Zz+1 − Zz−1)(Θθ+1 −Θθ−1)

nc+1∑i=1

(Vti)nrθz

np∑j=1

(kθzxijλjξj)nθ±1, (3.43)

54

Ar,z±1 =Vb∆t

(rr+1 − rr−1)(Zz+1 − Zz−1)

nc+1∑i=1

(Vti)nrθz

np∑j=1

(kzrxijλjξj)nz±1, (3.44)

and

Aθ,z±1 =Vb∆t

rr(Zz+1 − Zz−1)(Θθ+1 −Θθ−1)

nc+1∑i=1

(Vti)nrθz

np∑j=1

(kzθxijλjξj)nz±1. (3.45)

3.3.1 Upstream Weighting

In equation (3.39), I have used one-point upstream weighting to approximate molar density,

(ξj)m±1/2, phase composition, (xij)m±1/2, and relative mobility, (λrj)m±1/2. For example,

upstream weighting for molar density is given by

(ξj)n(m−1/2) = (ξj)

n(m−1) if (Φj)m > (Φj)m, (3.46)

(ξj)n(m+1/2) = (ξj)

n(m) if (Φj)m > (Φj)m+1, (3.47)

and

(ξj)n(m−1/2) = (ξj)

n(m) if (Φj)m < (Φj)m, (3.48)

(ξj)n(m+1/2) = (ξj)

n(m+1) if (Φj)m < (Φj)m+1, (3.49)

where (Φj)m is the potential of fluid phase j at an element with index m, where m = r, θ,

z. This term is equal to the sum of capillary pressure, gravity potential force, and pressure,

namely,

(Φj)m = Pm + (Pc2j)m − gfγjD, (3.50)

where gf is a multiplier for gravity force; this parameter in field units is equal to 0.433

[psi/ft]. For the dispersion terms in equations (A.5) through (A.13), upstream weighting is

55

performed as

(φSjξjKrr,ij)r±1/2 =Dij

τ(φSjξj)r±/2

+(ξj)r±1/2

|uj |r±1/2(αlju

2rj + αtju

2θj + αtju

2zj)r±1/2, (3.51)

and

(φSjξjKrθ)r±1/2 = (αlj − αtj)(ξjuruθ)r±1/2

|uj |r±1/2, (3.52)

whereDij is molecular diffusion coefficient of component i in phase j, τ is tortuosity, urj , uθj ,

uzj are velocities of fluid phase j in radial, azimuthal, and vertical directions, respectively;

αlj and αtj are, respectively, logitudinal and transverse dispersivity of fluid phase j. In

equations (3.51) and (3.52), I use an average upstream velocity approximation to calculate

velocities at grid faces, viz.,

(uθj)(r±1/2,θ,z) =1

2

[(uθj)(r,θ+1/2,z) + (uθj)(r,θ−1/2,z)

];

if (Φj)r,θ,z > (Φj)r±1,θ,z, (3.53)

(uzj)(r±1/2,θ,z) =1

2

[(uzj)(r,θ+1/2,z) + (uzj)(r,θ−1/2,z)

];

if (Φj)r,θ,z > (Φj)r±1,θ,z, (3.54)

and

(uθj)(r±1/2,θ,z) =1

2

[(uθj)(r±1,θ+1/2,z) + (uθj)(r±1,θ−1/2,z)

];

if (Φj)r,θ,z < (Φj)r±1,θ,z, (3.55)

56

(uzj)(r±1/2,θ,z) =1

2

[(uzj)(r±1,θ+1/2,z) + (uzj)(r±1,θ−1/2,z)

];

if (Φj)r,θ,z < (Φj)r±1,θ,z. (3.56)

3.3.2 Fluid-Phase Transmissibility

The fluid-phase transmissibility terms in equation (3.39) are defined as

(Tj)nm±1/2 =

(krjξjµj

)nm±1/2

Tm±1/2 for m = r, θ, z, (3.57)

where krj , ξj , and µj are relative permeability, molar density, and viscosity of fluid phase

j, respectively. Fluid phase transmissibility, Tm±1/2, can be calculated in each of the r, θ,

and z directions by (Peaceman, 1977)

Tr±1/2 =±∆Θθ∆Zz

ln rr±1

rr±1/2

(krr)r±1+

lnrr±1/2

rr

(krr)r

, (3.58)

Tθ±1/2 =± log (

rr+1/2

rr−1/2)∆Zz

Θθ±1 −Θθ±1/2

(kθθ)θ±1+

Θθ±1/2 −Θθ

(kθθ)θ

, (3.59)

where ∆Zz = Zz+1/2 − Zz−1/2, ∆Θθ = Θθ+1/2 −Θθ−1/2, and

Tθ±1/2 =±∆Θθ

2 (r2r+1/2 − r

2r−1/2)

Zz±1 − Zz±1/2

(kzz)z±1+Zz±1/2 − Zz

(kzz)z

. (3.60)

In equations (3.58) through (3.60), (kll)l−1, (kll)l, and (kll)l+1 are, respectively, permeabil-

ities of grid blocks with indices l − 1, l, and l + 1, where l = r, θ, z.

57

3.3.3 Capillary-Pressure Term

The capillary-pressure term, Fcap, in equation (3.35) represents the sum of all capillary force

terms, namely,

Fcap =

nc+1∑i=1

(Vti)nrθz

np∑j=1

∆(xijTj)n∆(Pnc2j), (3.61)

where Vti is the partial derivative of total fluid volume with respect to component i, and

∆(xijTj)n∆(Pnc2j) = ∆r(xijTj)

n∆r(Pnc2j)+

∆θ(xijTj)n∆θ(P

nc2j) + ∆z(xijTj)

n∆z(Pnc2j). (3.62)

In equation (3.62), the spatial derivatives (divergence of capillary pressure gradient) is the

sum of terms in the radial, azimuthal, and vertical directions. Below, I expand the divergence

of capillary pressure gradient in the radial direction. The remaining spatial derivative terms

(azimuthal and vertical directions) can be expressed in a similar manner.

∆r(xijTj)n∆r(P

nc2j) =

(xijTj)nr+1/2 [(Pc2j)r+1 − (Pc2j)r]

n − (xijTj)nr−1/2 [(Pc2j)r − (Pc2j)r−1]

n+

2Vb(r2r+1 − r2

r−1)(Θθ+1 −Θθ−1)(krθxijλjξj)

nr+1 [(Pc2j)r+1,θ+1 − (Pc2j)r+1,θ−1]

n−

2Vb(r2r+1 − r2


nr−1 [(Pc2j)r−1,θ+1 − (Pc2j)r−1,θ−1]

n+

2Vb(r2r+1 − r2

r−1)(Zz+1 − Zz−1)(rkrzxijλjξj)

nr+1 [(Pc2j)r+1,z+1 − (Pc2j)r+1,z−1]

n−

2Vb(r2r+1 − r2


nr−1 [(Pc2j)r−1,θ+1 − (Pc2j)r−1,θ−1]

n+

2Vb(r2r+1 − r2

r−1)(Zz+1 − Zz−1)(rkrzxijλjξj)

nr−1 [(Pc2j)r−1,z+1 − (Pc2j)r−1,z−1]

n, (3.63)

where Vb is the bulk volume, xij is mole fraction of component i in fluid phase j, Pc2j is

capillary pressure of fluid phase j, λj is mobility of fluid phase j, ξj is molar density of fluid

phase j, Tj is transmissibility of fluid phase j; ll+1 and ll−1 (l = r, θ, and z) are, respectively,

58

coordinates of right and left neighbors in the specified direction; n is discretization time level,

and subscripts r, r + 1/2, and r − 1/2 are, respectively, the radial coordinates of center,

right boundary, and left boundary of the corresponding grid block.

3.3.4 Gravity Term

In equation (3.35), I introduce the net effect of gravity in equation (3.7) as Fgrav; this term

can be expressed as

Fgrav =

nc+1∑i=1

(Vti)nrθz

np∑j=1

∆(xijTjγj)n∆D, (3.64)

where γj is specific density of fluid phase j, and D is the depth of the block center. The

discretization scheme of equation (3.64) is similar to that elaborated in equation (3.63).

However, upstream weighting of fluid phase specific density is based on pore volume weight

averaging, to wit,

(γj)m±1/2 =(Vpγj)m + (Vpγj)m±1

(Vp)m + (Vp)m±1, (3.65)

where (Vp)m and (Vp)m±1 are the pore volume of blocks with indices m and m± 1, respec-

tively, where m = r, θ, and z.

3.4 Discretization of the Molar Mass Equation

The simulation algorithm implemented in this dissertation consists of implicit calculation

of pressure and explicit calculation of molar concentration for each component. Moles for

each component are calculated explicitly after solving the pressure equations. Section 2.6

describes the relationship obtained from mass conservation equation for the time variation

of molar concentration of components. This section discretizes equation (2.45) using a

59

finite-difference central scheme, i.e.,

Nn+1i = Nn

i + ∆t

np∑j=1

∆r(xijTj)n∆r(P

n+1 + Pnc2j)−∆r(xijTjγj)n∆rD+

np∑j=1

∆θ(xijTj)n∆θ(P

n+1 + Pnc2j)−∆θ(xijTjγj)n∆θD+

np∑j=1

∆z(xijTj)n∆z(P

n+1 + Pnc2j)−∆z(xijTjγj)n∆zD+

Vb

np∑j=1

(∆rJnr,ij + ∆θJ

nθ,ij + ∆zJ

nz,ij) + qi

, for i = 1, . . . , nc, nc + 1, (3.66)

where Ni is the number moles of component i, Tj is the transmissibility of fluid phase j, xij

is mole fraction of component i in fluid phase j, γj is specific density of fluid phase j, D is

depth, Vb is bulk volume, Pc2j is capillary pressure of fluid phase j and oil phase, qi is flow

rate of component i, Jr,ij , Jθ,ij , and Jz,ij are spatial derivatives of physical dispersion in the

radial, azimuthal, and vertical directions, respectively. Convection and physical dispersion

terms in equation (3.66) are expanded similar to equation (3.63).

3.5 Phase Behavior

This section summarizes the algorithm used to calculate the number of fluid phases and

their corresponding compositions including stability analysis, flash calculation for two-phase

equilibrium, and fluid phase identifications. The implemented algorithm for the enforcement

of phase equilibrium is similar to those of Chang’s (1990) and Perschke’s (1988).

3.5.1 Phase Stability

In the phase stability calculation, I search for a trial phase with composition −→x , which

minimizes the following equation (Chang, 1990; Firoozabadi, 1999):

∆G(−→x ) =

nc∑i=1

xi[µi(−→x )− µi(−→z )], (3.67)

60

where G is molar Gibbs free energy, µi is the chemical potential of component i, −→z is overall

hydrocarbon composition, and xi is molar fraction of component i in the trial fluid phase.

3.5.1.1 Tangent-Plane Distance Approach

In this method, I solve the following system of nonlinear equations for the independent

variable included in the phase stability condition, Xi, given by

lnXi + lnϕi(−→x )− ln zi − lnϕi(

−→z ) = 0, i = 1, . . . , nc, (3.68)

where ϕi is fugacity coefficient of component i and is defined by equation (2.90). Mole

fraction, xi, is related to the independent variable of phase stability, Xi, by

xi =Xi∑ncp=1Xp

, i = 1, . . . , nc. (3.69)

I use the method of successive substitution to solve equation (3.68) (Firoozabadi, 1999). In

this method, at each iteration, the independent variable of phase stability, Xi, is updated

by

Xnewi = exp

[ln zi + lnϕi(

−→z )]− lnϕi(

−→x )

, i = 1, . . . , nc, (3.70)

and subsequently xi is updated with equation (3.69). Depending upon the assumption for

feed composition (i.e., assuming liquid or vapor for feed composition), −→z , I choose one of

the following values as initial guess for Xi:

Liquid phase

Xi = ziKi, i = 1, . . . , nc, and (3.71)

Vapor phase

Xi =ziKi, i = 1, . . . , nc, (3.72)

61

where the equilibrium K-value of component i, Ki, is estimated from Wilson’s correla-

tion (Wilson, 1969), namely,

Ki =PciP

exp

[5.37(1 + ωi)(1−

TciT

)

], i = 1, . . . , nc, (3.73)

in which Pci and Tci are the critical pressure and temperature, respectively, of component i.

In the above equation, T and Tci are expressed in [R], and P and Pci are expressed in [psi].

3.5.1.2 Flash Calculation

Stability analysis indicates whether an specific composition of hydrocarbon components at

the desired pressure and temperature is stable or not. In the event that a mixture is found

unstable, it will be split into more than one phase. I have assumed that the mixture splits

into two phases: liquid and gas. Flash calculation computes the amount and composition of

each hydrocarbon phase. Equations for two-phase flash calculation are (Firoozabadi, 1999):

Equality of chemical potential or fugacity at equilibrium, namely,

fLi (T, P,−→x ) = fVi (T, P,−→x ), (3.74)

where P is the pressure of reference fluid pahse, fLi and fVi are the fugacities of

component i in liquid and gas, respectively.

Material balance for components, to wit,

zi = Fvyi + (1− Fv)xi, (3.75)

where xi is the molar fraction of component i in liquid phase, yi is the molar fraction

of component i in vapor phase, and Fv is vapor mole fraction, defined as

Fv =nv

nv + nL. (3.76)

62

where nv and nL are number of moles in vapor and liquid phases, respectively.

Sum of molar fractions of all fluid phases is equal to one, i.e.,

nc∑i=1

xi = 1, (3.77)

and

nc∑i=1

yi = 1. (3.78)

3.5.1.2.1 Successive Substitution Method

In this method, I search for a solution of vapor mole fraction, Fv, in an iterative manner.

For this purpose, I define the K-value of component i, Ki, as

Ki =yixi. (3.79)

By combining equations (3.75), (3.77) , (3.78), and (3.79), I obtain

xi =zi

1 + Fv(Ki − 1), (3.80)

yi = Kixi, (3.81)

and

h(Fv) =

nc∑i=1

zi(Ki − 1)

1 + Fv(Ki − 1)= 0, (3.82)

where the last equation is usually referred to as Rachford-Rice’s expression (Rachford and

Rice, 1952). Equlilibrium condition for fugacity, equation (3.74), and equations (3.79)

and (2.90) give

yiPϕvi = xiPϕ

Li ⇒ Ki =

yixi

=ϕLiϕvi

, (3.83)

63

where ϕLi and ϕvi are fugacity coefficient of component i in liquid and vapor phases, respec-

tively.

The procedure used to find the composition of fluid each phase,−→X and

−→Y, is as

follows:

1. Guess the initial values of Ki. I use the values estimated from Wilson’s correlation

(equation (3.73)) as initial guess.

2. Solve Rachford-Rice’s expression. I use Newton’s method to solve equation (3.82).

3. Calculate the composition of gas and liquid phases from equation (3.80) and (3.81).

4. Update Ki. I update the equilibrium ratio using Knewi = Kold

ifLifvi

.

3.5.1.2.2 Newton’s Method

Equations (3.74), (3.75), (3.77), and (3.78) define the two-phase flash calculation. I solve

2nc + 1 equations using Newton’s method to find 2nc + 1 unknowns, Fv,−→x , and −→y . This

method is explained in detail by Firoozabadi (1999).

3.5.1.3 Combination of the Successive Substitution Method and Newton’s Method

In this simulator, I use the successive substitution method to obtain the composition of

the fluid phases with a tolerance of 10−4 [dimensionless], then I apply Newton’s method to

obtain the composition of the fluid phases with a smaller tolerance (10−8) [dimensionless].

3.5.2 Phase Identification

The simulator developed in this dissertation is capable of simulating fluid flow of three co-

existing fluid phases: water, oil, and gas, where the aqueous phase is tracked separately.

When there are two hydrocarbon phases, the one with greater molar density is labeled

as oil and the remaining one as gas. If a single-phase hydrocarbon is found to be stable,

hydrocarbon phase is identified with one of the following schemes:

64

Depending on whethernc∑i=1

ziKi

>

nc∑i=1

ziKi, (3.84)

is true or not, then the fluid phase is labeled as gas or liquid, respectively, (Perschke,

1988).

Pedersen and Christensen (2007) suggested that if

v

b< const, (3.85)

then the fluid phase is liquid; otherwise it is gas. In equation (3.85), v is the molar

volume [ft3/lbm], and b is parameter b in the cubic EOS [dimensionless] (for instance

in equation (2.76)). For the PR-EOS, Pedersen and Christensen (2007) recommend

that const = 1.75 [ft3/lbm].

3.6 Boundary and Well Conditions

In this section, I describe different boundary conditions implemented for wells, including

injection with constant volume rate, injection with constant bottomhole pressure, production

with constant volumetric rate, and production with constant bottomhole pressure.

3.6.1 Boundary Conditions

The no-flow boundary condition is given by

−→n · −→u j = 0, (3.86)

where −→n is the unit normal vector along the no-flow boundary. In this model, I impose

equation (3.86) by setting to zero the following parameters: total diagonal transmissibility,

Tm±1/2 (m = r, θ, and z), in equation (3.57), and total off-diagonal transmissibilities, Alk

(l, k = r, θ, and z where l is different from k) in equations (3.40) through (3.45). For an

65

inflow boundary condition, the total injection rate of each component is specified, which

depends on the assumed well condition.

3.6.2 Well Models

In the developed model, sink or source terms in the mass conservation equations are repre-

sentative of wells in the reservoir. Wells are constrained with one of the following boundary-

condition constraints:

1. flow rate, and

2. bottomhole pressure.

The operating wells can also be monitored with another boundary condition; for instance,

an injection (or a production) well operating with constant flow rate can be monitored with

a constant bottomhole pressure. Volumetric flow rate and bottomhole pressure of the well

are related through the productivity index, namely,

Qj = PIj(Pwf − Pj), (3.87)

where PIj is productivity index [ft3/day/psi], Qj is volumetric flow rate [ft3/day], Pwf is

bottomhole pressure [psi], and Pj is pressure of fluid phase j [psi]. For a 3D cylindrical

model, the phase productivity index can be written as (Peaceman, 1977)

PIj =krr∆Zλrjfa

25.14872 (lnrorw

+ s), (3.88)

where

fa =∆Θ

2π, (3.89)

where s is skin factor, ∆Z is grid thickness, ∆Θ is cake-like slice shown in Figure 3.2(a), λrj

is relative mobility of fluid phase j, rw is the radius of wellbore [ft], 25.14872 is a consistency

66

conversion constant, and

ro = r1 exp(−0.5)× expln( r1rw )

( r1rw )2 − 1, (3.90)

where r1 is the radius of the first radial grid [ft].

3.6.2.1 Injection with Constant Volume Rate

Total volumetric flow rate, Qt, and total molar flow rate of hydrocarbon components, (qt)hyd,

are related by the total molar volume of injected hydrocarbon components, (vt)hyd,inj , i.e.,

(qt)hyd =Qt

(vt)hyd,inj, (3.91)

where total molar volume of the hydrocarbon components of injected fluid, (vt)hyd,inj , is

calculated by

(vt)hyd,inj = (1− fvξ2

)inj + (fvξ3

)inj , (3.92)

where ξ2 and ξ3 are, respectively, molar densities of oil, and vapor phase in the injected

fluid; and fv is molar fraction of vapor phase in the injected fluid. Furthermore, the flow

rate of each component is given by

qi = (zi)inj(qt)hyd, (3.93)

where zi is mole fraction of component i in the injected fluid. On the other hand, molar

flow rate of the water component is given by

(qt)nc+1 = (Qt)aqu(ξ1)inj , (3.94)

where ξ1 is molar density of water in the injected fluid. Moreover, values of (ξj)inj and

(fv)inj are determined by flash calculations for the injected fluid. Bottomhole pressure is

67

determined using the equation

Pwf = P +QtPIt

, (3.95)

where

PIt =

np∑j=1

PIj , (3.96)

and P is the pressure of the grid adjacent to the wellbore.

3.6.2.2 Injection with Constant Bottomhole Pressure

In this type of well constraint, the rate of fluid injection is controlled by the bottomhole

pressure, Pwf , molar fraction of water, (f1)inj , and the composition of hydrocarbon com-

ponents, (zi)inj . At each layer z, I compute flow rate for each component with the relation

qi∣∣z=[1− (f1)inj

](zi)inj qt

∣∣z, (3.97)

where (f1)inj is molar fraction of water, (zi)inj is mole fraction of component i in the injected

fluid, and qt is total molar volume rate. For the case of the water component,

qnc+1

∣∣z= (f1)inj qt

∣∣z. (3.98)

The total molar volume rate, qt, and total volume rate at depth z, Qt |z, are related by

qt∣∣z=

Qt∣∣z

(vt)inj, (3.99)

where

Qt∣∣z=

np∑j=1

PIj∣∣z

[Pwf |z − Pj |z], (3.100)

and

(vt)inj =

(f1

ξ1

)inj

+ [1− (f1)inj ]

[(1− fvξ2

)inj

+

(fvξ3

)inj

], (3.101)

68

where fv is molar fraction of the vapor phase, and ξ2 and ξ3 are the molar densities of

the oil and gas phases, respectively. By replacing the source term in the pressure equation

(equation (3.35)), with equations (3.97) and (3.98), I obtain a new pressure equation, namely,

An

LHPPn+1

+∆t

vt∣∣inj

×

nc∑i=1

Vti

(1− f1

∣∣inj

)zi∣∣inj

+Vt,nc+1 f1

∣∣inj

nrθz

× PIt∣∣nrθzPn+1rθz

= Bn

RHP +

np∑j=1

PIj (Pwf − Pc2j)

nrθz

×

∆t

vt∣∣inj

×

nc∑i=1

Vti

(1− f1

∣∣inj

)zi|inj + Vt,nc+1 f1

∣∣inj

nrθz

, (3.102)

where ALHP is left-hand side matrix of pressure equation, P is vector of pressures, and

BRHP is right-hand side vector of the pressure equation without a source or sink term

corresponding to a well. After solving equation (3.102), I calculate the flow rate of each

component using equations (3.97) through (3.101).

3.6.2.3 Production with Constant Volumetric Rate

In this type of well, the total volumetric production rate, Qt, is prescribed in the calculations.

The production rate for every component is calculated with

qi =qt |z ×

∑npj=2 ξjxijPIj |z∑np

j=1 ξjPIj |zfor i = 1, . . . , nc, (3.103)

and

qi |z =qt |z × ξ1PI1 |z∑np

j=1 ξjPIj |z, (3.104)

where total molar rate of production is calculated with

qt|z =Qt∑npj=1 ξjPIj |z∑ztz=zb

PIt|z, (3.105)

69

and ξj is molar density of fluid phase j, xij is molar fraction of component i in fluid phase

j, PIj |z is productivity index of fluid phase j at vertical location z, and PIt is total

productivity index defined as

PIt =

np∑j=1

PIj . (3.106)

3.6.2.4 Production with Constant Bottomhole Pressure

For this type of well, I assume that bottomhole pressure at the lower part of perforation is

known. Pressure at other layers is calculated based on the specific density of the fluid, i.e.,

Pwf |z = Pwf∣∣z=zb−γz∆D, (3.107)

where D is depth (positive downward as shown in Figure 3.1) and

γz =

∑npj=1 (γjPIj)z

PIt. (3.108)

For this type of well condition, the production rate for each component is obtained from

(qi) =

np∑j=2

(ξjxijPIj)(Pwf − Pj)z for i = 1, . . . , nc, (3.109)

and

(qnc+1) = (ξ1PI1)(Pwf − P1)z, (3.110)

where ξj is molar density of fluid phase j, Pwf is bottomhole pressure, and np and nc are

numbers of fluid phases and components, respectively. By substituting equations (3.109)

70

and (3.110) into equation (3.35), I obtain a new pressure equation, i.e.,

An

LHPPn+1

+ ∆t× nc∑i=1

Vti

np∑j=2

ξjxijPIj + Vt,nc+1ξ1PI1

nrθz

=

Bn

RHP + ∆t× nc∑i=1

Vti

np∑j=2

ξjxijPIj (Pwf − Pc2j)

+ Vt,nc+1ξ1PI1(Pwf − Pc21)

nrθz

. (3.111)

After solving equation (3.111), flow rate of each component is calculated using equations (3.109)

and (3.110).

3.7 Computation of Saturation

Subsequent to calculation of moles of each component in all grid blocks, I calculate the

saturation of each fluid phase. In this dissertation, S1, S2 and S3 are, respectively, saturation

of aqueous, oil, and gas phases. Saturation of the aqueous phase is calculated with

Sn+11 =

Nn+1w

(Vpξ1)n+1 , (3.112)

where (ξ1)n+1 is the molar density of water at time level n+ 1, (Vp)n+1 is the pore volume

at time level n + 1, and Nn+1w is total moles of the water component. On the other hand,

saturation of the gas phase is given by

Sn+13 =

(1− Sn+1

1

) (Fvξ3

)n+1

∑npj=2

Fjξj

, (3.113)

where Fv is molar fraction of the vapor phase, ξj is molar density of the fluid phase j, and

Fj is the molar fraction of the fluid phase j. Finally, saturation of oil is calculated with

Sn+12 = 1− Sn+1

1 − Sn+13 . (3.114)

71

3.8 Material Balance Error

In the discretization of the pressure equation (3.35), I include the material balance error as

Vt − Vp, where Vp is pore volume at the current pressure, and Vt is total fluid volume, i.e.,

Vt =

np∑j=1

njvj , (3.115)

where nj is total moles of all hydrocarbon components in fluid phase j, and vj is molar

volume of the fluid phase j. The material balance error arises due to numerical errors in

the calculation of pressure and saturation from the previous time step (Acs et al., 1985;

Spillette et al., 1973).

3.9 Automatic Time-Step Control

Selection of a proper time step determines the speed and stability of a numerical method.

The developed simulator applies the method of relative changes (Chang, 1990; Jensen, 1980)

to dynamically select time step. The procedure for updating the time step is as follows:

1. Obtain time step bounds

Initial time step, ∆tinit,

Maximum and minimum time step, ∆tmax, and ∆tmin,

Maximum relative change of pressure, ∆Plim,

Maximum change of saturation, ∆Slim,

Maximum relative change of volume error, ∆Vlim,

Maximum relative change of moles of a given component, ∆Nlim.

72

2. Calculate maximum changes in the following variables at times n and n + 1 for all

grid blocks:

∆Pmax = max(|Pn+1m − Pnm|Pn+1m

), (3.116)

∆Smax = max(|(Sj)n+1m − (Sj)

nm|), (3.117)

∆Vmax = max(|(Vt)n+1

m − (Vt)nm|

(Vt)n+1m

), (3.118)

∆Nmax = max(|(Ni)n+1

m − (Ni)nm|

(Ni)n+1m

), (3.119)

for m = 1, . . . , total number of grid blocks,

j = 1, . . . , np,

and i = 1, . . . , nc, nc + 1, and salt,

where Pnm is pressure at time level n in the m− th grid block, (Sj)nm is saturation of

fluid phase j at time level n in the m − th grid block, (Vt)nm is total fluid volume at

time level n in the m− th grid block, and (Ni)nm is total moles of component i at time

level n in the m− th grid block.

3. Update time step by

∆tnew = min(∆tP ,∆tS ,∆tV ,∆tN ), (3.120)

where

∆tP = ∆told∆Plim∆Pmax

, (3.121)

73

∆tS = ∆told∆Slim∆Smax

, (3.122)

∆tV = ∆told∆Vlim∆Vmax

, (3.123)

and

∆tN = ∆told∆Nlim∆Nmax

. (3.124)

4. Limit the updated time step

The updated time step should satisfy the following relation

∆tmin ≤ ∆tnew ≤ ∆tmax. (3.125)

3.10 Structure and Solution of the Pressure Equation

The matrix constructed for the left-hand side of equation (3.35) can be asymmetric and

non-diagonally dominant. Figure 3.4 shows the structure of a sparse matrix obtained for a

model with 4x2x3 grids in the radial, azimuthal, and vertical directions. When non-diagonal

terms in the permeability tensor are zero, the matrix for the pressure equation has three

diagonal bands for a one-dimensional (1D) model, five diagonal bands for a two-dimensional

(2D) model, and seven diagonal bands for a 3D model. However, when non-diagonal terms

in the permeability tensor are not zero, then the sparse matrix has 19 diagonal bands.

For the 1D model with three diagonal bands, I use the Thomas algorithm (Mitchell and

Griffiths, 1980) to solve the pressure equation. I implemented the package for iterative

solvers developed by Saad (2003) for 2D and 3D problems. This package includes a bi-

conjugate gradient method (BCG), a BCG stabilized method (BCGSTAB), and a transpose-

free quasi-minimum residual method. I implemented a BCGSTAB as the default iterative

solver. The package for iterative solvers also includes several preconditioners such as an

incomplete LU factorization with a dual truncation strategy (ILUT), an ILUT with column

74

x x x xx x x x x

x x x xx x x x

x x x x x

x x x xx x x x x

x x x x x xx x x x x

x x x x x


x x x x xx x x x x x

x x x x xx x x x x

x x x x x


x x x xx x x x x

x x x x

x x x xx x x x

x x x x xx x x x

Figure 3.4: Structure of the matrix for pressure equation (3.35) when constructed for amodel with 4x2x3 grids in radial, azimuthal, and vertical directions.

pivoting, an incomplete LU factorization with single dropping and diagonal compensation

(ILUD), and an ILUD with column pivoting (ILUDP) (Saad, 2003).

75

Chapter 4

Verification of the Simulator

This chapter conducts verification tests by comparing results obtained with the developed

method against two reservoir simulators commercialized by Computer Modeling Group Ltd.

(CMG). The developed simulator, referred to as UTFEC, is based on an equation-of-state

compositional algorithm; therefore, I chose the Generalized Equation-of-State Model Com-

positional Reservoir Simulator (GEM) for verification purposes. Furthermore, I use CMG-

WinProp to calculate input data for Steam, Thermal, and Advanced Processes Reservoir

Simulator (STARS); this enabled the verification of results obtained for aqueous salt con-

centrations.

4.1 Introduction

I perform verification of the simulator for different flow regimes, including gas-water, oil-

water, and gas-oil-water. In doing so, the following cases are studied: one-dimensional (1D)

radial, two-dimensional (2D) axisymmetric, 2D radial, and three-dimensional (3D) cylindri-

cal configurations. I also test several boundary conditions including injection with constant

flow rate, injection with variable flow rate, injection with constant bottom hole pressure,

production with constant flow rate, and production with constant bottomhole pressure. De-

76

50505050

40

50

40

50

40

50

40

50Rock Type 1Rock Type 2Rock Type 3

30

c [psi] 30

c [psi] 30

c [psi] 30

c [psi]

Rock Type 3

P c[psi]

10

20P c

10

20P c

10

20P c

10

20P cP

0 0.2 0.4 0.6 0.8 1W S i

0 0.2 0.4 0.6 0.8 1

W S i0 0.2 0.4 0.6 0.8 1

W S i0 0.2 0.4 0.6 0.8 1

W S iWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

(a)

1111

0.8

1

0.8

1

0.8

1

0.8

1

kro0.6

k r

0.6

k r

0.6

k r

0.6

k rk r krw

ro

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S iWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

(b)

Figure 4.1: Water-oil (a) capillary pressure and (b) relative permeability curves of rocktypes studied in this dissertation. Variables kro and krw are relative permeability of oil andwater, respectively.

pending on the goal of the verification case, I compare oil phase pressure, saturation of fluid

phases, components concentration, fractional flow of water, and gas-oil-ratio. For 2D and 3D

cases, I show the relative difference between oil phase pressures obtained with UTFEC and

a CMG simulator, and the absolute difference between saturations calculated using UTFEC

and a CMG simulator. In the calculation of the spatial distribution for salt concentration,

I assume physical dispersion, αl1 = α = 0, unless a nonzero value is specified.

4.2 Description of Case Studies

The following sections summarize the reservoir properties, rock types, and assumed compo-

nents with their physical properties, used in the construction of case studies for verification

purposes.

4.2.1 Rock Types

I test the developed simulator on three theoretical rock types. Figure 4.1 shows the assumed

capillary pressure and relative permeability curves for these rock types. Table 4.1 lists the

remaining properties of rock types including absolute permeability and porosity.

77

Table 4.1: Absolute permeability, porosity, residual water saturation, and residual oilsaturation for three synthetic rock types assumed in this chapter.

Unit Rock Type 1 Rock Type 2 Rock Type 3

Absolute Permeability md 10 100 500Porosity fraction 0.16 0.25 0.32Swir fraction 0.16 0.11 0.07Soir fraction 0.12 0.12 0.12

4.2.2 Components

For simulator verification, several hydrocarbon components are selected from CMG-WinProp:

Table 4.2 lists the properties of these components. In the simulator, water is a component

which only remains in the aqueous phase and does not enter hydrocarbon fluid phases. I

assume that the water component has the properties listed in Table 4.3.

4.3 One-Dimensional Simulations

Several cases for the verification of 1D fluid-flow simulations are studied in Pour (2008).

This dissertation documents 1D verification cases for two-phase flow, three-phase flow, and

dispersion of aqueous salt concentration.

4.3.1 Two-Phase Flow Simulations

Verification of simulations for two-phase flow regimes includes cases for gas-water and oil-

water. For verification of 1D case studies, I assume that the reservoir exhibits the properties

listed in Table 4.4.

4.3.1.1 Gas-Water Simulation

I consider a reservoir with the initial conditions listed in Table 4.5. The reservoir contains

hydrocarbon components C1, C2, and C3 with properties described in Table 4.2. In this

case study, the well constraint is 1 day of water injection (salt concentration is equal to 3

78

Tab

le4.2

:P

rop

erti

esof

hyd

roca

rbon

com

pon

ents

ass

um

edin

the

sim

ula

tor

veri

fica

tion

;Pcrit

,Tcrit

,ω

,Mw

,Vcrit

,an

dΨi

are

crit

ical

pre

ssu

re,

crit

ical

tem

per

atu

re,

ace

ntr

icfa

ctor,

mole

cula

rw

eight,

crit

ical

mola

rvolu

me,

an

dp

ara

chor

of

the

com

pon

ents

,re

spec

tive

ly.

IC4,

IC5,

and

FC

6th

rou

ghF

C18

are

pse

ud

oco

mp

on

ents

(Sou

rce:

CM

G-W

inP

rop

).

Nam

ePcrit

[atm

]Tcrit

[K

]ω

[]

Mw

[g/gm

ol]

Vcrit

[m3/kgm

ol]

Ψi

[dyn

es1/4/cm

1/4/lb

m]

C1

45.4

191

0.0

116

0.1

77

C2

48.2

305

0.1

30.1

0.1

5108

C3

41.9

370

0.1

544.1

0.2

150

IC4

3640

80.1

858.1

0.2

6182

IC5

33.4

460

0.2

372.2

0.3

1225

FC

632

.550

80.2

886

0.3

4250

FC

731

543

0.3

196

0.3

8278

FC

829

.157

10.3

5107

0.4

2309

FC

926

.9599

0.3

9121

0.4

7347

FC

10

2562

20.4

4134

0.5

2382

FC

18

15.6

760

0.7

5251

0.9

3660.7

79

Table 4.3: Assumed properties for the water component.

Property Unit Value

Water compressibility 1/psi 3.6× 10−6

Viscosity cp 1.0Density lb/ft3 62.4278

Table 4.4: Properties assumed in the description of the reservoir.


Wellbore radius ft 0.477Well outer radius ft 2000Rock compressibility 1/psi 4× 10−7

Reservoir temperature F 200Number of radial grids - 50

Table 4.5: Assumed initial reservoir properties for gas and water.

Property Unit Value

Pressure psi 1500Salt concentration kppm NaCl 168Sw fraction 0.25

Table 4.6: Summary of initial conditions assumed for the reservoir containing oil and waterfluid phases.

Property Unit Value

Pressure psi 3500Salt concentration kppm NaCl 168Sw fraction 0.25

80

[kppm NaCl]) at standard conditions with a constant flow rate of STW=10 [bbl/day]. The

formation has petrophysical properties of Rock Type 1, described in Table 4.1 and Figure 4.1.

I calculate radial distribution of water saturation and salt concentration at each sim-

ulation time and apply Archie’s equation (Archie, 1942) to calculate the radial distribution

of rock electrical resistivity, given by

Rt = Rw ·a

φm Snw, (4.1)

where Rt is true formation resistivity, a is tortuosity factor, m is cementation exponent, n

is saturation exponent, and Rw is connate water resistivity calculated with (Bigelow, 1992)

Rw =(

0.0123 +3647.5

C0.955salt

)· 81.77

T + 6.77, (4.2)

where Csalt is [NaCl] concentration in parts per million (ppm) and T is formation temper-

ature in F. In all of the subsequent case studies, I assume a = 1 and m = n = 2.

Figures 4.2(a), 4.2(b), 4.3(a), and 4.3(b) show the radial distribution of pressure,

water saturation, salt concentration, and electrical resistivity, respectively, calculated after

0.01, 0.1, and 1 day from the onset of injection. In this chapter, pressure is expressed in

[psi], water saturation is in [fraction], salt concentration is in [ppm NaCl], and electrical

resistivity is in [Ω.m].

A comparison of results calculated with UTFEC to those obtained with CMG sim-

ulators, GEM and STARS, (Figures 4.2 and 4.3) indicates a very good agreement between

results.

4.3.1.2 Oil-Water Simulation

For simulations of multi-phase fluid flow of oil and water, I assume a reservoir with the

initial conditions described in Table 4.6. It is assumed that in-situ oil is composed of hy-

drocarbon pseudo components FC10 and FC18 with an initial molar composition of 0.7 and

0.3, respectively. The well boundary condition is injection of oil with the composition (0.1

81

24002400

2200

2400CMG-GEMUTFEC2200

2400

2000

sure

[psi

]

2000

sure

[psi

]

0.01 day

1600

1800

Pres

s

1600

1800

Pres

s

1 day

0.1 day

5 10 15 20 251400

R di l Di t [ft]

5 10 15 20 25

1400

R di l Di t [ft]

Radial Distance [ft]Radial Distance [ft]

(a)

1001001

80

100

[%]

CMG-GEMUTFEC80

100

[%]

1

0.8

ction]

60

atur

atio

n [

60

atur

atio

n [

1 day

0.01 day0.6

ation [frac

20

40

Wat

er S

a

20

40

Wat

er S

a 1 day

0.1 day0.4

0.2ter S

atura

5 10 15 20 250

R di l Di t [ft]

5 10 15 20 25

0

R di l Di t [ft]

Wat


(b)

Figure 4.2: Rock Type 1: Comparison of calculated (a) pressure and (b) water saturationwith CMG-GEM and UTFEC along the radial direction at three different times after theonset of injection. Initial pressure = 1500 [psi], initial water saturation = 0.25, and initialcomposition (0.3, 0.6, and 0.1) for components (C1, C2, and C3). The maximum time ofwater injection is 1 day with a constant flow rate of 10 [STW/day].

2 x 105

2 x 105

1.5

2

ppm

]

1.5

2

ppm

]m

NaC

l]

1

1.5

ntra

tion

[p

1

1.5

ntra

tion

[p 1 day0.1 day

0.01 day

ation [ppm

0.5

alt C

once

n

CMG-GEM0.5

alt C

once

nCo

ncen

tr

5 10 15 20 250

R di l Di t [ft]

Sa

CMG-GEMUTFEC

5 10 15 20 250

R di l Di t [ft]

Sa

Salt


(a)

3535

25

30

35

m]

CMG-GEMUTFEC

25

30

35

m]

15

20

25

vity

[

. m

15

20

25

vity

[

. m

0.01 day

5

10

15

Res

istiv

5

10

15

Res

istiv

1 day0.1 day

5 10 15 20 250

5

R di l Di t [ft]

5 10 15 20 25

0

5

R di l Di t [ft]

1 day


(b)

Figure 4.3: Rock-type 1: Comparison of results for (a) salt concentration and (b) electri-cal resistivity calculated with CMG-STARS and UTFEC along the radial direction at threedifferent times after the onset of injection. Initial pressure = 1500 [psi], initial water satu-ration = 0.25, and initial composition (0.3, 0.6, and 0.1) of components (C1, C2, and C3).The maximum time of water injection is 1 day with a constant flow rate of 10 [STW/day].

82

3800380038003800

3700

3800

0.1 day

1 day3700

3800

0.1 day

1 day3700

3800

0.1 day

1 day3700

3800

0.1 day

1 day

3600

sure

[psi

]

0.01 dayy

3600

sure

[psi

]

0.01 dayy

3600

sure

[psi

]

0.01 dayy

3600

sure

[psi

]

0.01 dayy

3400

3500

Pres

s

2 daysCMG-GEM3400

3500

Pres

s

2 days3400

3500

Pres

s

2 days3400

3500

Pres

s

2 days

100 101 1023300

R di l Di t [ft]

CMG GEMUTFEC

100 101 1023300

R di l Di t [ft]

100 101 102

3300

R di l Di t [ft]

100 101 102

3300

R di l Di t [ft]

Radial Distance [ft]Radial Distance [ft]Radial Distance [ft]Radial Distance [ft]

(a)

1001001

80

100

[%]

CMG-GEMUTFEC80

100

[%]

1

0.8

ction]

60

atur

atio

n

0.01 day

0.1 day 1 d

60

atur

atio

n

0.01 day

0.1 day 1 d

0.6

ratio

n [fra

20

40

Wat

er S

a 0.1 day 1 day

2 days20

40

Wat

er S

a 0.1 day 1 day

2 days

0.4

0.2

ater Satur

100 101 1020

R di l Di t [ft]

2 days

100 101 1020

R di l Di t [ft]

2 days

Wa


(b)

Figure 4.4: Rock Type 2: Comparison of calculated (a) pressure and (a) water saturationwith CMG-GEM and UTFEC along the radial direction at different times. The boundarycondition is 1 day injection of oil with a composition (0.1, 0.9) of FC10 and FC18, and witha bottomhole pressure constraint of 3800 [psi]. After injection, fluid withdrawal takes placefor 1 day with a constant flow rate of 5 [bbl/day].

and 0.9) imposed by constraining the bottomhole pressure to 3800 [psi]; injection time is

1 [day]. Subsequently, a fluid withdrawal boundary condition is imposed on the wellbore.

Fluid withdrawal proceeds for 1 day with a constant flow rate equal to 5 [bbl/day]. Fig-

ures 4.4 and 4.5 compare results obtained with UTFEC and CMG-GEM for Rock Type 2

at different times. Results obtained with UTFEC closely match those obtained with the

commercial simulator

4.3.2 Three-Phase Flow Simulations

In this section, I compare results obtained for the case of three-phase fluid flow simulations.

Three hydrocarbon components:

For the first case, I consider the existence of three hydrocarbon components: C1, C3,

and FC7 and assume an initial composition of (0.4, 0.3, 0.3) in the reservoir. In order

to enforce three-phase flow, I assume a pressure of 800 [psi] and a temperature of

200 [F] in the phase envelope so that the composition separates into two coexisting

hydrocarbon phases. Initial water saturation is assumed equal to 0.25. Remaining pa-

83

121212

10

12

m]

CMG-GEMUTFEC10

12

m] 10

12

m]

6

8

vity

[

. m

0.01 day

0.1 day6

8

vity

[

. m

0.01 day

0.1 day6

8

vity

[

. m

0.01 day

0.1 day

2

4

Res

istiv 1 day

2 days2

4

Res

istiv 1 day

2 days2

4

Res

istiv 1 day

2 days

100 101 1020

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

Radial Distance [ft]Radial Distance [ft]Radial Distance [ft]

Figure 4.5: Rock Type 2: Comparison of results for resistivity calculated with CMG-GEM and UTFEC along the radial direction at different times. The boundary condition is1 day injection of oil with a composition (0.1, 0.9) of FC10 and FC18, and with a bottomholepressure constraint of 3800 [psi]. After injection, fluid withdrawal takes place for 1 day witha constant flow rate of 5 [bbl/day].

rameters necessary for the description of the reservoir are the same as those described

in previous simulations. The boundary condition is 1 day of injection of oil with a

composition (0.1, 0.3, 0.6) of components (C1, C3, and FC7) imposed by a constrain-

ing bottomhole pressure of 1300 [psi]. Figures 4.6 and 4.7 compare results obtained

with the UTFEC and CMG-GEM; a very good agreement is observed between results

obtained using the two numerical fluid-flow simulators.

Four hydrocarbon components:

In this case, it is assumed that four hydrocarbon components, namely, C1, C2, FC6,

and FC7 with an initial composition (0.55, 0.35, 0.05, 0.05) exist in the reservoir. In

order to enforce three-phase flow, pressure is assumed equal to 800 [psi] and temper-

ature is equal to 200 [F] in the phase envelope for this composition (Figure 4.10).

Initial water saturation is equal to 0.25. Remaining parameters necessary for the de-

scription of the reservoir are the same as those described in the previous simulation.

The boundary condition is 1 day of injection of oil with a composition (0.15, 0.15, 0.35,

0.35) of components (C1, C2, FC6, and FC7), and with a constraint of bottomhole

pressure equal to 1800 [psi]. Figures 4.8 and 4.9 compare results obtained with the

84

13001300

1200


1300

1100

sure

[psi

]

1 day

1100

sure

[psi

]

1 day

900

1000

Pres

s

0 01 d

0.1 day

1 day

900

1000

Pres

s

0 01 d

0.1 day

1 day

100 101 102800

R di l Di t [ft]

0.01 day

100 101 102800

R di l Di t [ft]

0.01 day


(a)

1001001

80

100

[%]

CMG-GEMUTFEC80

100

[%]

1

0.8

actio

n]

60

atur

atio

n

0.1 day1 day60

atur

atio

n

0.1 day1 day0.6

ratio

n [fra

20

40

Wat

er S

a

0.01 day

20

40

Wat

er S

a

0.01 day0.4

0.2

Water Satu

100 101 1020

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

W


(b)

Figure 4.6: Rock Type 3: Comparison of calculated (a) pressure and (b) water saturationwith CMG-GEM and UTFEC along the radial direction at three different times after theonset of injection. The boundary condition is 1 day of injection of oil with a composition(0.1, 0.3, 0.6) of components (C1, C3, and FC7) imposed by a constraining bottomholepressure of 1300 [psi].

1001001

80

100

%]

CMG-GEMUTFEC80

100

%]

1

0.8

tion]

60

urat

ion

[%

60

urat

ion

[%

0.6

tion [fract

20

40

Oil

Satu

0.01 day 0.1 day 1 day20

40

Oil

Satu

0.01 day 0.1 day 1 day0.4

0.2

Oil Saturat

100 101 1020

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

O


(a)

1001001

80

100

%]

CMG-GEMUTFEC80

100

%]

1

0.8

ction]

60

tura

tion

[%

60

tura

tion

[%

0.6

ation [frac

20

40

Gas

Sat

0.01 day0.1 day

1 day

20

40

Gas

Sat

0.01 day0.1 day

1 day0.4

0.2Gas Satura

100 101 1020

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

G


(b)

Figure 4.7: Rock Type 3: Comparison of calculated (a) oil and (b) gas saturations withCMG-GEM and UTFEC along the radial direction at three different times after the onsetof injection. The boundary condition is 1 day injection of oil with a composition (0.1, 0.3,0.6) of components (C1, C3, and FC7) imposed by a constraining bottomhole pressure of1300 [psi].

85

18001800

1600


1800

1400

sure

[psi

]

1 day

1400

sure

[psi

]

1 day

1000

1200

Pres

s

0 01 day

0.1 day

1 day

1000

1200

Pres

s

0 01 day

0.1 day

1 day

100 101 102800

R di l Di t [ft]

0.01 day

100 101 102800

R di l Di t [ft]

0.01 day


(a)

1001001

80

100

[%]

CMG-GEMUTFEC80

100

[%]

1

0.8

actio

n]

60

atur

atio

n

60

atur

atio

n

0.6

ratio

n [fra

20

40

Wat

er S

a

0.01 day

0.1 day

1 d20

40

Wat

er S

a

0.01 day

0.1 day

1 d

0.4

0.2

Water Satur

100 101 1020

R di l Di t [ft]

1 day

100 101 1020

R di l Di t [ft]

1 dayW


(b)

Figure 4.8: Rock Type 2: Comparison of (a) pressure and (b) water saturation calculatedwith CMG-GEM and UTFEC along the radial direction at three different times after theonset of injection. The boundary condition is 1 day injection of oil with a composition(0.15, 0.15, 0.35, 0.35) of components (C1, C2, FC6, and FC7), and with a constraint ofbottomhole pressure equal to 1800 [psi].

developed simulator and CMG-GEM. UTFEC’s results closely match those obtained

with the commercial simulator for three-phase flow.

4.3.3 Variable Flow Rate

For the simulations of time-variable flow rate, I consider a reservoir with initial conditions

described in Table 4.6. The reservoir is assumed to contain the hydrocarbon component

FC6 and water. Figure 4.11 displays the flow rate imposed at the sand face. Figures 4.12

and 4.13 compare results calculated with the developed simulator to those obtained with

CMG simulators, GEM and STARS, at different times after the onset of injection. Com-

parisons indicate that results calculated using UTFEC agree very well with those obtained

with the commercial simulator.

4.3.4 Dispersion of Salt Concentration

In this section, I verify the implementation of dispersion of aqueous salt concentration.

Physical dispersion for fluid flow in porous media is a scale-dependent phenomenon (see

86

1001001

80

100

%]

CMG-GEMUTFEC80

100

%]

1

0.8

tion]

60

urat

ion

[%

0 1 d

1 day60

urat

ion

[%

0 1 d

1 day0.6

tion [fract

20

40

Oil

Satu

0.01 day

0.1 day

20

40

Oil

Satu

0.01 day

0.1 day0.4

0.2Oil Saturat

100 101 1020

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

O


(a)

1001001

80

100

%]

CMG-GEMUTFEC80

100

%]

1

0.8

tion]

60

tura

tion

[%

0.01 day

60

tura

tion

[%

0.01 day

0.6

ation [frac

20

40

Gas

Sat

y

0.1 day

1 day20

40

Gas

Sat

y

0.1 day

1 day

0.4

0.2

Gas Satura

100 101 1020

R di l Di t [ft]

1 day

100 101 1020

R di l Di t [ft]

1 dayG


(b)

Figure 4.9: Rock type 2: Comparison of (a) oil and (b) gas saturations calculated withCMG-GEM and UTFEC along the radial direction at three different times after the onsetof injection. The boundary condition is 1 day injection of oil with a composition (0.15, 0.15,0.35, 0.35) of components (C1, C2, FC6, and FC7), and with a constraint of bottomholepressure equal to 1800 [psi].

25002500

2000

2500

2000

2500

1500

sure

[psi

]

1500

sure

[psi

]

Reservoir Initial C di i

500

1000

Pres

s

Phase Envelope500

1000

Pres

s Condition

0 50 100 150 200 2500

T t [F]

pCritical Point

0 50 100 150 200 2500

T t [F]

Temperature [ F]Temperature [ F]

Figure 4.10: Phase envelope for four hydrocarbon components C1, C2, FC6, and FC7 witha composition of (0.55, 0.35, 0.05, and 0.05).

87

1212

10

12

ay]

10

12

ay]

6

8

te [b

bl/d

a

6

8

te [b

bl/d

a

2

4Fl

ow R

at

2

4Fl

ow R

at

0.0001 0.001 0.1 30

2

Ti [d ]

0.0001 0.001 0.1 3

0

2

Ti [d ]

Time [day]Time [day]

Figure 4.11: Assumed time-variation of injection flow rate.

35603560

3540

3550

3560CMG-GEMUTFEC

3540

3550

3560

Increasinginvasion

3520

3530

3540

sure

[psi

]

3520

3530

3540

sure

[psi

] invasiontime

3500

3510

3520

Pres

s

3500

3510

3520

Pres

s

100 101 1023490

3500

R di l Di t [ft]

100 101 102

3490

3500

R di l Di t [ft]


(a)

35503550

3540


3550

3530

sure

[psi

]

3530

sure

[psi

]

3510

3520

Pres

s

0.012589

0.125893510

3520

Pres

s

0.012589

0.12589

100 101 1023500

R di l Di t [ft]

1.2589

100 101 1023500

R di l Di t [ft]

1.2589


(b)

Figure 4.12: Variable Flow Rate: Comparison of radial profiles of pressure at differenttimes after the onset of injection. Panel (a) shows that pressures increase at the beginning ofinjection and panel (b) shows that pressures decrease with time after 0.002 day of injection.Dynamic flow rate corresponding to those simulations is shown in Figure 4.11. Invasiontimes are (a) = [0.011, 0.108, 1.088, 10.877, 108.771] seconds and (b) = [0.012, 0.126, 1.259]days.

88

1001001

80

100CMG-GEMUTFEC80

100

Increasing

1

0.8

n actio

n]

60

ratio

n [%

]

60

ratio

n [%

] Increasinginvasiontime

0.6

Saturatio

ratio

n [fra

20

40

Satu

r

20

40

Satu

r 0.4

0.2

Water

Water Satur

100 101 1020

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

W


(a)

2 x 105

2 x 105

1.5

2

[ppm

]

1.5

2

[ppm

]m

NaC

l]

1ntra

tion

[

1ntra

tion

[

Increasingation [ppm

0.5

alt C

once

CMG-GEM0.5

alt C

once invasion

time

Concen

tr

100 101 1020

R di l Di t [ft]

Sa

CMG GEMUTFEC

100 101 1020

R di l Di t [ft]

Sa

Salt


(b)

Figure 4.13: Rock Type 2: Comparison of radial profiles of (a) water saturation and (b)salt concentration at different times after the onset of injection. Figure (4.11) shows theimposed flow rate at different invasion times. Radial profiles are shown at invasion times= [0.011, 0.109, 1.088, 10.877, 108.771, 1087.715, 10877.1552, 108771.552] seconds after theonset of injection.

106106

104

10

]

104

10

]

100

102

rsivity [ft]

100

102

rsivity [ft]

10-2

100

Dispe

Transmission DataEcho Data (Lab)10-2

100

Dispe

10-2 100 102 104 10610-4

Di t [ft]

Echo Data (Lab)Echo Data (SWTT)

10-2 100 102 104 10610-4

Di t [ft]

Distance [ft]Distance [ft]

Figure 4.14: Dispersivity data measured for different rock types and different scales (Plotadapted from John (2008)). This figure compares dispersivities measured from laboratoryecho tests, field scale echo tests (single well transmission test), and with the traditionalforward flow method.

89

Figure 4.14). Therefore, verification case studies include tests on a range of dispersivity val-

ues which may occur in near-wellbore simulations. Table 4.7 is a summary of geometrical,

fluid, petrophysical, and Brooks-Corey’s properties assumed in the simulations described

in this section. Dispersion effects are quantified for two different rock types: a permeable

formation and a tight formation. Table 4.8 summarizes the petrophysical and fluid proper-

ties for the selected rock types assumed in the simulations of dispersion in porous media.

Figures 4.15(a) and 4.15(b) show the relative permeability and capillary pressure curves,

respectively, assumed for the synthetic case studies. I examine the dispersion effect for two

situations: (i) invading water has larger salinity than connate water and (ii) invading water

has lower salinity than connate water.

All the simulations assume that formation oil is composed of pseudo components

FC10 and FC18 with a composition of (0.70 and 0.3). Table 4.2 summarizes the properties of

the assumed pseudo components. For the calculation of viscosity, I modify the parameters

included in Lohrenz et al.’s viscosity correlation, equation (2.60), with new values listed

in Table 4.9.

4.3.4.1 Case 1: Rock Type I

This case considers the injection of fresh water into a high-permeability formation saturated

with brine; a fraction of saturating water is assumed to be movable. The assumptions in this

test are as follows: (a) permeability is equal to 500 [md], (b) porosity is equal to 0.32, (c)

residual water saturation is equal to 0.7, (d) total water saturation is equal to 0.20 (movable

water exists in the formation), and (e) salinity of connate water is 168 [kppm NaCl]. The

boundary condition is injection of water at a constant rate of 0.5 [bbl/day] for a period of 1

[day]. Injection water has a salinity of 3 [kppm NaCl]. Figure 4.16(a) compares the radial

distribution of water saturation calculated with UTFEC to that obtained with STARS.

Lake and Hirasaki (1981) and Sternberg and Greenkorn (1994) showed that macro-

scopic heterogeneities, layering, and cross-flow can lead to large dispersivities in field-scale

studies. Taylora and Howard (1987) showed that dispersion incrementally increases with

90

distance traveled by a tracer. Figure 4.14 shows that dispersion is a scale-dependent phe-

nomenon. Therefore, I test the developed simulation method for different values of disper-

sivity, including the following cases: αl1 = 0, αl1 = 0.2, and αl1 = 1 [ft] (equations 2.38

through 2.43). Figure 4.16(b) compares radial distribution of salt concentration calculated

with UTFEC to those obtained with CMG-STARS. For all dispersivity values a very good

agreement exists between results obtained with UTFEC and commmerical simulator.

4.3.4.2 Case 2: Rock Type II

In this section, I study the injection of saline water into a low-permeability formation. It

is assumed that the formation is saturated with oil and that residual water has very low

salinity.

Assumptions made in this verification test are as follows: (a) permeability is equal

to 0.1 [md], (b) porosity is equal to 0.5, (c) water saturation is equal to 0.20, (d) residual

water saturation is equal to 0.20, and (e) salinity of connate water is 3 [kppm NaCl]. The

boundary condition is injection of water at a constant rate of 0.5 [bbl/day] for a period of

1 [day]. Injection water has a salinity of 168 [kppm NaCl]. Figure 4.17(a) compares radial

distributions of water saturations calculated with UTFEC and STARS. Similar to Case 1,

I assume the following values for dispersivity: αl1 = 0, αl1 = 0.2, and αl1 = 1 [ft] (equa-

tions 2.38 through 2.43). Figure 4.17(b) compares radial distributions of salt concentration

calculated with UTFEC to those obtained with STARS. Cases 1 and 2 indicated a very

good agreement between results obtained with the two numerical algorithms.

4.4 Two-Dimensional Simulations

In this section, I verify results obtained for 2D multi-phase fluid-flow simulations. Studied

cases include 2D axisymmetric (vertical wells), 2D radial-azimuthal (vertical wells), and 2D

radial-azimuthal (horizontal wells) simulations.

91

Table 4.7: Summary of geometrical, fluid, petrophysical, and Brooks-Corey’s propertiesassumed in the simulations described in Section 4.3.4.

Variable Unit Value

Wellbore radius ft 0.477

Formation outer boundary ft 500

Formation thickness ft 1.0

Number of radial grids - 50

Initial formation pressure psi 3500

Injection flow rate bbl/day 0.5

Injection time day 1.0

Rock compressibility 1/psi 4.0E-7

Water compressibility 1/psi 3.60E-6

Oil viscosity cp 2.0

Formation water salinity kppm NaCL 3 or 168

Temperature F 200

Table 4.8: Summary of petrophysical and fluid properties for different rock types assumedin Section 4.3.4.

Variable Unit Rock Type I Rock Type II

Permeability md 500 0.1

Porosity fraction 0.32 0.05

residual water saturation fraction 0.07 0.2

residual oil saturation fraction 0.15 0.25

Table 4.9: Summary of parameters assumed in the Lohrenz et al.’s (1964) viscosity corre-lation 2.60 for the simulations described in Section 4.3.4.


a0 dimensionless 0.4223a1 dimensionless 0.023364a2 dimensionless 0.58533a3 dimensionless −0.040758a4 dimensionless 0.0093324

92

111

0.8

1

0.8

1

0.8

1

kro

0.6

k r

0.6

k r

0.6

k r krw

0.2

0.4

0.2

0.4

0.2

0.4

0 0.2 0.4 0.6 0.8 10

W t S t ti0 0.2 0.4 0.6 0.8 1

0

W t S t ti0 0.2 0.4 0.6 0.8 1

0

W t S t tiWater SaturationWater SaturationWater SaturationWater Saturation [fraction](a)

150150150

100

c [psi

]

100

c [psi

]

100

c [psi

]

Rock Type II

50

P c

50

P c

50

P c

0 0.2 0.4 0.6 0.8 1W t S t ti

0 0.2 0.4 0.6 0.8 1W t S t ti

0 0.2 0.4 0.6 0.8 1W t S t ti

Rock Type I

Water SaturationWater SaturationWater SaturationWater Saturation [fraction](b)

Figure 4.15: Rock Fluid Properties: (a) water-oil relative permeability and (b) capil-lary pressure curves assumed for Rock Type I (solid lines) and Rock Type II (dotted lines);krw and kro are relative premeabilities of water and oil, respectively.

11

0.8

1

n

0.8

1

n

UTFECCMG-STARS

actio

n]

0.6

Satu

ratio

n

0.6

Satu

ratio

nratio

n [fra

0.2

0.4

Wat

er S

0.2

0.4

Wat

er S

Water Satur

100 1010

100 1010

W


(a)

2 x 105

2 x 105

1.5

2

ppm

]

1.5

2

ppm

]pm

NaC

l]

1ntra

tion

[p

1ntra

tion

[pratio

n [pp

0.5

Salt

Con

ce

0.5

Salt

Con

ce

= 0.0 [ft]

t Con

cent

100 1010

S

100 1010

S

= 0.2 [ft] = 1.0 [ft]Sa

lt


(b)

Figure 4.16: Radial distributions of (a) water saturation and (b) salt concentration calcu-lated after 1 day from the onset of water injection with a constant flow rate of 0.5 [bbl/day].The dashed blue and solid red curves identify water saturation calculated with UTFEC andCMG-STARS, respectively. Initial water saturation is equal to 0.20 and residual water sat-uration is equal to 0.07. Connate water salinity equals 168 [kppm NaCl] and invading-watersalinity equals 3 [kppm NaCl]. The formation exhibits the petrophysical properties of RockType I (described in Table 4.8 and Figure 4.15).

93

11

0.8

1n

0.8

1n

UTFECCMG-STARS

actio

n]

0.6

Satu

ratio

n

0.6

Satu

ratio

nratio

n [fra

0.2

0.4

Wat

er S

0.2

0.4

Wat

er S

Water Satur

100 1010

100 1010

W


(a)

2 x 105

2 x 105

1.5

2

ppm

]

1.5

2

ppm

]

= 0.0 [ft] = 0.2 [ft] = 1.0 [ft]

pmNaC

l]

1ntra

tion

[p

1ntra

tion

[pratio

n [pp

0.5

Salt

Con

ce

0.5

Salt

Con

cet C

oncent

100 1010

S

100 1010

S

Salt


(b)

Figure 4.17: Radial distributions of (a) water saturation (b) salt concentration calculated1 day after the onset of water injection with a constant rate of 0.5 [bbl/day]. The dashed blueand solid red curves identify water saturation calculated with UTFEC and CMG-STARS,respectively. Initial water saturation is equal to 0.20 which is equal to residual water.Connate water salinity equals 3 [kppm NaCl] and invading-water salinity equals 168 [kppmNaCl]. The formation exhibits the petrophysical properties of Rock Type II (described inTable 4.8 and Figure 4.15).

4.4.1 Two-Dimensional Axisymmetric Simulations

The developed algorithm is capable of simulating injection and fluid withdrawal for different

types of fluid under a 2D axial-symmetric configuration. In this section, the method is tested

for the case of fluid sampling from a formation which has been invaded with water-base mud

(WBM) prior to fluid pumpout. It is assumed that the radial invasion depth is equal to 2.5

[ft] whereas the spatial distribution of fluid saturation is piston like.

Formation oil is composed of hydrocarbon components C2, IC5, FC8, and FC18 with

corresponding properties described in Table 4.2. I assume that initial molar concentration of

each hydrocarbon is 0.25. Table 4.10 summarizes the geometrical and numerical parameters

assumed for simulations of fluid withdrawal. Reservoir rock and rock-fluid properties, in

addition to initial and boundary conditions assumed in the following simulations are sum-

marized in Table 4.11. Fluid sampling takes place with a constant flow rate of 10 [bbl/day]

from the lower section of the formation for 12 [hours].

Figures 4.18(a) and 4.19(a) show the spatial (radial and vertical directions) distri-

94

Sw [fraction]pth [ft]

Dep

Radial Distance [ft]

(a)

x

tion]

w,GEM

[fract

pth [ft]

w,UTFEC–S wDep

S w


(b)

Figure 4.18: 2D Axisymmetric Model: Spatial distributions of (a) water saturationcalculated with UTFEC and (b) difference between water saturations calculated with UT-FEC and CMG-GEM. Initially, it was assumed that the formation was invaded to a radialdepth of 2.5 [ft] before the onset of fluid sampling. Sampling takes place between the depthsof 2139 to 2140 [ft] at a constant rate of 10 [bbl/day] for 12 [hrs]. In the above figures, ra-dial and vertical distances are displayed in logarithmic and linear scales, respectively. Theformation exhibits petrophysical properties of Rock Type I described in Table 4.1 and Fig-ure 4.1.

butions of water saturation and pressure calculated using UTFEC. Figure 4.18(b) shows the

spatial distribution of the difference between water saturation calculated with UTFEC and

that obtained with CMG-GEM. Similarly, Figure 4.19(b) shows the spatial distribution of

the relative difference between pressure calculated using UTFEC and that obtained with

CMG-GEM. The variation of fractional flow corresponding to sampled fluid is calculated

using UTFEC and is compared to that obtained with CMG-GEM in Figure 4.20. Fig-

ures 4.18(b), 4.19(b), and 4.20 indicate that simulation results obtained with UTFEC agree

well with those obtained with commercial fluid-flow software for this 2D axisymmetric case.

4.4.2 Two-Dimensional Radial Simulation

In this section, I test the developed simulator for modeling fluid sampling within an az-

imuthal section of the wellbore perimeter. In doing so, I consider a 2D radial model where

pressure and fluid properties may vary in the radial and azimuthal directions. I assume an

oil-saturated formation which was invaded with WBM to a radial length of 2.5 feet (piston-

95

P [psi]

[ft]

pth [ft]

Z Dep


(a)

x

ction]

pth [ft]

)/P G

EM [frac

Dep

UTFEC–P G

EM)

(PU


(b)

Figure 4.19: 2D Axisymmetric Model: Spatial distributions of (a) pressure calculatedwith UTFEC and (b) relative difference between pressures calcualted with UTFEC andCMG-GEM. Initially, it was assumed that the formation was invaded to a radial depth of2.5 [ft] before the onset of fluid sampling. Sampling takes place between the depths of 2139to 2140 [ft] at a constant rate of 10 [bbl/day] for 12 [hrs]. In the above figures, radial andvertical distances are displayed in logarithmic and linear scales, respectively. The formationexhibits petrophysical properties of Rock Type I described in Table 4.1 and Figure 4.1.

11

0.8

1

ow, F

w

0.8

1

ow, F

w CMG‐GEMUTFEC

0.6

ctiona

l Flo

0.6

ctiona

l Flo

0.4

ater Frac

0.4

ater Frac

0 0.1 0.2 0.3 0.4 0.50.2

Time (days)

Wa

0 0.1 0.2 0.3 0.4 0.5

0.2

Time (days)

Wa

Time [days]Time (days)Time (days)Time [days]

Figure 4.20: 2D Axisymmetric Model: Time evolution of fractional flow of water,Fw, for fluid sampled at the sand face during fluid withdrawal. The maximum differencebetween calculated fractional-flow curves obtained with UTFEC and CMG-GEM is lessthan 4 × 10−3. The formation exhibits petrophysical properties of Rock Type I describedin Table 4.1 and Figure 4.1. Figures 4.18(a) and 4.19(a) show spatial distributions of watersaturation and pressure corresponding to this sampling process.

96

Table 4.10: Summary of geometrical and numerical parameters assumed for the numericalsimulation described in Section 4.4.1.

Property Unit Value


Number of vertical grids - 10


Drainage radius ft 1600

Grid size (radial) ft logarithmically spaced

Grid size (vertical) ft 1.0

Table 4.11: Summary of formation rock, rock fluid properties, initial conditions, andboundary conditions assumed in sections 4.4.1 and 4.4.2.

Property Unit Value

Porosity fraction 0.25Horizontal permeability md 100Vertical permeability md 100Rock compressibility 1/psi 4.0 ×10−7

Initial pressure psi 3500Initial salt concentration kppm NaCl 168Fluid withdrawal flow rate (BHF) bbl/day 10.0Fluid withdrawal time day 0.5

97

Table 4.12: Summary of geometrical and numetrical parameters assumed in the simulationsdescribed in Section 4.4.2.

Property Unit Value


Number of azimuthal grids - 10




Grid size (azimuthal) degrees 18.0

like invasion). Reservoir oil is assumed to have a composition of 0.25 with four pseudo

components, C2, IC5, FC8, and FC18. Table 4.2 lists properties of the assumed pseudo

components. The formation has a permeability of 100 [md] and a porosity of 0.25 [fraction].

Remaining petrophysical properties are those of Rock Type 3 described in Table 4.1 and Fig-

ure 4.1. Table 4.12 summarizes the assumed geometrical and numerical parameters for the

simulation of fluid withdrawal. Table 4.11 also summarizes the reservoir rock properties,

initial condition, and boundary conditions assumed in the simulation of this section. Fluid

sampling takes place with a constant flow rate of 10 [bbl/day] within the first azimuthal

angle for 12 hours.

Figures 4.21(a) and 4.22(a) show the spatial distributions of water saturation and

pressure calculated using UTFEC. In these horrizontal cross sections, the axes center (0, 0)

is located at the well center; X and Y are relative distances from the well center. Fig-

ure 4.21(b) shows the spatial distribution of the difference between water saturation cal-

culated with UTFEC and that obtained with CMG-GEM. Similarly, Figure 4.22(b) shows

the spatial distribution of relative difference between pressure calculated with UTFEC and

that obtained with CMG-GEM. Figure 4.23 compares the fractional flow calculated with

UTFEC to that obtained with CMG-GEM. Figures 4.21(b), 4.22(b), and 4.23 indicate that

results obtained with UTFEC are in good agreement with those obtained with CMG-GEM

for this 2D simulation case.

98

5Sw [fraction]

5

0

Y [ft]

Y

‐5‐5 0 5

[f ]X [ft]

(a)

x5

ction]

5

w,GEM

[frac

0

Y [ft]

w,UTFEC–S wY

S

‐5‐5 0 5

[f ]X [ft]

(b)

Figure 4.21: 2D Radial Model: Spatial distributions (planar view) of (a) water satura-tion obtained with UTFEC and (b) the difference between water saturations calculated withUTFEC and CMG-GEM. It was assumed that the formation was invaded to a radial lengthof 2.5 [ft] before the onset of sampling. Fluid sampling takes place within an azimuthalangle from 0 to 18 at a constant flow rate of 10 [bbl/day] for 12 [hrs]. The formation ex-hibits a permeability of 100 [md] and a porosity of 0.25 [fraction]. Remaining petrophysicalproperties are those of Rock Type 3, described in Table 4.1 and Figure 4.1.

P [psi]55

t] 0Y [ft]

Z [ft Y

‐5‐5 0 5

[f ]X [ft]X [ft]

(a)

x5

tion]

5

t] /PGEM

[fract

0

Y [ft]

Z [ft

TFEC–P G

EM)/Y

(PUT

‐5‐5 0 5

[f ]X [ft]X [ft]

(b)

Figure 4.22: 2D Radial Model: Spatial distributions (planar view) of (a) pressureobtained with UTFEC and (b) the relative difference between pressures calculated withUTFEC and CMG-GEM. It was assumed that the formation was invaded to a radial lengthof 2.5 [ft] before the onset of sampling. Fluid sampling takes place within an azimuthalangle from 0 to 18 at a constant flow rate of 10 [bbl/day] for 12 [hrs]. The Formationexhibits a permeability of 100 [md] and a porosity of 0.25 [fraction]. Remaining petrophysicalproperties are those of Rock Type 3, described in Table 4.1 and Figure 4.1.

99

100100

1

10

w, F

w

CMG‐GEMUTFEC

1

10

w, F

w

10-1

tion

al Flow

10-1

tion

al Flow

10-2

Water Fract

10-2

Water Fract

0 0.1 0.2 0.3 0.4 0.510-3

Ti (d )

W 0 0.1 0.2 0.3 0.4 0.5

10-3

Ti (d )

W


Figure 4.23: 2D Radial: Time evolution for water fractional flow, Fw, of the fluid sam-pled at the sand face during fluid pumpout. Maximum difference between simulation resultscalculated with UTFEC and CMG-GEM is less than 2×10−3. Formation exhibits a perme-ability of 100 [md] and a porosity of 0.25 [fraction]. Remaining petrophysical properties arethose of Rock Type 3 described in Table 4.1 and Figure 4.1. Figures 4.21(a) and 4.22(a)show spatial distributions of water saturation and pressure corresponding to this samplingprocess.

4.4.3 Two-Dimensional Horizontal-Well Simulations

Similar to fluid-flow simulations in vertical wells, the developed algorithm employs cylindri-

cal coordinates to perform simulation for cases of horizontal wells. To the author’s knowl-

edge, there are no other numerical methods formulated in cylindrical coordinates docu-

mented in the open technical literature which can calculate fluid distributions in the vicinity

of a horizontal well.

For verification purposes, a 2D (XZ) model is constructed in Cartesian coordinates

with proper adjustments performed on grid blocks (nullified wellbore grids) and well defi-

nitions to accurately reproduce simulations in a cylindrical framework. I use CMG-GEM

to execute fluid-flow simulations of the described model in Cartesian coordinates. Very

fine meshing is implemented with both models constructed with UTFEC and CMG-GEM

simulators; this necessity is due to the importance of wellbore geometry in near-wellbore

simulations and also for reliable comparison of simulations performed in two different co-

ordinate systems. The model in cylindrical coordinates (UTFEC) includes 200 radial and

360 azimuthal grids whereas the model in Cartesian coordinates (CMG-GEM) consists of

100

Table 4.13: Summary of geometrical, fluid, and petrophysical properties assumed in thesimulations described in Section 4.4.3.

Variable Unit Value



Formation thickness ft 1.0

Formation horizontal permeability md 100

Formation vertical permeability md 100

Formation porosity fraction 0.25

Formation oil density lb 54.64

Number of radial grids (UTFEC) - 200

Number of azimuthal grids (UTFEC) - 360

Number of grids in X-direction (CMG) - 300

Number of grids in Z-direction (CMG) - 300


Initial water saturation fraction 0.38

Injection time day 10



Formation water salinity kppm NaCL 168

Temperature F 200

300x300 grids. Among all grids for the CMG model, 3930 of them are used to reproduce

the wellbore geometry. A single well in UTFEC model is reproduced with an injector well

imposed on 284 finite-difference grids across the perimeter of the wellbore.

The following verification case studies are similar to cases described by Alpak et

al. (2003). I define a “base” case with the geometrical, fluid, and petrophysical properties

described in Table 4.13.

Two types of oil are considered as in-situ fluid, Table 4.14 pressure-volume-temperature

(PVT) properties of the assumed oils for equation-of-state (EOS) calculations. Figure 4.24

shows the assumed oil-water relative permeability curves. Similar to Alpak et al. (2003), I

assume that capillary pressure is zero.

101

Table 4.14: Summary of PVT properties of in-situ hydrocarbon components assumed inthe EOS calculations described in Section 4.4.3.

Property Unit Oil1 Oil2

Critical temperature K 622.1 622.1

Critical pressure atm 25.01 25.01

Acentric factor - 0.4438 0.4438

Critical molar volume m3/kgmol 0.521 0.521

Molecular weight g/mol 160.3 138.87

Density lb/ft3 54.64 47.34

Viscosity cp 0.67 0.156

API fraction 30 55

0 80 8

0.6

0.8

0.6

0.8

kro

0.4k r 0.4k r

0.20.2krw

0 0.2 0.4 0.6 0.8 10

W t S t ti0 0.2 0.4 0.6 0.8 1

0

W t S t tiWater SaturationWater SaturationWater Saturation [fraction]

(a)

Figure 4.24: Water-oil relative permeability curves assumed for the base case correspond-ing to simulations described in Section 4.4.3; krw and kro are relative permeabilities of waterand oil fluid phases, respectively.

102

4.4.3.1 Case I

For the first case, I consider a formation with equal horizontal and vertical permeabilities of

1000 [md] and a porosity of 0.35 [fraction]. Initially, the formation exhibits water saturation

equal to residual water saturation, Sw = Swirr = 0.38 [fraction]. The saturating oil is “Oil1”

with corresponding properties listed in Table 4.14. Formation petrophysical properties are

those of the base case. The boundary condition is injection of water with a constant flow

rate of 0.00475 [bbl/day] (average of dynamic flow rate of invasion). Figure 4.25(a) shows

the spatial (cross section of a plane perpendicular to the well axis) distribution of water

saturation obtained with UTFEC after 10 days from the onset of water injection. The

model in Cartesian coordinates is simulated using CMG-GEM; subsequently, results are

transformed into cylindrical coordinates. Figure 4.25(b) shows the spatial distribution of the

difference between water saturations calculated with the two methods: UTFEC and CMG-

GEM. Figure 4.25(b) shows that the maximum of the absolute difference between results

obtained with the two numerical methods is less than 5× 10−3; this indicates reliability of

simulation results obtained with the developed algorithm.

4.4.3.2 Case II

For the second case, the formation is assumed to be saturated with a lighter oil (Oil2 as

defined in Table 4.14) than in previous case. The formation exhibits equal horizontal and

vertical permeabilities of 500 [md] and a porosity of 0.32 [fraction]. Remaining petrophysical,

initial, and boundary conditions are the same as those assumed for Case I (Section 4.4.3.1).

Figure 4.26(a) shows the spatial (cross section of a plane perpendicular to the well axis)

distribution of water saturation obtained with UTFEC after 10 days from the onset of

water injection. Figure 4.26(b) shows the spatial distribution of the difference between

water saturations calculated obtained with UTFEC and CMG-GEM. Figure 4.26(b) shows

that maximum difference between the results obtained with the two numerical methods is

less than 4×10−3; results confirm the reliability of simulation results obtained with UTFEC.

103

10Sw [fraction]

5

10

0

Z [ft]

‐5

Z

‐10‐10 0 10‐5 5

X [ft]

(a)

10x

5

10

ction]

0

Z [ft]

w,GEM

[frac

‐5

Z

w,UTFEC–S w

‐10‐10 0 10‐5 5

S

X [ft]

(b)

Figure 4.25: 2D Radial Horizontal Well: 2D spatial (cross section of a plane perpen-dicular to the well axis) distributions of (a) water saturation obtained with UTFEC and (b)the difference between water saturations calculated using CMG-GEM and UTFEC after10 days from the onset of water injection with a constant flow rate of 0.00475 [bbl/day].Initially, water saturation is equal to residual water saturation, Swi = Swirr = 0.38 [frac-tion]. Saturating oil exhibits a specific density of 0.87 and formation permeability is equalto 1000 [md]. Remaining properties of the formation are those of the base case describedin Table 4.13 and Figure 4.24.

4.4.3.3 Case III

Having tested fluid-flow simulations in rock formations with isotropic permeability, I ver-

ify results obtained in formations with anisotropic permeability. The formation exhibits

horizontal permeability equal to 100 [md] and permeability anisotropy ratio equal to 10,

Raniso = 10. Remaining petrophysical, initial, and boundary conditions are the same as

those described for Case I (Section 4.4.3.1). Saturating oil is “Oil1” with a density of 54.64

[lb/ft3]; assumed PVT properties are listed in Table 4.14. The boundary condition is in-

jection of water with a constant flow rate of 0.095 [bbl/day] (average flow rate for dynamic

invasion).

Figure 4.27(a) shows the spatial (cross section of a plane perpendicular to the well

axis) distribution of water saturation obtained with UTFEC after 10 days from the onset of

water injection. Figure 4.27(b) shows the spatial distribution of the difference between cal-

culated water saturations obtained with UTFEC and CMG-GEM. Figures 4.25 through 4.27

104

10Sw [fraction]

5

10t] 0Z [ft]

Z [ft

‐5

Z

‐10‐10 0 10‐5 5

X [ft]X [ft]

(a)

10x

5

10

ction]

t] 0Z [ft]

w,GEM

[frac

Z [ft

‐5

Z

w,UTFEC–S w

‐10‐10 0 10‐5 5

S

X [ft]X [ft]

(b)

Figure 4.26: 2D Radial Horizontal Well: 2D spatial (cross section of a plane per-pendicular to the well axis) distributions of (a) water saturation obtained with UTFECand (b) the difference between water saturations calculated with CMG-GEM and UTFECafter 10 days from the onset of water injection with a constant flow rate of 0.00475 [bbl/day].Saturating oil exhibits specific density of 0.76 and formation permeability is equal to 500[md]. Remaining properties of the formation are those of the base case. This case study isdescribed in Section 4.4.3.2.

confirm that the developed method is reliable when calculating fluid-flow distributions in

the vicinity of a horizontal well.

4.5 Three-Dimensional Cylindrical Simulation

One of the main applications of the developed method is simulation of fluid sampling with

a point sink probe, for instance, a probe-type formation-tester. Figure 4.28 shows the ge-

ometrical configuration of a deviated well model (for vertical wells, θw = 0) in cylindrical

coordinates, as implemented in the formulations of this dissertation for fluid-flow modeling

in the vicinity of the wellbore. The graph displays a probe-type formation tester deployed

on the boundary of the well. Figure 4.29 describes the configuration of a simple probe is

assumed in the simulations conducted in Sections 4.5.1 and 4.5.2.

The developed method, UTFEC, simulates formation-tester measurements such as

fractional flow, pressure, gas-oil-ratio (GOR), density, and viscosity of sampled fluid. I test

105

10Sw [fraction]

5

10t]t] 0Z [ft]

Z [ft

Z [ft

‐5

Z

‐10‐10 0 10‐5 5

X [ft]X [ft]X [ft]

(a)

10

5

10

on]

t] 0Z [ft]

GEM

[fracti

Z [ft

‐5

Z

UTFEC–S w

,G

‐10‐10 0 10‐5 5

S w,

X [ft]X [ft]

(b)

Figure 4.27: 2D Radial Horizontal Well: 2D spatial (cross section of a plane perpen-dicular to the well axis) distributions of (a) water saturation obtained with UTFEC and (b)the difference between water saturations calculated with CMG-GEM and UTFEC after 10days from the onset of water injection with a flow rate of 0.095 [bbl/day]. Horizontal per-meability is equal to 100 [md] and Raniso = 10. Remaining petrophysical properties of theformation are those of base case described in Table 4.13 and Figure 4.24. Section 4.4.3.3describes this case study.

UTFEC for fluid withdrawal using a probe-type formation tester placed on the wellbore

perimeter at a specified depth and azimuthal angle.

Simulations documented below make the following assumptions: the radial length

of mud-filtrate invasion prior to fluid sampling is 2.5 [ft]; composition of reservoir oil is 0.25

[fraction] of four pseudo components: C2, IC5, FC8, and FC18 with properties described in

Table 4.2. Table 4.15 summarizes the geometrical, petrophysical, and numerical parameters

assumed in the simulation of fluid withdrawal. The formation exhibits water-oil relative

permeability and capillary pressure curves of Rock Type 3, described in Figure 4.1. Addi-

tionally, the boundary condition is fluid sampling at a constant flow rate of 10 [bbl/day]

within azimuthal angles of 0 - 18 for a period of 12 hours.

I describe simulation results and verifications for two cases, namely, fluid sam-

pling from an oil-saturated formation after water-base mud-filtrate (WBMF) invasion (Sec-

tion 4.5.1) and fluid sampling from a hydrocarbon-saturated formation after oil-base mud-

filtrate (OBMF) invasion (Section 4.5.2).

106

^ qkv1

θ

^n

k 2

Layer 1 qmfkh1

XY

θw

h

kh2

kv2

Layer 2

qmf

Probe krr,ijk

kzz,ijk

kθθ,ijk

X

Dep

th

zpqmf

Plane view

θjLayer 3 rkh3

kv3

X

Y

Figure 4.28: 3D Cylindrical Model: Geometrical description of a deviated well (forvertical wells, θw = 0) in cylindrical coordinates used in the formulation of fluid-flow equa-tions described in this dissertation. In this graph, r, θj , and z designate the radial location,azimuthal angle, and vertical location, respectively; n is the unit normal vector to the bed-ding plane, h is bed thickness, zp is the vertical distance from probe to a bed boundary,θw is wellbore deviation angle measured from the bedding normal vector, n, and qmf ismud-filtrate flow rate; krr, kθθ, and kzz are diagonal terms of the permeability tensor aftertransformation to cylindrical coordinates.

107

Side ViewProbe Dimensions

0.45

n]

0.3

0 15

r = 0.3 [in]

tive

dept

h [in

Top View

0.15

0

Rel

at-0.15

-0.3

-0.45[in]

(a)

Top ViewProbe Face

0 66

0.6

0

-0.6

3

0

-36

4

2

0

2

4

0.6-0.6 0 0.6

[in]-3 0 3

[in]

(b)

Figure 4.29: Description of the probe-type formation tester assumed in the numericalsimulations of fluid withdrawal performed with the developed algorithm.

108

4.5.1 Sampling after WBMF Invasion into an Oil-Bearing Forma-

tion

This case verifies results obtained for fluid withdrawal with a probe-type formation tester

from a rock formation which has been previously invaded with WBM. The formation ex-

hibits the properties described in Section 4.5. Piston-like invasion to a radial length of 2.5

[ft] is assumed prior to fluid pumpout. Fluid withdrawal takes place within azimuthal an-

gles 0 − 18 and depths of 2133 − 2135 [ft]. Figures 4.30(a) and 4.31(a) show the spatial

distributions of water saturation and pressure calculated with UTFEC. The spatial distri-

bution of the difference between water saturations calculated with UTFEC and CMG-GEM

is shown in Figure 4.30(b). Figure 4.31(b) shows the spatial distribution of the relative

difference between pressure calculated with UTFEC and that obtained with CMG-GEM.

The comparison indicates that results obtained with the two numerical methods agree well

and that UTFEC simulations are reliable.

Figure 4.32 compares the time evolution of water fractional flow, Fw, of sampled

fluid at the sand face during fluid pumpout. The maximum difference between fractional

flow of water calculated with UTFEC and that obtained with CMG-GEM is approximately

5× 10−3.

4.5.2 Sampling after OBMF Invasion into a Gas-Bearing Formation

Analogous to Section 4.5.1, this case verifies fluid distributions and measurements simulated

during fluid pumpout with a probe-type formation tester from a formation which has been

previously invaded with oil-base mud (OBM). Formation properties are those described

in Section 4.5. It is assumed that the formation was previously invaded to a radial length

of 2.5 [ft] with OBM prior to fluid pumpout. Fluid sampling takes place within azimuthal

angles 0 − 18 and a depth of 2133− 2135 [ft].

Figures 4.34(a), 4.34(b), and 4.35(a) show the spatial distributions of concentrations

of C1, FC18, and pressure calculated with UTFEC, respectively. Figure 4.35(b) shows the

109

Table 4.15: Summary of geometrical, petrophysical, and numerical properties/parametersassumed in the simulations described in Section 4.5.1.

Property Unit Value



Formation depth ft 2140

Porosity fraction 0.25

Horizontal permeability md 100

Vertical permeability md 100

Rock compressibility 1/psi 4.0 ×10−7

Initial pressure psi 3500

Initial salt concentration kppm NaCl 168

Fluid withdrawal flow rate (BHF) bbl/day 10.0

Fluid withdrawal time hours 12





Grid size (azimuthal) degrees 18.0

Grid size (vertical) ft 1.0

Fluid withdrawal flow rate [bbl/day] 10

Sampling section (azimuthal angle) degrees 0 − 18

Sampling section (vertical depth) ft 2133− 2135

110

Sw [fraction]h [ft]

Dep

th

X [ft] Y [ft]

(a)

tion]

–S w

,GEM

[fract

h [ft]

S w,UTFEC–

Dep

th

X [ft] Y [ft]

(b)

Figure 4.30: 3D Cylindrical Vertical Well: 3D spatial distributions of (a) watersaturation and (b) the difference between water saturations calculated with CMG-GEMand UTFEC after 12 hours from the onset of fluid sampling. Fluid withdrawal takes placethrough azimuthal angles 252 to 288 [degrees]. Formation petrophysical properties are thoseof Rock Type 1 described in Table 4.1. The formation was invaded with WBM to a radiallength of 2.5 [ft] prior to fluid withdrawal. Fluid sampling takes place with a constant flowrate of 10 [bbl/day].

111

P [psi]h [ft]

Dep

th

X [ft] Y [ft]

(a)

ction]

h [ft]

M)/P G

EM [frac

Dep

th

(PUTFEC–P G

EM

X [ft] Y [ft]

(

(b)

Figure 4.31: 3D Cylindrical Vertical Well: 3D spatial distributions of (a) pressureobtained with UTFEC and (b) the relative difference between pressures calculated withCMG-GEM and UTFEC after 12 hours from the onset of fluid sampling. Fluid withdrawaltakes place through azimuthal angles 252 to 288 [degrees]. Formation petrophysical prop-erties are those of Rock Rype 1 described in Table 4.1. The formation was invaded withWBM to a radial length of 2.5 [ft] prior to fluid withdrawal. Fluid sampling takes placewith a constant flow rate of 10 [bbl/day].

112

spatial distribution of the relative difference between pressures calculated with UTFEC and

CMG-GEM. Figure 4.33 compares the time evolution of sampled-fluid GOR at the sand

face during pumpout. The maximum relative difference between GOR curves calculated

with UTFEC and that obtained with CMG-GEM is lower than 1%.

4.6 Summary and Conclusions

This chapter described the verification of UTFEC for different applications and physical

phenomena using simulated case studies. The simulation cases were designed to include

multi-phase fluid-flow regimes including gas-water, oil-water, and gas-oil-water for different

rock types, and include various boundary conditions in simulation of fluid injection and

production. UTFEC was verified for multi-dimension simulations, including 1D radial,

2D axis-symmetric, 2D radial, 3D vertical, and 2D horizontal wells. Verifications were

performed against two commercial, petroleum industry standard, reservoir simulators. The

results indicated that simulations obtained with UTFEC were comparable to the industry

standards. Moreover, I verified the implemented model for physical dispersion of aqueous

salt. For different dispersivity values, spatial distribution of salt concentrations obtained

with UTFEC were compared against those of CMG-STARS.

To secure accurate results, the material balance error is set to 10−4. In the cal-

culation of the spatial distribution of salt concentration, the controlling parameter is set

to 10−3. Numerical stability in simulation of cases with high capillary pressure and large

density contrasts, required smaller numerical controllers (e.g., 10−5 for the material bal-

ance error), slowing the simulations. In simulating cases of two-phase flow in vertical wells

(single-phase hydrocarbon) involving a small number of grid blocks (e.g., less than 1000

grid blocks), I found that the simulation time was chiefly spent on matrix construction (ap-

proximately 45%) and numerical solution(approximately 40%). However, with an increase

in the number of grid blocks the ratio of numerical solver time to total simulation time

significantly increased.

113

11

0 8

0.9

1

w, F

w

CMG‐GEMUTFEC

0 8

0.9

1

w, F

w

0 6

0.7

0.8

tion

al Flow

0 6

0.7

0.8

tion

al Flow

0 4

0.5

0.6

Water Fract

0 4

0.5

0.6

Water Fract

0 0.1 0.2 0.3 0.4 0.5

0.4

Ti [d ]

W

0 0.1 0.2 0.3 0.4 0.5

0.4

Ti [d ]

W

Time [days]Time [days]

Figure 4.32: 3D Cylindrical Vertical Well: Time evolution of the fractional flow ofwater, Fw, for fluid sampled at the sand face during fluid withdrawal.

14001400

1000

1200

1400

CF/SBO

] CMG-GEMUTFEC

1000

1200

1400

CF/SBO

]

600

800

1000

o, GOR [SC

600

800

1000

o, GOR [SC

200

400

600

s‐Oil Ra

tio

200

400

600

s‐Oil Ra

tio

0 0.1 0.2 0.3 0.4 0.50

200

Ti (d )

Gas

0 0.1 0.2 0.3 0.4 0.5

0

200

Ti (d )

Gas


Figure 4.33: 3D Cylindrical Vertical Well: Time evolution of GOR for the fluidsampled at the sand face during pumpout.

114

ction]

h [ft]

on of C

1 [frac

Dep

th

Concen

tratio

X [ft] Y [ft]

C

(a)

ction]

h [ft]

n of FC 1

8 [frac

Dep

th

oncentratio

n

X [ft] Y [ft]

Co

(b)

Figure 4.34: 3D Cylindrical Vertical Well: 3D spatial distribution of hydrocarboncomponents (a) C1 and (b) FC18 obtained with UTFEC after 0.5 days from the onset offluid sampling through azimuthal angles 0 to 18 and at a depth of 2133− 2135 [ft]. Theformation was previously invaded with OBMF to a radial length of 2.5 [ft]. Fluid samplingtakes place with a constant flow rate of 10 [bbl/day]. Due to symmetry, a half-cylindermodel is used in the numerical simulation.

115

P [psi]

h [ft]

Dep

th

X [ft] Y [ft]

(a)

ction]

h [ft]

M)/P G

EM [frac

Dep

th

(PUTFEC–P G

EM

X [ft] Y [ft]

(

(b)

Figure 4.35: 3D Cylindrical Vertical Well: 3D spatial distribution (a) pressure ob-tained with UTFEC and (b) the relative difference between pressures calculated using usingUTFEC and CMG-GEM after 12 [hrs] from the onset of fluid sampling through azimuthalangles 0 to 18 and at a depth of 2133− 2135 [ft]. The formation was previously invadedwith OBMF to a radial length of 2.5 [ft]. Fluid sampling takes place with a constant flowrate of 10 [bbl/day]. Due to symmetry, a half-cylinder model is used in the numericalsimulation.

116

Chapter 5

Simulation of Mud-Filtrate

Invasion

This chapter discusses the effect of mud-filtrate invasion on fluid-flow simulations. I imple-

ment a mudcake model in the developed three-dimensional (3D) compositional simulator

(UTFEC) described in previous chapters. The mudcake model allows one to study the pro-

cess of mud-filtrate invasion and subsequent variations in the distribution of fluid around

the wellbore; it relates mudcake overbalance pressure to the dynamic variations of mud-

cake thickness, mudcake permeability, and mudcake porosity. This chapter studies mudcake

growth formulations, verification with previous publications, and undertakes several case

studies on water- and oil-base mud-filtrate invasion, including the effect of dispersion on the

radial distribution of aqueous salt concentration in conjunction with water-base mud-filtrate

invasion.

5.1 Formulation

The process of mud-filtrate invasion dynamically couples mud and rock properties. Mud and

rock properties control mudcake growth during mud-filtrate invasion. Dewan and Chenev-

117

ert (2001) performed laboratory experiments of water-base mud (WBM) invasion to study

both mudcake buildup and mud-filtrate invasion. They proposed that mudcake permeability

and mudcake pressure differential are related through the equation

kmc(t) =kmc0P vmc(t)

, (5.1)

where t is time, kmc0 is mudcake reference permeability, Pmc is mudcake pressure differential,

and v is a compressibility exponent which varies in the range from 0.4 to 0.9. Moreover,

Dewan and Chenevert (2001) introduced an expression for the time evolution of mudcake

porosity, given by

φmc(t) =φmc0P v·δmc (t)

, (5.2)

where φmc0 is mudcake reference porosity and δ is a multiplier for the porosity exponent

which varies in the range from 0.1 to 0.2.

Chin (1995) introduced a relation for the time evolution of mudcake thickness as-

suming that solid particles in the mud do not enter the formation, given by

rmc(t)|(θ,z) · drmc|(θ,z) =fs

(1− fs)[1− φmc(t)|(θ,z)

] · kmc|(θ,z)Pmc(t)|(θ,z)dtµf (t)

, (5.3)

where fs is mud solid fraction, rmc is mudcake thickness, drmc is differential mudcake

thickness, dt is differential time, and µf is mud-filtrate viscosity.

Wu et al. (2004) implemented a mudcake model with a black-oil simulator. Re-

cently, Pour (2008) implemented a similar mudcake model in a 1D equation-of-state (EOS)

compositional fluid-flow simulator. In this dissertation, I generalize the previous work and

implemente a model for mudcake growth with a 3D cylindrical equation-of-state (EOS)

compositional simulator. I assume that mudcake initially has a thickness of 10−9 [in] and

that fluid flow in the mudcake grid (first grid) includes only one fluid phase. The procedure

118

adopted for the simulation is as follows:

1. Calculate mudcake pressure differential at each depth:

Mudcake pressure differential is given by

Pmc|(θ,z) = (Pw − P2)|(θ,z), (5.4)

where Pw is sandface pressure and P2 is pressure of the second grid block in the radial

direction.

2. Update mudcake properties:

Equations (5.1) and (5.2) update mudcake permeability and mudcake porosity.

3. Calculate mudcake thickness:

Equation (5.3) gives the mudcake thickness at the current mudcake condition.

4. Update transmissibilities:

Variations in mudcake overbalance pressure alter mudcake permeability. Having up-

dated the permeability of the first radial grid (mudcake), diagonal transmissibilities

of the first and second grids in the radial direction are updated using equations (3.58)

through (3.60), i.e.,

Tr±1/2 =±∆θ∆z

ln rr±1

rr±1/2

(krr)r±1+

lnrr±1/2

rr

(krr)r

,

Tθ±1/2 =± log (

rr+1/2

rr−1/2)∆z

Θθ±1 −Θθ±1/2

(kθθ)θ±1+

Θθ±1/2 −Θθ

(kθθ)θ

,

and

Tθ±1/2 =±∆θ

2 (r2r+1/2 − r

2r−1/2)

Zz±1 − Zz±1/2

(kzz)z±1+Zz±1/2 − Zz

(kzz)z

.

5. Update phase productivity index:

The thickness and permeability of the first grid (mudcake) varies with time, hence,

119

the phase productivity index changes to a new value given by equation (3.88), namely,

PIj =krr∆z∆θλrj

25.14872× π ×(

lnrorw

+ s

) .

6. Solve the pressure equation.

7. Calculate the flow rate.

8. Update concentrations and all other properties and advance the simulation by one

time step.

5.2 Validation of the Simulations

In this section, I simulate an experiment performed with field Mud 97074 (Dewan and Chen-

evert, 2001). Table 5.1 describes the physical properties of mudcake and reservoir. Figure 5.1

compares the volume of injected filtrate obtained with UTFEC against experimental data.

At t = 0, there is approximately 0.433 [cm3] of difference between filtrate volume calculated

with numerical simulations and experimental data, which may be associated with “spurt

loss”1. Figure 5.1 shows a good agreement between numerical simulations and experimental

data.

In addition, I verify end-points of the flow rates obtained for the simulation of

mud-filtrate invasion. At the onset of mud-filtrate invasion, the flow rate is equal to that

without mudcake. Table 5.2 summarizes the assumed parameters for the description of both

mudcake and reservoir in the simulations of mud-filtrate invasion. Figures 5.3(a) and 5.3(b)

compare flow rates obtained with numerical simulations in the presence of mudcake and

those calculated without presence of mudcake; They show that flow-rate values at the onset

of invasion are approximately the same.

1The instantaneous volume (spurt) of liquid which passes through a filter prior to mudcake stabiliza-tion (Bourgoyne et al., 1986).

120

Table 5.1: Summary of assumed mudcake parameters used in the numerical simulation ofmud-filtrate invasion (field Mud 97074) (Dewan and Chenevert, 2001; Wu, 2004).


Solid fraction, fs fraction 0.231

Mudcake reference permeability, kmc0 md 0.003

Mudcake reference porosity, φmc0 - 0.59

Compressibility exponent, v - 0.63

Porosity exponent, δ - 0.1

Mudcake thickness limit in 0.25

Initial reservoir pressure psi 4000

Pressure at the sandface psi 4300

Reservoir thickness ft 0.0449


Simulation CPU time minute 36

66

5

6

cm3 ]

ExperimentSimulation

5

6

cm3 ]

3

4

f Filtrate [

3

4

f Filtrate [

1

2

Volume of

1

2

Volume of

0 1 2 3 4 5 60

S R t f Ti [ i 0.5]

V

0 1 2 3 4 5 6

0

S R t f Ti [ i 0.5]

V

Square Root of Time [min0.5]Square Root of Time [min0.5]

Figure 5.1: Comparison of volume of filtrate obtained with numerical simulations us-ing UTFECS against that measured in the laboratory with field Mud 97074 (Dewan andChenevert, 2001).

121

350] 350] 350] 350

ssure [psi] 350

ssure [psi] 350

ssure [psi]

300

alance Pres

300

alance Pres

300

alance Pres

250

ke Overba

250

ke Overba

250

ke Overba

10-5 10-3 10-1 100200

Ti [d ]

Mucak

10-5 10-3 10-1 100

200

Ti [d ]

Mucak

10-5 10-3 10-1 100

200

Ti [d ]

Mucak

Time [day]Time [day]Time [day]

Figure 5.2: Time variation of mudcake overbalance pressure. The formation exhibits theproperties of Rock Type 3 described in Table 4.1 and Figure 4.1.

Wu (2004) found that, shortly after the onset of invasion, mudcake overbalance

pressure reaches a pressure close to the initial pressure difference between reservoir and

sandface. Figure 5.2 shows a similar trend in the time variation of mudcake overbalance

pressure. I use Darcy’s equation to approximate the final value of flow rate, to wit,

q =KmcA

µf

Pmcdr

, (5.5)

where Kmc is absolute permeability of mudcake, A is cross-sectional area (equal to 2πr∆Z);

µf is filtrate viscosity, Pmc is mudcake pressure differential, and dr is mudcake thickness.

For this simulation, flow rate is approximately equal to

q =0.03

3500.4 [md]

1[cp]× (2π × 0.5[ft]× 1[ft])× 350[psi]

0.412 [ft]

×1.06232× 10−14 ft2

md

1.67868× 10−12 cppsi×day

= 0.6013ft3

day= 0.1071

bbl

day,

which is close to the value of the late-time flow rate shown in Figures 5.3(a) and 5.3(b).

These figures indicate that flow rates calculated for cases with presence of mudcake are

consistently lower than those obtained for cases without presence of mudcake; this is due to

very low permeability of mudcake.

122

Table 5.2: Summary of assumed mudcake properties in the numerical simulations of mud-filtrate invasion.


Solid fraction, fs fraction 0.06

Mudcake reference permeability, kmc0 md 0.03

Mudcake reference porosity, φmc0 fraction 0.25

Compressibility exponent, v - 0.4

Porosity exponent, δ - 0.1

Mudcake thickness limit in 0.4

Initial reservoir pressure psi 3650

Pressure at the sandface psi 4000

Simulation time day 3

104104

103

10

ay]

Without MudcakeWith Mudcake103

10

ay]

101

102

ate [bbl/da

101

102

ate [bbl/da

100

101

Flow

Ra

100

101

Flow

Ra

10-13 10-11 10-9 10-7 10-5 10-3 10-110010-1

Ti [d ]10-13 10-11 10-9 10-7 10-5 10-3 10-1100

10-1

Ti [d ]Time [day]Time [day]

(a)

104104

103

10

ay]


10

ay]

101

102

ate [bbl/da

101

102

ate [bbl/da

100

101

Flow

Ra

100

101

Flow

Ra

10-13 10-11 10-9 10-7 10-5 10-3 10-110010-1

Ti [d ]10-13 10-11 10-9 10-7 10-5 10-3 10-1100

10-1

Ti [d ]Time [day]Time [day]

(b)

Figure 5.3: Time variation of mud-filtrate flow rate after the onset of invasion into aformation with (a) Rock Type I and (b) Rock Type III (rock types are described in Table 4.1and Figure 4.1). For each rock type, the following cases are considered: presence of mudcakeand no mudcake at the well boundary.

123

Table 5.3: List of parameters assumed in the description of the reservoir.



Wellbore outer radius ft 2000

Rock compressibility 1/psi 4× 10−7

Reservoir temperature F 200


5.3 Simulations of the Process of Mud-Filtrate Invasion

In this section, I explain the effect of mudcake growth on flow rate, pressure, saturations,

salt concentration, and radial resistivity profiles. I compare simulations with and without

presence of mudcake for the cases of two- and three-phase fluid flow.

5.3.1 Case Study of Two-Phase Flow

This case simulates mudcake growth and mud-filtrate invasion in the presence of two-phase

fuid flow. The oil phase consists of FC6, whereas the aqueous-phase is composed of water

and salt. Table 5.3 summarizes the parameters assumed for the description of the reservoir.

Initial reservoir pressure is 3500 [psi], water saturation is 0.25 [fraction], and salt concen-

tration is 168 [kppm NaCl]. For the wellbore boundary condition, I assume a bottomhole

pressure constraint equal to 3800 [psi] and filtrate water with salt concentration equal to 3

[kppm NaCl]. Figure 5.4 compares mudcake growth and mud-filtrate flow rates for different

values of mudcake reference permeability.

Figure 5.5 shows the effect of implementing a mudcake model on the radial distribu-

tions of pressure and water saturation in the vicinity of the wellbore. Figure 5.5(a) indicates

that as invasion time progresses, reservoir pressure decreases to its initial formation pres-

sure. Figure 5.5(b) shows that simulated radial fronts of water saturation are shallower than

those without the presence of mudcake.

124

0 40 4

0.3

0.4

[inch] Kmc0 = 3 md

Kmc0 = 0.30.3

0.4

[inch]

0.2

Thickness [ mc0

Kmc0 = 0.03

0.2

Thickness [

0.1

Mud

cake

T

0.1

Mud

cake

T

10-6 10-5 10-4 10-3 10-2 10-10

Ti [d ]

M

10-6 10-5 10-4 10-3 10-2 10-10

Ti [d ]

M


(a)

102102102

101

10

ay] 101

10

ay] 101

10

ay]

100

ate [bbl/da

100

ate [bbl/da

100

ate [bbl/da

10-1

Flow

Ra

Kmc0 = 3 md

Kmc0 = 0.3 10-1

Flow

Ra

10-1

Flow

Ra

10-6 10-5 10-4 10-3 10-2 10-110-2

Ti [d ]

Kmc0 = 0.03

10-6 10-5 10-4 10-3 10-2 10-110-2

Ti [d ]

10-6 10-5 10-4 10-3 10-2 10-1

10-2

Ti [d ]

Time [day]Time [day]Time [day]

(b)

Figure 5.4: Time variation of (a) mudcake thickness and (b) flow rate after the onsetof invasion for different values of reference mudcake permeability. The invaded formationexhibits the petrophysical properties of Rock Type 1 described in Table 4.1 and Figure 4.1.

38003800

3750

3800

1 d


3800

1 d

3650

3700

sure [p

si]

0 01 d

0.1 day

1 day

3650

3700

sure [p

si]

0 01 d

0.1 day

1 day

3550

3600Press 0.01 day

3550

3600Press 0.01 day

100 101 1023500

R di l Di t [ft]

100 101 102

3500

R di l Di t [ft]


(a)

1001001

80

100

80

1001

0.8

actio

n]

40

60

ration

[%]

0.1 day

1 day

40

60

ration

[%]

0.1 day

1 day0.6

ratio

n [fra

20

40

Satur

0.01 day

y

20

40

Satur

0.01 day

y0.4

0.2

Water Satu

100 101 1020

R di l Di t [ft]

y

100 101 1020

R di l Di t [ft]

y

W


(b)

Figure 5.5: Two-Phase Flow: Comparison of radial distributions of (a) pressure and (b)water saturation at different times after the onset of invasion for two cases: (i) withoutpresence of mudcake and (ii) with presence of mudcake. Initial P=3500 [psi], Sw = 0.25[fraction]. Well constraint is 1 day of WBMF invasion with BHP=3800 [psi]. Mudcakereference permeability, Kmc0=0.3 [md], mudcake reference porosity, φmc0=0.3 [fraction],and solid fraction, fs=0.06 [fraction]. The invaded formation exhibits the petrophysicalproperties of Rock Type 1 described in Table 4.1 and Figure 4.1.

125

5.3.2 Case Study of Three-Phase Flow

In this section, I illustrate the effect of mudcake growth for the case of three-phase fluid flow.

I assume an initial composition (0.4, 0.3, and 0.3) of pseudo components (C1, C3, FC7);

the aqueous phase is composed of water and salt. Table 4.4 recapitulates the remaining

parameters for the description of the reservoir. Initial reservoir pressure is 500 [psi], water

saturation is 0.25 [fraction], and salt concentration is 168 [kppm NaCl]. I assume a bot-

tomhole pressure of 1300 [psi], injecting water with a salt concentration of 3 [kppm NaCl].

Figure 5.6 compares mudcake growth calculated for different values of mudcake reference

permeability to the corresponding injection fluid-flow rates.

Figures 5.7 through 5.9 show the effect of accounting for mudcake growth in the

simulations of mud-filtrate invasion and compare them to results obtained for the case of no

mudcake. Figure 5.7 indicates that reservoir pressure decreases to initial formation pressure

after approximately 1 day. Figures 5.7 through 5.9(a) show that fronts of water, oil, and

gas saturation, as well as salt concentration obtained with the simulation of mud-filtrate

invasion are shallower than those obtained without presence of mudcake.

From the calculated radial distributions of water saturation and salt concentration,

Archie’s equation (Archie, 1942) yields the corresponding radial distribution of rock electrical

resistivity, given by

Rt = Rw ·a

φm Snw, (5.6)

where Rt is true formation resistivity, a is tortuosity factor, m is cementation exponent, n

is saturation exponent, and Rw is connate-water resistivity calculated with (Bigelow, 1992)

Rw =(

0.0123 +3647.5

C0.955salt

)· 81.77

T + 6.77, (5.7)

where Csalt is [NaCl] concentration in parts per million (ppm) and T is formation tem-

perature in F. Table 5.4 lists the assumed parameters in Archie’s equation and reservoir

temperature to calculate spatial distribution of rock electrical resistivity throughout this

126

Table 5.4: List of parameters assumed in this chapter for Archie’s (1942) equation tocalculate rock electrical resistivity.


Archie’s tortuosity/cementation factor (a) dimensionless 1

Archie’s cementation exponent (m) dimensionless 2

Archie’s saturation exponent (n) dimensionless 2

Reservoir temperature F 200

0 40 4

0.3

0.4

[inch] Kmc0 = 3 md

Kmc0 = 0.3 0.3

0.4

[inch]

0.2

Thickness [ mc0

Kmc0 = 0.03

0.2

Thickness [

0.1

Mud

cake

T

0.1

Mud

cake

T

10-8 10-6 10-4 10-3 10-2 10-10

Ti [d ]

M

10-8 10-6 10-4 10-3 10-2 10-10

Ti [d ]

M


(a)

104104

2

10

ay]

2

10

ay]

102

ate [bbl/da

102

ate [bbl/da

100Flow

Ra

Kmc0 = 3 md

Kmc0 = 0.3 100

Flow

Ra

10-8 10-6 10-4 10-3 10-2 10-110-2

Ti [d ]

Kmc0 = 0.03

10-8 10-6 10-4 10-3 10-2 10-110-2

Ti [d ]


(b)

Figure 5.6: Three-Phase Flow: Time variations of (a) mudcake thickness and (b) flowrate after the onset of invasion for different values of reference mudcake permeability. The in-vaded formation exhibits the petrophysical properties of Rock Type 3 described in Table 4.1and Figure 4.1.

chapter. Figure 5.9(b) shows the radial distributions of electrical resistivity calculated for

two cases of invasion: with and without mudcake. Based on the simulations considered in

this chapter, it is observed that the conductive annulus for the simulation of invasion with-

out mudcake exhibits a low resistivity and spreads to a large radial distance. Figure 5.7(b)

shows the effect of presence of mudcake during invasion into a high-permeability formation

(k = 500 [md]); water saturation does not increase to 1− Sor; whereas in the simulation of

invasion without mudcake, water saturation increases approximately to the level of 1−Sor.

Therefore, the near-wellbore electrical resistivity for the case with mudcake is higher than

that without mudcake (see Figure 5.9(b)).

127

14001400

1200

1400Without Mudcake With Mudcake1200

1400

800

1000

sure [p

si]

1 day800

1000

sure [p

si]

1 day

600

800

Press

0.01 day0.1 day

600

800

Press

0.01 day0.1 day

100 101 102400

R di l Di t [ft]

100 101 102

400

R di l Di t [ft]


(a)

1001001

80

100

[%]

1 day80

100

[%]

1 day

1

0.8

actio

n]

40

60

aturation [

0.01 day0.1 day

1 day

40

60

aturation [

0.01 day0.1 day

1 day

0.6

ratio

n [fra

20

40

Water Sa

Without Mudcake20

40

Water Sa 0.4

0.2

Water Satu

100 101 1020

R di l Di t [ft]

With Mudcake

100 101 1020

R di l Di t [ft]

W


(b)

Figure 5.7: Three-Phase Flow: Comparison of radial profiles of (a) pressure and (b) wa-ter saturation calculated at different times after the onset of invasion for two cases: withoutand with presence of mudcake. Initial P=500 [psi], Sw = 0.25 [fraction], temperature, T=200[F], and composition (0.4, 0.3, and 0.3) for pseudo components (C1, C3, and FC7). Wellconstraint is 1 day of water-base mud invasion with BHP=1300 [psi]. Mudcake referencepermeability, Kmc0=0.3 [md], mudcake reference porosity, φmc0=0.3 [fraction], and solidfraction, fs=0.06 [fraction]. The invaded formation exhibits the petrophysical properties ofRock Type 3 described in Table 4.1 and Figure 4.1.

1001001

80

100

%]

Without Mudcake With Mudcake80

100

%]

1

0.8

tion]

40

60

uration [%

40

60

uration [%

0.6

tion [fract

20

40

Oil Sat

20

40

Oil Sat 0.4

0.2

Oil Satura

100 101 1020

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

O


(a)

1001001

80

100

%]

Without Mudcake With Mudcake80

100

%]

1

0.8

tion]

40

60

turation

[%

1 day40

60

turation

[%

1 day

0.6

ation [frac

20

40

Gas Sat

0.01 day0.1 day

y

20

40

Gas Sat

0.01 day0.1 day

y0.4

0.2

Gas Satura

100 101 1020

R di l Di t [ft]

100 101 102

0

R di l Di t [ft]

G


(b)

Figure 5.8: Three-Phase Flow: Comparison of radial profiles of (a) oil and (b) gassaturation calculated at different times after the onset of invasion for two cases: withoutand with presence of mudcake. Initial P=500 [psi], Sw = 0.25 [fraction], temperature, T=200[F], and composition (0.4, 0.3, and 0.3) for pseudo components (C1, C3, and FC7). Wellconstraint is 1 day of water-base mud invasion with BHP=1300 [psi]. Mudcake referencepermeability, Kmc0=0.3 [md], mudcake reference porosity, φmc0=0.3 [fraction], and solidfraction, fs=0.06 [fraction]. The invaded formation exhibits the petrophysical properties ofRock Type 3 described in Table 4.1 and Figure 4.1.

128

x 105x 105

1.5

2 x 10pp

m]

1.5

2 x 10pp

m]

m

NaC

l]

1

1.5

ntration

[p

1 day1

1.5

ntration

[p

1 dayation [ppm

0.5

Salt Con

cen

0.01 day

0.1 day0.5

Salt Con

cen

0.01 day

0.1 day

Without MudcakeConcen

tr

100 101 1020


S

y

100 101 1020


S

yWith Mudcake

Salt


(a)

10210210

m]

Without MudcakeWith Mudcake

10

m]

101

vity [ . m

101

vity [ . m

100

Resistiv

0.01 day

0.1 day 1 day100

Resistiv

0.01 day

0.1 day 1 day

100 101 10210‐1

R di l Di t [ft]

y

100 101 10210‐1

R di l Di t [ft]

y


(b)

Figure 5.9: Three-Phase Flow: Comparison of radila profiles of (a) salt concentrationand (b) electrical resistivity calculated at different times after the onset of invasion for twocases: without and with presence of mudcake. Initial P=500 [psi], Sw = 0.25 [fraction],temperature, T=200 [F], and composition (0.4, 0.3, and 0.3) for pseudo components (C1,C3, and FC7). Well constraint is 1 day of water-base mud invasion with BHP=1300 [psi].Mudcake reference permeability, Kmc0=0.3 [md], mudcake reference porosity, φmc0=0.3[fraction], and solid fraction, fs=0.06 [fraction]. The invaded formation exhibits the petro-physical properties of Rock Type 3 described in Table 4.1 and Figure 4.1.

5.3.3 Comparison of Oil- and Water-Base Mud-Filtrate

In this dissertation the same formulation (see Section 5.1) is assumed for invasion of OBM

and WBM. Productivity index (equation (3.88)) couples mudcake and reservoir properties.

From the definition of productivity index, namely,

PIj =krr∆z∆θkrj

25.14872× πµj ×(

ln (rorw

) + s

) , (5.8)

invasion flow rates depend on the variables krr, krj , and µj . Remaining parameters in equa-

tion (5.8) are constant and independent of the filtrate type. Inasmuch as flow in mudcake is

assumed to be single phase, it follows that, krj = 1. On the other hand, the variation of krr

is approximately the same for OBM and WBM. Therefore, in the calculation of mud-filtrate

flow rate, µj is the variable which is different for OBM and WBM. I simulate OBM invasion

for two cases: (i) OBM filtrate exhibits a viscosity lower than water, and (ii) OBM filtrate

129

0 40 4

0.3

0.4[in

ch] Oil‐Base Mud

Water‐Base Mud0.3

0.4[in

ch]

0.2

Thickness [

0.2

Thickness [

∆tWBM/∆tOBM ≈2

0.1

Mud

cake

T

0.1

Mud

cake

T

10-7 10-5 10-3 10-1 1000

Ti [d ]

M

10-7 10-5 10-3 10-1 1000

Ti [d ]

M


(a)

106106

104

10

ay]

Oil‐Base MudWater‐Base Mud104

10

ay]

102

ate [bbl/da

102

ate [bbl/da

100

Flow

Ra

100

Flow

Ra

qOBM/qWBM ≈2

10-13 10-11 10-9 10-7 10-5 10-3 10-110010-2

Ti [d ]

10-13 10-11 10-9 10-7 10-5 10-3 10-1100

10-2

Ti [d ]

qOBM/qWBM ≈2


(b)

Figure 5.10: Time variation of (a) mudcake thickness and (b) flow rate after the onsetof invasion of water-base and oil-base mud (µo = 0.5 [cp]). The invaded formation has thepetrophysical properties of Rock Type 3 described in Table 4.1 and Figure 4.1.

exhibits a viscosity higher than water.

The simulation results shown in Figure 5.10 correspond to the case where oil has a

viscosity of 0.5 [cp], whereas simulation results shown in Figure 5.11 correspond to invasion of

OBM with a viscosity higher than water, µo = 2 [cp]. It is found that mudcake stabilization

time is inversely correlated with mud-filtrate viscosity. For instance, in the case shown

in Figure 5.11(a), mudcake reaches its limiting thickness after ∆tWBMstabilization = 1.066

[day] for WMB invasion and ∆tOBMstabilization = 2.122 [day] for OBM invasion. In that

case, mud-filtrate flow rate for WBM and OBM invasion are qWBMstabilization = 0.1182

and qWBMstabilization = 0.0593 [bbl/day], respectively. However, often OBMs have high

viscosity values and consequently OBM invasion flow rates are lower than those for WBM;

this behavior is consistent with the findings of Salazar et al. (2009).

5.3.4 Mud-Filtrate Invasion In Vertical Wells

I perform simulations of mud-filtrate invasion in a vertical well assuming that the invaded

formation is saturated with gas and that the drilling fluid is oil-base mud. The invaded

formation exhibits petrophysical properties of Rock Type 5-III described in Table 5.5. A

130

0 40 4

0.3

0.4

[inch] Oil‐Base Mud

Water‐Base Mud0.3

0.4

[inch]

0.2

Thickness [

0.2

Thickness [

∆tOBM/∆tWBM ≈2

0.1

Mud

cake

T

0.1

Mud

cake

T

10-7 10-5 10-3 10-1 1000

Time [day]

M

10-7 10-5 10-3 10-1 1000

Time [day]

M


(a)

104104

2

10

ay]

Oil‐Base MudWater‐Base Mud

2

10

ay]

102

ate [bbl/da

102

ate [bbl/da

100

Flow

Ra

100

Flow

Ra

qWBM/qOBM ≈2

10-13 10-11 10-9 10-7 10-5 10-3 10-110010-2

Ti [d ]

10-13 10-11 10-9 10-7 10-5 10-3 10-1100

10-2

Ti [d ]

qWBM/qOBM ≈2


(b)

Figure 5.11: Time variation of (a) mudcake thickness and (b) flow rate after the onsetof invasion of water-base and oil-base mud (µo = 2.0 [cp]). The invaded formation has thepetrophysical properties of Rock Type 3 described in Table 4.1 and Figure 4.1.

200200200200

150

200

150

200

150

200

150

200RT 5‐III

100

c [psi]

100

c [psi]

100

c [psi]

100

c [psi] RT 5‐II

50

P c

50

P c

50

P c

50

P c

RT 5 I

0 0.2 0.4 0.6 0.8 1W S i

0 0.2 0.4 0.6 0.8 1W S i

0 0.2 0.4 0.6 0.8 1W S i

0 0.2 0.4 0.6 0.8 1W S i

RT 5‐IRT: Rock Type

Water SaturationWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

(a)

11111111

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1kro

0.6

k r

0.6

k r

0.6

k r

0.6

k r

0.6

k r

0.6

k r

0.6

k r

0.6

k r krw

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S iWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

(b)

Figure 5.12: Water-oil (a) capillary pressure and (b) relative permeability curves of rocktypes studied in Sections 5.3.4 and 5.3.5. Variables kro and krw are relative permeability ofoil and water, respectively. Rock types 5-I, 5-II, and 5-III are identified with square, circle,and star markers, respectively.

131

120120120120

100

120

100

120

100

120

100

120

60

80

c [psi]

60

80

c [psi]

60

80

c [psi]

60

80

c [psi]

RT 5‐III

20

40

P c

20

40

P c

20

40

P c

20

40

P c

RT 5‐II

0 0.2 0.4 0.6 0.8 1

20

W S i0 0.2 0.4 0.6 0.8 1

20

W S i0 0.2 0.4 0.6 0.8 1

20

W S i0 0.2 0.4 0.6 0.8 1

20

W S i

RT 5‐IRT: Rock Type

Water SaturationWater SaturationWater SaturationWater SaturationGas Saturation [fraction]

(a)

1111

0.8

1

0.8

1

0.8

1

0.8

1

krg

0.6

k r

0.6

k r

0.6

k r

0.6

k r

kro

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S iWater SaturationWater SaturationWater SaturationWater SaturationGas Saturation [fraction]

(b)

Figure 5.13: Gas-oil (a) capillary pressure and (b) relative permeability curves of rocktypes studied in Sections 5.3.4 and 5.3.5. Variables kro and krg are relative permeability ofoil and gas, respectively. Rock types 5-I, 5-II, and 5-III are identified with square, circle,and star markers, respectively.

Table 5.5: Absolute permeability, porosity, residual water saturation, and residual oilsaturation for three synthetic rock types assumed in Sections 5.3.4 and 5.3.5. Figure 5.12shows the relative permeability and capillary pressure curves corresponding to these rocktypes.

Unit Rock Type 5-I Rock Type 5-II Rock Type 5-III

Absolute Permeability md 500 10 1Porosity fraction 0.32 0.15 0.05Swir fraction 0.07 0.11 0.20Soir fraction 0.15 0.20 0.25

132

2500Sw [fraction]

2500

26000 6

0.7

0 6

0.7

2700

pth [ft]

0.5

0.6

0.5

0.6

pth [ft]

2800

2900

Dep

0.40.4Dep

100 101

2900

3000

0.30.3

10 10Radius [ft]Radial Distance [ft]

(a)

2500Sg [fraction]

2500

2600 0.6

2700

pth [ft]

0.4

pth [ft]

2800

2900

Dep

0.2

Dep

100 101

2900

3000 010 10

Radius [ft]Radial Distance [ft]

(b)

Figure 5.14: Spatial (radial and vertical directions) distributions of (a) water saturationand (a) gas saturation after three days from the onset of oil-base mud-filtrate invasioninto a formation with petrophysical properties of Rock-Type 5-III. Overbalance pressure isassumed equal to 300 [psi], and mudcake reference permeability is 0.03 [md] (described inTable 5.5 and Figure 5.12).

water-gas contact is located at the depth of 2950 [ft], and the formation is at capillary-

gravity equilibrium prior to invasion. OBMF exhibits a viscosity of 2 [cp] and invasion time

is 3 [days]. Mudcake reference permeability is assumed equal to 0.03 [md]. Figure 5.14 shows

the spatial (radial and vertical directions) distribution of water and gas saturations after 3

days from the onset of invasion. In the vicinity of the well, gas is completely replaced with

oil. It is assumed that residual gas between oil and gas is equal to zero (see Figure 5.13).

Figure 5.14(a) also shows the effect of the capillary transition zone; in the vertical direction,

water saturation decreases to residual water saturation. Radial length of invasion in the

described model is approximately the same along the vertical direction.

5.3.5 Mud-Filtrate Invasion in Deviated Wells

In this section, I simulate invasion of water-base mud into a formation with three petro-

physical layers. I assume that the formation is penetrated with a well deviation angle of

45 [degrees] through three petrophysical layers. Figure 4.28 describes the deviated well in

133

cylindrical coordiantes used for the model of this section; similar to that graph, from the

top to bottom layer, formation layers exhibit, respectively, petrophysical properties of Rock

Types 5-I, 5-II, and 5-III. (rock types are described in Table 5.5 and Figures 5.12 and 5.13).

The boundary condition throughout the well is invasion of water under a bottomhole pres-

sure equal to 300 [psi] for 10 days. Mudcake reference permeability is equal to 0.03 [md].

The numerical model consists of 80 radial, 42 azimuthal, and 30 vertical grid blocks. Ini-

tially, each layer is assumed to have water saturation equal to the residual saturation of

the specified rock type; Rock Type 5-I with Swir = 0.07 [fraction], Rock Type 5-I with

Swir = 0.11 [fraction], and Rock Type 5-I with Swir = 0.20 [fraction]. Saturating oil has

a specific density of 0.76; connate water has a salinity of 160 [kppm NaCl] and the salinity

of invading water is 3 [kppm NaCl]. Figures 5.15 and 5.15 show two views of the spatial

distributions of water saturation and salt concentration; X and Y axes are parallel to those

described in Figure 4.28. In Figures 5.15 and 5.15, the y axis of the graph corresponds to

true vertical depth. Figure 5.15(a) shows that the top petrophysical layer, which consists of

a high-permeability rock (Rock Type 5-I) is significantly affected by gravity; there is axial

symmetry in the distribution of water saturation around the wellbore. On the other hand,

the low permeability rock (Rock Type 5-III) is not remarkably affected by gravity and water

saturation has displaced the original saturating fluid to almost the same radial length.

5.3.6 Physical Dispersion During Mud-Filtrate Invasion

In this section, I document the effect of physical dispersion on the radial distribution of

aqueous salt concentration during mud-filtrate invasion. As shown in Section 4.3.4, physical

dispersion leads to radial spreading of aqueous salt concentration. In formation evaluation,

radial spreading of salt concentration impacts the radial distribution of electrical resistivity.

When studying mud-filtrate invasion, it is customary to encounter situations wherein

the invading water is saltier or fresher than connate water. For each situation, I calculate

radial distribution of water saturation, salt concentration, and rock electrical resistivity

for different dispersivity values. Due to the scale dependency of physical dispersion (Sec-

134

Sw [fraction]

3115

3120 0.7

0.8

RT I3125

3130VD [f

t]

S0.5

0.6RT I

RT II

3135

TV

Sw

0.3

0.4

RT III

-20 -10 0

3140

3145 0.1

0.2

RT: Rock Type

Y-dir [ft]Y [ft]

(a)

Sw [fraction]

3120

312

0.7

0.8

RT I

3125

3130VD [f

t]

S0.5

0.6

RT II3130

3135

TV

Sw

0.3

0.4

RT III

-10 -5 0 5 103140

0.1

0.2

θw =45°

X-dir [ft]X [ft]

(b)

Figure 5.15: Spatial distributions of water saturation after 10 days from the onset ofwater-base mud-filtrate invasion into a formation with three petrophysical layers (verticalaxis is the true vertical depth). Wellbore deviation angle, θw, is equal to 45 [degrees].Overbalance pressure is assumed to be 300 [psi]. The petrophysical properties of top, middle,and bottom layers are those of rock 5-I, 5-II, and 5-III, respectively (described in Table 5.5and Figure 5.12). Prior to WBM invasion, water saturations in all layers were assumedequal to residual saturation. Saturating oil has an API of 55°.

x 104

3115

312012

14

16

RT I3125

3130VD [f

t]

ppm

NaC

l]

8

10

12RT I

RT II

3135

TV

Csa

lt [p

4

6

8

RT III

-20 -10 0

3140

3145

2RT: Rock Type

Y-dir [ft]Y [ft]

(a)

x 104

3120

312 12

14

16

RT I

3125

3130VD [f

t]

ppm

NaC

l]8

10

12

RT II3130

3135

TV

Csa

lt [p

4

6

8

RT III

-10 -5 0 5 103140

2θw =45°

X-dir [ft]X [ft]

(b)

Figure 5.16: Spatial distributions of salt concentration after 10 days from the onset ofwater-base mud-filtrate invasion into a formation with three petrophysical layers (verticalaxis is the true vertical depth). Wellbore deviation angle, θw, is equal to 45 [degrees]. Thepetrophysical properties of top, middle, and bottom layers are those of rock 5-I, 5-II, and5-III (described in Table 5.5 and Figure 5.12), respectively. Prior to WBM invasion, watersaturations in all layers were assumed equal to residual saturation. Connate water has asalinity equal to 160 [kppm NaCl], whereas invading water has a salinity equal to 3 [kppmNaCl].

135

tion 4.3.4), a range of dispersivity values is chosen for the study. In the subsequent simu-

lations, mud filtrate penetrates to a depth of approximately 5 [ft] and dispersivity (αl) can

be as large as 1 [ft] (see Figure 4.14).

5.3.7 Injection of the Fresh Water

I study the injection of fresh water into a formation with salty connate water. It is assumed

that salinity of connate water is equal to 168 [kppm NaCl], whereas the salinity of invading

water is equal to 3 [kppm NaCl]. Section 4.3.4 describes the assumed formation properties

as well as the initial formation conditions prior to the onset of mud-filtrate invasion. Fig-

ures 5.17(a) and 5.17(b) show the radial distributions of water saturation after 1 day from

the onset of water injection into formations with rock types I and II, respectively. Injection

rate of 0.5 [bbl/day] is approximately the average flow rate for 1 day of invasion under an

overbalance pressure of 300 [psi] and mudcake permeability of 0.3 [md]. Porosity of Rock

Type I is 0.320.05 = 6.4 times greater than that of Rock Type II; therefore, the radial length

of invasion into a formation with Rock Type I is shorter than for the case of Rock Type

II. Figure 5.17(a) corresponds to Rock Type I, with permeability of 500 [md] and capil-

lary pressure end-point of 7 [psi], whereas Figure 5.17(b) shows the water saturation for

Rock Type II, with permeability of 0.1 [md] and capillary pressure end-point of 170 [psi].

Figure 5.17 indicates that capillary pressure for Rock Type I leads to significant smoothing

(from the wellbore to radial length of 5 [ft]) in the radial profile of water saturation, whereas

capillary pressure for Rock Type II only causes 3 [ft] spreading in the radial distribution of

water saturation.

Figure 5.18 shows radial distributions of salt concentration and rock electrical resis-

tivity for Rock Type I; the dashed, dotted, and dashed-dotted curves correspond to radial

profiles for dispersivity values (in equations (2.38) through (2.43)) of αl1 = α = 0, α = 0.2,

and α = 1 [ft], respectively. Figure 5.18(a) shows that an increase in salt dispersivity leads

to nonlinear increments in the radial spreading of salt concentration. For instance, increas-

ing dispersivity from 0 to 0.2, increases two [ft] by the radial spreading of salt concentration.

136

I note that the radial spreading of salt concentration in Figure 5.18 for zero dispersivity is

due to numerical dispersion. Pour et al. (2011a) found that numerical dispersion in the sim-

ulations performed by the developed method is close to physical dispersion measured with

experimental measurements. In the calculation of electrical resistivity, the assumed Archie’s

parameters and reservoir temperature are those described in Table 5.4. Figure 5.18(b) shows

that, as dispersivity (physical dispersion) increases, the electrical conductive annulus be-

comes wider; a large dispersivity value (α = 1 [ft]) may decrease the electrical resistivity to

a value lower than the electrical resistivity of the uninvaded formation.

Similarly, Figure 5.19 shows radial distributions of salt concentration and rock elec-

trical resistivity for Rock Type II. Figures 5.18 and 5.19 show that salt dispersivity can

significantly affect the radial distribution of electrical resistivity. Figure 5.19(b) indicates

that, as dispersivity increases, the radial profile of electrical resistivity becomes smoother

whereas the amplitude of the electrical conductive annulus decreases.

5.3.8 Injection of Salty Water

In this section, I perform simulations for cases in which connate water is fresh (3 [kppm

NaCl]) whereas the invading water is salty (168 [kppm NaCl]). Analogous to Section 5.3.7,

I simulate the invasion of WBM with an average flow rate of 0.5 [bbl/day] into formations

with two Rock Types I (high-permeability formation) and II (high-permeability formation)

described in Table 4.8.

Formation properties and initial conditions are those assumed for Cases 1 and 2

(Section 4.3.4.1 and Section 4.3.4.2). Figure 5.17(a) and 5.17(b) show the radial distributions

of water saturation calculated after 1 [day] from the onset of water injection into formations

with petrophysical properties of the described rock types.

Figures 5.20 shows the radial distributions of salt concentration and rock electrical

resistivity for Rock Type I, whereas Figure 5.21 shows the radial distributions of salt concen-

tration and rock electrical resistivity for Rock Type II. In Figures 5.20 and 5.21, the dashed,

dotted, and dashed-dotted curves correspond to radial profiles for dispersivity values (in

137

11

0.8

1

n0.8

1

n

UTFECCMG-STARS

actio

n]

0.6

Satu

ratio

n

0.6

Satu

ratio

nratio

n [fra

0.2

0.4

Wat

er S

0.2

0.4

Wat

er S

Water Satur

100 1010

100 1010

W


(a)

11

0.8

1

n

0.8

1

n

UTFECCMG-STARS

actio

n]

0.6

Satu

ratio

n

0.6

Satu

ratio

nratio

n [fra

0.2

0.4

Wat

er S

0.2

0.4

Wat

er S

Water Satur

100 1010

100 1010

W


(b)

Figure 5.17: Radial distributions of water saturation calculated after 1 day from the onsetof water injection with a constant rate of 0.5 [bbl/day]. The dashed blue curve identifieswater saturation calculated with UTFEC, and the solid red curve identifies water saturationobtained with CMG-STARS. Initially, the invaded formation exhibits water saturation equalto 0.20 [fraction] and residual water saturations equal to (a) 0.07 [fraction] (Sw,movable = 0.13[fraction]) and (a) 20 [fraction] (Sw,movable = 0). Petrophysical properties of the invadedformation are those of (a) Rock Type I and (b) Rock Type II described in Table 4.8.

138

2 x 105

2 x 105

1.5

2

ppm

]1.5

2

ppm

]pm

NaC

l]

1ntra

tion

[p

1ntra

tion

[pratio

n [pp

0.5

Salt

Con

ce

0.5

Salt

Con

ce

= 0.0 [ft]0 2 [ft]t C

oncent

100 1010

S

100 1010

S

= 0.2 [ft] = 1.0 [ft]Sa

lt


(a)

1414

10

12

14

.m

]

10

12

14 = 0.0 [ft] = 0.2 [ft] = 1.0 [ft]

6

8

10

esis

tivity

[

6

8

10

4

6

lect

rical

Re

4

6

100 1010

2El

100 101

0

2


(b)

Figure 5.18: Radial distributions of (a) salt concentration and (b) electrical resistivity cal-culated after 1 day from the onset of water injection with a constant rate of 0.5 [bbl/day].The dashed, dotted, and dashed-dotted curves correspond to radial profiles for dispersivityvalues (in equations (2.38) through (2.43)): αl1 = α = 0, α = 0.2, and α = 1 [ft], re-spectively. Salt concentration in the invaded formation is 168 [kppm NaCl], whereas saltconcentration in the invading water is 3 [kppm NaCl]. Figure 5.17(a) shows the radialdistribution of water saturation corresponding to this case.

139

2 x 105

2 x 105

1.5

2

ppm

]1.5

2

ppm

]pm

NaC

l]

1ntra

tion

[p

1ntra

tion

[pratio

n [pp

0.5

Salt

Con

ce

0.5

Salt

Con

ce

= 0.0 [ft]0 2 [ft]t C

oncent

100 1010

S

100 1010

S

= 0.2 [ft] = 1.0 [ft]Sa

lt


(a)

2525

20

25

.m

]

20

25 = 0.0 [ft] = 0.2 [ft] = 1.0 [ft]

15

esis

tivity

[

15

5

10

lect

rical

Re

5

10

100 1010

El

100 1010


(b)

Figure 5.19: Radial distributions of (a) salt concentration and (b) electrical resistivitycalculated after 1 day from the onset of water injection with a constant flow rate of 0.5[bbl/day]. The dashed, dotted, and dashed-dotted curves correspond to radial profiles fordispersivity values (in equations (2.38) through (2.43)): αl1 = α = 0, α = 0.2, and α = 1[ft], respectively. Salt concentration in the invaded formation is 168 [kppm NaCl], whereassalt concentration in the invading water is 3 [kppm NaCl]. Figure 5.17(b) shows the radialdistribution of water saturation corresponding to this case.

140

equations (2.38) through (2.43)) of αl1 = α = 0, α = 0.2, and α = 1 [ft], respectively.

Figures 5.20 and 5.21 indicate that, when the invading water has higher salinity

than connate water, the effect of dispersion on the radial distribution of electrical resistivity

is negligible. Figures 5.18(b), 5.19(b), 5.20(a) and 5.21(a) indicate that the effect of disper-

sivity on the radial profile of salt concentration does not depend on the relative salinity of

invading water and connate water. However, near-wellbore electrical resistivity in one case

(invasion of fresh water) decreased to one-third and, whereas in another case (invasion of

salty water) it was negligible.


I implemented an experimentally tested method to model mudcake growth and mud-filtrate

invasion during drilling. The algorithm coupled the mudcake model with a reservoir sim-

ulator which enabled the simulation of the dynamic process of mud-filtrate invasion. The

developed algorithm was capable of simulating water- and oil-base mud-filtrate invasion into

formations saturated with different types of fluid including water, oil, and gas. Compari-

son of water and oil-base mud-filtrate invasion showed that invasion flow rate was inversely

proportional to mud-filtrate viscosity; for instance OBM with a viscosity two times greater

than WBM gave rise to an invasion flow rate equal to half of that for WBM.

This dissertation used a previously tested 1D mud-filtrate model for mudcake growth

developed by Pour (2008) and extended to 3D models when mudcake thickness and flow rate

varied around the perimeter of the wellbore. The algorithm coupled the mudcake model

with a 3D reservoir simulator to calculate fluid distribution around vertical and deviated

wells after invasion.

I studied the effect of physical dispersion on the radial distribution of aqueous salt

concentration during mud-filtrate invasion. The simulator calculated radial distributions of

salt concentration and rock electrical resistivity for different dispersivity values. Simulations

were performed for situations where the invading water was saltier or fresher than formation

141

connate water. It was found that (i) radial spreading of salt concentration was affected by

formation heterogeneity and dispersivity and (ii) near-wellbore electrical resistivity signifi-

cantly changed when the invading fluid was fresher than connate water.

142

2 x 105

2 x 105

1.5

2

ppm

]1.5

2

ppm

]

= 0.0 [ft] = 0.2 [ft] = 1.0 [ft]

pmNaC

l]

1ntra

tion

[p

1ntra

tion

[pratio

n [pp

0.5

Salt

Con

ce

0.5

Salt

Con

cet C

oncent

100 1010

S

100 1010

S

Salt


(a)

200200

150

200

.m

]

150

200

100

esis

tivity

[

100

50

lect

rical

Re

50 = 0.0 [ft]0 2 [ft]

100 1010

El

100 1010

= 0.2 [ft] = 1.0 [ft]


(b)

Figure 5.20: Radial distributions of (a) salt concentration and (b) electrical resistivity cal-culated after 1 [day] from the onset of water injection with a constant rate of 0.5 [bbl/day].The dashed, dotted, and dashed-dotted curves correspond to radial profiles for dispersivityvalues (in equations (2.38) through (2.43)): αl1 = α = 0, α = 0.2, and α = 1 [ft], re-spectively. Initially, formation is assumed to have water saturation equal to 0.20 [fraction]and residual water saturation is equal to 0.07 [fraction] (Sw,movable = 0.13 [fraction]). Saltconcentration in the invaded formation is 3 [kppm NaCl], whereas salt concentration in theinvading water is 168 [kppm NaCl]. Figure 5.17(a) shows the radial distribution of watersaturation corresponding to this case.

143

2 x 105

2 x 105

1.5

2

ppm

]1.5

2

ppm

]

= 0.0 [ft] = 0.2 [ft] = 1.0 [ft]

pmNaC

l]

1ntra

tion

[p

1ntra

tion

[pratio

n [pp

0.5

Salt

Con

ce

0.5

Salt

Con

cet C

oncent

100 1010

S

100 1010

S

Salt


(a)

200200

150

200

[ ]

150

200

[ ]

[ .m

]

100

Res

istiv

ity

100

Res

istiv

ity

Res

istiv

ity

50

Elec

tric

al R

50

Elec

tric

al R

= 0.0 [ft]0 2 [ft]El

ectr

ical

R

100 1010

E

100 1010

E

= 0.2 [ft] = 1.0 [ft]

E


(b)

Figure 5.21: Radial distributions of (a) salt concentration and (b) electrical resistivity cal-culated after 1 [day] from the onset of water injection with a constant rate of 0.5 [bbl/day].The dashed, dotted, and dashed-dotted curves correspond to radial profiles for dispersivityvalues (in equations (2.38) through (2.43)): αl1 = α = 0, α = 0.2, and α = 1 [ft], re-spectively. Initially, the formation is assumed to have a water saturation equal to residualsaturation (Swi = 0.20 [fraction] and Sw,movable = 0). Salt concentration in the invadedformation is 3 [kppm NaCl], whereas salt concentration in the invading water is 168 [kppmNaCl]. Figure 5.17(b) shows the radial distribution of water saturation corresponding tothis case.

144

Chapter 6

Simulation of Wettability

Alteration

Resistivity logs acquired in hydrocarbon-bearing formations invaded by oil-base mud (OBM)

often indicate abnormally high values of mobile water saturation. It is not possible to

explain such abnormally high values of water saturation with saturation-height analysis.

The common explanation invokes rock wettability alterations due to surfactants included

in oil-base mud-filtrate (OBMF). A quantitative study is needed to explain whether the

interaction of OBMF surfactants with water-wetted grains can cause a sufficiently large

increase in mobile water saturation in the near-wellbore region to affect resistivity logs.

In this chapter, I use the developed simulator to model the processes of mud-filtrate

invasion and ensuing wettability alteration once emulsifiers included in OBMF make contact

with grain surfaces. I assume a wettability alteration model in which the degree and type

of alteration are governed by the pore-volume concentration of emulsifier in OBMF within

the invaded formation.

145

6.1 Introduction

Oil-base mud-filtrate is partially miscible with original (in-situ) fluid in the reservoir and

often contains surfactants – cationic and anionic – to suspend fluid components in the

additive mixture. Surfactants present in OBM wet the surface of rock cuttings and facilitate

their removal from the wellbore.

Sharma and Wunderlich (1985) studied wettability alterations due to water-base

mud-filtrate (WBMF) invasion. From a series of experiments, they found that rock surfaces

with a strongly oil-wet condition became less oil-wet after becoming in contact with WBMF.

Later, Menezes et al. (1989) investigated the mechanisms that change the wettability of

sandstone upon interaction with OBMF. Their experiments showed that contact angle and

capillary pressure could change drastically after hydrocarbon components included in OBM

make contact with the rock surface. Ballard and Dawe (1988) studied the influence of

surfactants in OBMF on the wettability of glass surfaces. They showed that even small

concentrations of surfactants in mud filtrate could make rock surfaces to become more

oil wet. They concluded that wettability alteration leads to a significant decrease in water

saturation and found that residual water saturation tended to be lower for an oil-wet section

than for a water-wet section.

Yan et al. (1993) implemented the combined Amott1/USBM2 method to study al-

teration in rock surface wettability by calculating the variation of contact angle after OBMF

made contact with the rock surface. They showed that some wetting agents such as EZ Mul3

and DV-334 could significantly change the rock’s state of wettability. More recently, Gam-

bino et al. (2001) performed a series of experiments to study formation damage associated

with invasion of mud filtrate. They investigated different mechanisms during drilling and

cementing which led to formation damage. These mechanisms included wettability alter-

1The Amott wettability index is calculated from experiments for combined imbibition-displacement testwith refined oil and synthetic brine.

2United states bureau of mines (USBM) wettability Index is calculated from a series of experiments forforced water and oil displacement using a centrifuge.

3An emulsifier and oil-wetting agent.4An oil-wetting agent.

146

ation, kaolinite migration, and precipitation of insoluble salt. There are some published

studies about the effect of wettability alteration on well logs. Based on nuclear magnetic

resonance (NMR) logs, Chen et al. (2004) and Shafer et al. (2004) found that OBMF inva-

sion changed the wettability of rock surfaces. They noted that cores which were at residual

water saturation, Swirr, and were saturated with OBMF displayed much faster T1 and T2

relaxation times compared to those of rock saturated with bulk OBMF. It was also found

that temperature had a negligible effect on relaxation times. They determined that fast

relaxation times were due to changes of rock surface wettability in those cases where the

rock became more oil wet. Salazar and Torres-Verdın (2009) compared radial distribution of

fluid saturation associated with water-base mud (WMB) and OBM and showed that a water

bank could develop in the radial profile of water saturation as a consequence of wettability

alteration due to OBMF invasion.

In this chapter, I use the developed compositional fluid-flow simulator to study the

effect of wettability alteration on the spatial distribution of fluid saturation resulting from

OBMF invasion.

6.2 Physical Model

In the study of wettability alteration, I make the following assumptions: (1) the reservoir is

isothermal, (2) the reservoir is impermeable at a specified drainage radius, (3) there is no

chemical reaction or precipitation between fluid and rock, (4) the formation is slightly com-

pressible, (5) Darcy’s law for multiphase flow is valid, and (6) there is no mass transfer from

hydrocarbon components into the aqueous phase. Chapter 2 describes the mathematical

formulations for the equation-of-state fluid-flow simulator used to perform the calculations

reported in this chapter.

147

6.3 Mudcake Model

Mud-filtrate invasion is a dynamic process in which mudcake thickness, mudcake perme-

ability, and mudcake porosity vary with time of invasion. It is therefore necessary to couple

a reliable model of mudcake growth with a fluid-flow simulator which takes into account

static and dynamic petrophysical properties. Chapter 5 describes in detail the method of

modeling mud-filtrate invasion.

6.4 Wettability Alteration Model

The effect of wettability alteration on reservoir fluid flow is a research topic well studied

in connection with chemical flooding. Reservoir engineers calculate oil recovery for the

processes in which surfactants lead to wettability alteration (Delshad et al. (2006) and Fathi

Najafabadi et al. (2009)). In chemical flooding studies, rock surface wettability changes to a

water-wet condition. In contrast, during OBMF invasion rock surface wettability changes to

a more oil-wet state. Recently, Salazar and Martin (2010) studied the invasion of OBMF into

a tight-gas formation in Offshore Vietnam. They found that surfactants in OBMF gave rise

to wettability alteration that decreased near-wellbore residual water saturation. The effects

of wettability on relative permeability have been investigated by several authors. Owens

and Archer (1971) added surfactants to either oil or water in order to change wettability and

calculated the ensuing relative permeability of both phases. McCaffery and Bennion (1974)

calculated relative permeability data for different fluid phases with various contact angles

using a synthetic polytetra-fluoroethylene sample.

This chapter studies the process of OBMF invasion in a systematic manner by

including pertinent physical models to describe mudcake growth, miscibility of OBMF with

in-situ oil, and wettability alteration. In the study of wettability alteration, I assume that

surfactant concentration is greater than the critical micelle concentration. The developed

method allows simulation of multi-phase fluid flow with arbitrary relative permeability and

capillary pressure curves. Accordingly, the relative permeability in each grid block, krl,

148

is calculated via linear interpolation of relative permeability for two different conditions:

initial condition and completely altered wettability. In a similar manner, capillary pressure,

Pc2j , is obtained from linear interpolation between initial and final wetting state, i.e.,

krl = εkfinalrl + (1− ε)kinitialrl , (6.1)

and

Pc2j = εP finalc2j + (1− ε)P initialc2j , (6.2)

where ε is the scaling factor given by

ε =Cs

Cs + Ct, (6.3)

where Cs, and Ct are concentrations of the adsorbed and total surfactant, respectively.

Equation (6.3) decribes the wettability alteration of a rock surface after a fraction of sur-

factant is adsorbed by the rock surface.

6.5 Solution Approach

The simulation algorithm is based on solving pressure implicitly and calculating concen-

trations explicitly (IMPEC). Following the calculation of the overall composition in each

grid block, I use Gibbs’s stability criterion and determine the number of fluid phases. At

grids where two hydrocarbon phases co-exist, the composition of each phase is determined

with flash calculations. The solution method is based on finite-difference discretization in

cylindrical coordinates to solve the pressure and concentration equations (Equations (3.35)

and (3.66)). The discretization allows spatial variations of pressure, concentration, and

petrophysical properties in the radial, azimuthal, and vertical directions. A logarithmically

increasing distribution of radial grids is used to secure accurate and reliable simulations in

the near-wellbore region, where most of the fluid concentration variations take place.

149

Table 6.1: Summary of geometrical, fluid, petrophysical, and Brooks-Corey’s propertiesassumed in the simulations described in this chapter.

Variable Unit Value



Permeability md 50.00

Porosity fraction 0.25





Connate-water salinity ppm NaCL 161000

Temperature F 200

Total invasion time days 6

Archie’s tortuosity/cementation factor (a) dimensionless 1.00

Archie’s cementation exponent (m) dimensionless 2.00

Archie’s saturation exponent (n) dimensionless 2.00

6.6 Flow Rate of Mud-Filtrate Invasion

I investigate the effect of wettability alteration during OBMF invasion on the rate of filtrate

flowing into the invaded formation. A “base” case of study is introduced with a set of

numerical and physical properties for mudcake, mud-filtrate, formation, fluid phases, and

rock-fluid. The base case consists of a formation with a porosity equal to 0.25 [fraction] and

a permeability equal to 50 [md]. It is assumed that the formation is saturated with oil and

water is at residual saturation. Table 6.1 summarizes the remainder of rock petrophysical

properties assumed in simulations of the base case. In the following simulations, formation

oil is lumped into a set of pseudo-components and invading filtrate into another set of

pseudo-components. Table 6.2 lists PVT properties of the pseudo-components assumed in

the simulations.

The relation between rock and fluid is given by two types of relative permeability and

150

Table 6.2: Summary of PVT properties of in-situ hydrocarbon and mud-filtrate compo-nents assumed in equation-of-state calculations described in this chapter.

Property Unit C6C17 MC16


Critical pressure atm 27 12.834



Molecular weight g/mol 100.2 284.5

Viscosity cp 1.0 10.0

Table 6.3: Summary of Brooks-Corey’s properties assumed in the simulations described inthis chapter.

Variable Unit Water-Wet Oil-Wet



Endpoint of water relative permeability n/a 0.2 0.8

Endpoint of oil relative permeability n/a 1.0 0.6

Coefficient for capillary pressure, Pco psi 8 -15

Capillary pressure exponent, ep n/a 2.5 4.5

151

111

0.8

1

0.8

1

0.8

1krw, water‐wet

kro, water‐wet

0.6

k r

0.6

k r

0.6

k r

,

krw, oil‐wet

kro, oil‐wet

0.2

0.4

0.2

0.4

0.2

0.4

0 0.2 0.4 0.6 0.8 10

W S i

0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S iWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

Figure 6.1: Water-oil relative permeability curves assumed for water-wet and oil-wet con-ditions. Variables kro and krw are relative permeability of oil and water, respectively.

capillary pressure curves, i.e., water-wet and oil-wet. Table 6.3 summarizes the parameters

used to define relative permeability and capillary pressure curves. Figures 6.1 and 6.2,

respectively, show the base curves for relative permeability and capillary pressure in this

study. I note that Swirr,water−wet > Swirr,oil−wet, Sor,water−wet < Sor,oil−wet, whereas

capillary pressure for an oil-wet system is negative.

It is assumed that, at the onset of invasion, there is an overbalance pressure of 300

[psi] between formation and wellbore. Rock pressure increases as mud filtrate invades the

formation, whereby overbalance pressure decreases with time. As invasion proceeds, the

thickness of mudcake increases and mudcake permeability decreases until mudcake thick-

ness reaches a limiting value of 1 [cm]. Productivity index, given by equation (3.88), relates

mudcake properties to formation properties; it quantifies the role played by mudcake refer-

ence permeability, filtrate viscosity, and mudcake limiting thickness which control the flow

rate of invasion. Specifically, equation (3.88) shows that a reduction in productivity index

and/or overbalance pressure causes a reduction in the flow rate of mud-filtrate invasion

(Figures 6.3(b), and 6.4(b)).

152

202020

10

20

10

20

10

20Water‐WetOil‐Wet

‐10

0c [psi]

‐10

0c [psi]

‐10

0c [psi]

‐20

‐10P c

‐20

‐10P c

‐20

‐10P c

0 0.2 0.4 0.6 0.8 1

‐30

W S i

0 0.2 0.4 0.6 0.8 1

‐30

W S i0 0.2 0.4 0.6 0.8 1

‐30

W S iWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

Figure 6.2: Water-oil capillary pressure, Pcow, curves assumed for water-wet and an oil-wetrock surfaces.

1

Kmc0 = 0.3 md

K 0 03

0.7

ess

(cm

) Kmc0 = 0.03

Kmc0 = 0.003

ss [cm]

0.4

Thic

kne

Thickn

es

0.001 0.01 1 10 1000

0.1

Time (hours)

Time [hours]Time (hours)Time [hours]

(a)

100

101

m)

Kmc0 = 0.3 md

K = 0 03m]

10-1

10

(m3 /d

ay/m Kmc0 = 0.03

Kmc0 = 0.003

[m3 /day/

10-3

10-2

Flow

rate

low Rate

0.0001 0.01 1 100 10-4

Time (hours)

F

Time [hours]

F

Time (hours)Time [hours]

(b)

Figure 6.3: Calculated time variations of (a) mudcake thickness and (b) mud-filtrateflow rate after the onset of invasion for different values of mudcake reference permeability.Formation permeability is assumed equal to 300 [md]; remaining petrophysical propertiesare those of the base case.

153

1

Kmc0 = 0.3 md

K 0 = 0.03

0.7

ness

(cm

) Kmc0 0.03

Kmc0 = 0.003

ness [cm]

0 1

0.4

Thic

knTh

ickn

0.001 0.01 1 10 100

00.1

Time (hours)

Time [hours]

(a)

100

101

y/m

)

Kmc0 = 0.3 md

Kmc0 = 0.03m]

-2

10-1

e (m

3 /day mc0

Kmc0 = 0.003

[m3 /day/

10-2

Flow

rate

low Rate

0.0001 0.01 1 100

Time (hours)

Time [hours]

F

[ ]

(b)

Figure 6.4: Calculated time variations of (a) mudcake thickness and (b) mud-filtrateflow rate after the onset of invasion for different values of mudcake reference permeability.Formation permeability is assumed equal to 1 [md]; remaining petrophysical properties arethose of the base case.

6.6.1 Effect of Formation Permeability on the Flow Rate of Mud-

Filtrate Invasion

Permeability of mudcake is significantly smaller than that of conventional reservoirs. For

rocks with high permeability, the flow rate of invasion is linearly proportional to mudcake

reference permeability. However, in the case of tight formations, the flow rate of mud-filtrate

invasion depends on formation properties such as permeability, porosity, capillary pressure,

and relative permeability, as well as mudcake and filtrate properties. Figure 6.3 compares

mud-filtrate flow rates calculated for different mudcake reference permeabilities for the case

of formation permeability equal to 300 [md]. For a tight formation with permeability equal

to 1 [md], Figure 6.4 compares the flow rate of mud-filtrate invasion calculated for dif-

ferent mudcake reference permeabilities. For permeable formations, flow rates are parallel

while mudcake thickness increases to its limiting value (Figure 6.3(b)). In tight formations,

mudcake permeability is not the main factor controlling the flow rate of mud-filtrate inva-

sion (Figure 6.4(b)). Our calculations indicate that the flow rate of mud-filtrate invasion

decreases with time even after mudcake has reached its limiting thickness.

154

6.6.2 Effect of Mudcake Permeability on the Flow Rate of Mud-

Filtrate Invasion

In the invasion simulations (WBM or OBM), I observe that if the formation has high perme-

ability, mud-filtrate flow rate varies linearly with variations of mudcake reference permeabil-

ity (Figure 6.3(b)). However, for low-permeability formations (less than 1 md) petrophysical

properties such as permeability, porosity, and rock-fluid properties also influence the flow

rate of mud-filtrate invasion.

6.6.3 Effect of Wettability Alteration on the Rate of Mud-Filtrate

Invasion

During OBMF invasion, rock surface wettability changes; whereby, water and oil relative

permeabilities change. Numerical simulations indicate that the effect of wettability alter-

ation on the rate of mud-filtrate invasion is negligible. This chapter found that invasion

flow rates for cases in which rock surface wettability changes are the same as those with

no-wettability alteration (Figures 6.3(b) and 6.4(b)).

6.7 Wettability Alteration Effects on Saturation and

Resistivity

Invasion of OBMF gives rise to OBMF surfactant contact with the rock’s surface within the

invasion zone. Surfactants included in OBMF change the rock surface wettability from a

water-wet to a more oil-wet condition.

In the altered wettability condition, oil makes contact with the grain surface (prior

to that water was in contact with the grain surface). Consequently, some of the originally

immobile water becomes moveable, whereby residual water saturation decreases with respect

to that of the rock’s original state (Figures 6.1 and 6.2). During invasion, OBMF displaces

the excess moveable water into the formation; hence water saturation decreases near the

155

0.2

n2.4 hours1 dayct

ion]

0.15Satu

ratio

ny

3 days

aturation

ation [frac

Wat

er S

Water Sa

ter S

atura

0.2 0.3 0.4 0.50.1

Radial Distance (m)Radial Distance [ft]0.6 0.9 1.2 1.5

Wat

Radial Distance (m)Radial Distance [ft]

(a)

252.4 hours

.m

)m]

esis

tivity 1 day

3 days

istiv

ity (

stivity

[Ω.m

15

ectr

ical

Re

rical

Res

irical Resis

0.2 0.3 0.4 0.55

Ele

Elec

tr

0.6 0.9 1.2 1.5

Electr

Radial Distance (m)Radial Distance [ft]0.6 0.9 1.2 1.5

(b)

Figure 6.5: Radial distributions of (a) water saturation and (b) rock electrical resistivitycalculated at different times after the onset of invasion with OBMF containing surfactant.Initially, the formation is assumed to be water-wet with water saturation equal to residualsaturation (0.16 [fraction]). After wettability alteration, residual water saturation decreasesto 0.12 [fraction]. Mudcake reference permeability is assumed to be equal to 0.003 [md]and initial overbalance pressure is 300 [psi]. Table 6.1 lists the parameters used in Archie’sequation to calculate rock resistivities. Mud-filtrate viscosity is equal to 10 [cp]. Formationpetrophysical properties are those of the base case.

borehole wall with respect to its original value.

I study a synthetic case of OBMF invasion into a formation with connate water

saturation equal to residual water saturation. Mudcake is assumed to have a reference

permeability of 0.003 [md] and the overbalance pressure is 300 [psi]. Furthermore, I assume

the OBMF surfactants decrease the level of water saturation by 0.04 [fraction] (with respect

to the original water saturation). Figure 6.5(a) shows radial distributions of water saturation

simulated after 2.4 hours, 1 day, and 3 days from the onset of OBMF invasion. residual

water saturation (initially equal to 0.16 [fraction]) decreases in the invaded zone (to 0.12

[fraction]) with the excess water saturation (beyond new residual water saturation) becoming

moveable, thereby creating a water bank in the shallow radial zone. In the presence OBMF,

salt concentration does not radially change.

Section 5.3.6 shows that rock electrical resistivity depends on the distribution of wa-

ter saturation and salt concentration. I use equation (5.6) to calculate formation electrical

resistivity using radial distributions of water saturation and salt concentration. Figure 6.5(b)

156

shows radial distributions of calculated rock electrical resistivities at three different times

after the onset of OBMF invasion. The reduction in water saturation in wellbore vicinity

causes large electrical resistivity in the invaded zone. Figure 6.5(b) indicates that devel-

opment of a water bank in the radial distribution of water saturation in the invasion zone

gives rise to a conductive annulus in the radial profile for electrical resistivity.

6.7.1 Effect of Different OBM Emulsifiers on Wettability Alter-

ation

Menezes et al. (1989) and Yan et al. (1993) tested several types of mud filtrate with differ-

ent emulsifiers to find out their effect on contact angle and rock surface wettability. They

found that the corresponding variation of contact angle depended on the specific surfac-

tant included in OBM. Therefore, it is pertinent to conclude that the level of wettability

change depends on the specific composition of OBMF. Some emulsifiers, e.g. EZ, MUL, and

DV-33, change the rock’s surface wettability condition to a completely oil-wet condition,

whereas some others, e.g. DFL cause the rock surface to become less water-wet. In this

chapter, I consider four synthetic cases where initial (residual) water saturation decreases

to different extents, namely, 0.02, 0.04, 0.06, and 0.08 [fraction] (from the original value).

For each simulation, I assume a specific set of saturation-dependent relative-permeability

and capillary-pressure curves.

The simulations assume that only residual water saturation changes while the rest

of petrophysical properties, such as relative permeability and capillary pressure end points,

remain the same. Figure 6.6(a) shows radial distribution of water saturation simulated for

cases where residual water saturation decreases from 0.16 [fraction] to 0.14, 0.12, 0.10, and

0.08 [fraction]. As shown in Figure 6.6(b), the presence of an abnormal local water bank in

the simulated radial distributions of water saturation gives rise to an electrically conductive

annulus in the corresponding radial distributions of electrical resistivity.

157

0.25

0.3n

Swir=2 %

Swir=4 %nactio

n]

0 15

0.2

Satu

ratio Swir=6 %

Swir=8 %

Saturatio

ratio

n [fra

0.1

0.15

Wat

er S

Water

Water Satur

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.05

Radial Distance (m)

Radial Distance [ft]0.6 0.9 1.2 1.5 1.8 2.1 2.4

W

(a)

45

55 Swir=2 %

Swir=4 %( .m

)[Ω.m

]

25

35 Swir=6 %

Swir=8 %

esis

tivity

esistiv

ity [

15

25

ectr

ical

Re

ectrical Re

0.2 0.3 0.4 0.5 0.6 0.7 0.8

5

Radial Distance (m)

Ele


Ele

[ ]

(b)

Figure 6.6: Radial distributions of (a) water saturation and (b) rock electrical resistivitycalculated after 3 [days] of invasion with OBMF containing surfactant for different valuesof reference mudcake permeability. Initially, the formation is assumed to be water-wet withwater saturation equal to residual saturation (0.16 [fraction]). After wettability alteration,residual water saturation decrease to 0.14, 0.12, 0.10, and 0.08 [fraction]. Overbalancepressure is 300 [psi]. Table 6.1 lists the parameters used in Archie’s equation to calculateelectrical resistivity values. Formation petrophysical properties are those of the base case.

6.7.2 Effect of Mudcake Reference Permeability

Figures 6.3(b) and 6.4(b) show that the rate of mud-filtrate invasion depends on mudcake

permeability. I assume that initial formation pressure is 3500 [psi] and that water saturation

is at the residual condition. Furthermore, I assume that the decrease of residual water sat-

uration due to wettability alteration is the same for all three types of mudcake, and is equal

to 0.04 [fraction]. Figure 6.7(a) compares radial distributions of water saturation simulated

for three values of mudcake reference permeability. The higher the mudcake permeabil-

ity, the higher the flow rate of invasion and, consequently, the deeper the invasion. Using

Archie’s equation, I calculate the corresponding radial distributions of rock electrical resis-

tivity based on the simulated radial distributions of water saturation and salt concentration

(Figure 6.7(b)). Figures 6.5(b) and 6.7(b) provide examples for the occurrence of a electrical

conductive annulus in the invaded zone after invasion of OBMF containing surfactants.

Subsequent to the calculation of radial distribution of electrical resistivity, I use

158

0.25

Kmc0 = 0.3 md

Kmc0 = 0.03tion]

0atur

atio

n

0.2mc0

Kmc0 = 0.003

aturation

tion [fract

0

Wat

er S

a

0.15

Water Sa

ter S

atura

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.1

Radial Distance (m)


Wat

Radial Distance (m)Radial Distance [ft]

(a)

25

ity

2.4 hoursKmc0 = 0.3 md

Kmc0 = 0.03( .m

)Ω.m

]

15l Res

istiv

mc0Kmc0 = 0.003

esis

tivity

sistivity

[Ω

Elec

tric

alct

rical

Re

ctrical Res

0.2 0.3 0.4 0.5 0.6 0.7 0.8

5

Radial Distance (m)

Elec


Elec


(b)

Figure 6.7: Radial distributions of (a) water saturation and (b) rock electrical resistivitycalculated after 3 days from the onset of invasion with OBMF containing surfactant fordifferent values of reference mudcake permeability. Initially, the formation is assumed tobe water-wet with a water saturation equal to residual saturation (16%). After wettabilityalteration, residual water saturation decreases to 12%. Overbalance pressure is equal to 300[psi]. Table 6.1 lists the parameters used in Archie’s equation to calculate resistivity values.Petrophysical properties of the formation are those of the base case.

UTAPWeLS5 to simulate the corresponding array-induction apparent resistivity curves

(AIT6) for different radial lengths of investigation (AIT10, AIT20, AIT30, AIT60, and

AIT90). In doing so, I assume that the top and bottom of the simulated formations are

bounded by shale layers of electrical resistivity equal to 1 [Ω.m].

Figure 6.7 indicates that the radial length of invasion for OBMF with high mudcake

permeability (0.3 md) is relatively large. The effect of radially deep invasion is prominent

separation of simulated apparent resistivity logs. When mudcake permeability is low (0.003

[md]) the radial length of invasion is relatively small, whereby there is only an appreciable

separation between the shallowest-sensing apparent resistivity log (AIT10) and the remain-

der of the apparent resistivity logs (Figure 6.8(b)).

5The University of Texas at Austin’s Petrophysical and Well-Log Simulator6Mark of Schlumberger.

159

h

(m)

h (m

) RT10 RT20 RT30AIT10 AIT20 AIT30pth

Dep

th

Dep

th

RT60 RT90AIT60 AIT60

Dep

100 101

100 101

RT60 RT90AIT60 AIT60

100 101Induction Apparent Resistivity [Ω m]Induction Apparent Resistivity [Ω.m]

(a)

h (m

)

h (m

)pth

Dep

th

Dep

thDep

100 101

R i ti it ( )100 101

R i ti it ( )100 101Induction Apparent Resistivity [Ω m]Resistivity (.m)Resistivity (.m)Induction Apparent Resistivity [Ω.m]

(b)

Figure 6.8: Array-induction (AIT) apparent resistivity curves simulated for the case ofOBMF invasion with mudcake reference permeability values equal to (a) 0.3 and (b) 0.003md. Figures 6.7(a) and 6.7(b) show the corresponding radial distributions of water satura-tion and rock electrical resistivity.

6.8 Wettability Alteration in Oil- and Gas-Bearing For-

mations and Corresponding Effect on Water Satu-

ration and Electrical Resistivity

In this section, I compare the effect of OBMF invasion into oil- and gas-saturated rock for-

mations. I consider the three rock-types described in Table 6.4, and study invasion with one

set of values/properties for mudcake, mud filtrate, and overbalance pressure. Figures 6.9

and 6.10 show the relative permeability and capillary pressure curves, respectively, assumed

for the synthetic case studies. Simulations are performed under the assumption that the

reduction of residual water saturation due to wettability alteration is proportional to the

surface-to-volume ratio of the flow conduit. Therefore, the decrease of residual water satu-

ration due to wettability alteration is different for each rock type.

Table 6.4 summarizes the properties assumed for mudcake and formation rock. In

subsequent fluid-flow simulations, I assume that formation gas is composed of methane (C1),

in-situ oil is composed of pseudo components IC4 and FC7, and OBMF is composed of two

pseudo components, FC10 and FC18. The properties of the pseudo components are described

in Table 6.5. I use the developed simulator, UTFEC, to perform the simulations.

160

1111111

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1

Water‐WetOil‐Wet

0.6k r

0.6k r

0.6k r

0.6k r

0.6k r

0.6k r

0.6k r

krokrw

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S i0 0.2 0.4 0.6 0.8 10

W S iWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

Figure 6.9: Water-oil relative permeability curves assumed for Rock Type I (dashed lines),Rock Type II (solid lines), and Rock Type III (dotted lines) for two different wettabilityconditions. Blue and red curves identify water- and oil-wet conditions, respectively. residualwater-saturation for oil-wet conditions is smaller than that of water-wet conditions.

500500500500500500500

0

c [psi]

0

c [psi]

0

c [psi]

0

c [psi]

0

c [psi]

0

c [psi]

0

c [psi]

‐500P c ‐500P c ‐500P c ‐500P c ‐500P c ‐500P c ‐500P c

Water‐Wet

0 0.2 0.4 0.6 0.8 1

‐1000

W S i0 0.2 0.4 0.6 0.8 1

‐1000

W S i0 0.2 0.4 0.6 0.8 1

‐1000

W S i0 0.2 0.4 0.6 0.8 1

‐1000

W S i0 0.2 0.4 0.6 0.8 1

‐1000

W S i0 0.2 0.4 0.6 0.8 1

‐1000

W S i0 0.2 0.4 0.6 0.8 1

‐1000

W S i

Water WetOil‐Wet

Water SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

Figure 6.10: Water-oil capillary pressure curves assumed for Rock Type I (dashed lines),Rock Type II (solid lines), and Rock Type III (dotted lines) for two different wettabilityconditions. Blue and red curves identify water- and oil-wet conditions, respectively. In thecase of oil-wet conditions, oil is the wetting phase and capillary pressure becomes negative.

161

Table 6.4: Summary of petrophysical and fluid properties for different rock types assumedin the simulations of the process of mud-filtrate invasion described in this chapter.

Variable Unit Rock Type I Rock Type II Rock Type III

Permeability md 300 1.00 0.01

Porosity fraction 0.25 0.14 0.05

residual water saturation fraction 0.07 0.20 0.35

(water-wet condition)

residual water saturation fraction 0.03 0.10 0.15

(oil-wet condition)

residual oil saturation fraction 0.15 0.15 0.15

(water-wet condition)

residual oil saturation fraction 0.18 0.18 0.18

(oil-wet condition)

Salt concentration ppm NaCl 160,000 160,000 160,000

The assumed overbalance pressure for all cases is 300 [psi]. Table 6.6 describes

the corresponding properties assumed in the numerical simulations except for reference

permeability, which is equal to 0.3 [md].

In the following sections, invasion is studied in a vertical well where invasion time

is determined by drilling time; at the top layer invasion time is maximum whereas at bot-

tom layer, invasion time is minimum. Time of invasion and depth do not exhibit a linear

relationship.

6.8.1 Oil-Base Mud-Filtrate Invasion Into an Oil-Saturated Forma-

tion

The formation is assumed to be saturated with oil composed of pseudo components IC4

and FC7 with molar compositions equal to 0.4 and 0.6, respectively, whereas OBMF in-

cludes the pseudo components FC10 and FC18 with molar compositions equal to 0.2 and

0.8, respectively. Figures 6.11(a), 6.12(a), and 6.13(a) show spatial (radial and vertical

directions) distributions of water saturation along a vertical well after invasion of OBMF

162

Table 6.5: Summary of PVT properties for in-situ hydrocarbon and mud-filtrate compo-nents assumed in equation-of-state calculations described in this chapter.

Variable Unit C1 IC4 FC7 FC10 FC18

Critical temperature K 190.60 408.10 543.20 622.10 760.50

Critical pressure atm 45.40 36.00 30.97 25.01 15.65

Acentric factor - 0.008 0.176 0.308 0.444 0.757

Critical molar volume m3/kgmol 0.099 0.263 0.381 0.521 0.930

Molecular weight g/mol 16.04 58.12 96.00 134.00 251.00

containing surfactant. In Figures 6.11 through 6.13, invasion time corresponding to each

depth is the same for all rock types. I note that the simulated radial lengths of invasion in

tight formations are much shorter than those obtained for formations which exhibit higher

permeabilities. Initially, the formation is assumed to be water-wet and saturated with oil

(So = 1− Swirr). Fluid distributions are calculated along the depth direction. Invasion at

the top layer took place for 1 day, and at the bottom layer there was no invasion (tinv = 0).

Mudcake reference permeability is equal to 0.3 [md] and overbalance pressure is 300 [psi].

Moreover, the formation is bounded from top and bottom by impermeable layers.

Similar to previous cases, Figures 6.11(b), 6.12(b), and 6.13(b) indicate that the

presence of an anomalous water bank in the radial distribution of water saturation gives rise

to an electrically conductive annulus in the distribution of rock resistivity.

Having calculated the radial distribution of rock resistivity, the UTAPWeLS7 resis-

tivity simulation module is used to calculate the corresponding array-induction apparent-

resistivity curves. Figures 6.11(c), 6.12(c), and 6.13(c) show apparent-resistivity curves

calculated for different radial lengths of investigation (AIT10, AIT20, AIT30, AIT60, and

AIT90). Apparent resistivity logs were calculated with the assumption that the electri-

cal resistivity of upper and lower bounding shale layers is equal to 1 [Ω.m]. The separation

between apparent resistivity logs with different radial lengths of investigation is more promi-

nent in formations with higher permeability. Even though there is a conductive annulus in

7The University of Texas at Austin Petrophysical and Well-Log Simulator.

163

SwSw [fraction]

700

750

800

0.12

0.14Shale

800

850

900

pth [ft]

0.1n Time

950

1000

1050

De

0.06

0.08

Invasion

2 4 6

1100

1150

R di l Di [f ]

0.04Shale


(a)

RtRt [Ω.m]

700

750

800

300

350Shale

800

850

900

pth [ft]

200

250

950

1000

1050

De

100

150

2 4 6

1100

1150

R di l Di [f ]

50Shale


(b)

Obm into oil rt1Obm into oil rt1

700

750

800

700

750

800

AIT10 AIT20 AIT30

800

850

900

pth [ft]

800

850

900

pth [ft]

950

1000

1050

De 950

1000

1050

De

100 102

1100

1150

100 102

1100

1150

AIT60 AIT90

Resistivity [.m]Resistivity [.m]

(c)

Figure 6.11: Spatial (radial and vertical directions) distribution of (a) water saturation, (b)electrical resistivity, and (c) array-induction apparent resistivitys log calculated after inva-sion of OMBF containing surfactant into an oil-saturated formation. The formation exhibitspetrophysical properties of Rock Type I described in Table 6.4 and Figures 6.9 and 6.10.Archie’s properties for the calculation of electrical resistivity are those listed in Table 6.1.

SwSw [fraction]

700

750

800

0.26

0.28

0.3Shale

800

850

900

pth [ft]

0 2

0.22

0.24

0.26

n Time

950

1000

1050

De

0.16

0.18

0.2

Invasion

1 2 3 4 5

1100

1150

R di l Di [f ]

0.1

0.12

0.14

Shale


(a)

RtRt [Ω.m]

700

750

800

80

90

100Shale

800

850

900

pth [ft]

60

70

80

950

1000

1050

De

30

40

50

1 2 3 4 5

1100

1150

R di l Di [f ]

20

30

Shale


(b)

Obm into oil rt 2Obm into oil rt 2

700

750

800

700

750

800

AIT10 AIT20 AIT30

800

850

900

pth [ft]

800

850

900

pth [ft]

950

1000

1050

De 950

1000

1050

De

100 102

1100

1150

100 102

1100

1150

AIT60 AIT90


(c)

Figure 6.12: Spatial (radial and vertical directions) distribution of (a) water saturation, (b)electrical resistivity, and (c) array-induction apparent resistivity logs calculated after inva-sion of OMBF containing surfactant into an oil-saturated formation. The formation exhibitspetrophysical properties of Rock Type II described in Table 6.4 and Figures 6.9 and 6.10.Archie’s properties for the calculation of electrical resistivity are those listed in Table 6.1.

164

SwSw [fraction]

700

750

800

0 4

0.45

Shale

800

850

900

pth [ft] 0.35

0.4

n Time

950

1000

1050

De

0.25

0.3

Invasion

0.5 1 1.5

1100

1150

R di l Di [f ]

0.2

Shale


(a)

RtRt [Ω.m]

700

750

800

300

350Shale

800

850

900

pth [ft]

200

250

950

1000

1050

De

100

150

0.5 1 1.5

1100

1150

R di l Di [f ]

50

100

Shale


(b)

Obm into oil rt‐3Obm into oil rt 3

700

750

800

700

750

800

AIT10 AIT20 AIT30

800

850

900

pth [ft]

800

850

900

pth [ft]

950

1000

1050

De 950

1000

1050

De

100 102

1100

1150

100 102

1100

1150

AIT60 AIT90


(c)

Figure 6.13: Spatial (radial and vertical directions) distribution of (a) water saturation, (b)electrical resistivity, and (c) array-induction apparent resistivity logs calculated after inva-sion of OMBF containing surfactant into an oil-saturated formation. The formation exhibitspetrophysical properties of Rock Type III described in Table 6.4 and Figures 6.9 and 6.10.Archie’s properties for the calculation of electrical resistivity are those listed in Table 6.1.

the radial distribution of electrical resistivity, its effect is not visible on the simulated appar-

ent resistivity logs. AIT resistivity curves for shallow invasion (or early times of invasion)

show a reverse OBM effect where deep resistivity is smaller than shallow resistivity. The

OBM reversal effect is more prominent in formations with low permeability (between the

depths of 900-1000 [ft], as shown in Figure 6.13).

6.8.2 Oil-Base Mud-Filtrate Invasion into a Gas-Bearing Formation

I study synthetic cases for water-wet formations which are saturated with gas (C1). For

simulations of invasion, I assume that the invaded formation is at residual water saturation.

OBMF is composed of pseudo components FC10 and FC18 with molar compositions equal

to 0.2 and 0.8, respectively.

Figures 6.14(a), 6.15(a), and 6.16(a) show spatial (radial and vertical directions)

distributions of water saturation along a vertical well after invasion of OBMF containing

165

surfactant. Initially, the formation is assumed to be water-wet and saturated with gas

(Sg = 1 − Swirr). Fluid distributions are calculated along the depth direction. I assume

that invasion at the top layer took place for 2.4 hours while that at bottom layer was

negligible. Mudcake reference permeability is 0.3 [md] and overbalance pressure is 300 [psi].

Moreover, the formation is bounded at the top and bottom by shale layers.

The radial length of invasion into the permeable formation (k = 300 md) is much

longer than that in formations with low permeability, while the anomalous water bank due

to excess movable water is less prominent.

It can be observed that the radial length of invasion in low-permeability formations

(k < 0.1 md) is much shorter than for the case of high-permeability formations (k > 100

md).

Subsequent to simulating radial distributions of water saturation, I calculate the

corresponding radial distributions of rock electrical resistivity using Archie’s equation. As

shown in Figures 6.14(b), 6.15(b), and 6.16(b), analogous to invasion into oil-bearing for-

mations, the anomalous water bank in the radial distribution of water saturation gives rise

to an electrically conductive annulus in the radial distribution of electrical resistivity. Fig-

ures 6.14(c), 6.15(c), and 6.16(c) show the simulated AIT apparent resistivity logs associated

with the simulation of OBMF invasion into the three rock types (as described in Table 6.4

and Figures 6.9 and 6.10). I observe a prominent separation between calculated apparent

resistivity logs, shallow to deep, for all rock types. AIT resistivity curves for shallow invasion

(or early times of invasion) show a reverse OBM effect where deep resistivity is smaller than

shallow resistivity. Similar to the case of OBMF invasion into an oil-saturated formation,

the OBM reversal effect is more significant in formations with low permeability (between

the depths of 850-950 [ft] in Figure 6.16). Simulation results described in this section agree

with findings by Salazar and Martin (2010) from field measurements.

166

Table 6.6: Summary of mudcake and mud filtrate properties assumed in the simulationsof the process of mud-filtrate invasion.

Variable Unit Value

Mudcake reference porosity fraction 0.30

Mud solid fraction fraction 0.06

Mudcake maximum thickness cm 1.00

Mudcake compressibility exponent, ν fraction 0.40

Mudcake exponent multiplier, δ fraction 0.10

Oil-base mud-filtrate viscosity cp 4.00

Mudcake reference permeability md 0.03

Mud-filtrate salinity ppm NaCl 161000

SwSw [fraction]

700

750

0.08

0.09Shale

800

850

pth [ft]

0 06

0.07

n Time

900

950

1000

De

0.05

0.06

Invasion

2 4 6

1000

1050

R di l Di [f ]

0.03

0.04

Shale


(a)

RtRt [Ω.m]

700

750

300

350Shale

800

850

pth [ft]

200

250

900

950

1000

De

100

150

2 4 6

1000

1050

R di l Di [f ]

50

100

Shale


(b)

Obm into gas, rt1Obm into gas, rt1

700

750

700

750

AIT10 AIT20 AIT30

800

850

pth [ft]

800

850

pth [ft]

900

950

1000

De 900

950

1000

De

100 102

1000

1050

100 102

1000

1050

AIT60 AIT90


(c)

Figure 6.14: Spatial (radial and vertical directions) distribution of (a) water saturation, (b)electrical resistivity, and (c) array-induction apparent resistivity logs calculated after inva-sion of OMBF containing surfactant into a gas-saturated formation. The formation exhibitspetrophysical properties of Rock Type I described in Table 6.4 and Figures 6.9 and 6.10.Archie’s properties for the calculation of electrical resistivity are those listed in Table 6.1.

167

SwSw [fraction]

700

750

0.3

Shale

800

850

pth [ft]

0.25

n Time

900

950

1000

De

0.15

0.2

Invasion

0.5 1 1.5 2

1000

1050

R di l Di [f ]

0.1Shale


(a)

RtRt [Ω.m]

700

750

80

90

100Shale

800

850

pth [ft]

60

70

900

950

1000 De

30

40

50

0.5 1 1.5 2

1000

1050

R di l Di [f ]

10

20Shale


(b)


700

750

700

750

AIT10 AIT20 AIT30

800

850

pth [ft]

800

850

pth [ft]

900

950

1000

De 900

950

1000

De

100 102

1000

1050

100 102

1000

1050

AIT60 AIT90


(c)

Figure 6.15: Spatial (radial and vertical directions) distribution of (a) water saturation, (b)electrical resistivity, and (c) array-induction apparent resistivity logs calculated after inva-sion of OMBF containing surfactant into a gas-saturated formation. The formation exhibitspetrophysical properties of Rock Type II described in Table 6.4 and Figures 6.9 and 6.10.Archie’s properties for the calculation of electrical resistivity are those listed in Table 6.1.

SwSw [fraction]

700

750

0 4

0.45Shale

800

850

pth [ft] 0.35

0.4

n Time

900

950

1000

De

0.25

0.3

Invasion

0.5 1 1.5

1000

1050

R di l Di [f ]

0.2

Shale


(a)

RtRt [Ω.m]

700

750

300

350Shale

800

850

pth [ft]

200

250

900

950

1000

De

100

150

0.5 1 1.5

1000

1050

R di l Di [f ]

50

100

Shale


(b)


700

750

700

750

AIT10 AIT20 AIT30

800

850

pth [ft]

800

850

pth [ft]

900

950

1000

De 900

950

1000

De

100 102

1000

1050

100 102

1000

1050

AIT60 AIT90


(c)

Figure 6.16: Spatial (radial and vertical directions) distribution of (a) water saturation, (b)electrical resistivity, and (c) array-induction apparent resistivity logs calculated after inva-sion of OMBF containing surfactant into a gas-saturated formation. The formation exhibitspetrophysical properties of Rock Type III described in Table 6.4 and Figures 6.9 and 6.10.Archie’s properties for the calculation of electrical resistivity are those listed in Table 6.1.

168


In this chapter, I implemented a mud-filtrate invasion model for the invasion of oil-base mud

containing surfactants. Simulations indicate that, in highly permeable formations, invasion

of OBMF is primarily governed by mudcake and mud-filtrate properties such as mudcake

reference permeability, mudcake thickness, and mud-filtrate viscosity.

I studied rock wettability alterations due to emulsifiers and oil-wetting agents con-

tained in OBMF. Surfactants included in OBMF can change the rock’s surface wettability

from a water-wet to a neutral or oil-wet condition. This behavior causes a fraction of the

originally residual pore volume of connate water to become moveable, whereby the radial

distribution of water saturation exhibits variations different from those of the rock in its

original (uninvaded) state. The radial displacement of movable water by OBMF can give

rise to a radial zone (annulus) where water saturation is abnormally high (water bank),

which in turn causes the radial rock resistivity to be abnormally low (resistivity annulus).

Salt concentration during invasion of OBMF does not radially change; the anoma-

lous water bank gives rise to an electrically conductive annulus in the radial distribution

of electrical resistivity. For invasion with the same volume of filtrate into the invaded for-

mation, I showed that the radial distribution of water saturation changed with flow rate

of invasion. Relatively low flow rates of invasion give rise to prominent variations in the

radial distribution of water saturation because wettability alteration takes place through

the invasion zone and grain surfaces become oil-wet. For invasion with high flow rates, some

locations in the radial transition zone exhibit a mixed-wet condition. Consequently, the

amplitude of the anomalous water bank is smaller, but it is wider in the radial direction.

The degree of alteration of contact angle and wettability due to invasion with OBMF

depends on the strength of the emulsifier included in mud filtrate. In turn, these properties

impact the corresponding variation of residual water saturation. Simulations indicated that

the amplitude of the anomalous water bank decreased with a large variation of contact angle.

Apparent resistivity logs simulated for the case of OBMF invasion into an oil-saturated

permeable formation (permeability higher than 100 md) exhibited prominent, progressive

169

separation from shallow- to deep-sensing logs. In the case of low permeability formations,

shallow-sensing apparent resistivity exhibited higher values of apparent resistivity than the

remaining logs.

Simulations of OBMF invasion into an oil-saturated formation indicated that all

apparent resistivity logs exhibited measurable separation when mud filtrate invaded deeply

into the formation. Invasion of OBMF into gas-bearing formations was accompanied with

an anomalous water bank in the radial distribution of water saturation. In the latter case,

coexistence of three mobilities, i.e., gas, water, and oil, gave rise to two water banks. Sim-

ulated apparent resistivity logs across layers which exhibited shallow invasion showed a

reverse OBM effect where deep resistivity was larger than shallow resistivity. The reverse

OBM effect was more prominent in formations with low permeability. Simulated apparent

resistivity logs for this special case exhibited measurable and progressive separation from

shallow- to deep-sensing measurements.

170

Chapter 7

Simulation of Formation-Tester

Measurements Acquired in

Deviated Wells

Formation-tester measurements (FTM) acquired in thinly-bedded formations and in highly

deviated wells often show a large pressure drop during the drawdown period of a pressure-

transient test; such large pressure drop may indicate low permeability at the probe location.

Accurate analysis of FTMs requires simulating mud-filtrate invasion prior to the pressure-

transient test. In deviated wells, the interplay between gravity, capillary, and viscous forces

leads to a highly non-symmetric fluid distribution around the wellbore. It then becomes

crucial to perform fluid sampling at an optimum probe location around the perimeter of the

wellbore.

This chapter considers two topics: (a) verification of FTMs in benchmarck single-

phase flow, and (b) simulating invasion and subsequent FTMs in thinly-bedded formations.

The study about thinly-bedded formations quantifies the effect of bed boundaries, mud-

filtrate invasion, well deviation angle, and location of the probe on the borehole wall during

171

fluid sampling.

7.1 Introduction

Over the course of last four decades, formation testers (FT) have been used to measure

formation properties including pressure, permeability, and saturating fluid.

Different analytical and numerical methods have been used to analyze pressure tran-

sient tests acquired with a FT and to estimate formation properties. Most of the available

analytical methods are limited to pressure response due to a packer-type FT. Abbaszadeh

and Hegeman (1990) derived analytical expressions for the pressure variations during a

drawdown-buildup test in vertical, horizontal, and slanted wells. They derived analytical

solutions for different boundary conditions including no-flow and constant pressure at the

top and bottom of the formation. Abbaszadeh and Hegeman’s calculation method was based

on single-phase fluid flow in a reservoir with an infinite lateral boundary. Similar to Ab-

baszadeh and Hageman (1990), Cinco-Ley et al. (1975) introduced an analytical solution to

describe pressure-transient well tests assuming a line source. Analytical solutions proposed

by Kuchuk and Wilkinson (1991) and Ozkan and Raghavan (2000) were obtained in the

Laplace domain. Recently, Onur et al. (2004) suggested approximate analytical solutions

for pressure tests conducted with a dual packer-probe wireline formation tester (WFT) in

a deviated well. As with other analytical solutions, Onur et al.’s solution was valid only

for spherical single-phase fluid-flow regimes. Several researchers have attempted to apply

numerical methods to overcome the limitations of analytical expressions for pressure varia-

tions recorded at the borehole wall during a well test. Angeles et al. (2011) conducted one

of the first studies that used modeling of FTs in highly deviated wells to account for the

effect of mud-filtrate invasion. Angeles et al.’s model was constructed with non-orthogonal

corner-point grids in Cartesian coordinates. However, because their numerical algorithm did

not include non-diagonal terms in the permeability tensor, it was not recommended for its

applications in high-angle wells. On the other hand, accurate invasion simulation requires a

172

dynamic mudcake growth model coupled to a reservoir fluid-flow simulator. This necessity

becomes important in deviated wells where gravity segregation of fluids and anisotropy can

cause a significant eccentricity in the spatial distribution of mud filtrate in the vicinity of

the wellbore.

It is observed that WFT measurements obtained in thinly-bedded formations vary

when the tool is located at different locations with respect to bed boundaries. Several

researchers (Alpak et al., 2004; Suryanarayana et al., 2007; Wu et al., 2002) studied pressure-

transient well-test measurements when the probe was placed in the center of a permeable

bed. Previous researchers (Alpak et al., 2008; Proett et al., 2001b; Xu et al., 1992) noted

that when a probe straddles between a boundary separating low- and high-permeability

layers, it became significantly more difficult for the probe to efficiently secure a clean in-situ

sample. Moreover, the existence of a two-phase region in the vicinity of wellbore makes the

permeability measurement more complicated (Angeles, 2008; Hadibeik et al., 2009; Malik

et al., 2009b; Moinfar et al., 2010).

This chapter is devoted to simulations in highly-deviated wells. I study the effects of

bed boundaries and wellbore deviation on FTMs. The three-dimensional (3D) multi-phase

fluid-flow simulator (UTFEC) developed in Chapter 4 is applied to simulate the process

of mud-filtrate invasion and the acquisition of FTMs. First, a series of pressure tests are

performed in a water-saturated formation. When single-phase flow takes place, synthetic

pressure responses for different well deviation angles are calculated with UTFEC and are

compared to those obtained with an analytical expression. Next I study probe-type FTMs

acquired in thinly-bedded formations by simulating mud-filtrate invasion, pressure variations

during drawdown-buildup tests, and fluid sampling.

7.2 Mathematical Model

The numerical simulations in this chapter are performed using the cylindrical near-wellbore

fluid-flow method developed in the dissertation. Chapter 2 describes the assumptions and

173

mathematical formulations used in the development of UTFEC.

Pressure-transient tests conducted with a probe-type FT require a few minutes (0.5−

2 minutes) of fluid withdrawal and a few minutes (5 − 10 minutes) of pressure buildup.

Provided that fluid flow takes place in a spherical single-phase regime, analytical expressions

can be used to calculate pressure variations during fluid pumpout. I apply the analytical

solution suggested by Onur et al. (2004) to calculate pressure variations at the packer and

at an observation probe during drawdown-buildup tests with a dual packer-probe WFT.

Figure 7.1 describes the dual packer-probe WFT assumed in Onur et al.’s study. Assuming

spherical flow, the pressure variation at the packer center is given by (Onur et al., 2004)

∆pp(t) =141.2qµ

khlw

( lw√kh/kv2rsw

+ s)− 2453qµ

√φctµ

k3/2s

1√t, (7.1)

where kh and kv are horizontal and vertical permeabilities [md], respectively, q is sampling

flow rate [bbl/day], µ is fluid viscosity [cp], φ is formation porosity, ct is total compressibility

[1/psi], t is time [hours], s is skin factor, lw is packer half length [ft], and ks is spherical

permeability [md], defined as

ks = 3

√k2hkv. (7.2)

In equation (7.1), rsw is defined as equivalent spherical wellbore radius [ft], given by

rsw =2l′

w

ln(

√4l′2w + r′2w + 2l

′

w√4l′2w + r′2w − 2l′w

) +r′

w

l′w−√

4l′2w + r′2wl′w

, (7.3)

where l′

w is equivalent packer half length in an anisotropic formation, given by

l′

w = lw

√(kh/kv) cos2 θw + sin2 θw, (7.4)

174

kv^n

kh 2lww Probe

zo

Packer

h

zw

Figure 7.1: Description of a dual packer-probe WFT deployed in a deviated well. Theobservation probe does not withdraw fluid; it only measures fluid pressure variations. Inthis diagram, n is the unit normal vector to bedding plane, θw is wellbore deviation fromthe bedding normal vector, h is formation thickness (100 [ft]), 2lw is the length of the dualpacker (6 [in]), zo is distance of the observation probe from the packer center (6 [in]), andzw is the distance of the packer center from the lower bed boundary (200 [ft]).

and r′

w is equivalent wellbore radius in an anisotropic formation, given by

r′

w =rw2

(1 +

1√(kv/kh) sin2 θw + cos2 θw

), (7.5)

where rw is wellbore radius [ft], and θw is wellbore deviation angle. The pressure variation

at an observation probe located at a distance zo from the packer center (Figure 7.1) is given

by (Onur et al., 2004)

∆po(t) =141.2qµ

4√khkvl

′

w

ln(zo + lwzo − lw

)− 2453qµ√φctµ

k3/2s

1√t, (7.6)

where ∆po is the difference between pressure at the observation probe and the initial pressure

and q is sampling flow rate [bbl/day].

175

7.3 Simulations of Dual-Packer Formation-Tester Mea-

surements

Figure 7.1 describes the configuration of a WFT as deployed in a deviated well. Pressure

variations are recorded at the packer center as well as at an observation probe located

at zo= 6 [in] from the packer center. In Figure 7.1, n is the unit normal vector to the

formation bedding plane. I assume that the packer is located far enough away from both

vertical (zw=200 [ft]) and horizontal (Rdrainage = 1000 [ft]) boundaries. The packer size,

2lw, is assumed equal to 6 [in], and the pressure-transient well test consists of 1 minute

fluid withdrawal with a constant flow rate of 10 [bbl/day] followed by 9 minutes of pressure

buildup. I apply UTFEC to simulate pressure variations during drawdown-buildup tests in

the described model assuming that the formation is saturated with water, whereby pressure

variations can also be calculated using analytical formulas.

Figure 7.2 shows pressure variations simulated at the packer center and at the ob-

servation probe after 1 minute drawdown and 1 minute pressure buildup (total build up

time is 9 minutes). In Figures 7.2(a) and 7.2(b), the pressure response is recorded for a test

in a formation which exhibits a horizontal permeability of 5 [md], permeability anisotropy

of 10 (Raniso=10), and porosity of 0.12 [fraction]; these figures illustrate the effect of well

deviation angle on formation-tester measurements. Simulations show that when the packer

is located far away from the boundaries (i.e., pressure effects do not reach the boundaries)

the pressure drop measured at the packer is highest when the well is vertical. This trend,

however, is reversed in the bounded formations described in Figure 7.10. Figures 7.2(c)

and 7.2(d) show the effect of formation anisotropy on pressure variations acquired dur-

ing a drawdown-buildup test. I consider two formations with anisotropy ratios of 5 and

10, horizontal permeability of 500 [md] and porosity of 0.32 [fraction]. It is observed that a

larger pressure drop during drawdown corresponds to the test performed in a formation with

larger anisotropy. The comparisons described in Figure 7.2 indicate a very good agreement

between results obtained with UTFEC and analytical expressions.

176

10 cc/sec, k=5 md, kv= 0.5 md, por=0.1210 cc/sec, k 5 md, kv 0.5 md, por 0.12

35003500

3400

3500

3400

3500Raniso = 10

θw = 0°

3300

sure [p

si]

3300

sure [p

si]

Packer Pressure [UTFEC]Observation Probe [UTFEC]

3200

Press

3200

Press Exact SolutionAnalytical Solution

0.5 1 1.5 23100

Ti [ i ]

0.5 1 1.5 2

3100

Ti [ i ]

Time [min]Time [min]

(a)

10 cc/sec, k=5 md , kv= 0.5 md, por=0.12, DA=60

35003500

/ , , , p ,

3450

3500

3450

3500Raniso = 10

θw = 60°

3350

3400

sure [p

si]

3350

3400

sure [p

si]

3250

3300 Press

3250

3300 Press

Packer Pressure [UTFEC]Observation Probe [UTFEC]Exact SolutionAnalytical Solution

0.5 1 1.5 23200

3250

Ti [ i ]

0.5 1 1.5 2

3200

3250

Ti [ i ]

Exact SolutionAnalytical Solution


(b)10 cc/sec, k=500 md , kv= 50 md, por=0.32, DA=90/ , , , p ,

35003500

3499

3500

3499

3500Raniso = 10

θw = 90°

3498

sure [p

si]

3498

sure [p

si]

3497

Press

3497

Press Packer Pressure [UTFEC]

Observation Probe [UTFEC]Exact SolutionAnalytical Solution

0.5 1 1.5 23496

Ti [ i ]

0.5 1 1.5 2

3496

Ti [ i ]Time [min]Time [min]

(c)

10 cc/sec, k=500 md , kv= 100 md, por=0.32, DA=90/ , , , p ,

35003500

3499

3500

3499

3500Raniso = 5

θw = 90°

3498

sure [p

si]

3498

sure [p

si]

Packer Pressure [UTFEC]

3497

Press

3497

Press

[ ]Observation Probe [UTFEC]Exact SolutionAnalytical Solution

0.5 1 1.5 23496

Ti [ i ]

0.5 1 1.5 2

3496

Ti [ i ]Time [min]Time [min]

(d)

Figure 7.2: Comparison of pressure time variation at the packer center and an observationprobe calculated using UTFEC and those calculated with an analytical expression (Onuret al., 2004). Figure 7.1 describes the configuration of a dual packer used to conduct thepressure test. The formation is assumed to be saturated with water. It is assumed thatpacker pumps out fluid with a constant flow rate of 10 [cc/sec] with the pressure test con-sisting of 1 [min] fluid withdrawal followed by 9 [min] of pressure buildup. The observationprobe is located at 6 [in] above the packer center. Formation properties are as follows: (a)Rock Type II: kh = 5 [md], kv = 0.5 [md], and porosity = 0.12 [fraction], (b) Rock Type II:kh = 5 [md], kv = 0.5 [md], and porosity = 0.12 [fraction], (c) Rock Type I: kh = 500 [md],kv = 50, and porosity = 0.32 [fraction], and (d) Rock Type I: kh = 500 [md], kv = 100 [md],and porosity = 0.32 [fraction].

177

7.4 Petrophysical Properties

An empirical relationship is assumed to relate formation permeability with porosity. For

shaly sandstone, Dussan V. et al. (1994) proposed the following equation:

φ = 0.1 log10(kh) + 0.05, (7.7)

where kh is horizontal permeability, and φ is formation porosity. I use equation (7.7) to

define two rock types, Rock Types I and II, which exhibit consistency between formation

permeability and porosity (Table 7.1). Brooks-Corey’s equation (Brooks and Corey, 1964)

is employed to obtain saturation-dependent relations for rock-fluid properties in drainage

and imbibition processes (Figure 7.3). I calculate relative permeability of water with the

equation

krw = k0rw(Sewwn), (7.8)

and relative permeability of oil with

kro = k0ro(1− Senwwn ). (7.9)

For a drainage process, capillary pressure is given by

Pc = Pce

√φ

k(S− 1ep

wn ), (7.10)

whereas imbibition capillary pressure is given by

Pc = Pce

√φ

k(1− S

− 1ep

wn ). (7.11)

In equations (7.8) through (7.11), krw, kro, k0rw, and k0

ro are relative permeability of oil

and water, and endpoint values of water and oil relative permeabilities, respectively; Pce

is capillary entry pressure. Constants ew, enw, and ep are exponents of water relative

178

Table 7.1: Summary of petrophysical properties assumed for different rock types in thenumerical simulations described in this chapter.

Variable Unit Rock Type I Rock Type II

Permeability md 500 5

Porosity fraction 0.32 0.12



permeability, oil relative permeability, and capillary pressure, respectively. The normalized

water saturation, Swn, is given by

Swn =Sw − Swirr

1− Swirr − Sor, (7.12)

where Swirr and Sor are residual saturations for water and oil, respectively.

7.5 Mud-Filtrate Invasion in Deviated Wells

FTMs are often obtained hours or days after the onset of drilling; therefore, pressure and

fluid saturation around the wellbore are different from those far away from the wellbore.

Mud-filtrate invasion causes time and spatial variations in near-wellbore pressure, satura-

tion, and fluid properties. Therefore, proper estimation of formation properties based on

FTMs requires taking into account disturbances in in-situ fluid saturation due to mud-

filtrate invasion. Mud-filtrate invasion is a dynamic process in which mudcake thickness,

mudcake permeability, and mudcake porosity vary with time of invasion. The algorithm for

the simulation of mud-filtrate invasion is described in Chapter 5.

Invasion simulations are performed for two rock types defined in Table 7.1. Table 7.2

summarizes the geometrical, fluid, petrophysical, and Brooks-Corey’s properties assumed

in the simulations described in this study. In-situ oil is composed of pseudo components

FC10 and FC18 with compositions of 70 and 30, respectively. Pressure-volume-temperature

179

111111

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1

0.8

1

kkro

0.6

k r

0.6

k r

0.6

k r

0.6

k r

0.6

k r

0.6

k r

krw

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0.2

0.4

0 0.2 0.4 0.6 0.8 10

W t S t ti0 0.2 0.4 0.6 0.8 1

0

W t S t ti0 0.2 0.4 0.6 0.8 1

0

W t S t ti0 0.2 0.4 0.6 0.8 1

0

W t S t ti0 0.2 0.4 0.6 0.8 1

0

W t S t ti0 0.2 0.4 0.6 0.8 1

0

W t S t tiWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater SaturationWater Saturation [fraction]

(a)

150150150

R k T II

100

c [psi

]

100

c [psi

]

100

c [psi

]

Rock Type II

50

P c

50

P c

50

P c

0 0.2 0.4 0.6 0.8 1W t S t ti

0 0.2 0.4 0.6 0.8 1W t S t ti

0 0.2 0.4 0.6 0.8 1W t S t ti

Rock Type I

[ ]Water SaturationWater SaturationWater SaturationWater Saturation [fraction]

(b)

Figure 7.3: Water-oil (a) relative permeability and (b) capillary pressure curves assumedfor (i) Rock Type I (solid lines) and (ii) Rock Type II (dot-dashed lines). The symbolskrw and kro designate relative permeabilities of water and oil, respectively, fluid phases.Petrophysical properties of the rock types are given in Table 7.1.

180

Table 7.2: Summary of geometrical, fluid, petrophysical, and Brooks-Corey’s propertiesassumed in the simulations described in this chapter.

Variable Unit Value






Initial formation pressure at psi 3500

reference depth

Injection time day 5




Formation water salinity kppm NaCL 161

Temperature F 200

Coefficient for capillary pressure, Pco psi 8

Capillary pressure exponent, ep n/a 2.5

Total invasion time days 3

Average invasion flow rate bbl/day 0.158

Overbalance pressure psi 200

(PVT) properties of the aforementioned pseudo components are given in Table 7.3.

Figures 7.4 and 7.5 show spatial distributions of water saturation simulated after

5 days from the onset of water-base mud-filtrate (WBMF) invasion into formations with

petrophysical properties of Rock Types I and II, respectively. In all invasion simulations in-

vasion takes place under an overbalanced pressure of 200 [psi]. Figures 7.4(a) through 7.4(c)

show that in highly deviated wells, gravity segregation causes a significant asymmetry in the

spatial distribution of mud filtrate in the vicinity of the wellbore. However, in a formation

with low permeability (k=5 [md]), gravity is not a prominent force. Figure 7.5 indicates

that fluid distribution after invasion does not remarkably vary around the well when the

formation exhibits a low permeability.

181

3102

Aspect ratio x/y= 1

Sw [fraction]

3104

3106VD

[ft]

S0.6

0.8

3108

TV Sw

0.2

0.4

θw =45°

-45 -40 -35 -303110

Y-dir [ft]

Y [ft]

(a)

3102

Aspect ratio x/y= 1

Sw [fraction]

3104

3106

VD [f

t]

S0.6

0.8

3108

TV

Sw

0.2

0.4

θw =60°

-70 -65 -60 -553110

Y-dir [ft]

Y [ft]

(b)

Sw [fraction]310231043106

VD [f

t]

S0 40.60.8Aspect ratio x/y=2

Sw [fraction]

-220 -210 -200 -190 -180 -170

31083110

Y-dir [ft]

TV

Sw0.20.4

θw =80°

Y [ft][ ]Y [ft]

(c)

Figure 7.4: Spatial distribution of water saturation after 5 days from the onset of WBMFinvasion under an overbalance pressure of 200 [psi]. Wellbore deviations are (a) 45, (b)60, and (c) 80 degrees. It is assumed that mudcake reference permeability is 0.003 [md].Petrophysical properties of the formation are those of Rock Type I (described in Table 7.1and Figure 7.3).

182

3102

0 9Aspect ratio x/y= 1

Sw [fraction]

3104

3106VD

[ft]

S0 6

0.7

0.8

0.9

3108

TV Sw

0.4

0.5

0.6

θw =45°

-45 -40 -35 -303110

Y-dir [ft]

Y [ft]

(a)

3102

0 9Aspect ratio x/y= 1

Sw [fraction]

3104

3106

VD [f

t]

S0 6

0.7

0.8

0.9

3108

TV

Sw

0.4

0.5

0.6

θw =60°

-70 -65 -60 -553110

Y-dir [ft]

Y [ft]

(b)

Sw [fraction]310231043106

VD [f

t]

S0 6

0.8Aspect ratio x/y=2

Sw [fraction]

-220 -210 -200 -190 -180 -170

31083110

Y-dir [ft]

TV

Sw

0.4

0.6θw =80°

Y [ft][ ]Y [ft]

(c)

Figure 7.5: Spatial distribution of water saturation after 5 days from the onset of WBMFinvasion under an overbalance pressure of 200 [psi]. Wellbore deviations are (a) 45, (b)60, and (c) 80 degrees. It is assumed that mudcake reference permeability is 0.003 [md].Petrophysical properties of the formation are those of Rock Type II (described in Table 7.1and Figure 7.3).

183

Table 7.3: Summary of PVT properties and in-situ hydrocarbon components assumed inthe equation-of-state calculations described in this chapter (Source: CMG-WinProp).

Property Unit FC10 FC18


Critical pressure atm 25.01 15.65



Molecular weight g/mol 134 251

7.6 Simulation of Probe-Type FTMs Acquired in Thinly-

Bedded Formations

This section discusses pressure variation during drawdown-buildup tests and cleanup time

measured during fluid sampling in high-angle deviated wells. In subsequent sections, pressure-

transient tests are carried out using a simple-probe FT. Figure 7.6 describes the geometrical

and numerical modeling of a deviated well in cylindrical coordinates used in the following

simulations; r, θj , and z designate the radial location, azimuthal angle, and vertical location,

respectively; n is the unit normal vector to the bedding plane, h is bed thickness, zp is the

probe vertical distance from the lower horizontal boundary, and θw is the wellbore deviation

angle measured with respect to the bedding normal vector, n. Formation thickness is equal

to 10 [ft].

7.6.1 Drawdown-Buildup Test in Deviated Wells

The procedure for a pressure-transient test consists of 1 minute fluid withdrawal and 9

minutes of pressure buildup. For all cases, fluid pumpout takes place with a constant flow

rate of 1 [cc/sec]. To quantify the effect of a bed boundary, I simulate pressure-transient

tests separately at locations (1) and (2) (illustrated in Figure 7.6). Testing points (1) and

(2) are located at vertical distances of 0.5 and 5 [ft], respectively, from the lower shale

boundary. In all of the pressure-transient well tests (including those simulated under the

184

Shale

^nX

Y

hSand 2θj =90°

θj =180°

Probe

θw

h

kh

kv

Dep

th

Shale

1

Probeθj

zp

Y

X

Y

Figure 7.6: Geometrical description of a deviated well model in cylindrical coordinates.The variables r, θj , and z designate the radial location, azimuthal angle, and vertical loca-tion, respectively; n is the unit normal vector to the bedding plane, h is the bed thickness,zp is the probe vertical distance from the lower horizontal boundary, and θw is wellboredeviation from the bedding normal vector, n. Formation thickness is assume to be 10 [ft];probes 1 and 2 are located at vertical distances of 0.5 and 5 [ft], respectively, measured fromthe lower shale boundary.

assumption of no invasion and those simulated under the assumption of invasion before

the drawdown period) a larger pressure drop occurs when the probe is located at the bed

boundary (Figures 7.7 through 7.9). Figure 7.7 shows that pressure drops simulated at

the probe (test locations 1 and 2 in Figure 7.6) are higher for the cases in which WBM

invasion was considered before fluid pump out. Moreover, in formations invaded prior

to the pressure-transient test, pressure did not increase to its initial value at the end of

the pressure-buildup period (Figure 7.8), i.e., there exists a pressure difference between

initial pressure and near wellbore pressure at the end of buildup. I found that the pressure

difference (initial and near-wellbore pressure at the end of the buildup period) is equal

to the capillary pressure of the formation at the wellbore vicinity. When a zero or small

capillary pressure was assumed for the formation, the simulated pressure returned to the

original formation pressure. Figure 7.9 shows that permeability anisotropy increases the

pressure drop simulated during drawdown. Figure 7.10 compares pressure responses during

drawdown-buildup test at different well deviation angles; viz., 45, 60, and 80 [degrees]. The

test is performed assuming a formation which exhibits a permeability anisotropy ratio equal

185

Uninvaded Formation‐ Pressure test h θwhen θw = 80°

350035003500

3450

3500

3450

3500

3450

3500Raniso = 1

θw = 80°

3350

3400

sure

[psi

]

Probe at Point 1Probe at Point 2

3350

3400

sure

[psi

]

3350

3400

sure

[psi

]

3250

3300

Pre

ss Probe at Point 2

3250

3300

Pre

ss

3250

3300

Pre

ss

k = 5 [md]

0.5 1 1.5 23200

3250

Ti [ i ]

0.5 1 1.5 2

3200

3250

Ti [ i ]

0.5 1 1.5 2

3200

3250

Ti [ i ]

kh = 5 [md]φ = 12 [pu]

Time [min]Time [min]Time [min]

(a)

Invaded Formation‐ Pressure test h θwhen θw = 80°

3500350035003500Raniso = 1

θw = 80°

3000

sure

[psi

]

Probe at Point 1Probe at Point 2

3000

sure

[psi

] P

ress Probe at Point 2

Pre

ss

k = 5 [md]

0.5 1 1.5 22500

Ti [ i ]

0.5 1 1.5 2

2500

Ti [ i ]

kh = 5 [md]φ = 12 [pu]


(b)

Figure 7.7: Time variations of pressure simulated during drawdown-buildup tests withprobe-type FTs conducted at points 1 and 2 (described in Figure 7.6) within a thinly-bedded formation. In this graph, dashed lines and solid lines indentify pressure variationsat locations 1 and 2, respectively, in Figure 7.6. Prior to the pressure test the formation hasundergone (a) no invasion, and (b) WBM invasion. Petrophysical properties of the formationare those of Rock Type II (described in Table 7.1 and Figure 7.3). It is assumed that theformation exhibits an isotropic permeability, i.e., Raniso= 1. The wellbore inclination anglefrom the normal to bedding plane is assumed equal to 80 [degrees].

to 10. Petrophysical properties of the formation corresponding to simulations results in

Figures 7.10(a) and 7.10(b) are those of Rock Types I and II, respectively. Prior to the

drawdown test, the formation is WBM invaded as described in Section 7.5.

In Figure 7.10(a), the probe is located at the center of the formation (point 2),

whereas in the case of Figure 7.10(b), it is placed at the vertical vicinity of a horizontal

boundary (point 1) within a thinly-bedded formation. For both case studies corresponding

to Figures 7.10(a) and 7.10(b), comparison of the simulated pressure responses indicates

that the pressure drop increases as the wellbore deviation angle increases.

7.6.2 Cleanup Time and Fluid Sampling

In vertical wells, fluid cleanup time does not normally vary when performed at different

locations along the perimeter of the well. In deviated wells, however, fluid cleanup is achieved

at different times depending on probe location along the wellbore perimeter. This study

compares cleanup process performed at three locations along the wellbore perimeter: (i)

186

Comparison of pressure test for the ff f θeffect of invasion; θw = 80°

3500350035003500

Without invasion

Raniso = 1

3000

sure

[psi

]

3000

sure

[psi

] Raniso 1

θw = 80°

Pre

ss P

ress

With invasion

k = 5 [md]

0.5 1 1.5 22500

Ti [ i ]0.5 1 1.5 2

2500

Ti [ i ]

kh = 5 [md]φ = 12 [pu]


(a)


36003600

3400

3600

3400

3600

Without invasion Raniso = 1

3200

sure

[psi

]

3200

sure

[psi

] Raniso 1

θw = 30°

2800

3000

Pre

ss

2800

3000

Pre

ss

With invasion k = 5 [md]

0.5 1 1.5 22600

Ti [ i ]0.5 1 1.5 2

2600

Ti [ i ]

kh = 5 [md]φ = 12 [pu]


(b)

Figure 7.8: Comparison of pressure time variations recorded at probes 1 (dashed lines)and 2 (solid lines) in formations with two different initial conditions: (i) not invaded, and(ii) WBM invaded. Synthetic pressure responses are calculated in formations penetratedwith wells with deviation angles (a) 80 and (b) 30 degrees. Petrophysical properties of theformation are those of Rock Type II (described in Table 7.1 and Figure 7.3). It is assumedthat the formation exhibits an isotropic permeability, i.e., Raniso= 1. Figure 7.6 describesthe geometrical configuration of the synthetic model.


36003600

3200

3400

3600

3200

3400

3600

Without invasionRaniso = 10

2800

3000

3200

sure

[psi

]

2800

3000

3200

sure

[psi

] Raniso 10

θw = 30°

2400

2600

2800

Pre

ss

2400

2600

2800

Pre

ss

With invasionk = 5 [md]

0.5 1 1.5 22200

2400

Ti [ i ]0.5 1 1.5 2

2200

2400

Ti [ i ]

kh = 5 [md]φ = 12 [pu]


Figure 7.9: Comparison of pressure time variations recorded at probes 1 (dashed lines)and 2 (solid lines) in formations with two different initial conditions: (i) not invaded, and(ii) WBM invaded. Synthetic pressure responses are calculated in a formation with wellboredeviations of 30 degrees. It is assumed that the formation exhibits an anisotropic permeabil-ity of 10 (Raniso= 10). Figure 7.8(b) shows pressure time variations when Raniso= 1. Thepetrophysical properties of the formation are those of Rock Type II (described in Table 7.1and Figure 7.3). Figure 7.6 describes the geometrical configuration of the synthetic model.

187

Invaded Formation; Effect of deviation langle;

34903490

3485

3490

3485

3490Raniso = 10

3480

sure

[psi

]

DA = 80o

DA = 60o3480

sure

[psi

]

3475 Pre

ss DA = 45o

3475 Pre

ss

k = 500 [md]

0.5 1 1.5 23470

Ti [ i ]

0.5 1 1.5 2

3470

Ti [ i ]

kh = 500 [md]φ = 32 [pu]


(a)

Invaded Formation; Effect of deviation langle;

3500350035003500Raniso = 10

3000

sure

[psi

]

DA = 80o

DA = 60o

3000

sure

[psi

]2500

Pre

ss DA = 45o

2500 P

ress

k = 5 [md]

0.5 1 1.5 22000

Ti [ i ]

0.5 1 1.5 2

2000

Ti [ i ]

kh = 5 [md]φ = 12 [pu]


(b)

Figure 7.10: Comparison of pressure time variations recorded with a probe-type FT de-ployed in deviated wells. Synthetic pressure tests are conducted in three wellbore deviationangles of 45, 60, and 80 [degrees]. The pressure tests are conducted in deviated wells pene-trated into formations with petrophysical properties of (a) Rock Type I, and (b) Rock TypeII. In (a) probe is located at point 2, and in (b) probe is located in point 1; Figure 7.6describes the geometrical properties associated with this simulation. It is assumed thatthe formation has been previously invaded with WBM before the onset of pressure test.Figures 7.4 and 7.5 show the distribution of water saturation after WBMF invasion. It isassumed that formation exhibits an anisotropic permeability of Raniso= 10. Table 7.1 andFigure 7.3 describe petrophysical properties of each rock type.

188

θj = 0o; the probe faces down, and (ii) θj = 90o; the probe faces to the side, and (iii)

θj = 180o; the probe faces up.

The sampling operation is conducted with a simple-probe FT which has a radius

of 0.3 [in] (described in Figure 4.29). The pump withdraws fluid with a constant flow rate

of 80 [cc/sec]. Formation exhibits petrophysical properties of Rock Type I (described in

Table 7.1 and Figure 7.3). Fluid sampling takes place from a formation which has been

previously invaded with WBM for 5 days. The corresponding spatial distribution of water

saturation before the onset of fluid withdrawal is shown in Figure 7.4.

I conduct the pressure test in wells with deviation angles of 80 and 45 degrees

(Figures 7.11 and 7.12). Figure 7.11 shows the spatial distributions of water saturation

after 2.4 [hours] of fluid withdrawal when performed at locations on the wellbore perimeter.

As Figure 7.4 shows, gravity causes the denser invading fluid (water in this case) to segregate

downward; therefore, the depth of contamination at the top of the wellbore is shallower than

that below it. Consequently, cleanup is achieved faster when the probe faces up compared to

other circumferential angles. For instance, in a well with a deviation angle of 60 [degrees],

achieving a fluid sample with a 5% contamination requires 5 [min] of fluid withdrawal,

whereas the same level of cleanup requires 65 and 98 [min] when the probe faces to the side

or down, respectively (see Figure 7.13). In Figures 7.13 and 7.14, the rise in the fractional

flow at the early times of sampling is due to the inflow of water from top of the wellbore.

Comparison of Figures 7.14(a) and 7.14(b) shows that the shortest cleanup time occurs

for highly deviated wellbores, when probe is located at the top of the well; i.e., θj = 180

[degrees].

7.7 Summary and Discussion

This chapter described fluid-flow simulations in deviated wells and specifically simulated

FTMs acquired in thinly-bedded formations. I showed the importance of several parame-

ters in the analysis of FTMs including mud-filtrate invasion, wellbore deviation angle, bed

189

3102Sw [fraction]

3102

3103

3104 0.8

0.9Probe faces down

3105

3106

VD [f

t]

S0 5

0.6

0.7

Probe

3107

3108

TV

Sw

0.3

0.4

0.5

-200 -198 -196 -194 -192 -190

3109

3110 0.1

0.2θw = 80°

Y-dir [ft]Y [ft]

(a)

3102Sw [fraction]

3102

3103

3104 0.8

0.9Probe faces up

3105

3106

VD [f

t]

S0 5

0.6

0.7

Probe

3107

3108

TV

Sw

0.3

0.4

0.5

-200 -198 -196 -194 -192 -190

3109

3110 0.1

0.2θw = 80°

Y-dir [ft]Y [ft]

(b)

3102Sw [fraction]

3102

3103

3104 0.8

0.9Probe faces to the side

3105

3106

VD [f

t]

S0 5

0.6

0.7

Probe

3107

3108

TV

Sw

0.3

0.4

0.5

-200 -198 -196 -194 -192 -190

3109

3110 0.1

0.2θw = 80°

Y-dir [ft]Y [ft]

(c)

Sw [fraction]

3103

3104 0.8

0.9

Wellbore Probe faces to the side

3105

pth

[ft]

S0 5

0.6

0.7

Probe

W

3106

3107

Dep

Sw

0.3

0.4

0.5

0 1 2 3 4 5 6 7

3108

0.1

0.2θw = 80°

X-dir [ft]X [ft]

(d)

Figure 7.11: Spatial distribution of water saturation after 2.4 hours from the onset offluid sampling with a probe-type FT. Sampling takes place when the probe (a) faces down-ward, (c) faces to the side, (b) faces upward, and (d) faces to the side. Wellbore deviationis 80 degrees. Sampling takes place after WBMF invasion for 5 days; Figure 7.4(c) showsthe spatial distribution of water saturation before the onset of fluid sampling. Petrophys-ical properties of the formation are those of Rock Type I (as described in Table 7.1 andFigure 7.3).

190

3102Sw [fraction]

3102

3103

3104 0.8

0.9Probe faces down

3105

3106

VD [f

t]

S0 5

0.6

0.7

Probe

3107

3108

TV

Sw

0.3

0.4

0.5

-40 -38 -36 -34 -32 -30

3109

3110 0.1

0.2θw = 45°

Y-dir [ft]Y [ft]

(a)

3102Sw [fraction]

3102

3103

3104 0.8

0.9Probe faces up

3105

3106

VD [f

t]

S0 5

0.6

0.7Probe

3107

3108

TV

Sw

0.3

0.4

0.5

-40 -38 -36 -34 -32 -30

3109

3110 0.1

0.2θw = 45°

Y-dir [ft]Y [ft]

(b)

3102Sw [fraction]

3102

3103

3104 0.8

0.9Probe faces to the side

3105

3106

VD [f

t]

S0 5

0.6

0.7Probe

3107

3108

TV

Sw

0.3

0.4

0.5

-40 -38 -36 -34 -32 -30

3109

3110 0.1

0.2θw = 45°

Y-dir [ft]Y [ft]

(c)

Sw [fraction]

3103

3104 0.8

0.9

Wellbore Probe faces to the side

3105

pth

[ft]

S0 5

0.6

0.7

Probe

W

3106

3107

Dep

Sw

0.3

0.4

0.5

0 1 2 3 4 5 6 7

3108

0.1

0.2θw = 45°

X-dir [ft]X [ft]

(d)

Figure 7.12: Spatial distribution of water saturation after 2.4 hours from the onset offluid sampling with a probe-type FT. Sampling takes place when the probe (a) faces down-ward, (c) faces to the side, (b) faces upward, and (d) faces to the side. Wellbore deviationis 45 degrees. Sampling takes place after WBMF invasion for 5 days; Figure 7.4(a) showsthe spatial distribution of water saturation before the onset of fluid sampling. Petrophys-ical properties of the formation are those of Rock Type I (as described in Table 7.1 andFigure 7.3).

191

11

0.8

1

F w

Probe Face DownProbe Face SideProbe Face Up

0.8

1

F w

θw = 60° Probe faces down

Probe faces to the side

Probe faces upward

0.6

nal F

low

, F

Probe Face Up

0.6

nal F

low

, F Probe faces upward

0.2

0.4

Frac

tion

0.2

0.4

Frac

tion

5% Contamination

20 40 60 80 1000

Ti [ i ]

20 40 60 80 100

0

Ti [ i ]


(a)

10010010

F w

10

F w

θw = 60°

10-1

nal F

low

,

10-1

nal F

low

, Fr

actio

n

Probe Face DownProbe Face Side

Frac

tion

Probe faces down


5% Contamination

10-4 10-2 100 10210-2

Ti [ i ]

Probe Face SideProbe Face Up

10-4 10-2 100 10210-2

Ti [ i ]

Probe faces upward


(b)

Figure 7.13: Simulated time evolution of fractional flow of water during fluid withdrawalat different azimuthal angles plotted in a (a) linear-linear and (b) log-log scale. Wellboredeviation is equal to 60 degrees. Petrophysical properties of the formation are those of RockType I (described in Table 7.1 and Figure 7.3).

10010010

F w

10

F w

θw = 80°

10-1

nal F

low

,

10-1

nal F

low

, Fr

actio

n


Frac

tion

Probe faces down


5% Contamination

10-4 10-2 100 10210-2

Ti [ i ]


10-4 10-2 100 10210-2

Ti [ i ]

Probe faces upward


(a)

10010010

F w

10

F w

θw = 45°

10-1

nal F

low

,

10-1

nal F

low

, Fr

actio

n


Frac

tion

Probe faces down


5% Contamination

10-4 10-2 100 10210-2

Ti [ i ]


10-4 10-2 100 10210-2

Ti [ i ]

Probe faces upward


(b)

Figure 7.14: Simulated time evolution of fractional flow of water during fluid withdrawal atdifferent azimuthal angles plotted in a log-log scale. Wellbore deviations are (a) 80 and (b)45 degrees. The simulations were performed for a simple probe FT. Petrophysical propertiesof the formation are those of Rock Type I (as described in Table 7.1 and Figure 7.3).

192

thickness, location of the tool with respect to bed boundaries, and azimuthal orientation of

the probe during reservoir fluid sampling. All numerical simulations were conducted with

the UTFEC.

Dynamic mud-filtrate invasion simulations indicated a non-symmetric spatial dis-

tribution of invading fluid saturation around the wellbore. The eccentricity of the spatial

distribution of fluid around the wellbore was significant in high-angle wells penetrating

high-permeability rock formations. Pressure drops during drawdown tests taking place in

the vicinity of a bed boundary were greater than those acquired in the center of thinly-

bedded formation; this effect was enhanced with an increase in well deviation angle. If

bed-boundary effect were neglected the estimated formation permeability would be abnor-

mally low. Comparison of pressure drops corresponding to pressure tests simulated for an

invaded formation to those simulated for an uninvaded formation indicated that (i) pressure

drop in an invaded formation was greater than that in an uninvaded formation, and (ii)

an invaded formation required a longer pressure build-up time compared to an uninvaded

formation. This study showed that fluid clean-up time varied if a sampling probe was placed

at different locations around the perimeter of the wellbore. A probe placed at the top of

the wellbore achieved the fastest fluid cleanup when the invading fluid was denser than the

in-situ fluid.

193

Chapter 8

Summary, Conclusions, and

Recommendations

This chapter summarizes the important contributions achieved in the course of my Ph.D.

research, concludes the studies described in the dissertation, and provides recommendations

for future research.

8.1 Summary

I developed and successfully verified a new three-dimensional (3D) compositional fluid-flow

simulator in cylindrical coordinates specifically designed for analysis of near-wellbore prob-

lems. New capabilities of the developed simulator, UTFEC, are (i) accurate simulations of

mud-filtrate invasion in deviated wells performed by coupling a dynamic mudcake growth

model with a verified 3D fluid-flow simulator, and (ii) simulation of formation-tester mea-

surements in deviated wells with an efficient and accurate algorithm.

The method simulated simultaneous fluid flow of a maximum of three fluid phases

(water, oil, and gas). Fluid-flow formulations were based on the following assumptions: an

isothermal reservoir, no chemical reaction, negligible adsorption, an impermeable reservoir

194

at a great radial distance from the wellbore, validity of Darcy’s law for fluid flow through

porous media, a slightly compressible formation, and negligible dispersion in hydrocarbon

fluid phases. The fundamental pressure equation was based on the assumption that pore

volume contains the total volume of the fluid. It was assumed that rock pore volume was

a function of pressure whereas total fluid volume was a function of pressure and moles of

components. Oil pressure was used as reference in the pressure equation. The algorithm

enforced a general mass balance equation to calculate the moles of each component at a

given pressure. Calculations of moles for each component as well as the calculation of

saturations of fluid phases were carried out with an explicit method. Because this research

was targeted for deviated wells, a full-tensor permeability was implemented using mass

balance equations. Simulations were based on the method of implicit pressure and explicit

concentration (IMPEC) to solve the partial differential equation arising from the discretized

fluid-flow equations.

I implemented the following auxiliary relations to calculate ncnp+ 6np+ 2 unknown

parameters (nc and np are number of components and fluid phases, respectively) in the

volume constraint (pressure equation) and mass balance equations: saturation constraint,

porosity dependency on pressure, phase molar density, phase mass density, phase composi-

tion constraint, flow rate, phase pressure, phase viscosity, relative permeability, and phase

equilibrium.

Accordingly, the viscosity of the aqueous phase was calculated using McCain’s rela-

tion whereas the viscosity of each hydrocarbon fluid phase was calculated using Lohrenz et

al.’s (1964) relations. In a two-phase flow regime, relative permeability of each fluid phase

was calculated using Brook-Corey’s (1964) parametric model. For a three-phase flow regime,

relative permeability of the oil phase was calculated using Stone’s model 2. I modeled hydro-

carbon components and fluid phases based on Peng-Robinson’s equation of state. No mass

transfer was assumed between the hydrocarbon components and the aqueous phase. The

aqueous phase consisted of water and salt components. Moreover, the water component did

not affect phase behavior. From the equality of fugacities in thermodynamic equilibrium, I

195

obtained an analytical expression for partial derivatives of total fluid volume introduced in

the pressure equation.

Calculations of phase stability were performed based on the tangent-plane distance

approach. When two hydrocarbon fluid phases were detected, a flash calculation deter-

mined the composition of each hydrocarbon phase. In UTFEC, the flash calculations were

performed using a combination of successive substitutions and Newton-Raphson’s method.

UTFEC was verified with analytical solutions and commercial reservoir simulators.

Several case studies were designed to verify simulations of multi-phase fluid flow regimes

including gas-water, oil-water, and gas-oil-water for different rock types. Case studies in-

cluded different types of boundary conditions for fluid injection and production. Simulations

of formation-tester measurements in deviated wells were verified against those obtained us-

ing analytical expressions. Analytical expressions included solutions for pressure tests per-

formed in formations with permeability anisotropy. This procedure allowed the verification

of the full-tensor permeability formulation in cylindrical coordinates. In all verification case

studies, very good agreement was observed between results obtained with UTFEC and those

yielded by both analytical solutions and commercial simulators.

In vertical wells, separation of apparent resistivity log responses occurs in the pres-

ence of radial variations in water saturation and salt concentration. Radial variations of

salt concentration, however, depend on dispersion. Understanding the behavior of apparent

resistivity logs required quantification of salt dispersion. Therefore, a model for physical

dispersion was implemented for aqueous salt concentration. Checks on the reliability and

accuracy of the method were performed against the commercial reservoir simulator CMG-

STARS. I used the two numerical methods (UTFEC and CMG-STARS) to calculate the

radial distribution of salt concentration subject to different values of physical dispersion.

My approach coupled an experimentally validated mudcake model with the devel-

oped reservoir simulator. Reservoir and mudcake models were coupled using the productivity

index. The algorithm allowed different mudcake growth rates at different azimuthal angles

around the perimeter of the wellbore. Subsequently, sensitivity analyses were conducted on

196

various parameters governing the presence of water- and oil-base mud-filtrate invasion.

Invasion of oil-base mud can lead to wettability alteration. Therefore, I imple-

mented a wettability alteration model to enable the time and space evolution of wettability

conditions, including relative permeability and capillary pressure. I found that wettability

alteration measurably affected the spatial distribution of fluid saturation in the vicinity of

the wellbore and, therefore, apparent resistivity logs.

Finally, UTFEC was applied to analyze formation-tester measurements (FTM) ac-

quired in deviated wells and penetrating thinly-bedded formations. The method calculated

spatial distributions of mud-filtrate saturation in deviated wells for rock formations with

high and low permeabilities. The study of formation-tester measurements quantified the

effects of invasion, anisotropy, pressure-test location with respect to bed boundaries, and

location around the perimeter of the wellbore.

8.2 Conclusions

The following are the most important conclusions stemming from this dissertation:

1. A new, 3D, IMPEC-type, EOS, compositional, fluid-flow simulator in cylindrical co-

ordinates was developed for near-wellbore problems. Central processing unit (CPU)

times associated with the simulations were as follows: (i) 1D cases for 3 days of

WBM/OBM invasion with 50 radial grids were between 0.1− 2 minutes; (ii) 2D cases

for 3 days of WBM/OBM invasion with 50× 30 grids were between 0.5− 10 minutes;

(iii) base packer-type formation-tester simulations with 50 × 30 grids were between

0.5−5 minutes; (iv) 3D cases for 3 days of WBM/OBM invasion with 50×30×10 grids

were between 1 − 30 minutes; and (v) base probe-type formation-tester simulations

with 50× 30× 10 grids were between 1− 10 minutes.

2. Suggested default numerical and phase behavior controllers/parameters were chosen to

maximize numerical stability for typical petrophysical and phase behavior properties.

The material balance error was set to 10−4 to secure accurate results. When calculat-

197

ing spatial distribution of salt concentration, the numerical controlling parameter for

salt concentration was equal to 10−3.

3. The required random access memory (RAM) was below 1MB for the cases described in

this dissertation. For instance, a case with 50× 50× 20 grids required 100kb of RAM.

To simulate a case of two-phase flow in a deviated well (single-phase hydrocarbon) with

a small number of grids (less than 1000 grids), the simulation time was chiefly spent

on matrix construction (approximately 60%) and numerical solution (approximately

30%). However, the ratio of numerical solver time to total simulation time significantly

increased with an increase in the number of grid blocks.

4. UTFEC has the following limitations: (i) when a single hydrocarbon phase is stable

in high temperatures and pressures, the phase behavior package fails to distinguish

between gas and oil. This technical problem prompted modifications to the constant

in equation (3.85); (ii) in the cases with high capillary pressure and large density

contrasts, numerical stability required small numerical controllers (e.g., 10−5 for the

material balance error) and this slowed down the numerical simulations; (iii) in simula-

tions involving deviated wells, horizontal bed boundaries did not conform to cylindrical

grids; and (iv) in deviated wells where horizontal boundaries obliquely crossed numer-

ical grids, an assignment of a single rock type (described in Section 3.1) decreased the

accuracy of simulation results.

5. Analysis of the effect of physical dispersion on the radial distribution of salt concen-

tration showed that electrical resistivity varied significantly for different dispersivities

when the invading fluid had lower salinity than in-situ water. Conversely, the effect of

salt dispersion was negligible when the invading fluid had greater salinity than con-

nate water. However, in general, high dispersivity decreased the separation between

shallow and deep apparent resistivity logs.

6. Numerical simulations indicated that in highly permeable formations invasion was

primarily governed by mudcake and mud properties such as mudcake permeability,

198

mudcake thickness, and mud-filtrate viscosity. On the other hand, in tight forma-

tions, invasion was mainly controlled by mud-filtrate, mud-filtrate viscosity, formation

permeability, relative permeability, porosity, and capillary pressure.

7. Surfactants included in oil-base mud-filtrate (OBMF) can change a rock’s surface

wettability from water-wet to neutral, or to an oil-wet condition. This behavior caused

a portion of the originally residual pore volume of connate water to become moveable,

whereby the radial distribution of water saturation exhibited variations different from

those of the rock in its original (uninvaded) state. The radial displacement of movable

water by OBMF gave rise to a radial zone (annulus) where water saturation was

abnormally high (water bank), which in turn caused the radial rock resistivity to be

abnormally low (resistivity annulus).

8. Simulated apparent resistivity logs for the case of OBMF invasion into an oil-saturated

permeable formation exhibited progressive separation from shallow- to deep-sensing

logs. In the case of deep OBM invasion, apparent resistivity from shallow-sensing

arrays exhibited higher values than the deeper-sensing array-induction logs. However,

for shallow invasion with OBMF, a reversal effect of OBM was observed; i.e., shallow

apparent resistivity was lower than deep apparent resistivity. This OBM reversal effect

was emphasized in tight and gas-bearing formations.

9. Mud-filtrate invasion in high-angle deviated wells caused a non-symmetric distribution

of the invading fluid around the perimeter of the wellbore. Eccentricity of fluid distri-

bution around the perimeter of the wellbore significantly increased in (i) formations

with high permeability, and (ii) high-angle wells.

10. Pressure-transient tests showed that the pressure drop for formation tests simulated

in the vicinity of a bed boundary was higher than that simulated in the center of

the formation. This bed-boundary effect was emphasized with an increase in well-

bore inclination. When the effect of bed boundaries was neglected, the estimated

formation permeability was abnormally low. Pressure-transient tests simulated in a

199

WBM-invaded formation were affected by capillary pressure; in these cases, fluid pres-

sure did not return to the initial fluid pressure at the end of a short buildup period.

11. The study of the effect of mud-filtrate invasion on FTMs simulated in high-angle wells

indicated that (a) pressure drop during drawdown in a WBM-invaded formation was

greater than that simulated in an uninvaded formation, and (b) build-up time in a

WBM-invaded formation was significantly longer than for the case of an uninvaded

formation.

12. Analysis on fluid sampling (acquired in deviated wells) using probe-type formation

testers showed that contamination clean-up time varied when the probe was placed at

different locations around the perimeter of wellbore. For all wellbore deviation angles,

the shortest cleanup was achieved when the probe was placed at the top of the well

and the invading fluid had a larger density than the in-situ fluid.

8.3 Recommendations for Future Work

The following list provides suggestions for extension and improvement of the simulator

developed in this dissertation:

1. The new compositional fluid-flow simulator was developed with an object-oriented

structure and avoided repeapted calculations. Computational times for simulations

of mud-filtrate invasion in most of the case studies was satisfactory. For instance,

depending on the assumptions, the simulation for a 1D case of one day OBM invasion

required between five seconds to two minutes. Simulations of FTMs required a large

number of grid blocks with small grid sizes (smaller than 0.1 [ft]) in the vicinity of the

wellbore. Stability analysis for numerical calculations based on an IMPEC scheme in-

dicated that time step depended on the size of the smallest grid block. Hence, stability

imposed a small increment in the time step when large variations existed in primary

variables such as pressure and concentrations of hydrocarbon components, water, and

200

salt. The following are several salient recommendations to decrease simulation times:

Implementation of fully implicit or sequential implicit compositional formula-

tions. In general, the time step used in conjunction with a fully implicit scheme

does not depend on grid-block sizes. In specific problems using small grid blocks,

and in the presence of large variations (in primary variables such as pressure,

concentrations, fluid phase saturations, and material balance error), the fully

implicit scheme allows larger time steps than the IMPEC method. A sequen-

tial implicit method can also accommodate larger time steps than those allowed

by IMPEC. In addition, this scheme does not have the complexities (analytical

derivatives in the Jacobian matrix) involved in a fully implicit method.

Coupling UTFEC with a streamline method. It was found that a finite-different

method in conjunction with a streamline-based algorithm can achieve stable sim-

ulations 5 to 10 times faster than the original uncoupled codes (Hadibeik et al.,

2011). However, there are several drawbacks in streamline methods including

(i) lower accuracy compared to finite-difference method, (ii) neglecting gravity

effects, (iii) instability due to lack of connection between pressure variations and

fluid movement.

Implementation of a faster and more specialized numerical solver. In order to

solve linear system of equations arising from fluid flow equations, the following it-

erative solvers were implemented in this dissertation: (i) a stabilized bi-conjugate

gradient method, (ii) a transpose-free quasi-minimum residual method, (iii) a

full orthogonalization method, (iv) a generalized minimum residual method, and

(v) a flexible version of generalized minimum residual method. I found that the

stabilized bi-conjugate gradient method was the fastest. The applied solver was

borrowed from the iterative solver package developed by Saad (2003). Implemen-

tation of a fast linear solver was not investigated in this research.

2. UTFEC can be coupled with a wellbore model (Frooqnia et al., 2011) to simulate

201

production logging measurements. For instance, various compartments of a reservoir

may contain fractures. The existence of fractures can significantly affect the produc-

tion flow rate. Using UTFEC, it is possible to assign very small grids with very large

permeabilities at fracture locations. Yet, a comprehensive simulator should include an

option for simulating naturally fractured reservoir as well as formations with induced

fractures. Implementation of dual porosity or dual permeability models, as well as

discrete fracture models are hence recommended for future endeavors.

3. UTFEC implemented cylindrical coordinates for all simulation studies including cases

of vertical, horizontal, and deviated wells. A cylindrical coordinate system provides

wellbore conformal gridding suitable for near-wellbore simulations. However, for sim-

ulations involving fluid flow in deviated and horizontal wells, cylindrical grids are not

conformal with horizontal petrophysical bed boundaries. Incorporating unstructured

gridding into the simulation would allow conformality with both wellbore geometry

and horizontal and vertical bed boundaries.

4. UTFEC can be coupled with an inversion algorithm to perform history matching of

field data sets. This extension would investigate the practicality and limitations of the

simulator for solving inverse problems. Specifically, UTFEC can be used to simulate

mud-filtrate invasion in high-angle wells and subsequently calculate borehole responses

including electrical resistivity, sonic, and nuclear measurements.

5. The finite-difference discretization uses a first-order method which can cause numerical

dispersion and error in the calculations. Application of higher-order upwind discretiza-

tion schemes would increase the accuracy and stability of such simulations.

6. The physical model in this dissertation assumed an isothermal condition in the reser-

voir. Incorporating an energy equation to the current system of equations (described

in Chapter 2) would enable the simulator to approach the following problems: (i)

invasion of mud-filtrate with a temperature different from that of in-situ fluids, (ii)

202

production from zones with different temperatures, and (iii) variation of formation

temperature during production time.

203

Nomenclature

Abbreviations

1D One-dimensional

2D Two-dimensional

3D Three-dimensional

AIT Array-induction tool (mark of Schlumberger)

CMG Computer modeling group Ltd

EOS Equation of state

PR-EOS Peng-Robinson’s equation of state

FT Formation tester

FTM Formation-tester measurement

GOR Gas-oil-ratio, [SCF/STB]

GEM Generalized equation-of-state model compositional reservoir simu-

lator (a CMG software)

IMPEC Implicit pressure and explicit concentration

NMR Nuclear magnetic resonance

204

OBM Oil-base mud

OBMF Oil-base mud filtrate

pu Porosity unit

PVT Pressure-volume-temperature

SRK Soave-Redlich-Kwong equation of state

STARS Steam, thermal, and advanced processes reservoir simulator (a CMG

software)

su Saturation unit

TVD True vertical depth

UTAPWelS University of Texas at Austin’s petrophysical and well-log simulator

UTFEC University of Texas at Austin’s formation evaluation compositional

fluid-flow simulator

WBM Water-base mud

WBMF Water-base mud filtrate

WinProp Phase behavior and property program (a CMG software)

Greek symbols

αlj Longitudinal dispersivity of fluid phase j, [ft]

αtj Transverse dispersivity of fluid phase j, [ft]

∆Nlim Maximum relative change of component moles, [dimensionless]

∆po Difference between pressure at the observation probe and the initial

pressure, [psi]

205

∆pp Difference between pressure at packer center and initial pressure,

[psi]

∆Plim Maximum relative change of pressure, [dimensionless]

∆Slim Maximum change of saturation, [fraction]

∆tinit Initial time step, [day]

∆tmax Maximum time step, [day]

∆tmin Minimum time step, [day]

∆Vlim Maximum relative change of volume error, [dimensionless]

∆ Gradient operator

δ Multiplier for the porosity exponent, [dimensionless]

δik Binary interaction between components i and k

ηj Mixture viscosity parameter of hydrocarbon phase j, [1/cp]

ε Scaling factor for wettability alteration, [dimensionless]

γj Specific weight of fluid phase j, [dimensionless]

λrj Mobility with respect to the reference fluid phase (in this disserta-

tion, the oil phase) [1/cp]

µf Mud-filtrate viscosity, [cp]

µi Chemical potential of component i, [psi ft3/lbm]

µj Viscosity of fluid phase j, [cp]

µ∗j Viscosity of hydrocarbon phase j at low pressure, [cp]

µw,14.7 Viscosity of the aqueous phase at atmospheric pressure, [cp]

206

µw Viscosity of the aqueous phase, [cp]

µi Viscosity of component i at low pressure, [cp]

ω Acentric factor of hydrocarbon component, []

Ωa Parameter of Peng-Robinson’s equation of state, [dimensionless]

Ωb Parameter of Peng-Robinson’s equation of state, [dimensionless]

φ Porosity, [fraction]

φ0 Porosity at reference reservoir pressure, [fraction]

Φj Potential of fluid phase j, [psi]

φmc0 Mudcake reference porosity, [fraction]

φmc Mudcake porosity, [fraction]

Ψi Parachor of component i, [dynes1/4/cm1/4/lbm]

ϕi Fugacity coefficient of component i, [dimensionless]

Dij Molecular diffusion coefficient of component i in fluid phase j, [ft2/day]

σj1j2 Interfacial tension of phase j1 and phase j2, [dynes/cm]

τ Tortuosity of the porous medium, [dimensionless]

θw Wellbore deviation angle, [radians]

ξ1 Water molar density, [lbm/ft3]

ξ01 Water molar density at reference pressure, [lbm/ft3]

ξj Molar density of fluid phase j, [lbm/ft3]

ξjr Reduced molar density of hydrocarbon phase j, [dimensionless]

207

ζi Viscosity parameter of component i, [1/cp]

Roman Letters

ALHP Left-hand side matrix of pressure equation (page 53)

A Equation-of-state parameter

a Constant in the equation of state

aik Equation-of-state parameter

BRHP Array at right-hand side of pressure equation without source or sink

term corresponding to a well (page 53)

B Equation-of-state parameter

b Constant in the equation of state

bik Equation-of-state parameter

c1 Compressibility of water at reference pressure, [1/psi]

cf Compressibility of the formation, [1/psi]

Cs Concentration of the adsorbed surfactant, [dimensionless]

Ct Concentration of the total surfactant, [dimensionless]

Cpc Constant of capillary pressure function, [psi√md/dynes/cm]

Csalt Concentration of salt in the aqueous phase, [ppm NaCl]

CtA Total compressibility of the aquifer which is the sum of water com-

pressibility and rock compressibility, [1/psi]

ct Total formation compressibility, [1/psi]

D Depth, [ft]

208

dt Differential time, [day]

ej1j2 Exponent of relative permeability function of fluid phase j1 flowing

with fluid phase j2

ej Exponent of relative permeability function of fluid phase j

Epc Exponent of capillary pressure function, [dimensionless]

Fcap Net effect of capillary pressure term in the pressure equation (page 53),

[ft3/day]

Fdisp Net effect of dispersion term in the pressure equation (page 53),

[ft3/day]

Fgrav Net effect of gravity term in the pressure equation (page 53), [ft3/day]

−→F ij Flux of component i in fluid phase j, [lbm/day/ft2]

fa Fraction of the azimuthal completion of the well in a given grid

block

Fi Flux term of component i, [lbm/ft3]

fLi Fugacity of component i in the liquid phase, [psi]

fVi Fugacity of component i in the gas phase, [psi]

fs Mud solid fraction, [fraction]

Fv Mole fraction of vapor

fij Fugacity of component i in fluid phase j, [psi]

Fw Fractional flow, [fraction]

G Gibbs free energy

k Permeability tensor, [md]

209

Kij Dispersion tensor, [ft2/day]

Dij Molecular diffusion coefficient of component i in fluid phase j, [ft2/day]

Ki Equilibrium K-value of component i

kmc0 Mudcake reference permeability, [md]

kmc Mudcake permeability, [md]

krj1j2 Relative permeability of fluid phase j1 flowing with fluid phase j2,

[dimensionless]

Krθ,ij , Krz,ij , Kθz,ij Non-diagonal terms of dispersion tensor, [ft2/day]

krθ, krz, kθz Non-diagonal terms in permeability tensor, [md]

krj Relative permeability of fluid phase j, [dimensionless]

k0rj Endpoint relative permeability of fluid phase j, [dimensionless]

Krr,ij , Kθθ,ij , Kzz,ij Diagonal terms of dispersion tensor, [ft2/day]

krr, kθθ, kzz Diagonal terms in permeability tensor, [md]

ks Spherical permeability, [md]

l′

w Equivalent half length of packer in an anisotropic formation, [ft]

lw Half length of a packer, [ft]

−→n Unit normal vector to a boundary

nc Number of hydrocarbon components

Ni Number of moles of component i, [lbm]

nj Total moles of all components in fluid phase j, [lbm]

nL Number of moles in liquid phase, [lbm]

210

np Number of fluid phases

nv Number of moles in vapor phase, [lbm]

Nw Total moles of water component, [lbm]

nkj Moles of component k in hydrocarbon phase j, [lbm]

Nsalt Number of moles of salt, [lbm]

Nwater Number of moles of water, [lbm]

P Array of pressures in grid blocks (page 53)

P Pressure, [psi]

P 0 Reference reservoir pressure, [psi]

P 01 Reference pressure for water compressibility, [psi]

Pj Pressure of fluid phase j, [psi]

Pr Pressure of the reference fluid phase (oil phase is the reference pres-

sure in this dissertation), [psi]

Pw Pressure at the sandface, [psi]

PAq Aquifer pressure at the external boundary, [psi]

Pave Average formation pressure at the aquifer-reservoir boundary, [psi]

Pc2j Capillary pressure of fluid phase 2 and fluid phase j, [psi]

Pce Capillary entry pressure, [psi]

Pci Critical pressure of component i, [psi]

Pcrj Capillary pressure between fluid phase j and pressure of the refer-

ence fluid phase, [psi]

211

Pmc Mudcake pressure differential, [psi]

Pwf Bottomhole pressure, [psi]

PIj Productivity index of fluid phase j, [ft3/day/psi]

PIt Total productivity index, [ft3/day/psi]

qi Molar flow rate of component i, [lbm/day]

Qj Volumetric flow rate, [ft3/day]

Qt Total volumetric flow rate, [ft3/day]

qt Total molar flow rate, [lbm/day]

drmc Differential mudcake thickness, [ft]

R Universal gas constant, [psi ft3/lbm/R]

r1 Radius of the first radial grid, [ft]

Ri Source term of component i, [lbm/ft3]

Rt True formation electrical resistivity, [Ω.m]

Rw Connate-water electrical resistivity, [Ω.m]

rw Radius of wellbore, [ft]

r′

w Equivalent wellbore radius in an anisotropic formation, [ft]

rmc Mudcake thickness, [ft]

rsw Effective spherical wellbore radius, [ft]

zo Distance of an observation probe from the center of a packer, [ft]

Raniso Permeability anisotropy ratio, [dimensionless]

212

Sj Normalized saturation of fluid phase j, [dimensionless]

−→S Saturation of all fluid phases, [fraction]

s Skin factor, [dimensionless]

Sj Saturation fluid phase j, [fraction]

Sjr Residual saturation of fluid phase j, [fraction]

Sj1rj2 Residual saturation of fluid phase j1 flowing with fluid phase j2,

[fraction]

T Temperature, [F]

t Time, [day]

Tci Critical temperature of component i, [R]

Tj Transmissibility of fluid phase j, [lbm/day/psi]

TriTTci

, the reduced temperature of component i, [dimensionless]

uj Superficial velocity of fluid phase j, [ft/day]

urj , uθj , uzj Velocities of fluid phase j in the r, θ, and z directions, respectively,

[ft/day]

v Compressibility exponent for dynamic mudcake properties, [dimen-

sionless]

v Molar volume, [ft3/lbm]

V1 Volume of the aqueous phase, [ft3]

v1 Molar volume of water, [ft3/lbm]

Vb Bulk volume, [ft3]

213

Vj Volume of fluid phase j, [ft3]

vj Molar volume of fluid phase j, [ft3/lbm]

Vp Pore volume, [ft3]

V 0p Pore volume at reference pressure, [ft3]

Vt Total fluid volume, [ft3]

Vci Critical molar volume of component i, [ft3/lbm]

Vci Critical molar volume of component i, [m3/kgmol]

Vti Partial derivative of total fluid volume with respect to moles of

component i, [ft3/lbm]

Wi Accumulation term of component i, [lbm/ft3]

Wti Molecular weight of component i, [lbs/lbm]

−→X Molar fraction of hydrocarbon components, [mole fraction]

−→x Phase composition, [mole fraction]

−→x j Composition of fluid phase j, [mole fraction]

−→X j Molar composition of components in fluid phase j, [mole fraction]

xij Mole fraction of component i in fluid phase j

xsalt,1 Molar fraction of salt in the aqueous phase, [fraction]

Z Compressibility factor, [dimensionless]

Zj Compressibility factor of fluid phase j, [dimensionless]

Subscript

i Component index

214

j Fluid phase index

l Index of grid block number (l=r, θ, z)

215

Appendix A

Discretization of Physical

Dispersion Terms

Chapter 3 discusses the computational approach to solve pressure and material balance

equations. Fluid-flow equations consists of potential (the net effect of pressure, capillary,

and gravity forces) and dispersion terms. This appendix uses the finite-difference algorithm

to discretize dispersion terms in fluid-flow equations.

In Section 3.2, I discretize the dispersion term in equation (3.7) as

−→∇ · φξjSjKij∇xij = JRij + JΘij + JZij , (A.1)

where

JRij =1

r

∂

∂r

[φξjSj

(Krr,ijr

∂xij∂r

+Krθ,ij∂xij∂θ

+Krz,ijr∂xij∂z

)], (A.2)

JΘij =1

r

∂

∂θ

[φξjSj

(Krθ,ij

∂xij∂r

+Kθθ,ij

r

∂xij∂θ

+Kθz,ij∂xij∂z

)], (A.3)

and

JZij =∂

∂z

[φξjSj

(Krz,ij

∂xij∂r

+Kθz,ij

r

∂xij∂θ

+Kzz,ij∂xij∂z

)]. (A.4)

216

Subsequently, I use the central-difference scheme to discretize equations (A.2) through (A.4)

to obtain

Jrr,ij =4

Υr+1/2 −Υr−1/2

[(ΥKrr,ijφξjSj)(r+1/2)

(xij)r+1,θ,z − (xij)r,θ,zΥr+1 −Υr

−

(ΥKrr,ijφξjSj)(r−1/2)

(xij)r,θ,z − (xij)r−1,θ,z

Υr −Υr−1

], (A.5)

Jrθ,ij =2

Υr+1 −Υr−1

[(Krθ,ijφξjSj)(r+1)

(xij)r+1,θ+1,z − (xij)r+1,θ−1,z

Θθ+1 −Θθ−1−

(Krθ,ijφξjSj)(r−1)

(xij)r−1,θ+1,z − (xij)r−1,θ−1,z

Θθ+1 −Θθ−1

], (A.6)

and

Jrz,ij =2

Υr+1 −Υr−1

[(Krz,ijφξjSj)(r+1)

(xij)r+1,θ,z+1 − (xij)r+1,θ,z−1

Zz+1 − Zz−1−

(Krz,ijφξjSj)(r−1)

(xij)r−1,θ,z+1 − (xij)r−1,θ,z−1

Zz+1 − Zz−1

], (A.7)

and for equation (A.4) I obtain

Jθr,ij =2

Θθ+1 −Θθ−1

[(Krθ,ijφξjSj)(θ+1)

(xij)r+1,θ+1,z − (xij)r−1,θ+1,z

Υr+1 −Υr−1−

(Krθ,ijφξjSj)(θ−1)

(xij)r+1,θ−1,z − (xij)r−1,θ−1,z

Υr+1 −Υr−1

], (A.8)

Jθθ,ij =1

Υr

1

Θθ+1/2 −Θθ−1/2

[(Kθθ,ijφξjSj)(θ+1/2)

(xij)r,θ+1,z − (xij)r,θ,zΘθ+1 −Θθ

−

(Kθθ,ijφξjSj)(θ−1/2)

(xij)r,θ,z − (xij)r,θ−1,z

Θθ −Θθ−1

], (A.9)

217

Jθz,ij =1

rr

1

Θθ+1 −Θθ−1

[(Kθz,ijφξjSj)(θ+1)

(xij)r,θ+1,z+1 − (xij)r,θ+1,z−1

Zz+1 − Zz−1−

(Kθz,ijφξjSj)(θ−1)

(xij)r,θ−1,z+1 − (xij)r,θ−1,z−1

Zz+1 − Zz−1

], (A.10)

and finally for equation (A.4) I obtain

Jzr,ij =1

Zz+1 − Zz−1

[(Krz,ijφξjSj)(z+1)

(xij)r+1,θ,z+1 − (xij)r−1,θ,z+1

rr+1 − rr−1−

(Krz,ijφξjSj)(z−1)

(xij)r+1,θ,z−1 − (xij)r−1,θ,z−1

rr+1 − rr−1

], (A.11)

Jzθ,ij =1

rr

1

Zz+1 − Zz−1

[(Kθz,ijφξjSj)(z+1)

(xij)r,θ+1,z+1 − (xij)r,θ−1,z+1

Θθ+1/2 −Θθ−1−

(Kθz,ijφξjSj)(z−1)

(xij)r,θ+1,z−1 − (xij)r,θ−1,z−1

Θθ+1 −Θθ−1

], (A.12)

Jzz,ij =1

Zz+1/2 − Zz−1/2

[(Kzz,ijφξjSj)(z+1/2)

(xij)r,θ,z+1 − (xij)r,θ,zZz+1 − Zz

−

(Kzz,ijφξjSj)(z−1/2)

(xij)r,θ,z − (xij)r,θ,z−1

Zz − Zz−1

]. (A.13)

218

Appendix B

Permeability Tensor

Transformation

In UTFEC, the formation permeability tensor is diagonal, that is,

k =

kx 0 0

0 ky 0

0 0 kv

. (B.1)

Numerical modeling of horizontal and deviated wells in cylindrical coordinates requires

an expression of the permeability tensor in a coordinate system conformal with the finite

difference grid. Below, I describe the details of the transformation of coordinate systems.

The superficial velocity of fluid phase j in a homogeneous and anisotropic porous

medium is calculated using Darcy’s law

−→uj =krjk

µ· ∇Φj , (B.2)

219

where Φj is the total potential of fluid phase j and k is permeability tensor described as

k =

krr krθ krz

krθ kθθ kθz

krz kθz kzz

. (B.3)

where krr, kθθ, and kzz are diagonal entries of permeability tensor and krθ, krz, and kθz are

non-diagonal entries of the permability tensor.

The rotation matrix for a counterclockwise rotation around the z axis is given by (Ar-

fken et al., 2005)

Rz(θj) =

cos θj sin θj 0

− sin θj cos θj 0

0 0 1

, (B.4)

where θj is the rotation angle around the z axis. When two coordinate systems are related

by a rotation matrix, then coordinates of a point in space can be transformed from one

coordinate system to the other by

−→X = Rz(θj)

−→X ′ ⇐⇒

−→X ′ = Rz(−θj)

−→X. (B.5)

Differentiating the displacement transformation (equation (B.5)) with respect to time, I

obtain the velocity transformation between the two coordinate systems, i.e.,

−→u′j = Rz(−θj)−→uj , (B.6)

Similarly, the potential gradients are related by

∇Φj = Rz(θj)∇Φ′j . (B.7)

220

Substituting equations (B.6) and (B.7) into equation (B.2) gives

u′j = Rz(−θj)krjk

µ·Rz(θj)∇Φ′j . (B.8)

By comparing equations (B.8) and (B.2), I obtain the transformed permeability tensor as

k′ = Rz(−θj)kRz(θj). (B.9)

Analogously, the transformed permeability tensor after rotation around the y axis is given

by

k′ = Ry(−θw)kRy(θw), (B.10)

where θw is the wellbore inclination angle. From equations (B.9) and (B.10), the perme-

ability tensor after two consecutive rotations of coordinate central axis is obtained by

k′ = Rz(−θj)Ry(−θw)kRy(θw)Rz(θj). (B.11)

Assuming rock permeability tensor given by equation (B.1), then the permeability tensor

entries after two sequential rotations, θw around y axis and θj around z axis, are given by

221

krr = kx(cos θj · cos θw)2 + kz(sin θw · cos θj)2 + ky(sin θj)

2, (B.12)

krθ = − sin θj · cos θj · kx(cos θw)2 + kz(sin θw)2 + ky(cos θj · sin θj), (B.13)

krz = cos θj · cos θw · sin θw(ky − kx), (B.14)

kθθ = kx(sin θj · cos θw)2 + ky(cos θj)2 + kz(sin θj · sin θw)2, (B.15)

kθz = sin θj · cos θw · sin θw(kx − kz), (B.16)

kzz = kx(sin θw)2 + kz(cos θw)2. (B.17)

222

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