EFFICIENT SIMULATION OF THERMAL ENHANCED …fn674bf7126/Thesis...EFFICIENT SIMULATION OF THERMAL...

EFFICIENT SIMULATION OF THERMAL ENHANCED OIL

RECOVERY PROCESSES

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ENERGY

RESOURCES ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Zhouyuan Zhu

August 2011

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/fn674bf7126

© 2011 by Zhouyuan Zhu. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/fn674bf7126

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Margot Gerritsen, Primary Adviser


Anthony Kovscek, Co-Adviser


Marco Thiele

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

Simulating thermal processes is usually computationally expensive because of the

complexity of the problem and strong nonlinearities encountered. In this work, we

explore novel and efficient simulation techniques to solve thermal enhanced oil recov-

ery problems. We focus on two major topics: the extension of streamline simulation

for thermal enhanced oil recovery and the efficient simulation of chemical reaction

kinetics as applied to the in-situ combustion process.

For thermal streamline simulation, we first study the extension to hot water flood

processes, in which we have temperature induced viscosity changes and thermal vol-

ume changes. We first compute the pressure field on an Eulerian grid. We then solve

for the advective parts of the mass balance and energy equations along the individual

streamlines, accounting for the compressibility effects. At the end of each global time

step, we account for the nonadvective terms on the Eulerian grid along with gravity

using operator splitting. We test our streamline simulator and compare the results

with a commercial thermal simulator. Sensitivity studies for compressibility, gravity

and thermal conduction effects are presented.

We further extended our thermal streamline simulation to steam flooding. Steam

flooding exhibits large volume changes and compressibility associated with the phase

behavior of steam, strong gravity segregation and override, and highly coupled energy

and mass transport. To overcome these challenges we implement a novel pressure up-

date along the streamlines, a Glowinski θ-scheme operator splitting and a preliminary

streamline/finite volume hybrid approach. We tested our streamline simulator on a

series of test cases. We compared our thermal streamline results with those computed

by a commercial thermal simulator for both accuracy and efficiency. For the cases

iv

investigated, we are able to retain solution accuracy, while reducing computational

cost and gaining connectivity information from the streamlines. These aspects are

useful for reservoir engineering purposes.

In traditional thermal reactive reservoir simulation, mass and energy balance equa-

tions are solved numerically on discretized reservoir grid blocks. The reaction terms

are calculated through Arrhenius kinetics using cell-averaged properties, such as av-

eraged temperature and reactant concentrations. For the in-situ combustion process,

the chemical reaction front is physically very narrow, typically a few inches thick. To

capture accurately this front, centimeter-sized grids are required that are orders of

magnitude smaller than the affordable grid block sizes for full field reservoir models.

To solve this grid size effect problem, we propose a new method based on a non-

Arrhenius reaction upscaling approach. We do not resolve the combustion front on

the grid, but instead use a subgrid-scale model that captures the overall effects of

the combustion reactions on flow and transport, i.e. the amount of heat released, the

amount of oil burned and the reaction products generated. The subgrid-scale model

is calibrated using fine-scale highly accurate numerical simulation and laboratory ex-

periments. This approach significantly improves the computational speed of in-situ

combustion simulation as compared to traditional methods. We propose the detailed

procedures to implement this methodology in a field-scale simulator. Test cases il-

lustrate the solution consistency when scaling up the grid sizes in multidimensional

heterogeneous problems. The methodology is also applicable to other subsurface

reactive flow modeling problems with fast chemical reactions and sharp fronts.

Displacement front stability is a major concern in the design of all the EOR

processes. Historically, premature combustion front break through has been an issue

for field operations of in-situ combustion. In this work, we perform detailed analysis

based on both analytical methods and numerical simulation. We identify the different

flow regimes and several driving fronts in a typical 1D ISC process. For the ISC

process in a conventional mobile heavy oil reservoir, we identify the most critical

front as the front of steam plateau driving the cold oil bank. We discuss the five

main contributors for this front stability/instability: viscous force, condensation, heat

conduction, coke plugging and gravity. Detailed numerical tests are performed to test

v

and rank the relative importance of all these different effects.

vi

Acknowledgements

First of all, I would like to express my appreciation to my advisors: Margot Gerritsen,

Anthony Kovscek, Marco Thiele and Louis Castanier. They provided me with great

experience, guidance and continuous support during my doctoral study. I want to

thank Professor Louis Durlofsky for serving on my committee. I would also like to

thank Professor Biondo Biondi for kindly chairing my defense.

I owe many thanks to my research group members in the SUPRI-C and SUPRI-A

group, including Alexandre Lapene, Wenjuan Lin, Murat Cinar, and Golnaz Alipour.

I would like to thank Mohammad Bazargan for providing the three reaction model. I

want to thank former and fellow students, including Morten Kristensen, Qing Chen,

Tianhong Chen, Jeremy Kozdon, Hui Zhou and many others. I would also like to

thank other alumnus Yuguang Chen, Tom Tang, Yuanlin Jiang, Yaqing Fan.

I would also like to thank all the other faculties, staff, and fellow students of

the Department of Energy Resources Engineering for making my doctoral study so

fruitful and enjoyable. Special thanks to Robert Lindblom, who is the coolest geologist

teacher I ever met.

I would like to thank my mentors and friends at BP, Steven Vittoratos and Chris

West, with special thanks to Steven who brought me into the Canadian heavy oil

community. I also would like to thank industry affiliates and mentors who provided

great help and advices during the course of my doctoral research work: Paul Naccache,

David Law, Paul Hammond, Franck Monmont, and many others.

This work was prepared with the support of the Stanford Graduate Fellowship

and financial aid from Schlumberger Ltd. The support of the Stanford University

Petroleum Research Institute (SUPRI-A and SUPRI-C) Industrial Affiliates is also

vii

acknowledged.

Finally, I would like to thank my family for their support during these years. I

want to especially thank my mother, Hui Jiang, who has devoted tremendous support

and encouragement for my education in the last twenty years. I would also like to

thank my cousin Mingyu Zhu and my uncle Hang Jiang who gave much support

during my PhD studies.

viii

Contents

Abstract iv

Acknowledgements vii

1 Introduction 1

1.1 Heavy Oil and Thermal Enhanced Recovery . . . . . . . . . . . . . . 1

1.2 Motivations and Objectives . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature Review 5

2.1 Streamline Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Streamline Simulation Concept and Basics . . . . . . . . . . . 5

2.1.2 Advantages and Disadvantages of Streamline Simulation . . . 6

2.1.3 Streamline Simulation for Complex Physics Processes . . . . . 9

2.2 In-situ Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 In-situ Combustion Process . . . . . . . . . . . . . . . . . . . 11

2.2.2 Field Applications of ISC . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Numerical Simulation of In-situ Combustion . . . . . . . . . . 15

2.2.4 Challenges in Combustion Reaction Modeling . . . . . . . . . 16

2.2.5 Alternative Approaches . . . . . . . . . . . . . . . . . . . . . . 16

2.2.6 Thermal Front Stability for ISC . . . . . . . . . . . . . . . . . 18

3 Governing Equations 21

3.1 General Conservation Equations . . . . . . . . . . . . . . . . . . . . . 21

ix

3.2 Phase Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Fluid Property Calculations . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Primary Equations and Variables . . . . . . . . . . . . . . . . . . . . 26

4 Thermal Streamline Simulation for Hot Water Flood 28

4.1 Streamline Simulation Framework . . . . . . . . . . . . . . . . . . . . 28

4.2 Streamline Formulation for Hot Water Flood . . . . . . . . . . . . . . 29

4.3 Specific Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.2 Operator Splitting for Non-advective Processes . . . . . . . . . 32

4.3.3 Treatment of Compressibility . . . . . . . . . . . . . . . . . . 33

4.4 Hot Water Flood Results . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4.1 Incompressible Hot Water Flood . . . . . . . . . . . . . . . . . 34

4.4.2 Compressible Hot Water Flood . . . . . . . . . . . . . . . . . 39

4.4.3 Gravity Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4.4 Heat Conduction Effects . . . . . . . . . . . . . . . . . . . . . 43

4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Thermal Streamline for Steam Flood 45

5.1 Streamline Simulation Framework . . . . . . . . . . . . . . . . . . . . 45

5.2 Streamline Formulation for Steam Flood . . . . . . . . . . . . . . . . 46

5.3 Specific Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3.1 1D Pressure and Volumetric Flux Update Approach for Large

Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3.2 Glowinski θ-Scheme Operator Splitting Approach for Non-advective

Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.3 Preliminary Hybrid Simulation . . . . . . . . . . . . . . . . . 56

5.4 Steam Flood Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.4.1 Heterogeneous Quarter-Five Spot Steam Flood . . . . . . . . . 58

5.4.2 Cyclic Steam Stimulation . . . . . . . . . . . . . . . . . . . . 62

5.4.3 Heterogeneous Multi-well Pattern Steam Flood . . . . . . . . 64

x

5.4.4 Vertical Cross Section Steam Flood . . . . . . . . . . . . . . . 69

5.5 Discussion and Applications . . . . . . . . . . . . . . . . . . . . . . . 72

5.5.1 Streamline Simulation as Fast Proxy: Cost Comparison . . . . 72

5.5.2 Optimization and Flux Patterns . . . . . . . . . . . . . . . . . 73


6 In-situ Combustion Simulation 76

6.1 Kinetic Reaction Models . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2 1D Combustion Tube Simulation . . . . . . . . . . . . . . . . . . . . 79

6.3 Sensitivity Studies on 1D ISC Problem . . . . . . . . . . . . . . . . . 85

6.4 Grid Size Effects and their Cause . . . . . . . . . . . . . . . . . . . . 89

6.4.1 Grid Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.4.2 The Cause of Grid Size Effects . . . . . . . . . . . . . . . . . . 91

6.4.3 Why not Adaptive Mesh Refinement? . . . . . . . . . . . . . . 92

6.4.4 The Need for Upscaling . . . . . . . . . . . . . . . . . . . . . 93


7 Upscaling for In-situ Combustion Reactions 95

7.1 Non-Arrhenius Reaction Modeling . . . . . . . . . . . . . . . . . . . . 96

7.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2.1 Pseudo Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2.2 Implementation in Commercial Software . . . . . . . . . . . . 100

7.3 Upscaling Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3.1 1D Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3.2 2D Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.3.3 3D Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.4 Discussion and Applications . . . . . . . . . . . . . . . . . . . . . . . 108

7.4.1 Valid Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4.2 Sub-grid Scale Heterogeneity and its Effects on Reaction Up-

scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111


xi

8 Front Stability Study for In-situ Combustion 121

8.1 1D Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.2 Contributors for Front Stability/Instability . . . . . . . . . . . . . . . 123

8.3 Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.3.1 Minimizing Numerical Errors . . . . . . . . . . . . . . . . . . 126

8.3.2 Small Sub-grid Scale Front Stability . . . . . . . . . . . . . . . 131

8.3.3 Large Field-scale Front Stability . . . . . . . . . . . . . . . . . 136


9 Future Directions 139

9.1 Thermal Streamline Simulation . . . . . . . . . . . . . . . . . . . . . 139

9.1.1 Non-advective Forces . . . . . . . . . . . . . . . . . . . . . . . 139

9.1.2 Guidelines for Commercial Code Development . . . . . . . . . 139

9.1.3 Thermal Streamline Simulation for SAGD? . . . . . . . . . . . 140

9.2 In-situ Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.2.1 History Matching of Field-scale ISC process . . . . . . . . . . 142

9.2.2 Calibrating the Operational Range for Sustaining ISC Combus-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.2.3 Grid Size Effects for Other Thermal or Non-Thermal EOR Pro-

cesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9.2.4 How to Use In-situ Combustion as a Gravity Drainage Process

in Fractured Carbonate Reservoir? . . . . . . . . . . . . . . . 145

10 Conclusions 150

A Simulation Inputs 154

B STARS Input File for Three Reaction Model 158

C Eclipse Input File for Three Reaction Model 166

D STARS Input File for Upscaled Three Reaction Model 179

E SAGD Input File 187

xii

F SAGD Input File for Grid Size Effects Study 193

Bibliography 203

xiii

List of Tables

4.1 Errors of different viscosity properties compared with reference result

(400× 400 STARS result) using L2 norm . . . . . . . . . . . . . . . . 37

4.2 Effects of thermal compressibility on the production (surface condition) 41

4.3 Heat Pe number and its influence on the simulation result. . . . . . . 44

5.1 Errors of thermal streamline and STARS compared to reference (360×180 STARS result) using the relative L2 norm. . . . . . . . . . . . . . 61

5.2 Sensitivity study on choice of θ. The simulation results close to each

other by using different θ. . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Errors of thermal streamline and STARS compared to reference (200×200 STARS result) using L2 norm. . . . . . . . . . . . . . . . . . . . 69

7.1 Explanation of major variables in the upscaling pseudo code. . . . . . 101

7.2 Explanation of major functions in the upscaling pseudo code. . . . . . 102

7.3 Summary of Sub-grid Scale Heterogeneity ISC Tests . . . . . . . . . . 118

A.1 Reservoir properties for hot water flood . . . . . . . . . . . . . . . . . 154

A.2 Viscosity relationships for hot water flood . . . . . . . . . . . . . . . 155

A.3 Fluid parameters for incompressible hot water flooding . . . . . . . . 155

A.4 Coefficients of density calculations in compressible hot water flood . . 155

A.5 Reservoir properties for steam flood . . . . . . . . . . . . . . . . . . . 155

A.6 Well control for steam flood . . . . . . . . . . . . . . . . . . . . . . . 156

A.7 Fluid parameters for steam flood . . . . . . . . . . . . . . . . . . . . 156

A.8 Reservoir properties for cyclic steam stimulation . . . . . . . . . . . . 156

xiv

A.9 Fluid properties in three reaction ISC model. . . . . . . . . . . . . . . 157

A.10 Rock properties in ISC simulation. . . . . . . . . . . . . . . . . . . . 157

A.11 Kinetics parameters for three reaction ISC model. . . . . . . . . . . . 157

A.12 Reaction stoichiometry for three reaction ISC model. . . . . . . . . . 157

xv

List of Figures

2.1 Streamline simulation framework, with the four major steps in a global

time step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Different zones in the field in-situ combustion process, courtesy of US

Department of Energy, Bartlesville, Oklahoma. . . . . . . . . . . . . . 13

2.3 Different zones in 1D in-situ combustion process. . . . . . . . . . . . 13

4.1 The permeability field in quarter five-spot hot water flood test case. . 34

4.2 The water saturation results in quarter five-spot hot water flood test

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 The temperature results in quarter five-spot hot water flood test case. 35

4.4 The pressure results in quarter five-spot hot water flood test case. . . 35

4.5 The comparison between 1D fully implicit (FIM) transport solver and

1D single point upwind explicit (SPU) transport solver for thermal SL

simulation (50X50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.6 The surface cumulative production results for M=10 quarter five-spot

hot water flood test case. . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.7 The surface cumulative production results for M=100 quarter five-spot

hot water flood test case. . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.8 The surface cumulative production results for M=1000 quarter five-

spot hot water flood test case. . . . . . . . . . . . . . . . . . . . . . . 39

4.9 The saturation results in compressible quarter five-spot hot water flood

test case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.10 The temperature results in compressible quarter five-spot hot water

flood test case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

xvi

4.11 The surface cumulative production comparison results in compressible

quarter five-spot hot water flood test case. . . . . . . . . . . . . . . . 41

4.12 Saturation results in gravity test case 1 (quarter five-spot hot water

flood with dipping). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.13 Saturation results in gravity test case 2 (quarter five-spot hot water

flood with dipping). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1 Steam condensation causes volume change and flux reduction right at

the steam front. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Streamline segments constructed from the injector to the producer. . 50

5.3 1D streamline pressure and volumetric flux update approach. Trans-

missibility and block volume of each segment is calculated for the 1D

transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.4 Streamline (s, n, m) coordinates and cross sectional area along the

streamline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.5 1D pressure update improvements in capturing transient pressure change

within a global time step, (M=1000 quarter five spot hot water flood

test case). With two BHP controlled wells, classical SL underestimates

the breakthrough. Both SL predict-correct [71] and SL with pressure

update improves the hot water breakthrough results, compared to fine-

scale reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.6 Pressure transient behavior inside a global time step (M=1000 quarter

five spot hot water flood test case). The pressure stitched together

from all the 1D pressure solves (B) is similar to the pressure we get at

the beginning of the next global time step (C). . . . . . . . . . . . . 54

5.7 Schematic of coupled energy-mass transport. Changing the total en-

ergy of a grid block changes the fluid volume inside, thus causing fluid

transport to adjacent cells. . . . . . . . . . . . . . . . . . . . . . . . . 55

5.8 Glowinski θ-scheme operator splitting. Most of the transport is solved

along 1D streamlines [0 ∼ (1− θ)∆t]. Small amount of advective flux

is used to correct the volume changes in heat conduction/gravity step. 56

xvii

5.9 An equivalent representation of Glowinski θ-scheme operator splitting.

Most of the transport is solved along 1D streamlines. Small amount

of advective flux is used to correct the volume changes in heat conduc-

tion/gravity step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.10 Quarter five spot steam flood permeability field and initial streamline

shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.11 Viscosity curve for water and oil. . . . . . . . . . . . . . . . . . . . . 59

5.12 Quarter five spot temperature and gas saturation at 3000 days (pre

breakthrough). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.13 Quarter five spot water and oil saturation at 3000 days (pre break-

through). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.14 Sensitivity study on the choice of θ. The simulation results are close

to each other by using different θ. . . . . . . . . . . . . . . . . . . . . 62

5.15 Hybrid procedures in cyclic steam stimulation (SL → FV → SL) and

the initial streamline shape. . . . . . . . . . . . . . . . . . . . . . . . 63

5.16 Cyclic steam stimulation temperature result. . . . . . . . . . . . . . . 64

5.17 Cyclic steam stimulation oil saturation result. . . . . . . . . . . . . . 65

5.18 Cyclic steam stimulation production profile (SL and FV). . . . . . . . 65

5.19 2D multiple well test case permeability field. . . . . . . . . . . . . . . 66

5.20 2D multiple well test case initial streamline shape. . . . . . . . . . . . 66

5.21 Multiple well test case temperature and oil saturation at 1500 days

(pre breakthrough). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.22 Multiple well test case field production history (pre breakthrough). . 68

5.23 Multiple well test case temperature and oil saturation at 2100 days

(post breakthrough). . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.24 Multiple well test case field production history (post breakthrough). . 69

5.25 2D vertical cross section steam flood test case 1 (homogeneous perme-

ability field). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.26 2D vertical cross section steam flood case 2 (heterogeneous permeabil-

ity field). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.27 Flux pattern map, by Thiele and Batycky [106]. . . . . . . . . . . . . 74

xviii

6.1 Gas composition from ramped temperature oxidation experiments. . . 77

6.2 Relative permeability (water-oil and liquid-gas, respectively) and oil

viscosity curve used in the simulation of Hamaca oil 1D combustion

tube experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.3 Temperature, oil saturation and pressure profiles in 1D combustion

tube simulation after 498 min of air injection. . . . . . . . . . . . . . 81

6.4 Typical temperature profile history from a 1D combustion tube exper-

iment, from Lapene et al., [66] . . . . . . . . . . . . . . . . . . . . . . 81

6.5 Water, oil and gas production rates in 1D combustion tube simulation. 83

6.6 Produced gas composition analysis in 1D combustion tube simulation. 84

6.7 Oil bank and combustion front distance-time diagram from 1D com-

bustion tube simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.8 1D ISC simulation: peak temperature and combustion front velocity

as a function of the injection air flux. . . . . . . . . . . . . . . . . . . 85

6.9 1D ISC simulation: oil saturation profile as a function of the initial oil

saturation Soi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.10 Temperature profiles as a function of the initial water saturation Swi

of 0.10, 0.24 and 0.50 from left to right. . . . . . . . . . . . . . . . . . 88

6.11 Combustion tube simulation of a mobile heavy oil with high initial oil

saturation Soi at 378min. . . . . . . . . . . . . . . . . . . . . . . . . 89

6.12 Characteristics of different fronts in 1D ISC problem: scenario 1 (bi-

tumen like oil viscosity and oil bank stays inside the steam plateau). . 90

6.13 Characteristics of different fronts in 1D ISC problem: scenario 2 (mo-

bile heavy oil viscosity and leading edge of steam plateau follows behind

the oil bank). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.14 Basic assumption for thermal reservoir simulation: instantaneous mix-

ing in each cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.15 Grid-size effects in 1D combustion tube simulation. We require at

least 200 grid blocks to achieve convergent solution in this 0.8m long

combustion tube simulation. . . . . . . . . . . . . . . . . . . . . . . . 92

xix

7.1 The critical physics for 1D ISC. When the reaction front sweeps through

the reservoir, a certain amount of oil becomes fuel and burns (usually

x = 5 ∼ 10% So). The rest of the mobile oil is displaced further

downstream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.2 ISC upscaling work flow. . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.3 1D ISC upscaling tests: kinetic and upscaled model using different grid

resolutions. The coarse scale upscaled model matches the fine-scale

kinetics based reference, while the coarse scale kinetic model fails. . . 103

7.4 1D ISC upscaled model with different equivalent fuel amount Sofuel.

The front location has a direct relationship with the fuel amount Sofuel. 103

7.5 ISC upscaling in horizontal 2D 1/4 five spot case: O2 fraction in gas

phase with different grid resolution and both kinetic and upscaled model.105

7.6 ISC upscaling in horizontal 2D multi well case: coarse grid upscaled

model and fine grid kinetic model. . . . . . . . . . . . . . . . . . . . . 105

7.7 ISC upscaling in horizontal 2D heterogeneous case: coarse grid up-

scaled model and fine grid kinetic model. . . . . . . . . . . . . . . . . 106

7.8 ISC upscaling in 2D vertical case: coarse grid upscaled model and

fine-grid kinetic model. . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.9 3D field-scale case using the upscaled reaction model. Consistency is

achieved between different grid resolutions. . . . . . . . . . . . . . . 109

7.10 Permeability and oil saturation results for 2D case with uncorrelated

white noise permeability field. The combustion front propagates stably. 112

7.11 2D case with correlated heterogeneity (low perm in the middle). We

observe some oxygen bypass and change in equivalent fuel amount. . 113

7.12 2D case with correlated low permeability squares. We calculate oxygen

efficiency Eu = 0.90 and fuel amount Sofuel = 9.7%. . . . . . . . . . 114

7.13 2D case with horizontal layered permeability. We observe oxygen by-

pass and change in equivalent fuel amount. . . . . . . . . . . . . . . . 115

7.14 2D case with heterogeneity (layer 1 of SPE 10 [23]). The oxygen effi-

ciency is Eu = 0.91 and fuel amount Sofuel = 8.6%. . . . . . . . . . . 116

xx

7.15 2D case with channelized heterogeneity (layer 51 of SPE 10 [23]). The

oxygen efficiency is Eu = 0.85 and fuel amount Sofuel = 13%. . . . . 117

7.16 Different scales in measurements, geomodeling and reservoir simula-

tion, lecture notes of ERE 241 Seismic Reservoir Characterization. . . 120

8.1 Different fronts and flow regimes in 1D ISC process. . . . . . . . . . . 123

8.2 Perturbation length λ at leading edge of steam plateau, from [50]. . . 125

8.3 Trigger instability by changing the initial oil saturation at the boundary.127

8.4 Test of numerical errors with triggered instability. The effect of grid

size and 5-point versus 9-point scheme is illustrated. . . . . . . . . . . 129

8.5 Test of temporal numerical errors with automatic time-step selection

and restricted time step sizes. . . . . . . . . . . . . . . . . . . . . . . 130

8.6 Sensitivity to thermal conductivity. Heat conduction stabilizes the dis-

placement front in small lab-scale tests. Reduced thermal conductivity

cases show greater instability. . . . . . . . . . . . . . . . . . . . . . . 132

8.7 Nusselt number characterizes the heat conduction stabilizing effect,

compared to Fig. 8.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.8 Pore blocking with the Kozeny-Carman correlation. Very small changes

are observed compared to base case. . . . . . . . . . . . . . . . . . . 134

8.9 Pore blocking with an exaggerated permeability reduction. The com-

bustion front slows down only when implementing pore blocking of

more than one order of magnitude reduction in permeability. . . . . 135

8.10 Unstable displacement in 2D larger scale ISC process. Heat conduction

is incapable of dissipating the energy of large wavelength perturbations. 137

9.1 Total phase velocity vectors in 2D SAGD process. Convection cell/loop

current occur when tracing streamlines. . . . . . . . . . . . . . . . . 141

9.2 Grid size effects in 2D SAGD simulation (early time 360 days). The

simulation achieves convergent solution when using grid size 0.5m ×0.5m in this case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

xxi

9.3 Grid size effects in 2D SAGD simulation (late time 1100 days). The

simulation achieves convergent solution when using grid size 0.5m ×0.5m in this case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

9.4 Concept of ISC assisted gas oil gravity drainage in fractured media.

Vent well is added to achieve hydrostatic gravity drainage condition in

the reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.5 Gravity stable ISC process for a single block of fractured media. Be-

cause of the gas diffusion, we are able to combust into the matrix.

Most of the flue gas is produced from upper right corner. . . . . . . . 149

xxii

Chapter 1

Introduction

1.1 Heavy Oil and Thermal Enhanced Recovery

A large part of the world oil resource exists in the form of heavy oil, which is usu-

ally defined as oil with API gravity less than 22 and viscosity typically larger than

100cp. Estimated original oil in place of more than 1.8 trillion barrels is present in

Venezuela, 1.7 trillion barrels in Alberta, Canada, and 20- 25 billion barrels on the

North Slope of Alaska. The development of such resources by traditional methods

(primary depletion, waterflood) is often inefficient due to the high viscosity of the

heavy oil. At such high viscosities, the oil flows extremely slowly or not at all. For

example, the bitumen resources in Athabasca oil sands typically have an extremely

large viscosity of about 106cp.

Thermal recovery processes rely on viscosity reduction of the oil through heat that

is injected (steam or hot water injection) or generated in-situ (in-situ combustion),

and are well suited for efficiently unlocking these heavy oil resources. Accordingly

to current U.S. Department of Energy data, thermal enhanced recovery techniques

account for about 50% of the domestic Enhanced Oil Recovery (EOR) production.

Steam flooding, cyclic steam stimulation and hot water flooding are widely used,

but other processes, such as in-situ combustion (ISC) and increasingly steam-assisted

gravity drainage (SAGD) are applied and are attractive to recover heavy oil resources.

1

CHAPTER 1. INTRODUCTION 2

1.2 Motivations and Objectives

Planning and management of thermal EOR processes generally make extensive use

of reservoir simulation. Nearly all the commercial and academic thermal simula-

tors are traditional finite volume (FV) based codes that use either a fully implicit

(FIM) time stepping method or an adaptive implicit (AIM) method [6, 37, 101]. The

computational costs for simulating thermal processes are usually high because of the

complexity of the problem and strong nonlinearity encountered. As a result, it is

time consuming to run optimization and/or sensitivity studies on grids with desirable

numerical resolution. For problems such as ISC and SAGD, the simulations show

extremely large computational costs with very strict requirements on the sizes of

grid blocks, due to the need to capture/resolve accurately the narrow thermal fronts

existing in these processes. Field-scale ISC simulation is still impractical, due to

large computational costs associated with accurately resolving the inch-sized reactive

combustion fronts. There is, therefore, an urgent need to develop fast and efficient

numerical simulation methodologies for thermal EOR problems. Predictive mathe-

matical models and efficient simulators are needed to improve our understanding of

these thermal EOR processes and enable cost-effective design of these projects.

In this work, we focus on three major problems: the extension of streamline

simulation (SL) to thermal problems (hot water flood and steam flood), the efficient

simulation of field-scale ISC process through reaction upscaling, and the analysis of

thermal front stability in ISC. The detailed objectives are as follows:

1. For the thermal streamline simulation, we are seeking a fast and effective reser-

voir simulator that gives sufficient accuracy for use in reservoir simulation stud-

ies, such as ranking, optimization and history matching. This is a first time that

streamline simulation is extended to complex thermal problems such as steam

flood. Problems such as fluid compressibility, strong coupling and gravity effects

need to be addressed.

2. For the ISC simulation, our main objective is to find an efficient simulation

technique to upscale the reaction kinetics for full field-scale ISC simulation. To


achieve this goal, the key problem relies on how to capture accurately the thin

reactive combustion front. We design upscaled reaction models that are tailored

to field-scale ISC process, which take into account the multi-scale nature of the

process.

3. Driving front stability is an important part for designing EOR processes. Ther-

mal front stability criteria needs to be extended to the ISC process. The stabil-

ity analysis is based on both analytical analysis and numerical tests. The front

stability is essential for guiding the design of field-scale ISC process, in which

different injection/production or wet/dry combustion scenarios are evaluated.

1.3 Thesis Outline

This thesis is organized in six parts:

Chapter 2 summarizes the literature on the main research topics pertinent to

this work, thermal reservoir simulation, streamline simulation, in-situ combustion

processes and thermal front stability.

Chapter 3 gives the basic equations for thermal compositional problems, including

the basic mass and energy transport equations, phase equilibrium, fluid property

calculations and kinetic chemical reactions.

Chapter 4 shows the thermal streamline simulation for hot water flood, in which

we include small compressibility, heat conduction and gravity effects.

Chapter 5 discusses the further extension of streamline simulation to steam flood-

ing. We extend streamline simulation to problems with strong compressibility, com-

plex phase behaviors, nonlinear coupling, and large gravity segregation effects. We im-

plement several novel techniques, include 1D pressure/flux update along the stream-

line, Glowinski θ-scheme operator splitting and hybrid finite volume/streamline sim-

ulation. The results show thermal streamline simulation capable of retaining solution

accuracy while reducing computational cost.

Chapter 6 shows the formulation of general reactive thermal reservoir simulation

for in-situ combustion. We then analyze the actual cause for the grid-size effects in


in-situ combustion simulation. Based on this, we conclude that ISC is a multi-scale

process such that the reaction kinetics and flow transport exist in two completely

different scales.

Chapter 7 introduces the concept of upscaling the chemical reaction kinetics for

field-scale in-situ combustion simulations. We present the reaction upscaling method-

ology and workflow. We show test cases to demonstrate the consistency achieved by

reaction upscaling and the capability for handling field-scale problems.

Chapter 8 shows the reaction front stability study for in-situ combustion process.

Multiple fronts and flow regimes are identified for the ISC process. We further define

the five major factors affecting the front stability, viscous force, condensation, heat

conduction, coke plugging and gravity. Numerical simulation tests are performed to

explore and construct the stability criteria and regions.

In Chapter 9, we propose the future research directions for thermal streamline

simulation and in-situ combustion process simulation.

Chapter 2

Literature Review

2.1 Streamline Simulation

2.1.1 Streamline Simulation Concept and Basics

Most of the academic and industrial reservoir simulators are based on Eulerian meth-

ods, in which the grids are considered stationary and fluids are transported between

discrete cells [6]. All these simulators treat the pressure equation implicitly. They

usually use either fully implicit and adaptive implicit time stepping, or implicit pres-

sure explicit saturation/composition (IMPES/IMPEC) method [18]. The computa-

tional costs are often very high, when simulating large full field reservoir models

with large number of simulation grid blocks. For thermal enhanced recovery simula-

tions in particular, the computational costs are even greater compared to black oil or

compositional problems, because of the large number of unknowns, complex thermal

phase behavior, strong nonlinearities in the system and highly coupled nature of the

governing equations. This creates great difficulty for running optimization and/or

sensitivity studies on grids with desirable numerical resolution.

Streamline-based flow simulation is a fundamentally different simulation technique

than traditional Eulerian methods [71, 7]. Streamline is defined as the curve that is

instantaneously tangent to the velocity vector of the flow. Streamline simulation is

based on the observation that in heterogeneous reservoirs the time scale with which

5

CHAPTER 2. LITERATURE REVIEW 6

fluids flow along streamlines is often faster than the time scale at which the streamline

locations change significantly. This allows decoupling of the costly 3D transport

problem into a set of 1D advection problems along the streamlines. Streamline based

reservoir simulation is conducted based on a series of global time steps and a dual grid

approach [7]. During each global time step, the pressure equation is initially solved

implicity on the Eulerian grid, followed by streamline tracing on the corresponding

fixed total phase velocity field [92]. Flow transport is then calculated along each

streamline, by solving the corresponding fractional flow transport problems along

the 1D streamline grid. A coordinate transformation is conducted to transfer the

flow problem from physical arc-length coordinate to 1D time of flight (TOF) grid

[7]. The time of flight is the time required to reach a distance, s, along a streamline

based on the velocity field along the streamline. Eventually, the solution is mapped

back to the original Eulerian grid, which concludes a global time step. Streamline

methods were originally applied to incompressible, multi-phase flow problems without

gravity, capillary or diffusive terms in the transport equations. Gravity, capillary and

diffusive terms have since been included through the operator splitting techniques

[14, 13], usually at the end of a global time step. The entire work flow for a global

time step in streamline simulation is shown in Fig. 2.1

2.1.2 Advantages and Disadvantages of Streamline Simula-

tion

Streamline simulation is deemed as an effective and complementary technology to

traditional Eulerian finite volume based simulation. This is because streamline simu-

lation is particularly effective in solving large, geologically complex and heterogeneous

systems, where flow is dictated by well rates and positions, reservoir structure, per-

meability and porosity, fluid mobility and gravity. These are the problems for which

traditional finite volume simulation has difficulties. A thorough summary and review

of streamline simulation theory and application is provided by Thiele [105].

The most important advantage of streamline simulation is its inherent efficiency.

Streamline simulation has been successfully applied to reduce computational costs


Figure 2.1: Streamline simulation framework, with the four major steps in a globaltime step.


whilst retaining accuracy. The computational efficiency is achieved because the trans-

port problem is decoupled from the 3D grid and solved along each 1D streamline,

which can use well-established mathematical techniques. This follows an inherent di-

vide and conquer strategy, in which a high dimension fully coupled system is divided

into a series of smaller 1D systems and solved individually. Furthermore, the number

of streamlines typically increase linearly with the number of active cells. And the

streamlines only need to be updated infrequently, which guarantees the usage of large

global time steps and a smaller number of pressure updates. This is different from

IMPES/IMPEC type methods, whose time step sizes are constrained by the well-

known Courant-Friedricks-Lewy (CFL) stability restriction. This is why streamline

simulation can exhibit a close to linear scaling in run times as a function of active

cells in the model, according to Thiele [105].

The bulk of the computational work in streamline simulation is typically associated

with solving the relevant transport equations along the 1D independent streamlines for

compositional/thermal problems. This makes the streamline simulation architecture

inherently parallelizable. Recent work on a research streamline simulator provides

promising results in parallel speed-up [70]. Parallelization of a commercial streamline

simulator to multi-core architectures based on the OpenMP programming model and

its performance on various field examples have also been presented [9].

The computational efficiency of streamline simulation is especially suitable for

work flows involving many simulations as might be the case in ranking, history match-

ing and optimization problems. Commercial streamline simulation is applied for such

purposes [1]. One example is the reservoir history matching studies for Iraq’s South

Rumaila field that first used simple material-balance approach, followed by fast reser-

voir simulations with streamlines, and eventually full finite volume simulation in se-

quence to aid understanding of reservoir-flow behaviors [59].

A powerful application of streamline simulation is the capability to define dynamic

well allocation factors (WAFs) and injection efficiencies (IEs) using streamline derived

information [106]. Because WAFs and IEs are derived directly from streamlines, the

data reflect all the complexities impacting the dynamic behavior of the reservoir

model. IE quantifies how much oil can be recovered at the offset producing wells for


every unit of water injected by an injector. Improved water flood management can

be implemented by reallocating injection water from low-efficiency to high-efficiency

injectors.

Streamline simulation is a type of Euler-Lagrange method, in which the pressure

and transport equations are treated and solved in different stages. Compared to fi-

nite volume simulation, streamline simulation implements a dual grid and dual time

stepping approach. Therefore, one disadvantage is that it is not mass conservative,

because of the mappings between the Eulerian grid and 1D streamline grid. Improved

mapping methods have been proposed to control the mass balance errors [72]. De-

spite the lack of exact mass conservation, studies have shown that the mass balance

errors are usually relatively small and streamline methods predict the global sweep

in heterogeneous reservoirs effectively.

2.1.3 Streamline Simulation for Complex Physics Processes

Streamline/streamtube simulation was initially applied successfully to problems such

as water flooding, first contact miscible flooding and tracer flows [76, 8, 7]. Streamline

simulation results have shown advantages in retaining simulation accuracy, reducing

numerical dispersion and achieving speed-up against finite volume simulations. Ac-

cordingly, commercial streamline simulators are designed to handle black oil, first

contact miscible injection, and tracer flows [1].

Since then, streamline simulation has also been applied for gas injection pro-

cesses and enhanced gas condensate processes, which includes full equation of state

(EOS) compositional effects [107, 29, 45, 56, 99]. For multi-contact miscible gas

injection, high resolution 1D solutions are constructed either by Method of Charac-

teristics (MOC) analytical solutions or higher order accurate numerical discretization

schemes, such as total variation diminishing schemes (TVD) and essentially non-

oscillatory schemes (ENO) [71, 100]. These techniques more accurately represent

the composition paths in phase space for 1D displacements, and hence, better pre-

dict local displacement efficiency than conventional first order numerical simulation.

While the MOC analytical solutions are attractive, they can only be used along


SL that have constant initial and boundary conditions and cases where the stream-

lines are fixed over all time. This is the major limitation for mapping analytical

solutions along the streamline. Streamline simulation enables the usage of the high

accuracy 1D solution techniques for solving the 3D reservoir processes, because the

flow transport is solved along 1D streamlines. The simulation results have shown

improvements in capturing the viscous fingering effects and reducing the numerical

smearing/dispersion in simulating the 3D miscible/immscible gas injection and gas

condensate processes. Streamline simulation effectively controls the numerical dis-

cretization errors and makes reliable predictions of the local displacement efficiency

and global sweep efficiency.

Chemical EOR processes, for example polymer flooding, can also be modeled

effectively by streamline simulation [104]. Because the flow transport is solved along

each streamline, it is convenient to include accurate 1D solutions using locally refined

grids blocks, which can potentially help to control the grid size effects (numerical

errors due to using large grid blocks) encountered in chemical EOR simulations [111].

Streamline simulation has also been extended to model fractured media dual

porosity single permeability problems [36]. The matrix-fracture interactions are in-

cluded in this work for two-phase transport in fractured systems. Fluid transfer be-

tween fracture and matrix enters as source/sink terms in the 1D transport equations

along the streamlines.

Traditional streamline simulation has been based on the incompressible flow as-

sumption [7], in which the incompressible total phase velocity field is fixed during

a global time step. This becomes a strong restriction in cases when we need to in-

clude fluid and rock compressibility. Consequently, streamline simulation has been

extended to account for compressible fluid and rock properties [88, 12]. In the first

work [21, 88], the total phase velocity field is still fixed during a global time step. The

divergence of the velocity field is added into the 1D transport equation as source/sink

terms along each streamline. In this way, the changes in the fluid volume with pres-

sure is captured, accounting for the compressibility effects. In the second work [12],

a compressible formulation for streamline simulation is presented, that also considers

API gravity tracking. Compared to the previous work by Cheng [21], cumulative


volume is used in the 1D transport, instead of time of flight, which reduces mass-

conservation errors and further allows the update of fluid phase velocities along each

individual streamline. Test cases have shown its capability to handle significant liquid

compressibility.

To the best of the author’s knowledge, very little work has been conducted in

extending streamline simulation to heavy-oil or thermal enhanced recovery problems.

Streamline simulation is investigated as a potential method for heavy-oil waterflood

performance forecasting [87]. The authors show that streamline simulation is a pow-

erful tool for quick screening of heavy-oil waterflood projects, estimating oil recovery,

and water breakthrough time. The overall streamline results are similar to that of

FV simulation, with reduced grid orientation effects. In the recent work by Pasarai

and Arihara [89], the authors propose an extension of streamline methods to hot

water flood, which solves the energy equation along streamlines. The algorithm first

computes the mass transport equations based on FV simulation and the current tem-

perature. It then solves the energy equation along the 1D streamlines. The work may

be further improved by fully solving both mass and energy transport along streamlines

and rigorously accounting for the compressibility effects and non-advective forces.

There is an urgent need for fast approximate solvers for thermal simulation prob-

lems. The essence of this work is to explore the extension of streamline simulation

to thermal enhanced recovery processes, which include hot water flooding and steam

flooding. Our aim is to design a fast and effective simulator that gives sufficient

accuracy for use in reservoir simulation studies, such as ranking, well placement op-

timization and history matching.

2.2 In-situ Combustion

2.2.1 In-situ Combustion Process

Among all the thermal recovery processes, ISC has been a source of interest for a

long time. In-situ combustion (ISC) is the process of injecting air (or air enriched

with oxygen) into oil reservoirs to oxidize part of the crude oil and enhance recovery


through the heat and pressure produced [93, 96]. In contrast to other thermal recovery

processes, the main energy required to displace the oil in ISC is generated in the

reservoir from the heat released by combustion reactions between oxygen and fractions

of the crude oil. In-situ combustion has been successfully applied to both light oil

and heavy oil recovery.

A typical field-scale forward ISC process is shown in Fig. 2.2. The oil is initially

ignited around the injection well. Continuous air injection keeps the combustion front

propagating towards the production well. As air is injected and the combustion front

moves in the reservoir, several zones are identified. Starting from the injector, the

first zone is the burned zone in which the reservoir volume has already been swept

by the combustion front. The burned zone typically contains the injected air and

almost no hydrocarbons. The combustion front has the highest temperature and

this is the place where most of the chemical reactions are taking place. Most of the

heat is generated in this narrow zone, typically several inches thick. The injected

oxygen reacts with the hydrocarbons, generating flue gas and steam. Hydrocarbons

contacted by the leading edge of the high temperature zone undergo thermal cracking

and vaporization. Mobilized light components are transported downstream where

they mix with the original crude. The heavy residue, which is normally referred to

as solid coke, is deposited on the core matrix and is the main fuel source for the

combustion process. Downstream of the vaporization zone is the steam plateau that

is formed from water of combustion and vaporization of formation water. Further

downstream, the steam condenses into a hot water bank when the temperature drops

below the steam saturation temperature. The leading edge of the hot water bank

is the primary area of oil mobilization where the oil is banked by the hot water. A

schematic of the 1D ISC process is also shown in Fig. 2.3.

2.2.2 Field Applications of ISC

Since its discovery in the early 20th century, ISC as an enhanced oil recovery process

has experienced a checkered history [96]. It was initially successfully applied as an

effective thermal recovery technique in California heavy-oil fields in the 1950s and


Figure 2.2: Different zones in the field in-situ combustion process, courtesy of USDepartment of Energy, Bartlesville, Oklahoma.

Figure 2.3: Different zones in 1D in-situ combustion process.


1960s, such as the West Newport ISC project [96]. The success achieved in California

ISC projects brought the process to Canada during the 1970s, 1980s and 1990s. Some

successful projects were demonstrated in the Lloydminster heavy-oil accumulations

in Canada, such as Murphy Oil’s Eyehill thermal project [81], Amoco’s Morgan cyclic

combustion project [75] and Husky’s ISC projects [65]. ISC has also been successfully

used as a follow-up process to oil sand cyclic steam stimulation in BP Canada’s Wolf

Lake bitumen thermal project [48]. This project is well known as the pressure-up

blowdown combustion process in a channeled reservoir with existing thermal com-

munication with adjacent wells. In spite of these successes, there were many more

failures. For example, in the Viking Kinsella Wainwright B project, the combustion

front failed to sustain and switched over to the undesirable low temperature oxidation

(LTO) mode, when the air injection rate was decreased [17]. ISC processes in Cana-

dian heavy oils and oil sands rarely reached its theoretical potential. Comprehensive

review of the field trials, successes and failures, and operational issues in Canada are

summarized [110, 80].

Currently the successful heavy oil ISC commercial projects in operation include

the Suplacu de Barcau field in Romania and the Balol/Santhal fields in India. The

Suplacu field is a conventional shallow heavy-oil field, that started in-situ combustion

process in 1965. The project implements a line-drive configuration and now has an

approximately 11 km long combustion front that is gradually driven down-dip within

the reservoir. As the combustion front moves close to the down-dip row of producers,

the producers are converted to new injectors and the old injector rows are converted

to water injectors. The details of the Suplacu field operations are summarized in [91].

The Balol/Santhal ISC projects started in 1990. They are heavy/viscous oil reservoirs

in the heavy-oil belt of Mehsana, India. Similarly, they implemented air injection from

up-dip of the reservoir and drive the combustion front down-dip towards the producer.

Both dry and wet combustion were used in this project [94].

In-situ combustion or air injection has also been applied for light-oil reservoirs,

which is often known as the High Pressure Air Injection (HPAI) process [46, 79, 42].

One of the most well known early light oil in-situ combustion project is the double

displacement process in Amoco’s West Hackberry project [41]. HPAI has also been


widely used as an enhanced oil recovery technique for carbonate reservoirs. It is the

distant second most widely used EOR technique to CO2 injection for carbonate reser-

voirs in the United States [73]. Most of the light-oil HPAI processes are operated at

the low temperature oxidation mode without high temperature combustion. These

processes mostly generate an immiscible flue gas drive, or serve as a pressure main-

tenance technique during the gravity drainage process. In this work, we will focus

mainly on the heavy-oil in-situ combustion process, where the reactions are intended

to operate in the exothermic high temperature oxidation (HTO) mode. The high

temperature oxidation and low temperature oxidation modes are discussed in detail

in section 6.1.1.

2.2.3 Numerical Simulation of In-situ Combustion

In traditional thermal reservoir simulation, conservation equations are solved numer-

ically on discretized reservoir grid blocks [27, 112, 61]. The governing mass and

energy transport equations for reactive thermal compositional problems include ex-

tra source/sink terms to represent the chemical reaction effects. The reaction source

and sink terms are calculated cell by cell using Arrhenius kinetics during each time

step, as shown in Eq. 3.16. This is the common reaction modeling technique in most

commercial simulators [37, 101].

Different Arrhenius kinetic reaction models have been developed for ISC simula-

tions. All these reaction models are based on multiple oil pseudo components and on

grouping the chemical reactions into a series of pseudo reactions. Parameters in these

models are usually derived from laboratory experiments of kinetic cells [24]. One of

the earliest and most commonly used models is the so-called Crookston model, also

known as the minimal model [30]. It comprises two oil components (heavy oil HO

and light oil LO) and four chemical reactions. Both HO and LO directly combust

with oxygen. The component HO pyrolyzes generating coke. Eventually, coke burns

completely with oxygen. Other models such as the Belgrave model [11] and SARA

model [43] group pseudo components of the crude oil based on the SARA fractions

(Saturate, Aromatic, Resin and Asphaltene). These methods also employ a series of


reactions, including coking (pyrolysis), low temperature oxidations, and high temper-

ature oxidations. The reactions generate some intermediate compound products and

eventually fully combust to generate heat. Another more simplified three reaction

model has also been proposed, which includes only one oil component and three se-

quential kinetic reactions (pyrolysis, LTO and HTO) [33]. The details of LTO and

HTO reaction modes are provided in section 6.1.1.

2.2.4 Challenges in Combustion Reaction Modeling

Most ISC simulation studies are performed on small laboratory scale problems, such

as history matching of kinetic cells and combustion-tube experiments. It is well

known that when increasing the length scale to full field reservoir models, severe

grid-size effects are encountered in in-situ combustion simulations [61, 112, 28, 47, 53].

This means the simulation results, especially the combustion front speed and peak

temperature, become functions of the grid block sizes.

To mitigate the grid-size effects in ISC simulations, various empirical and heuristic

techniques have been developed. In many cases, the reaction kinetics parameters are

changed or tuned to force the oil to burn in a large field-scale grid block [47]. This

can be achieved, for example, by adjusting the activation energy to small or even zero

values, which lowers the threshold crude-oil burning energy requirement. In some

cases, the temperature values in Arrhenius kinetics are also adjusted to make the

oil much easier to burn [28]. In another work [53], the oil phase flux has also been

modified/enhanced in an attempt to force more oil to flow out of the grid blocks

where combustion reactions are taking place. All these parameter adjustments have

to be repeated every time the grid size is changed. A comprehensive summary of all

these techniques is presented in [74].

2.2.5 Alternative Approaches

Alternatives to the Arrhenius kinetics based reservoir simulation first include the

various types of analytical solutions. Empirical and analytical models, such as the

Ramey and Gates model, were developed for prediction of ISC recovery performance


[44]. The Ramey and Gates model is essentially based on the material balance cal-

culations that estimates the amount of oil and water displaced from the volume of

reservoir swept by the high temperature oxidation front. Many variants have been

proposed since then [64], with the material balance main concept unchanged. A re-

view of these approaches and corresponding economic considerations for ISC process

is summarized by Nodwell [85]. Another set of analytical solutions were derived by

Akkutlu [3]. The authors proposed an analytical solution based on a constant pre-laid

fuel (coke) amount, and coke burning reactions with the injected oxygen.

There also exist many alternatives to Arrhenius kinetics in reservoir simulation.

A thin-flame technique is suggested for in-situ combustion simulation in a 1D system

[32]. In this work, the reaction front velocity is assumed to be constant. A coordinate

transformation is conducted to track the front at the same speed, thus changing the

governing equations to a steady-state boundary value problem. Furthermore, a trial

for field-scale history match is also performed for the Suplacu field pilot in Romania

[91]. Constant coke deposition and fuel amount is assumed in this simulation study,

that greatly reduces the grid-size effects. The fuel burning reaction is still based on

Arrhenius kinetics, with temperature modified according to method by Coats [28].

Finally, a field-scale in-situ combustion simulator is developed using a moving front

representation by Hwang [52]. The front is viewed not only as a moving heat source,

but also as a displacement pump enhancing oil flow. In this simulator, detailed

kinetics and thermodynamic relations are not used. Instead it approximates the

overall effects of combustion with a few measurable parameters. All these approaches

share one thing in common, that is to simplify the traditional kinetics based simulation

and introduce approximations valid at larger scale.

Finally, specialized numerical techniques such as adaptive mesh refinement [22, 84]

and operator splitting [61] have also been proposed or applied to ISC modeling. These

methods lead to some improvements. For example, simulation accuracy is clearly

improved through the use of operator splitting technique to capture the fast kinetic

reactions separately from the flow transport using higher order ordinary differential

equation (ODE) solvers [61]. The dynamic gridding option has been implemented in

commercial thermal simulator CMG STARS [101]. The gridding identifies the moving


front through large gradients of specific properties (temperatures, fluid saturations

and compositions). In the front vicinity, it will de-amalgamate the original coarse

cells, and later on re-amalgamate them once the front has passed through. Though

test cases of ISC process, it is shown that dynamic gridding is capable to divide the

CPU time of thermal simulations by a factor of 2 to 3 [22]. The simulator, however,

still lacks the capability to scale up the grid sizes several orders of magnitude to coarse

grid full field reservoir simulation. A field-scale predictive ISC simulator is therefore

urgently needed.

2.2.6 Thermal Front Stability for ISC

Front stability is an important issue when designing an EOR processes. In the classical

steam flooding problems, the thermal front stability criterion has been well established

[15]. In a 2D horizontal problem, the main factors affecting steam flood front stability

include: pressure gradient (viscous force), steam condensation, and heat conduction.

Among them, the pressure gradient contrast serves as a destabilizing effect, because

of the adverse mobility ratio of steam driving the hot water and the viscous oil

bank. The condensation of steam into hot water serves as a stabilizing effect. A

detailed formulation of front-stability criterion is presented for the idealized problem

of steam driving hot water in the porous media in [15]. Heat conduction also plays an

important role in stabilizing the steam front, which is dependent on the perturbation

wavelength λ [78]. For small scale perturbations, heat conduction quickly dissipates

the heat of the steam finger, thus stabilizing the front. On the other hand, for

larger scale perturbations, the stabilizing effects is less significant. This actually

serves as one of the main mechanisms for using steam injection as EOR method for

highly heterogeneous reservoirs such as fractured carbonate and fractured diatomite

formations [50]. A combined study of analytical methods and thermal simulation to

examine frontal stability in high-porosity diatomite rocks is presented in [50].

One of the early work of examining the front stability for wet/dry in-situ combus-

tion process is by Armento et al., [5]. An idealized model is introduced to analyze the

combustion zone stability. On the upstream size, it is the burned zone with air and


water. In the downstream side, it is the region with combustion generated steam/flue

gas mixture. The similar analytical approach for steam front stability [78] is utilized

here for studying the combustion zone stability. The drawback of this study is that

it does not take into account the existence of heavy oil in the in-situ combustion

process. Furthermore, multiple fronts besides the combustion zone exist in the ISC

process, as it will be shown later in chapter 8. A more detailed study is required to

fully understand the front stabilities.

In the recent work by Javad et al., [54], a 2D vertical physical experiment was im-

plemented in order to study the combustion front stability in a top down combustion

configuration. Top-down combustion is a process in which air is injected at the top of

the reservoir through a horizontal well or a number of vertical wells and fluids are pro-

duced from the horizontal producers placed near the bottom [26]. It was found that

the combustion front first spreads horizontally then moves down dip stably. These

experiments shows that gravity also helps with the front stability. This is consistent

and analogous with the classical vision of gravity stabilizing effects in steam based

gravity drainage process [5]. Furthermore, a concept of front self-correction is de-

scribed in this work by two effects: pore blocking and local drop in air flux [54]. It

was mentioned ”pore blocking: when an air finger develops, displacement of liquid

phase by the combustion front will build up a localized fluid bank which will in turn

reduce the gas mobility to such an extent that air will be diverted to other locations

on the combustion front” and ”local drop in air flux: finger formation in the com-

bustion front, enlarges front area locally and drops air flux in the finger. Lower flux

slows down combustion reactions in the finger and postpones its progress”. Further

study needs to be performed to study the detailed mechanism whether ISC front is

self-correcting or not.

Historically, premature combustion front breakthrough has also been an impor-

tant issue for field operations of in-situ combustion. For example, it is very often

advised that the combustion process should start from the top of the reservoir and

use line drive configuration to push everything downward towards production wells

[85]. This is mainly due to the fact that in a dipping reservoir, gravity greatly con-

tributes to stabilizing the ISC fronts. This technique has been applied to both heavy


oil and light oil ISC process, such as Suplacu de Barcau field in Romania and the

West Hackberry project [41, 91]. Furthermore, cyclic combustion technique has been

applied extensively in field operations in Canada in order to overcome premature

break through in the highly heterogeneous channel sand in the Lloydminster region

[74]. The BP Canada’s Wolf Lake pressure-up blowdown combustion process is also

designed to over come the over early breakthrough in heated channels from previous

cyclic steam stimulation [48]. Unfortunately, the early breakthrough still causes well

failures in part of the project. This shows the importance of understanding the front

stability in ISC process.

Overall, thermal front stability criteria needs to be extended to the ISC process.

The stability analysis is based on both analytical analysis and numerical tests in

this work. The front stability is essential for guiding the design of field-scale ISC

process, in which different injection/production or wet/dry combustion scenarios will

be evaluated for field operations.

Chapter 3

Governing Equations

3.1 General Conservation Equations

The governing transport equations for general thermal, compositional porous media

flow include Nc mass conservation equations and one energy conservation equation.

The mass balance equation for the i-th component is given by

∂

∂t

np∑j=1

φyijρjSj +∇ ·np∑j=1

yijρjuj = q̃i, (3.1)

where np is the number of total phases, φ is the porosity of the porous medium, yij is

the mole fraction of component i in phase j (Nc∑i=1

yij = 1), ρj is the phase molar density,

Sj the volumetric phase saturation (np∑j=1

Sj = 1), uj the velocity of phase j, and q̃i the

mass source or sink term.

The energy conservation equation is given by

∂

∂t

(1− φ) Ur +np∑j=1

φρjSjUj

+∇ ·np∑j=1

hjρjuj +∇ · (−α∇T ) = q̃h, (3.2)

where Ur is the internal energy of rock, Uj is the internal energy of phase j, hj is the

enthalpy of phase j and α the heat conductivity, and q̃h is the heat source or sink

term. Ut = Ur +np∑j=1

φρjSjUj is the total energy of the grid block.

21

CHAPTER 3. GOVERNING EQUATIONS 22

The volume conservation expresses that the fluid must fill the pore space, that is,

Vp =np∑j=1

Vj, (3.3)

where Vj is the volume of phase j, and Vp is the volume of the pore space.

Total fluid volume balance leads to the pressure equation

ct∂P

∂t+

nc∑i=1

∂Vt

∂Ni

∇ · np∑j=1

yijρjuj

+∂Vt

∂Ut

∇ · np∑j=1

hjρjuj

=∂Vt

∂Ut

q̃h +nc∑i=1

∂Vt

∂Ni

q̃i.

(3.4)

Here, ct is the total compressibility and q̃i and q̃h are the source and sink terms for

composition and energy. The partial derivative ∂Vt

∂Niand ∂Vt

∂Utexpress the dependency

of the total fluid volume on composition and total internal energy, respectively.

For simplicity, we first consider the case without capillary effects. The multi-phase

extension of Darcy’s law gives

uj = −krj

µj

k (∇P − ρjg∇D) , (3.5)

with P the pressure, k the permeability tensor, krj the relative permeability of phase

j, µj the phase viscosity of phase j, ρj the mass density and D the depth of the

reservoir.

3.2 Phase Equilibrium

We group all the components into two categories: water-like (aqueous) and oil-like

(oleic). Aqueous components exist in both gas and water phase, while oleic compo-

nents exist in the oil and gas phase. In addition to the conservation equations, we

assume the system is in instantaneous thermodynamic equilibrium. This is expressed

by the equality between chemical potentials, or equivalently fugacities,

f oi = f g

i ,

fwk = f g

k ,(3.6)


with f oi and f g

i the fugacity of the i-th oleic component in the oil phase and gas phase,

fwk and f g

k the k-th aqueous component in the water and gas phase. The fugacity

equality equation Eq. 3.6 is usually solved using the cubic equation of state (EOS)

approach. The phase equilibrium calculations are also known as flash calculations,

with details found in [77]. We introduce a simplified ideal mixture approach in which

phase equilibrium is modeled using the K-value approach [101],

yi,o ·Ki = yi,g,

yk,w ·Kk = yk,g,(3.7)

with Ki and Kk being the K-value for i-th and k-th component. The K-value is

pressure-temperature dependent and calculated by the Crookston correlation [101],

Ki =KV 1

Pexp(

KV 4

T −KV 5

). (3.8)

with KV 1, KV 4, KV 5 being the constants in Crookston K-value correlation.

For most heavy oil applications, the crude oil has significant amount of heavy

components. Because heavy oil reservoirs are usually shallow in depth, the operating

pressure is often very low. Therefore, the fluid system is relatively far away from

its critical point. The K-value approach is a valid approximation for most of the

heavy oil applications, and is also implemented in most commercial thermal reservoir

simulators [101, 37]


3.3 Fluid Property Calculations

For each component the fluid enthalpy and latent heat calculation are performed in

the same way as in [101], that is,

hig(T ) =T∫

Tref

Cgi(T ) · dT,

hiv(T ) = HV Ri · (Tc − T )0.38,

hil(T ) = hig(T )− hiv(T ).

(3.9)

hiv is the latent heat for the i-th component. hil and hig are the liquid phase enthalpy

and gas phase enthalpy for each component, respectively. HV Ri is a constant for

latent heat calculations.

We calculate the liquid phase (aqueous and oleic phase) molar densities for each

component using the commonly used relationship

ρil (T, P ) = ρscil exp [cil (P − Psc)− ail (T − Tsc)] , (3.10)

where cil and ail are the compressibility and thermal expansion coefficients for i-th

component. The Redlich-Kwong EOS is used to calculate the gas phase density for

each component [101]. It is given by,

ρig = P/(RTZi),

Z3i − Z2

i + (A−B2 −B)Zi − AB = 0,

A = 0.42748 · ( PPc

) · (Tc

T)2.5,

B = 0.08664 · ( PPc

) · (Tc

T).

(3.11)

We assume ideal mixing when calculating the phase density and phase enthalpy:

1/ρj =Nc∑i=1

(yij/ρij)

hj =Nc∑i=1

yij · hij

(3.12)


The viscosities of the i-th component in the water and oil phases are given by the

correlation [93]

µil = Ail exp(Bil/T ), (3.13)

where Ail and Bil are the empirical parameters for the temperature dependent vis-

cosities. Later, the water and oil phase viscosities are calculated by the logarithmic

mixing rule,

ln(µl) =nc∑i=1

yil · ln(µil). (3.14)

The viscosities of gas phase is calculated simply as a function of temperature T as,

µg = 0.0136 + 3.8 · 10−5 · T, (3.15)

with the units for µg in cp, and the unit for T in oC.

3.4 Chemical Reactions

When chemical reactions are present, such as in the case of ISC process, the governing

mass and energy transport equations become,

∂Ci

∂t+∇ · qi = Qwell

i + Qreaci ,

∂Ut

∂t+∇ ·

(qh,adv + qh,cond + qh,loss

)= Qh,well + Qh,reac,

(3.16)

where Ci is the mass composition of the i-th component, Ut is the total internal

energy, qi and qh,adv are the advective mass and energy fluxes, qh,cond and qh,loss

are the terms for heat conduction and heat losses, and Qwelli and Qh,well are the well

terms. The flux terms qi, qh,adv and qh,cond are calculated as previously. The phase

equilibrium and the fluid property calculations are the same as before.

The extra terms for chemical reactions are the source/sink terms Qreaci and Qh,reac.

We assume all chemical reactions to be kinetically driven. The reactions are mod-

eled using standard Arrhenius kinetics, as in most commercial reservoir simulators

[101, 37]. Both homogeneous and heterogeneous reactions are allowed. Nr chemical

reactions are assumed in the simulation. We denote the stoichiometry matrix for


these Nr chemical reactions by A, with Air the stoichiometry coefficient for compo-

nent i in the r-th reaction. We use negative values of Air for participating reactants,

and positive values for reaction products. The mass conservation of the r-th chemical

reaction gives ∑i

Air ·Mi = 0 (r = 1, · · · , Nr), (3.17)

where Mi is the molecular weight of the i-th component.

By using Arrhenius kinetics to express the rate of each reaction, the reaction

source/sink term for the i-th component is written as

Qreaci =

Nr∑r=1

Air · ar · exp(−Ear

Rg · T) ·∏l

(Cl)nrl , (3.18)

where Ear is the activation energy for the r-th reaction, ar is a constant known

as the pre-exponential frequency factor for the r-th reaction, Rg is the ideal gas

constant, T is the local absolute temperature, nrl is the reaction order and Cl is the

l-th reactant concentration. Generally, two main types of chemical reactions exist in

ISC: cracking/pyrolysis and full/partial oxidation reactions. For oxidation reactions,

usually the partial pressure of oxygen PO2 is used in the rate expressions, instead of

reactant concentrations in Eq. 3.18. The heat source/sink term is also calculated

using Arrhenius kinetics, and is written as,

Qh,reac =Nr∑r=1

∆Hr · ar · exp(−Ear

Rg · T) ·∏l

(Cl)nrl , (3.19)

where ∆Hr is the reaction enthalpy for the r-th reaction, that is the heat generated

from this reaction.

3.5 Primary Equations and Variables

We solve the governing equation numerically using fully implicit discretization [6].

The natural choice of primary equations are the mass balances Eq. 3.1, the energy

balance Eq. 3.2, and the volume balance Eq. 3.3. There are two main categories


of choosing primary variables: natural variables and overall quantity variables. Nat-

ural variables are based on saturations and mole fractions, whereas overall quantity

variables are based on total component moles and total energy/enthalpy. The main

feature of using natural variables is that we need to switch variables when the phase

state changes in a grid block. Using overall quantities as primary variables, we avoid

complicated variable switching for the phase changes. In this work, we implement

the overall quantity variables as primary variables. They include: Nc overall compo-

nent concentrations Ci, and total energy Ut and pressure P . For the calculation of

phase saturation, mole fractions and other fluid and rock properties, we implement

the isenthalpic/isothermal flash calculation method [103].

Chapter 4

Thermal Streamline Simulation for

Hot Water Flood

4.1 Streamline Simulation Framework

We first give the general framework of our thermal streamline simulator for hot water

flood, which is also illustrated in Fig. 2.1. Further details on the formulation and

individual stages are discussed in subsequent sections.

1. Given boundary conditions (well conditions) and current spatial distributions

of pressure and saturations, the pressure equation Eq. 3.4 is solved (implicitly)

for a global time step on the three-dimensional Eulerian grid. With the pressure

known, the velocity is explicitly computed using Darcy’s law Eq. 3.5.

2. Given the total phase velocity field

ut =np∑j=1

uj, (4.1)

tracing is performed with Pollock’s analytical tracing method [92].

3. The advective parts of the mass balance equations and energy equation Eq. 4.8

are solved along the 1D streamlines using an appropriate numerical method and

28

CHAPTER 4. THERMAL STREAMLINE SIMULATION FOR HOT WATER FLOOD29

for the same global time step used in the pressure solve. During this global time

step, the streamlines locations, the velocity and pressure field are held fixed.

This is slightly different than the approach we implement later on for steam

flood, in which we only fix the streamline locations and update the velocity and

pressure along each 1D streamline. We will discuss this in detail in chapter 5.

4. At the end of the global time step, the newly computed solution variables are

mapped back from the streamlines to the Eulerian grid. The non-advection

effects gravity and heat conduction are accounted for on the Eulerian grid using

an operator splitting approach. This finishes a global time step. The process is

now restarted from step 1.

The streamline simulation uses two distinct time steps: a global time step between

pressure updates, and local time steps used when solving for the advective processes

along the 1D streamlines and the non-advection steps on the Eulerian grid.

4.2 Streamline Formulation for Hot Water Flood

We simplify the hot water flood problem to two phases immiscible water-oil system.

The mass and energy conservation can be simplified as

∂∂t

φρwSw +∇ · ρwuw = q̃w,∂∂t

φρoSo +∇ · ρouo = q̃o,

∂∂t

((1− φ) Ur +

∑j=w,o

φρjSjUj

)+∇ · ∑

j=w,ohjρjuj +∇ · (−α∇T ) = q̃h,

(4.2)

and the pressure equation simplifies to

ct∂P

∂t+∇ · (ρouo)

ρo

+∇ · (ρwuw)

ρw

+∂Vt

∂Ut

∇ · ∑j=w,o

hjρjuj

=q̃w

ρw

+q̃o

ρo

+∂Vt

∂Ut

q̃h. (4.3)

If compressibility effects and thermal expansion of water and oil are sufficiently


small to be neglected, the pressure equation becomes

∇ · (kλt∇P ) = 0, (4.4)

where λt = krw

µw(T )+ kro

µo(T )is the total fluid mobility, k is the permeability tensor, and

the fluid viscosities µw (T ) and µo (T ) are functions of temperature, according to Eq.

3.13. The mass transport can be further simplified as

φ∂Sj

∂t+∇ · uj = 0. (4.5)

Both the compressible and incompressible formulations are implemented in this

work. We initially start with the simplified incompressible hot water flood and then

include the compressibility effects. The major thermal effects for thermal enhanced

recovery process include reduced oil viscosity, volume expansion, wettability changes

and oil and water interfacial tension changes [93]. The viscosity reduction is gener-

ally the primary mechanism for hot water flood, followed by volume expansion. We

currently do not consider the wettability and interfacial tension effects. We assume

incompressible rock in this study for simplicity.

By rearranging the governing mass and energy equations and splitting the gravity

segregation flux from the pressure driven flux, we get the governing equations along

each 1D streamline, given by

φ ∂∂t

ρwSw + ut∂∂s

fwρw + fwρw (∇ · ut) + ∂Gw

∂z= 0,

φ ∂∂t

ρoSo + ut∂∂s

foρo + foρo (∇ · ut) + ∂Go

∂z= 0,

∂∂t

(Ut) + ut∂∂s

∑j=w,o

fjρjhj +∑

j=w,ofjρjhj (∇ · ut) + ∂Gh

∂z+∇ · (−α∇T ) = 0,

(4.6)

where g is gravity constant, fw and fo are the fractional flow functions, Gw, Go and Gh

are the gravity segregation terms for mass and energy transport. Spatial derivatives

in these equations are taken with respect to the streamline arc length s. Following

the traditional streamline formulation, we can instead use the time-of-flight (TOF)

τ , given by

τ =∫ φ

ut

ds (4.7)


Operator splitting technique is implemented to solve the advective part of the trans-

port equations and to leave the gravity segregation and conduction terms to be solved

later on. The 1D transport equations along streamlines can then be written as

∂∂t

ρwSw + ∂∂τ

fwρw + fwρw(∇·ut)

φ= 0

∂∂t

ρoSo + ∂∂τ

foρo + foρo(∇·ut)

φ= 0

∂∂t

(Ut/φ) + ∂∂τ

∑j=w,o

fjρjhj +∑

i=w,ofjρjhj

(∇·ut)φ

= 0

(4.8)

In the compressible case, the nonzero divergence of the velocity is taken into account

as source or sink terms in the mass and energy transport equations, given respectively,

by fjρj(∇·ut)

φand

∑j=w,o

fjρjhj(∇·ut)

φ.

We use both a first order explicit and a first order implicit method to solve these

1D transport equations along the TOF grid Eq. 4.8 combined with the standard first

order upstream weighting scheme (SPU).

4.3 Specific Techniques

4.3.1 Mappings

Solution variables are mapped from the streamlines to the Eulerian grid at the end

of each global time step, and back from the Eulerian grid to the new streamlines

after the pressure update at the start of the next global time step. Care must be

taken when designing mappings as they can lead to smoothing of the solutions, as

well as mass balance errors. Mapping errors are not as great a concern for temperture

(or energy) as for saturations (or compositions): the temperature field is generally

smooth, whereas the saturation and composition may contain sharp gradients. In

order to alleviate the numerical diffusion that is often introduced by the mapping

from the pressure grid to the streamlines, we apply a piecewise linear interpolation of

the variables based on TVD (Total Variation Diminishing) slope limiting as proposed

by [72]. The same strategy is applied when mapping the solution variables from the

initial irregular TOF grid to the regularized TOF grid. When mapping back from


streamlines to the pressure grid, we employ the standard running sum approach [7]:

for any pressure grid cell we compute the solution variables as a weighted average

over the N streamlines crossing the cell, that is,

Ccell =N∑

i=1

αiCi, (4.9)

where Ci is a solution variable. The weights αi are chosen to reflect the relative

volumetric contribution of streamline i to the cell [7]. We choose αi = qiτi/(∑

qiτi),

with qi the volumetric flux of the streamline and τi the time of flight of the streamline

intersect that grid block [7]. In the incompressible case, qi is usually calculated as

a portion of the total volumetric flux from the injection well. It is also possible

to calculate qi from the production well. To calculate qi from the injector side or

the producer side is still open research topic. For compressible flow, the flux qi is

calculated according to q = qoe(∇·ut)τ

φ , which accounts for the compressibility of the

total phase velocity field [88].

4.3.2 Operator Splitting for Non-advective Processes

At the end of a global time step, we solve the conduction term of the energy equation

on the Eulerian grid, while holding the other solution variables, including pressure

and saturations/compositions, constant in each grid block. Gravity effects are also

accounted for at this time using operator splitting [14],

φ ∂∂t

ρwSw + ∂Gw

∂z= 0,

φ ∂∂t

ρoSo + ∂Go

∂z= 0,

∂∂t

Ut + ∂Gh

∂z+∇ · (−α∇T ) = 0.

(4.10)

After these corrections, the fluid volume will generally not fit the cell pore volume.

We estimate the magnitude of these volume discrepancies and include them as source

or sink terms in the 3D pressure equation in the subsequent global time step. The


discretized form of the pressure equation is now

ctP n+1 − P n

∆t=

∆V

∆t+∇ ·

(Tn∇ · P n+1

), (4.11)

where ∆V is the amount of volume mismatch between fluid volume and pore volume.

The operator splitting applied to the energy balance equations introduces a split-

ting error. This splitting error for the energy equation is acceptable provided the

processes considered are advection dominated. Sensitivity tests are conducted in the

next section to verify the applicability of operator splitting to hot water flooding.

4.3.3 Treatment of Compressibility

The streamline approach followed here can be interpreted as a streamtube approach,

in which each streamline has an associated volume. Streamlines are equivalent to

streamtubes if one considers the relationship between a streamtube and the central

streamline within a streamtube. As mass and energy are moved along the streamlines,

the fluid shrinks or expands because of temperature changes increasing or decreasing

the volumetric flux. We model the volume discrepancy using a dimensionless velocity

approach similar to the one suggested by [71] [35]. We set

∂∂t

(Ci) + ∂∂τ

(udFi) + 1φ

(∇ · ut) · Fi = 0,

ud,k+ 12

= ud,k− 12

+ ε∆τ∆t

(Vfluid/Vcell − 1) ,(4.12)

where Vfluid represents the fluid volume and Vcell is the block pore volume at time tn.

For hot water flooding, this dimensionless velocity approach is effective because

the volume discrepancy is relatively small. And the fluid properties are only weak

functions of absolute pressure.

4.4 Hot Water Flood Results

We test our thermal streamline simulator for hot water injection in a 2D quarter five-

spot configuration. The heterogeneous domain is 500× 500 m2. The permeability is


Figure 4.1: The permeability field in quarter five-spot hot water flood test case.

Figure 4.2: The water saturation results in quarter five-spot hot water flood test case.

shown in Fig. 4.1. We use the quadratic relative permeability curves krw = Sw2 and

kro = So2 for water and oil phases, respectively. The rock and reservoir parameters

are listed in Table A.1 in the appendix. We compare simulation results of the thermal

streamline simulator with a commercial thermal simulator.

4.4.1 Incompressible Hot Water Flood

Since the compressibility of water and oil liquid is usually small, we first assume the

liquids are incompressible. We use the viscosity relationship, given in Equation Eq.

3.13. We start the simulation with an initial oil saturation equal to 0.9. The initial

temperature of the oil in the reservoir is set to be 20oC, and the temperature of

injected hot water at 80oC. As we heat up the fluid, the oil viscosity will be reduced


Figure 4.3: The temperature results in quarter five-spot hot water flood test case.

Figure 4.4: The pressure results in quarter five-spot hot water flood test case.

dramatically, therefore making the mobility ratio M = µw/µo decrease from M = 10

at 20oC to M = 1 at 80oC. The other fluid properties for water and oil and well

settings are listed in Table A.2 and Table A.3.

The base pressure grid in the streamline simulator is chosen as 50 × 50. For

comparison, we show FV simulator results for grids varying from 50×50 to 400×400.

Comparisons of the water saturation, temperature, pressure field and cumulative

production are shown in Fig. 4.2, Fig. 4.3 and Fig. 4.4. Here, we used 90 streamlimes

and 20 global time steps for a total simulation time of 50000 days, equivalent to

0.29 pore volume injected (PVI). Because of the large heat capacity of the rock, the


Figure 4.5: The comparison between 1D fully implicit (FIM) transport solver and1D single point upwind explicit (SPU) transport solver for thermal SL simulation(50X50).

temperature front is moving slower than the leading edge of the water. We observe

water breakthrough first and the temperature front lags behind.

The results show that our streamline method is capable of generating results

of comparable accuracy with FV simulator. Furthermore, the streamline method

is especially capable of reducing numerical diffusion and accurately resolving the

channeling through the domain as we can see from the comparison in saturation of

FV simulation and streamline simulation. Also, since the fluid transport is calculated

along the optimal streamline direction, it can potentially help to alleviate the grid

orientation effects in FV simulation.

We note that we ran the simulations for both FIM and the explicit time step-

ping method along streamlines. The results are shown in 4.5. FIM leads to slightly

increased diffusion, but otherwise results are comparable.

The sensitivity of the solution to the number of global time steps is demonstrated

in Table 4.1. The table shows that the accuracy of the method depends on the

number of global time steps chosen. Too many pressure updates lead to increased

mapping errors. On the other hand, too few pressure updates degrade the accuracy

of the transport calculations since we are not capturing the temporal velocity field

sufficiently well.

We also tested the sensitivity of the streamline simulator accuracy to the viscosity

relationships µj = Aj exp(Bj/T ). The water viscosity is set as constant with respect

to temperature. Of course, as the viscosity increases, the reservoir needs a larger


Table 4.1: Errors of different viscosity properties compared with reference result(400× 400 STARS result) using L2 norm

M=10 (20oC) to M=1 (80oC) PVI= 0.32

Number of global time steps 2 5 10 20 40Streamline saturation error 18.1% 12.8% 10.1% 9.1% 11.4%Streamline temperature error 12.1% 8.0% 6.5% 7.1% 7.2%FV saturation error (50× 50) with reference 10.0%FV temperature error (50× 50) with reference 5.9%

M=100 (20oC) to M=1 (80oC) PVI= 0.18


M=1000 (20oC) to M=10 (80oC) PVI= 0.10


driving force and hence a higher pressure difference between injection and production

wells. We note that as the the exponential viscosity relationship becomes more non-

linear, FV simulator also requires smaller time steps. Both 50×50 thermal streamline

results and 50× 50 FV simulator results are compared with the reference result. The

relative error is calculated according to e = ‖δX‖‖X‖ . The surface cumulative production

is also plotted.

As we can see, the increasing exponents in the temperature dependent viscosity

relationship µj = Aj exp(Bj/T ) leads to larger errors in the streamline simulation

as the mobility ratio increases with temperature. Keeping the total velocity fixed

during a global time step in such highly viscous oil cases leads to more errors. Also

since we are using the initial velocity field un for the transport between time level

tn and tn+1, we are underestimating the water production in this hot water injection


Figure 4.6: The surface cumulative production results for M=10 quarter five-spot hotwater flood test case.

Figure 4.7: The surface cumulative production results for M=100 quarter five-spothot water flood test case.


Figure 4.8: The surface cumulative production results for M=1000 quarter five-spothot water flood test case.

case. We can improve the accuracy by increasing the number of pressure updates.

Another approach would be to account for total velocity changes by solving the 1D

pressure equation along the 1D streamline. We will show further improvements by

using pressure/flux update along 1D streamlines in later chapter.

4.4.2 Compressible Hot Water Flood

For hot water flooding problems with low compressibility, we use the dimensionless

velocity approach discussed previously. The initial oil saturation in this case is set at

1.0. All other variables are the same as in the previous test case. The temperature

dependent viscosity relationship is the same as the previous test case. The fluid

density calculations parameters are listed in Table A.4 in the appendix. The reference

pressure is Psc = 100kPa and the reference temperature is Tsc = 273.15K.

The results are shown below in Fig. 4.9, Fig. 4.10 and Fig. 4.11. The compressible

streamline simulation is able to account for the effect of volume changes and yields

results with similar accuracy as compared with the 400× 400 FV simulation results.

Again, we observe water breakthrough to the producer first and the temperature front

lags behind.

We have also explored the sensitivities of larger thermal compressibility for hot


Figure 4.9: The saturation results in compressible quarter five-spot hot water floodtest case.

Figure 4.10: The temperature results in compressible quarter five-spot hot water floodtest case.


Figure 4.11: The surface cumulative production comparison results in compressiblequarter five-spot hot water flood test case.

Table 4.2: Effects of thermal compressibility on the production (surface condition)

Small compressibility: cw = co = 1× 10−7/kPa, aw = ao = 1× 10−4/K

Oil Prod (104m3) Water Prod (104m3) Water Inj (104m3)SL (50× 50) 3.218 0.183 3.373STARS (50× 50) 3.121 0.193 3.299STARS (200× 200) 3.312 0.359 3.669

Large compressibility: cw = co = 5× 10−6/kPa, aw = ao = 5× 10−3/K

Oil Prod (104m3) Water Prod (104m3) Water Inj (104m3)SL (50× 50) 3.126 0.143 2.872STARS (50× 50) 3.068 0.166 2.852STARS (200× 200) 3.266 0.304 3.143


Figure 4.12: Saturation results in gravity test case 1 (quarter five-spot hot water floodwith dipping).

Figure 4.13: Saturation results in gravity test case 2 (quarter five-spot hot water floodwith dipping).

water flooding. Based on the previous case, we have increased the water and oil

compressibility to cw = co = 5× 10−6/kPa, aw = ao = 5× 10−3/K, which is a highly

compressible system. The production and injection results of SL and FV are shown in

Table 4.2. Once again, the thermal streamline simulation yields a comparable solution

to FV simulation. Further more, we can observe the thermal compressibility effects

in the oil production from our results. Compared with the small compressibility case,

the highly compressible system needs less amount of water injected to produce the

equivalent amount of oil at the producer, due to thermal expansion effects.

4.4.3 Gravity Effects

Gravitational effects are included through operator splitting [14]. We test the ap-

proach on a tilted (30 degree) reservoir for incompressible fluids. The results are

shown below in Fig. 4.12 and Fig. 4.13.


The result has shown by changing the depth of the reservoir, we have accelerated

and decelerated the flood. By using the operator splitting technique, we are able

to capture the gravity. Here the water and oil density difference is relatively small.

Therefore the gravity segregation effect between phases is weak.

4.4.4 Heat Conduction Effects

The streamline method is best suited for simulation of an advection dominated flow

process. Non-advective effects (gravity segregation, heat conduction and capillary)

are accounted for using operator splitting. As the heat Peclet number (Pe) decreases,

we expect the performance of the streamline simulator to worsen. The heat Peclet

number is the ratio between heat advection and conduction. We set Pe = L×u( α

Cv), where

L is the distance between the injector and producer, α the average heat conductivity,

Cv the average total heat capacity, and u the average velocity along the diagonal

direction. We have tested the original incompressible test problem for different Pe

numbers (Pe = 1.5, Pe = 15, Pe = 150) by arbitrarily changing the heat conductivity

of the fluid and rock. Again, both the 50× 50 streamline and 50× 50 FV simulation

results are compared with the refined 400 × 400 FV simulation results. The results

are listed in Table 4.3.

For cases with Pe = 15 and Pe = 150, the SL errors are comparable with the

FV simulation for both saturation and temperature fields because the flow is more

advection dominated. For the smaller Pe number (Pe = 1.5), the SL errors in the

temperature is larger than the FV temperature error (8.4% for SL compared to 5.3%

for FV). This is because of the splitting when solving the heat conduction and heat

convection on different fractional time steps. The saturation results show a weaker

relationship with the Pe number. This is because mass transport is purely advective

flow, thus, it is less sensitive to the operator splitting.


Table 4.3: Heat Pe number and its influence on the simulation result.

Pe = 1.5Streamline saturation error 9.4%FV saturation error (50× 50) with reference 8.4%Streamline temperature error 8.4%FV temperature error (50× 50) with reference 5.3%

Pe = 15Streamline saturation error 9.4%FV saturation error (50× 50) with reference 8.7%Streamline temperature error 9.1%FV temperature error (50× 50) with reference 7.8%

Pe = 150Streamline saturation error 9.3%FV saturation error (50× 50) with reference 10.8%Streamline temperature error 11.9%FV temperature error (50× 50) with reference 11.1%

4.5 Concluding Remarks

In this chapter, we have presented a thermal streamline simulation framework for two

phase hot water flood with the thermal effects of temperature dependent viscosity

and thermal compressibility. Our method is based on an operator splitting approach

where the advective parts of the governing equations are solved separately along the

streamlines with the other effects solved on the original Eulerian grid. The streamline

results are compared with a commercial FV simulator. We have shown the method

is capable of producing comparable results at a lower computational cost. Various

sensitivity tests were performed to study the accuracy and robustness of the method.

Our results provide guidelines for implementing the thermal streamline simulation for

hot water flood as a fast proxie with reasonable accuracy.

Chapter 5

Thermal Streamline for Steam

Flood

5.1 Streamline Simulation Framework

In this chapter, we focus on the extension of streamline simulation to steam flooding

of heavy oils. Steam floods pose severe challenges to traditional streamline methods

because of the large volume changes and compressibility associated with the phase

behavior of steam, the coupled mass and energy transport, plus the gravity segre-

gation and override. To account for the volume changes and compressibility, which

are particularly strong when steam condenses to water, we implement a pressure and

flux update along the 1D streamlines. To more accurately capture the strong cou-

pling between energy and mass transport, we employ a Glowinski θ-scheme operator

splitting [109]. We also utilize a preliminary hybrid FV/SL approach to deal with

short simulation periods when strong non-advective forces dominates, for example the

soaking period in cyclic steam stimulation. All these new improvements to traditional

streamline simulation are presented and discussed in this chapter.

We again first give the general framework of our thermal streamline simulation

for steam floods. The key steps are as follows:

1. Given boundary conditions (well constraints) and initial saturation/pressure

45

CHAPTER 5. THERMAL STREAMLINE FOR STEAM FLOOD 46

distributions, the pressure equation accounting for fluid and rock compressibility

Eq. 5.3 is solved on the 3D static Eulerian grid. With the pressure known, the

total phase velocity is explicitly computed at each grid interface using Darcy’s

law Eq. 3.5.

2. With the total phase velocity ut = ug +uo +uw known, streamlines are traced

using Pollock’s method [92]. Solution variables are mapped from the static

Eulerian grid onto the streamlines.

3. The advective parts of the mass and energy equations Eq. 5.2 are solved along

the streamlines along the arc length coordinates. The solution is moved forward

for a global time step by taking streamline specific time steps in a FIM 1D solver.

The key is that pressure and flux along the 1D streamlines are updated at each

local time step to account for steam compressibility and condensation (volume

changes). We are in essence solving a series of 1D, full-physics steam flooding

problems during a global time step. This is different than the previous approach

for hot water flood and is discussed in more detail in later section.

4. At the end of the global time step, the newly computed solution variables are

mapped back from the streamlines to the Eulerian grid. The gravity and heat

conduction effects are accounted for on the Eulerian grid using a Glowinski

θ-scheme operator splitting approach, detailed below.

5. We now return to step 1 to start the next global time step.

5.2 Streamline Formulation for Steam Flood

Steam flooding is a general multi-component compositional thermal simulation prob-

lem. In this work, we implement 1 aqueous component (water) and no oleic compo-

nents. Water exists in aqueous and gas phases, while the no oleic components exist in

oil and gas phases. The mass and energy transport equations are listed in Eq. 3.1 and

Eq. 3.2. The volume conservation equation is Eq. 3.3. The fluid phase equilibrium


for this multi-component system is expressed as,

xw = 1,no∑i=1

xi = 1,

Kw · xw +no∑i=1

Ki · xi = 1,

(5.1)

where xw is the water mole fractions in aqueous phase, xi is the oleic component mole

fractions in oil phase. K-values Kw and Ki are calculated by Eq. 3.8. We implement

flash calculations using a modified Rachford-Rice equation to account for the water

component in three phase equilibrium [67]. Other properties such as density, enthalpy

and viscosity are calculated according to the previous expressions Eq. 3.9, Eq. 3.10,

Eq. 3.11, Eq. 3.13 and Eq. 3.15.

In this work, we also utilize a simplified steam flood model. We assume that we

only have one water (component 1) and one nonvolatile hydrocarbon (component 2).

This is a valid simplification when the oil is relatively non-volatile. The water exists

in both the water and gas phase (steam), while the hydrocarbon exists only in the

oil phase. A K-value approach is used to model the water/steam phase behavior, i.e.

yw = Kw · xw. We use the standard K-value for water in the commercial simulator

[101].

The mass and energy conservation can now be simplified to

∂∂t

φ (ρwSw + ρgSg) +∇ · (ρwuw + ρgug) = q̃1,∂∂t

φρoSo +∇ · ρouo = q̃2,

∂∂t

((1− φ) Ur +

∑j=w,o,g

φρjSjUj

)+∇ · ∑

j=w,o,ghjρjuj +∇ · (−α∇T ) = q̃h.

(5.2)

The pressure equation becomes

ct∂P∂t

+

(∂Vt

∂N1

(∇ · ∑

j=w,gρjuj

)+ ∂Vt

∂N2(∇ · ρouo)

)+ ∂Vt

∂Ut

(∇ · ∑

j=w,o,ghjρjuj

)= ∂Vt

∂Utq̃h +

∑i=1,2

∂Vt

∂Niq̃i,

(5.3)


where N1 represents the water composition, and N2 represents the non-volatile hy-

drocarbon composition.

We rearrange the governing mass and energy equations and split the gravity

segregation flux from the pressure driven flux. We define total phase velocity as

ut = uw + uo + ug, where the phase velocities uj is defined as previously in Eq. 3.5.

We also define the aligned components of the phase velocities as u′j = fjut, where fj

is the fractional flow function of the j-th phase. Consequently, the pressure driven ad-

vective fluxes (Fi for the i-th component and Fh for energy) and gravity segregation

fluxes (Gi for the i-th component and Gh for energy) can be written as,

Fi =∑

j=o,w,gρju

′jyij,

Gi =∑

j=o,w,gρj

(uj − u′

j

)yij,

Fh =∑

j=o,w,ghjρju

′j,

Gh =∑

j=o,w,ghjρj

(uj − u′

j

).

(5.4)

With the advective fluxes written along the 1D streamline: in the arc length

coordinate s, the governing equations could be rewritten as,

∂∂t

Ci + ∂∂s

Fi + ∂∂z

Gi = 0,∂∂t

Ut + ∂∂s

F h + ∂∂z

Gh +∇ · (−α∇T ) = 0,(5.5)

where Ci is the i-th component accumulation term, Fi and Gi are the advective and

gravity segregation fluxes of i-th component, Ut is the total energy accumulation term,

F h and Gh are the advective and gravity segregation enthalpy fluxes, and ∇·(−α∇T )

is the heat conduction term. We solve the advective fluxes Fi and F h along the 1D

streamlines in SL step 3. The non-advective fluxes Gi, Gh and heat conduction are

solved later on the original Eulerian grid in SL step 4.


Figure 5.1: Steam condensation causes volume change and flux reduction right at thesteam front.

5.3 Specific Techniques

5.3.1 1D Pressure and Volumetric Flux Update Approach

for Large Compressibility

In previous work by [21], compressibility effects are accounted for by adding source/sink

terms along the streamlines. The volumetric flux along each 1D streamline is held

fixed during a global time step but it is no longer uniform as it would be for a strictly

incompressible case. A two/three phase fractional flow problem is then solved along

each streamline by accounting for these (fixed) source/sink terms. These terms are

updated at each new 3D pressure solution. In the work by [12], the transport along

1D streamlines is solved with the spatially varying volumetric flux accounted for ex-

plicitly along each streamline. The volumetric flux is also assumed to remain constant

over a global time step. However, for steam flooding the volumetric flux along the

streamline will reduce dramatically at the steam front due to condensation as shown

in Fig. 5.1. In other words, the volumetric flux can and will change dramatically dur-

ing the 1D transport step. Assuming a constant volumetric flux (constant source/sink

terms) over a single global time step as the work by [21] and [12] will necessarily mis-

calculate the speed of the steam front and/or require excessively small global time

steps. To retain the ability to take large global time steps, it is necessary to account

for the dependency of the volumetric flux during the 1D transport solve. To do so, we

have implemented a new approach to update pressure and temperature at each mini

time step along a streamline and consequently update the volumetric flux during the


Figure 5.2: Streamline segments constructed from the injector to the producer.

Figure 5.3: 1D streamline pressure and volumetric flux update approach. Transmis-sibility and block volume of each segment is calculated for the 1D transport.

transport calculations along each streamline.

In our new approach, we fix the streamline locations during a global time step,

but update the 1D pressure and volumetric flux to account for volume changes. We

must take into account the shrinking and expanding of the fluid as it moves along

the streamlines as illustrated in Fig. 5.2. To do so, we define the cross sectional area

A and block volume V =∫

A · ds along the streamlines, see Fig. 5.3. We rewrite the

divergence operation for total phase velocity ut as

∇ · ut =∂ut

∂s+

ut

A

∂A

∂s, (5.6)

where s is the local streamline arc length coordinate, illustrated in Fig. 5.4.

Thend(Aut)

Aut

=∫ ∇ · ut

ut

ds =∫∇ · utdτ ′, (5.7)

where τ ′ is the travel time, defined as τ ′ =∫ 1

utds. Note τ ′ does not include the

porosity φ , which is different than the definition of time of flight (TOF) in traditional

streamline simulation [8]. This treatment is intended to include rock compressibility.


Figure 5.4: Streamline (s, n, m) coordinates and cross sectional area along the stream-line.

After integration, we get

Qi+ 12

Qi− 12

=Ai+ 1

2uti+ 1

2

Ai− 12uti− 1

2

= exp

∫ si+1

2

si− 1

2

∇ · utdτ ′

. (5.8)

The cross sectional areas can now be integrated one by one from i− 12

to i+ 12

according

to Eq. 5.8 as ut is known along the streamlines.

The block volume Vi are now calculated by

Vi =∫ s

i+12

si− 1

2

A · ds =∫ s

i+12

si− 1

2

A · ut · dτ ′ =∫ s

i+12

si− 1

2

d(Aut)

(∇ · ut). (5.9)

The transmissibilities between cells i and i + 1 are calculated by

Ti+ 12

=

(∫ si+1

s

1

(k · A)ds

)−1

, (5.10)

where k are the permeability values derived from the Eulerian background grid. If

anisotropic permeability fields are present, a coordinate transformation is needed to

calculate the permeability values along the streamline directions.

With volumes and transmissibilities known along each 1D streamline, we can now

solve the Nc mass transport equations, coupled with one energy transport and the

volume conservation equation along the streamline. We do this fully implicitly. The


equations are given by

Viφi

(Cn+1

i,k − Cni,k

)=(Fi− 1

2,k + Fi+ 1

2,k

)·∆tlocal,

Vi

(Un+1

ti − Unti

)=(F h

i− 12

+ F hi+ 1

2

)·∆tlocal,

Vi = Vwi + Voi + Vgi.

(5.11)

Vi is the block pore volume of segment i along the streamline. Vwi, Voi and Vgi are the

fluid volume of water, oil and gas phases in segment i. ∆tlocal is the local time step. φi

is the porosity. Ci is the mass composition in block i. Uti is the total energy density

of block i. Fi+ 12,k and F h

i+ 12

are the mass advective flux for k-th component and the

energy advective flux, respectively, between segments i and i+1. They are calculated

according to the transmissibility calculated previously Eq. 5.10. We typically merge

very small grid blocks to regularize the grid and improve the conditioning of the 1D

transport problem.

Note that rock compressibility is taken into account through the porosity φi term

in Eq. 5.11. The Nc mass compositions, the total energy and the pressure are chosen

as the primary variables aligned with the given Nc +2 governing equations. An exact

pressure-energy flash is performed to calculate the secondary variables, which are the

saturations Sj, the mole fractions yij and the phase densities ρj [103]. At the end of

1D transport solve, we map the 1D streamline pressure back onto the Eulerian grid.

This stitched pressure profile is only used as an initial guess of the pressure at the

beginning of the next global time step.

In the previous chapter, we have shown the larger errors in streamline simulation

when simulating a high initial mobility ratio water flooding problem. This is mainly

due to the fact that in traditional streamline simulation, we need to fix the velocity

field during a global time step. In some sense, it is similar as forward Euler type time

stepping that only utilizes the initial velocity/pressure field. Reducing the global

time step sizes may help to improve this problem to some extent. The accumulative

mapping errors due to large number of mappings, however, eventually makes reducing

global time step size impractical. Updating the 1D flux/pressure along streamlines,

on the other hand, helps to mitigate the errors and capture the velocity and pressure


transient behavior inside a global time step.

To demonstrate this, we show results for a quarter five-spot hot water flood with

an initial mobility ratio of M = 1000, as illustrated in Fig. 5.5. We have two bottom

hole pressure controlled wells. In a high mobility ratio water flood problem, the

averaged field fluid mobility increases, as more and more mobile hot water is injected.

If we implement two pressure controlled wells, this will lead to a larger total flow

rate between injector and producer over the time. Classical streamline simulation

underestimates this water breakthrough because it uses the initial velocity/pressure

field. We update the pressure/flux along 1D streamlines during each global time step,

and also test the predictor-corrector method described in [71]. The predictor-corrector

method [71] is essentially utilizing the velocity field at the middle, instead of the

beginning of a global time step. Eventually, as we can see in Fig. 5.5, by comparing

to the 200 × 200 reference, using both the 1D pressure update and predict-correct

help to improve the accuracy, while classical streamline method underestimates the

break through.

The updated transient pressure field along 1D streamlines reflects the mobility

changes in heavy/viscous oil flooding. And this gives a better understanding of the

actual pressure changes. We can also observe this from the pressure profiles inside a

global time step as in Fig. 5.6. Here the first plot (A) is the pressure at the beginning

of the global time step. At the end of 1D transport solve, we map the 1D streamline

pressure back onto the Eulerian grid to get the second pressure field (B). At the start

of the next global time step, we solve the 3D pressure equation and get the third

pressure field (C). As we can see, the pressure field stitched together from all the 1D

pressure profiles (B) is similar to the pressure field we get at the beginning of the next

global time step (C). Indeed, by updating the 1D pressure we are able to capture the

pressure transients within a global time step.


Figure 5.5: 1D pressure update improvements in capturing transient pressure changewithin a global time step, (M=1000 quarter five spot hot water flood test case). Withtwo BHP controlled wells, classical SL underestimates the breakthrough. Both SLpredict-correct [71] and SL with pressure update improves the hot water breakthroughresults, compared to fine-scale reference.

Figure 5.6: Pressure transient behavior inside a global time step (M=1000 quarterfive spot hot water flood test case). The pressure stitched together from all the 1Dpressure solves (B) is similar to the pressure we get at the beginning of the next globaltime step (C).


Figure 5.7: Schematic of coupled energy-mass transport. Changing the total energy ofa grid block changes the fluid volume inside, thus causing fluid transport to adjacentcells.

5.3.2 Glowinski θ-Scheme Operator Splitting Approach for

Non-advective Processes

At the end of a global time step, we solve for the heat conduction part of the energy

equation on the static Eulerian grid. This part is represented by the term ∇·(−α∇T )

in equation Eq. 3.2. In steam flood problems, the fluid flow and energy are highly

coupled. When the energy of the fluid inside a cell changes, the fluid volume can

change because of phase behavior (condensation and vaporization). This is especially

the case in step 4 in section 4.1 at the steam flood condensing front when we diffuse

and lose heat of the fluid system, shown in Fig. 5.7. Therefore, if we are not allowing

the fluid in each individual cell to flow to account for the volume changes, we will

make significant volume balance errors, which leads to a volume imbalanced fluid

system at the beginning of the next global time step. This causes great numerical

difficulty.

To reduce volume balance errors in this operator splitting step, we employ the

Glowinski θ-scheme [109] shown in Fig. 5.8. An equivalent form of the Glowinski

θ-scheme is also shown in Fig. 5.9. In the conventional operator splitting scheme,

all of the advective flux is accounted for in the 1D streamline advective solves (SL

step 3) and none in the Eulerian grid update (SL step 4). In Glowinski’s θ−scheme,

a small fraction (θ) of the advective flux is kept in the heat conduction step. This is

desirable when large volume changes occur because of phase behavior: the fluids can

redistribute themselves and volume balance errors are reduced before the next global

time step is started.


Figure 5.8: Glowinski θ-scheme operator splitting. Most of the transport is solvedalong 1D streamlines [0 ∼ (1 − θ)∆t]. Small amount of advective flux is used tocorrect the volume changes in heat conduction/gravity step.

Gravity segregation effects are also accounted using this Glowinski θ-scheme oper-

ator splitting method Fig. 5.8. In the traditional streamline method, gravity segrega-

tion effects, resulting from density differences between phases, are taking into account

by solving for flow along gravity lines [7]. Again, gravity segregation, in which rising

steam can condense when it contacts colder areas of the formation, can lead to large

volume changes and fluid redistribution.

The choice of θ is process dependent and must be deduced heuristically. It is

dependent on the heat Peclet number Pe (the dimensionless number which represents

the heat advection effects against the heat conduction effects) and the gravity number

Ng (the dimensionless number which represents the gravity segregation forces against

the viscous forces). It will also be impacted by the details of the static geological

model. We typically use a large θ for processes with small Pe and large Ng, that is,

for problems with strong heat conduction and gravity segregation. We use small θ for

large Pe and small Ng when heat conduction and gravity segregation are relatively

weak. This is the case for most realistic steam flood problems when the steam gravity

override has been fully developed.

5.3.3 Preliminary Hybrid Simulation

Streamline simulation is based on an operator-splitting technique, in which we solve

the advective flow along the streamlines, and non-advectve flow on the Eulerian grid

later at the end of a global time step. Therefore, streamline method is well suited for

advection dominated flow problems, because the splitting error is relatively small. For


Figure 5.9: An equivalent representation of Glowinski θ-scheme operator splitting.Most of the transport is solved along 1D streamlines. Small amount of advective fluxis used to correct the volume changes in heat conduction/gravity step.

some particular thermal recovery processes, we do have periods dominated by non-

advective forces. For example, the initial steam circulation periods in SAGD process

are heat conduction dominated, with the purpose of creating thermal communication

between the two horizontal wells [19]. The soaking period in cyclic steam stimulation

process is also heat conduction dominant. The injected steam releases its heat to

the formation, thus making it easy for the oil to flow back into the producer in the

later production stage. In these scenarios, streamline simulation will have difficulty

handling the strongly non-advective flows.

Another type of problems challenging for streamline are cases with strong gravity

segregation effects. Streamline simulation combines the water, oil and gas phase

velocities together to solve the transport along total phase velocity streamlines [14, 7].

Gravity segregation between the three phases typically are accounted for at the end

of global time step using operator splitting. The splitting error may lead to larger

errors for problems that have strong segregation between different phases. Strong

gravity segregation is very often seen at the beginning of gas/steam flooding process,

but quickly diminishes as the injected low density gas/steam rises to the top of the

formation [57]. Therefore, it is these initial periods that often pose problems for the

accuracy of streamline simulation.

In recent work by Kumar [62], a SL/FV hybrid simulation approach has been

proposed. The hybrid approach consisted of using FV simulation when the cross flow

was strong and shifting to the streamline method when the cross flow was negligible.

We use a similar concept here for short simulation periods with strong non-advection

effects. A heuristic criterion was proposed for the crossover from initial FV to later


Figure 5.10: Quarter five spot steam flood permeability field and initial streamlineshape.

SL, based on the cross product of gas and oil phase velocities by Kumar [62]. As

we will see later, we also use FV simulation for the relatively short soaking periods

in cyclic steam stimulation and the initial steam rising periods in steam flooding

simulation. In this way, we can combine the relative advantages of SL and FV during

the course of the whole simulation to achieve both accuracy and efficiency.

5.4 Steam Flood Results

5.4.1 Heterogeneous Quarter-Five Spot Steam Flood

We first test our thermal streamline simulator for steam injection in a 2D quarter five-

spot heterogeneous reservoir. The reservoir properties are given in appendix Table

A.5 and the permeability field with associated streamlines in Fig. 5.10. The reservoir

is 150m by 150m, discretized by 50× 50 grid blocks. We have a bottom hole pressure

controlled injector and a bottom hole pressure controlled producer as described in

appendix Table A.6.

We implement the simplified two component steam flood model given in section

4.2, with the oil and water phase viscosity profiles shown in Fig. 5.11. At an initial

reservoir temperature of 50oC, the oil viscosity is µo = 105cp and the water viscosity

is µw = 0.5cp. At a steam chamber temperature of 276oC, µo = 0.18cp and µw =


Figure 5.11: Viscosity curve for water and oil.

0.075cp. The injected steam quality (the mass of steam divided by the total mass of

hot water and steam) is fs = 0.8. Other fluid properties are given in appendix Table

A.7. For each component the fluid enthalpy and latent heat calculation is performed

in the same way as previously, with Cg(T ) = CPG1 + CPG2 · T + CPG3 · T 2 +

CPG4·T 3. We simulate continuous steam injection for 3000 days, which corresponds,

approximately to a recovery factor (% of OOIP) of 18%. We use 30 global time steps,

with each global time step equal to 100 days. In this simulation, the finite volume

simulation uses 342 total time steps. For the operator splitting, θ is chosen as a

constant of 0.1, which works well for this advection-dominated case. We study the

sensitivity of θ in detail later. We use both our streamline simulator and a commercial

finite volume simulator [101]. A fine-grid (200 × 200) finite-volume solution is used

as reference solution. Results are shown in Fig. 5.12 and Fig. 5.13.

We compare the relative errors in the temperature and oil saturation profiles.

Both 50 × 50 thermal streamline results and 50 × 50 finite volume simulator results

are compared with the reference 200×200 solution for three different simulation times

Table 5.1. The relative error is calculated according to error = ‖δX‖‖X‖ , where x is the

solution variable, and the norm is the standard L2 norm.

The results show that our streamline simulator has comparable accuracy to the

finite volume simulator at the same grid resolution (50 × 50). We can observe the


Figure 5.12: Quarter five spot temperature and gas saturation at 3000 days (prebreakthrough).

Figure 5.13: Quarter five spot water and oil saturation at 3000 days (pre break-through).


Table 5.1: Errors of thermal streamline and STARS compared to reference (360×180STARS result) using the relative L2 norm.

Simulation Time 1000 days 2000 days 3000 daysStreamline oil saturation error 3.5% 4.5% 4.2%Streamline temperature error 14.3% 11.8% 9.8%FV oil saturation error (50× 50) with reference 5.7% 6.2% 5.1%FV temperature error (50× 50) with reference 16.7% 10.8% 11.2%

shape of the steamed zone, the hot water condensate bank outside of the steamed

zone, and the swept zone in the oil saturation profile.

We perform further sensitivity studies on the choice of θ. We test the same

quarter five spot steam flood problem for 1000 days with 10 global time steps, using

θ = 0.05, 0.1, 0.2. We compare the simulation results in between as shown in Fig.

5.14. We found the simulation results are very close to each other. As we can see the

steam zone sizes and temperature profiles are very similar in Fig. 5.14. The relative

difference in temperature results based on L2 norm and Inf norm is also relatively

small (within∼ 1%), shown in Table 5.2. Note here our main purpose is to use thermal

streamline simulation as a proxy. Steam flood is an advection dominated process, in

which heat conduction serves as a second order term compared to advection flux.

Therefore, we find the splitting error to heat conduction and the choice of θ is not

giving rise to large changes in the streamline result. In practice, we want to keep θ as

low as possible to keep most of the computations in 1D solve along streamlines and

minimize the computational costs. On the other hand, if we take θ to be close to 0,

the scheme becomes close to classical operator splitting. As we discussed previously,

classical operator splitting will cause a volume imbalanced system, because of the

coupled mass and energy transport. Thus, we can not taken extremely small θ so

that the advection flux becomes insufficient to compensate volume changes caused by

conduction. In practice, it often leads to numerical convergence issues when solving

the non-advective flux (step C in Fig. 5.8).


Figure 5.14: Sensitivity study on the choice of θ. The simulation results are close toeach other by using different θ.

Table 5.2: Sensitivity study on choice of θ. The simulation results close to each otherby using different θ.

(θ = 0.05)− (θ = 0.1) (θ = 0.2)− (θ = 0.1)Relative Difference (L2 norm) 0.3% 1.2%Relative Difference (Inf norm) 0.5% 1.4%

5.4.2 Cyclic Steam Stimulation

We test the thermal streamline method for single well cyclic steam stimulation prob-

lem. This case is intended to fully demonstrate the capability of handling compress-

ibility effects, since at any time we are only injecting or producing at a single well.

In this test, we have a 2D reservoir discretized by 51 × 51 grid blocks, of size 102m

by 102m. The reservoir properties are given in appendix Table A.8. We implement

homogeneous permeability in this case K = 500mD. We have a single well in the

middle that will serve as both steam injector and oil producer. We inject steam for

60 days, let it soak and release the heat for 100 days, and produce from the same

well for another 60 days. During the injection stage, we have a bottom hole pressure

controlled injector injects steam at 6000kPa. In the production stage, we control the

producer total bottom hole liquid production rate at 54m3/day as in appendix Table

A.6. We take 3 global time steps in the injection stage and 2 global time steps in

the production stage. We implement the same simplified two component steam flood


Figure 5.15: Hybrid procedures in cyclic steam stimulation (SL → FV → SL) andthe initial streamline shape.

model as in section 4.2.

As discussed previously, in the soaking period, streamline will have difficulty han-

dling the strongly heat conduction dominated flow. Therefore, we implement a SL-FV

hybrid approach to use SL for injection and production stages and FV for the soaking

period Fig. 5.15. When injecting or producing, the streamline will either start from

the injector well and end at the boundary of the reservoir, or start from the boundary

and converge to the central producer Fig. 5.15. This is due to the compressibility

effects. Our streamline simulator is able to fully capture this and stop the streamline

tracing at the certain locations when reaching the local source/sink generated from

the compressible velocity field.

The resulting temperature and oil saturation profiles are shown in Fig. 5.16 and

Fig. 5.17. We also perform the same problem with the commercial FV simulator

[101] (9 point stencil discretization scheme applied here to reduce the grid orientation

effects). The profiles show that our streamline simulator has comparable accuracy to

the FV simulator. We can observe a high temperature zone in the injection stage.

During soaking, the temperature diffuses and steam condenses to warm up the for-

mation around the well. When the producer is opened again, the oil flows out of the

reservoir with reduced viscosity, and the reservoir cools down. Streamline simulation


Figure 5.16: Cyclic steam stimulation temperature result.

is capable of handling the primary steam injection and primary production scenarios,

which are completely compressibility driven. Without the help of 1D pressure/flux

update, this will be an extremely difficult problem for traditional streamline methods.

5.4.3 Heterogeneous Multi-well Pattern Steam Flood

We also test the thermal streamline simulator on a heterogeneous steam flood problem

with multiple wells. We again use the two component simplified steam flood model.

The reservoir rock properties, the fluid properties, and well controls are the same as

before. We place two pressure controlled steam injectors and six pressure controlled

producers as shown in Fig. 5.19. The 2D reservoir is now 180m by 360m, and is

discretized into 60 × 120 grid blocks. Grid refinement to 180 × 360 is performed in

finite volume simulator to obtain a reference solution. The permeability field is shown

in Fig. 5.19.

We first run the steam flood for 1500 days, which corresponds to a recovery factor

of approximately 7%. In the streamline simulation, the global time step size is chosen

as a constant 100 days. Here, the streamline simulator uses 15 global time steps. For

the operator splitting, θ is again chosen as constant 0.1. The temperature profiles


Figure 5.17: Cyclic steam stimulation oil saturation result.

Figure 5.18: Cyclic steam stimulation production profile (SL and FV).


Figure 5.19: 2D multiple well test case permeability field.

Figure 5.20: 2D multiple well test case initial streamline shape.


Figure 5.21: Multiple well test case temperature and oil saturation at 1500 days (prebreakthrough).

and oil saturation profiles are shown in Fig. 5.21. The water and oil production

histories are plotted in Fig. 5.22. We continue the simulation to 2100 days, at

which time the steam has broken through at one of the producers. The streamline

simulator has 21 global time steps, while the finite volume simulation uses 207 FIM

time steps. The temperature and oil saturation profiles for this time are shown in

Fig. 5.23. The water and oil production histories are also plotted in Fig. 5.24.

After steam breakthrough, the water production increases quickly, indicating that

steam has connected the injector and producer. Oil is still produced together with

the steam through the steam drag mechanism. Heat conduction plays an important

role in steam drag, as discussed in the analytical model by Edmunds in [38]. In Fig.

5.23, we observe the steam zone around the post breakthrough producer is slightly

smaller than the FV simulation. This may be due to the fact that heat conduction

(non-advective force) becomes a more important effect in post breakthrough steam

drag process than pre breakthrough.

We compare the errors in the temperature and oil saturation for the coarser 120×


Figure 5.22: Multiple well test case field production history (pre breakthrough).

Figure 5.23: Multiple well test case temperature and oil saturation at 2100 days (postbreakthrough).


Figure 5.24: Multiple well test case field production history (post breakthrough).

Table 5.3: Errors of thermal streamline and STARS compared to reference (200×200STARS result) using L2 norm.

Simulation Time 1500 days 2100 daysStreamline oil saturation error 1.7% 11.5%Streamline temperature error 5.4% 19.4%FV oil saturation error (50× 50) with reference 1.4% 2.1%FV temperature error (50× 50) with reference 3.9% 6.4%

60 grids relative to the 360×120 reference solution in Table 5.3. Again, our streamline

simulator generates results of similar accuracy to the finite volume simulator. The

predicted breakthrough times are also acceptable.

5.4.4 Vertical Cross Section Steam Flood

We test the thermal streamline simulator in a vertical cross section to demonstrate its

ability to handle gravity override. We also use the simplified steam flood model in this

case, with the same fluid and reservoir rock properties as before. The reservoir is 12m


Figure 5.25: 2D vertical cross section steam flood test case 1 (homogeneous perme-ability field).

thick and 120m long, discretized by 8 × 20 grid cells. Two pressure controlled wells

are perforated at the bottom of the reservoir with pressure of 6000kPa and 4500kPa

respectively, as shown in Fig. 5.25. We use a homogeneous field permeability in this

case, with K = 2000mD. The resulting temperature profiles at 150 days are shown

in Fig. 5.25. We start simulation from the initial condition shown in Fig. 5.25.

Comparison is made to the commercial FV simulator [101]. We use a 9 point stencil

scheme to reduce grid orientation effects. The use of 9 point stencil is a temporary

solution. Rigorous multi-D upwinding [60] is more desired to help controlling grid

orientation effects. As we can see, the thermal streamline simulation is capable of

generating results similar to that of the FV simulator for this test case. We can

observe the shape of the steam chamber and the steam gravity override.

We further test a heterogeneous 2D vertical reservoir. The size of the vertical

reservoir is 15m thick and 120m long, discretized by 10 × 20 grid cells. A reference

solution is achieved with further grid refinement 20 × 40. We implement the same

pressure controlled wells at the bottom of the reservoir. The heterogeneous perme-

ability field is shown in Fig. 5.26. The resulting temperature and oil saturation

profiles at 150 days are also shown in Fig. 5.26. We again compare the thermal SL


Figure 5.26: 2D vertical cross section steam flood case 2 (heterogeneous permeabilityfield).


results with the commercial FV simulator [101]. The thermal streamline method gen-

erates comparable results with the FV simulator. We can clearly observe the gravity

override, which is very similar as the previous case.

5.5 Discussion and Applications

5.5.1 Streamline Simulation as Fast Proxy: Cost Comparison

One advantage of streamline simulation over traditional finite volume simulation is

its inherent computational efficiency [105]. To demonstrate this potential for ther-

mal simulation study, we present the profiling results for the multi-well steam flood

example shown previously.

For commercial FV simulator, we have 7,200 grid blocks. The commercial simu-

lator takes 207 time steps, thus 207 fully coupled solves with all the 7200× (Nc + 2)

unknown variables involved (Nc is the number of components).

The thermal streamline simulation uses 7,200 grid blocks on the Eulerian grid. In

this case, we currently use θ = 0.1 for the operator splitting (a smart selection of the

choice of θ to reduce spending on operator splitting is our ongoing work). Thus, for

the operator splitting part we assume 10% comparable cost as the FV simulation,

since we are solving for θ ·∆t time period. Compared to the rest 90% FV simulation

cost, we have 21 global time steps in streamline simulation, which implies 21 pressure

solves (much cheaper than FIM solves with all the unknowns). Inside each global

time step, we also have a set of independent inherently parallel 1D solves along the

streamlines. We included 410 streamlines, each with only 68 grid blocks on average.

Note these 1D solves are very low dimensional 1D problems, which can be solved

and parallelized very effectively [9]. In previous works, we have shown the speed-up

we achieve against FV simulations when simulating advection only problems such

as water flooding [9, 105]. If we assume a conservative 5× speed-up for this 90%

calculations, we will achieve about 3.6× overall speed-up. As we move to simulation

of full field reservoir model with many wells and millions cells, the large FV simulation

solves become intractable because of the increasing number of the unknowns and the


large condition numbers. By using streamline simulation, we can significantly reduce

the size of the problem by solving the transport along individual 1D streamlines.

Thus we are expecting more improvements for realistic full field reservoir simulation.

5.5.2 Optimization and Flux Patterns

One application of streamline simulation is its capability of outlining the flow patterns

[105]. Streamlines offer an immediate snapshot of the flow field clearly showing where

the flow is coming from and where the flow is going to. This can be see from the

flux pattern map Fig. 5.27 generated from commercial streamline simulator 3DSL

[1]. Same flux pattern can be easily constructed for thermal problems, i.e. hot water

flood and steam flood.

Another application of streamline method is the pattern balancing and flood man-

agement through streamline derived information [106]. Through streamline tracing,

well allocation factor and injection efficiency can be calculated, which provide quan-

titative measure of the connectivity in the reservoir. The well allocation factor is

calculated according to the flux from the injector to each corresponding producer.

The injection efficiency is calculated by the total offset oil production divided by the

injection rate for each injector. The same strategy can be used here for thermal flood-

ing problems. General well management will allocate more steam/hot water injection

rate to injectors with higher injection efficiencies.

If one already has a model based on a thermal FV simulator and just wants

to explore the connectivity and improve flood efficiency via the streamline-derived

information, we can also directly trace streamlines on the velocity field output from

the FV simulator. We will implement the same Pollock’s tracing method on the

velocity field calculated in FV simulation [92]. The same logic of well allocation factor

and injection efficiency based flood management will be implemented accordingly to

thermal problems.


Figure 5.27: Flux pattern map, by Thiele and Batycky [106].


We have extended streamline simulation to thermal recovery steam floods. We imple-

mented three techniques that address the large volume changes and strong couplings

inherent to steam floods: a 1D pressure update along the streamline, a Glowinski

θ-scheme operator splitting approach and preliminary hybrid SL-FV simulation. Our

test cases, which include a quarter five spot steam flood, a steam solvent flood, a

cyclic steam stimulation, a vertical cross section steam flood with gravity override,

and a multi-wells example, show that this steam flood streamline simulation is a vi-

able and accurate alternative to traditional finite volume approaches. We also show

for particular numerical issues such as grid orientation effects, SL can perform better

than normal FV simulation based on two point flux since we take into account the

upstream directions for the transport.

As in other streamline simulation application, the reduction of the 3D trans-

port problem to a series of 1D streamline systems is computationally attractive, in

particular because the streamline problems can be parallelized effectively [9]. The re-

sults presented in this paper are particularly encouraging as they show that thermal

streamline simulation may serve as a fast and effective proxy for reservoir simulation

studies, such as well placement, optimization and history matching. For example,

a reliable, fast proxy is particularly useful in the context of work flows associated

with the uncertainty of multiple geological models. Being able to screen earth models


efficiently is a key component of modern reservoir engineering work flows and by ex-

tending streamline simulation to thermal floods the computational cost of such work

flows can be substantially reduced. The unique streamline-derived dynamic connec-

tivity data remains a key element and can be used to improve pattern conformance

and improved steam injection in the same as streamlines are used to improve water

floods [106]. It is important to emphasize that our work is not to be understood as a

substitute to traditional simulation approaches. Instead, we see thermal streamline

simulation as complimentary tool to traditional finite-volume approaches by extend-

ing well-established uncertainty work flows to thermal problems as well as introducing

dynamic, streamline-based connectivity for improved oil recovery.

Chapter 6

In-situ Combustion Simulation

In thermal reactive reservoir simulation, conservation equations are solved numerically

on discretized reservoir grid blocks [27, 61]. Crude oil consists of thousands of complex

chemical species. When the crude oil reacts, we expect tens of thousands of chemical

reactions. When modeling in a numerical reservoir simulator, we need to lump the

components into several pseudo components and group the reactions into a limited

number of pseudo reactions. Accordingly, different kinetic reaction models have been

developed for ISC simulations.

6.1 Kinetic Reaction Models

The parameters for proposed kinetic models are usually derived from kinetic cell

laboratory experiments of ramped temperature oxidation (RTO) [24, 40]. In such

experiments, a small sample mixture of oil, water, sand, and clay is placed in the

kinetic cell that is subjected to controlled linear ramped heating while oxygen is in-

jected at a constant rate from the bottom of the cell. The produced gas composition

is obtained using a gas analyzer. A typical produced gas and ramped temperature

profile for a heavy oil is shown in Fig. 6.1. We observe two peaks in oxygen consump-

tion that correspond to two increases in the temperature profiles deviating from the

linear profile. Starting from the temperature of about 300oC, we initially have the

76

CHAPTER 6. IN-SITU COMBUSTION SIMULATION 77

Figure 6.1: Gas composition from ramped temperature oxidation experiments.

pyrolysis/cracking reaction that creates solid compound coke and some light hydro-

carbon components. Coke, that has very high molecular weight, serves as the main

source of fuel for the later combustion reactions. The first peak in temperature indi-

cates the low temperature oxidation (LTO) reactions, in which the coke is partially

oxidized, generating small amounts of heat. This is followed by a negative tempera-

ture gradient region, where the reaction rate decreases as the temperature increases.

Eventually, the coke fully combusts at higher temperatures, a process that is referred

to as high temperature oxidation (HTO). HTO is represented by the second peak

in the temperature profile. In the HTO region, the amount of oxygen consumption

is comparable to the production of carbon oxides. This implies a full combustion

reaction that generates CO2, CO, water, and a large amount of heat [24, 40].

One of the most commonly used kinetic reaction models is the Crookston model,

also known as the minimal model [30], given by

HO + O2 → CO + CO2 + H2O, (Heavy oil combustion)

LO + O2 → CO + CO2 + H2O, (Light oil combustion)

HO → Coke + LO, (Cracking/Pyrolysis)

Coke + O2 → CO + CO2 + H2O. (Coke combustion)

(6.1)


It comprises two oil components (heavy oil HO and light oil LO) and four chemical

reactions. Both HO and LO directly combust with oxygen to generate heat. The

component HO also pyrolyzes, generating coke and LO. This LO generation represents

an oil upgrading effect. Eventually, coke burns completely with oxygen, generating

heat. One disadvantage of the Crookston model is that it does not explicitly represent

the two peaks of LTO and HTO reactions as seen in RTO experiment gas analysis.

This is because we only have one step coke combustion, plus LO and HO direct

combustion. With its simple form and the early appearance, however, the Crookston

model is still very widely used in the literature.

We also introduce another three reaction model [33]. We create three reactions to

represent the pyrolysis, the low temperature oxidation and high temperature oxida-

tion, given by,

Oil + O2 → Coke1, (Pyrolysis)

Coke1 + O2 → Coke2 + CO + CO2 + H2O, (LTO)

Coke2 + O2 → CO + CO2 + H2O, (HTO)

(6.2)

where Coke1 and Coke2 are distinct fuel species. The advantage of the three reaction

model is that it captures fully the distinct LTO and HTO peaks. The disadvantage is

that we have only one dead oil component, thus no compositional effects or oil upgrad-

ing through combustion. We could combine the advantages of both the Crookston

and the three reaction model by adding an LTO intermediate step in the Crookston

model, or by adding light end products in the three reaction model’s pyrolysis re-

action. In the original Crookston and three reaction model, however, they are not

included [30, 33].

Other models such as the Belgrave model [11] and SARA model [43] group crude

oil pseudo components based on the SARA fractions (Saturate, Aromatic, Resin and

Asphaltene). These methods also employ a series of reactions, including coking (py-

rolysis), low temperature oxidation (LTO), and high temperature oxidation (HTO).

The main purpose of these models is to group the complex chemical reactions accord-

ing to the SARA fractions. This is because the different SARA fractions go through

different types of reactions when oxidized. Suggested by Freitag et al.[43], the SARA


model may be extended to different crude oils based on their SARA fractions. The

disadvantage of such models is that eventually when implementing in numerical sim-

ulators, the PVT property of the crude oil is not trivial to derive because the pseudo

component lumping is based on SARA fractions. This is because the SARA fractions

include hydrocarbons with wide range of carbon numbers, from the most light ends

all the way to very heavy hydrocarbons.

In our current ongoing research [25], a new kinetic reaction model is also proposed.

The reactions include: precipitation of coke, LTO reaction, change of coke active sites,

and HTO reaction. In the work by Lapene [68], a new kinetic reaction model with full

PVT characterization of Zuata crude oil is presented. It lumps the crude into eight

pseudo components (C1, C2−11, C12−16, C17−21, C22−27, C28−35, C36−49 and C50+). The

chemical reactions include the typical pyrolysis, LTO and HTO reactions.

We can choose different kinetic reaction models, depending on the complexity

we require and the experimental information we have. For example, for a bitumen

like heavy oil such as Hamaca, the oil composition is very heavy. The use of three

reaction model may be applied. For Athabasca bitumen, if we desire more detailed

intermediate steps in pyrolysis, LTO and HTO reactions, the Belgrave model serves

as a suitable choice.

In all the kinetics models, the common feature is that the crude oil is composed of

several components, and the chemical reactions are grouped into a series of reactions.

These components react, form intermediate products (oxidized compound or coke)

and eventually combust completely with oxygen. Each of these reactions has its own

kinetic parameters, that include stoichiometry coefficients, activation energy, pre-

exponential factor and reaction order. All these reactions follow the same Arrhenius

kinetics law as in Eq. 3.18 and Eq. 3.19.

6.2 1D Combustion Tube Simulation

We perform numerical simulation for a 1D dry combustion tube problem to demon-

strate the typical combustion behavior of an 1D ISC process. The three reaction

model is used in this study. This is because we have much experience with this


Figure 6.2: Relative permeability (water-oil and liquid-gas, respectively) and oil vis-cosity curve used in the simulation of Hamaca oil 1D combustion tube experiment.

model. The parameters are also well calibrated to the Hamaca oil experimental re-

sults [66]. The simulation domain is 2.16m, discretized into 600 grid blocks, each

with size 0.36cm. The oil sample is Hamaca oil from Venezuela, with 8.5 API gravity.

The corresponding fluid and rock properties are shown in Table A.9 and Table A.10

in Appendix A. The relative permeability and viscosity curve of the oil are shown

in Fig. 6.2. The kinetic parameters for the three reaction model are listed in Table

A.11, Table A.12 in Appendix A. We inject air from the left side and produce from

the right side. Initial ignition heating is introduced close to injector. The injector is

set at a constant air injection rate of 3000cm3/min at standard conditions, while the

producer is set at a pressure control of 780kPa. We perform simulation runs without

lateral heat losses for simplicity. Initially, we have a water saturation of Swi = 0.24

and an oil saturation of Soi = 0.26 in the tube. We simulate the problem for a total

of 1200 minutes. The resulting temperature, pressure and oil saturation profiles at

498 minutes are shown in Fig. 6.3.

We identify several different flow regimes in this 1D ISC problem. The leading edge

of the high temperature front is the combustion zone. Temperature ramps up very

steeply to about 800oC. This indicates a very narrow reaction zone typically several

inches thick. Note here that the peak temperature is greater than in the physical

1D combustion tube experiment. This is because we currently do not include heat

losses in our model. A typical combustion tube experiment temperature profile is


Figure 6.3: Temperature, oil saturation and pressure profiles in 1D combustion tubesimulation after 498 min of air injection.

Figure 6.4: Typical temperature profile history from a 1D combustion tube experi-ment, from Lapene et al., [66]


shown Fig. 6.4, by Lapene et al., [66]. Upstream of the reaction zone is the burned

zone. In this numerical experiment we see no residual oil saturation, because the high

temperature combustion reaction consumes all the fuel. The rest of the oil has been

displaced further downstream. In the burned zone, the temperature remains quite

high, because of the heat generated previously by combustion. Downstream of the

reaction zone, we have the steam plateau. The temperature remains at around 120oC.

The oil banks in the steam plateau, and So is constant around 0.8. In front of the

steam plateau is the unaffected zone with an initial oil saturation of 0.26. Along this

combustion tube, the overall pressure drop is relatively small, within 100kPa. In this

case, we combust with a low initial oil saturation of 0.26. The fuel amount for most

ISC processes is around 5 ∼ 10% So. Therefore, the size of the oil bank is relatively

small in this case. Furthermore, the oil is an 8.5 API bitumen with an initial viscosity

of 2 × 105cp at 15oC. According to the viscosity-temperature table, the oil still has

about 100cp viscosity at the steam plateau temperature. Thus, the oil bank remains

inside the steam plateau. If a less viscous oil sample (for example, Lloydminster type

conventional heavy oil) is simulated with oil viscosity reduced to < 10cp at steam

plateau temperature, the oil bank is likely to move in front of the steam plateau.

This illustrates that the flow regime behavior is very case dependent, depending on

the properties of the oil, the operating pressure, the initial oil saturation, and the

relative permeability.

We also show the production profiles in Fig. 6.5 and Fig. 6.6. After the initial

ignition period, the gas production reaches the relatively constant value of about

3000cm3/min. That behavior corresponds to the air injection rate. Most of the gas

produced is N2, with some CO2, and a very small amounts of CO and bypassed O2.

This indicates a good burn with combustion reactions in the HTO mode. The HTO

reactions consume most of the O2 and generate large amounts of CO2 as shown in Fig.

6.1. Water is constantly produced after the initial ignition, because the initial water

saturation is above the connate water saturation Swc in the relative permeability

curve. At about 750 minutes, the oil bank eventually reaches the producer. We

observe a sudden increase in the oil production rate. The oil bank produces until

about 1000 minutes, when the combustion front reaches the producer and O2 breaks


Figure 6.5: Water, oil and gas production rates in 1D combustion tube simulation.

through. This ends the 1D combustion process.

We further show the measured combustion front and the leading edge of the oil

bank movement in Fig. 6.7. In this case, we use a constant air injection rate. For

both front locations, we observe a close to linear increase with respect to time. This

means the combustion front has reached a pseudo steady state propagation in this 1D

problem. Combined with the previous stable production gas analysis, shown in Fig.

6.6, we conclude that when the combustion front is propagating steadily through the

1D system, the front is consuming the same amount of fuel, ”x”, per unit reservoir

volume. The rest of the oil flows into the oil bank, which makes the oil bank grow

linearly with time. This behavior has also been called a ”bulldozing” effect in the

literature [110]. This constant fuel consumption is also consistent with the basic

assumption in the classical Gates and Ramey analytical approach [44].

There is a fundamental difference between ISC and a steam flood process [96].

In ISC, we generate heat by the exothermic reactions in the combustion zone. This

generates non-condensable flue gas. This flue gas is constantly flushed further down-

stream and produced. The flue gas and steam mixture pushes the oil bank towards

the producer. The steam condenses at the leading edge of steam plateau and releases

heat to warm up the formation. In ISC, we need to keep supplying oxygen to the

combustion front in order to sustain the reactions. A good venting/circulation of the


Figure 6.6: Produced gas composition analysis in 1D combustion tube simulation.

Figure 6.7: Oil bank and combustion front distance-time diagram from 1D combustiontube simulation.


Figure 6.8: 1D ISC simulation: peak temperature and combustion front velocity as afunction of the injection air flux.

oxygen/flue gas from injector to producer is crucial to complete the whole process.

In a steam flood process, however, steam carries heat directly into the cold reser-

voir. Once the steam reaches the leading edge of the steam zone, it releases its latent

heat to the formation and condenses to hot water. We do not need to keep the oxy-

gen/flue gas circulation as in the ISC process. In a reservoir with undersaturated oil

(above the bubble point), we will not have free gas production until the live steam

breaks through to the producer.

6.3 Sensitivity Studies on 1D ISC Problem

In this section, we perform sensitivity studies on the previous 1D combustion tube

simulation. We test the effects of boundary conditions (air flux rate) and initial

conditions (Soi and Swi). Again, we use the same three reaction kinetic model [33]. We

have extensive experience using this model. It is also well calibrated to the Hamaca

oil experimental data here in Stanford University [66]. The previous simulation input

in section 6.1 is used as the base case for this sensitivity study. We increase the


Figure 6.9: 1D ISC simulation: oil saturation profile as a function of the initial oilsaturation Soi.

length of the 1D combustion tube to 10m to better observe the front propagation

behavior. The same grid block size (0.36cm in x direction) is used here. A more

detailed sensitivity study on the key input parameters for the three reaction model

is also presented in [86]. That study is based on an experimental design technique to

create the response surface map for different initial conditions, boundary conditions,

and input parameters.

We first perform tests for different air flux rates. The results are shown in Fig.

6.8. We observe a close to linear relationship between the air flux and combustion

front velocity. This implies that a constant amount of fuel, ”x” is consumed, again

consistent with the classical Gates and Ramey analytical approach [44]. The peak

temperature, as expected, shows a slightly increasing trend with increasing air injec-

tion rates, starting from 750oC at 750cc/min to 950oC at 7500cc/min.

We also test the effect of initial oil saturation, Soi, in Fig. 6.9. The combustion


fronts move to very similar locations, implying a consistent fuel amount ”x” per

reservoir volume burned. The Soi does impact the size of the oil banks that are

determined by the difference Soi − Sofuel, in which Sofuel is the saturation amount

of oil burned as fuel. The consistent fuel amount is likely due to the fact that once

steady state propagation is achieved, the combustion front sees the same oil bank

(So = 0.8 here) coming towards it, if we consider a frame of reference that moves at

the same speed as the combustion zone. Consequently the oil goes through similar

temperature increase in the combustion zone. The trailing edge of the oil bank moves

at more or less the same speed, regardless of Soi. The only difference is the different

oil bank lengths in front of the combustion zone. In the special case of Soi = 0.05,

the fuel amount Sofuel is higher than Soi. There is no oil bank.

For varying initial water saturation Swi, we observe a weak sensitivity on the

combustion front speed and fuel amount, as shown in Fig. 6.10. The reaction front

moves to similar locations in all these cases. This may be explained by the fact that

liquid initial water is vaporized further downstream away from the combustion zone

through the steam plateau. Thus the chemical reactions are not quite affected by

Swi.

We also study the sensitivity of oil in-situ viscosity. The previous simulation study

is for a very viscous bitumen type crude oil. We also study the 1D combustion process

of a mobile heavy oil (typical in-situ viscosity ∼ 103cp). We use the three reaction

models [33] with modifications to a conventional heavy-oil combustion problem (Lloy-

dminster type) by changing the viscosity-temperature relationship. This is because

the three reaction model is simple and capable of capturing the first order physics

of pyrolysis, LTO and HTO reactions. Initially, we have saturations Soi = 0.75,

Swi = 0.15 and Sgi = 0.10. The original oil viscosity is µo = 4600cp, with a initial

temperature of Ti = 15oC. We show the temperature, oil saturation and pressure

profile in Fig. 6.11. We observe a relatively small-sized steam plateau in this dry

forward combustion simulation. We also see a relatively large pressure drop over the

cold oil bank, in which the gas saturation has been reduced to a certain value that

is very close to critical gas saturation in the gas-liquid relative permeability. In the

case of wet combustion, the steam plateau size is greater, because of the additional


Figure 6.10: Temperature profiles as a function of the initial water saturation Swi of0.10, 0.24 and 0.50 from left to right.

heat carried forward from the burned zone by the injected water.

From these numerical simulation sensitivities studies, we observe the same be-

havior as the classical Gates and Ramey assumption [44]. Furthermore, we plot the

characteristics of different fronts in the 1D ISC process. These fronts include: com-

bustion front, leading edge of the oil bank, leading edge of the steam plateau. We

have observed two basic scenarios. Scenario 1 is the case when we have a low initial oil

saturation and a bitumen-like oil with very high in-situ oil viscosity, typically > 104cp.

Scenario 2 is the case when we have a less viscous conventional heavy oil (initial oil

viscosity in the 103cp range). In scenario 1, because of the high oil viscosity, the oil

bank stays inside the steam plateau. The leading edge of the steam plateau moves

ahead to warm up the formation downstream of the high saturation oil bank. This

is the case for the simulation studies of sections 6.1 and 6.2. In scenario 2, the oil is

more mobile such that the banked oil moves downstream of the steam plateau. The


Figure 6.11: Combustion tube simulation of a mobile heavy oil with high initial oilsaturation Soi at 378min.

steam plateau heat front drives the banked oil downstream. This is the typical case

in the field when we perform ISC for a mobile heavy-oil reservoir [41, 75, 81].

6.4 Grid Size Effects and their Cause

6.4.1 Grid Size Effects

In traditional Arrhenius kinetics, the reaction term Qreaci is highly dependent on the

local temperature T and reactant concentration Ci and/or O2 partial pressure. The

basic assumption for first order finite volume simulation is instant equilibrium and

instantaneous mixing in each cell, as shown in Fig. 6.14. This means that all physical

properties are homogenized (averaged) inside each grid block. Thus, we have only

averaged temperature and reactant concentrations that determine the kinetic reaction

terms in Eq. 3.18.

In reality, the reaction front is typically several centimeters thick at most, in both

lab experiments and field operations [96]. In order to capture accurately the detailed


Figure 6.12: Characteristics of different fronts in 1D ISC problem: scenario 1 (bitumenlike oil viscosity and oil bank stays inside the steam plateau).

Figure 6.13: Characteristics of different fronts in 1D ISC problem: scenario 2 (mobileheavy oil viscosity and leading edge of steam plateau follows behind the oil bank).


Figure 6.14: Basic assumption for thermal reservoir simulation: instantaneous mixingin each cell.

reaction kinetics inside the combustion front, we need centimeter sized grid blocks.

We tested a typical 1D combustion tube problem (0.8m long tube) using different

numbers of grid blocks. In this simulation study, we used the Crookston model for

kinetic reactions, with the same reaction parameters and settings independent of grid

size. The number of grid cells ranges from 10 to 800, with corresponding grid cell sizes

from 8cm to 0.1cm. Results are shown in Fig. 6.15. The temperature profiles and the

combustion front locations are strongly dependent on grid size. To achieve a spatially

convergent solution in this case (based on ”eye ball norm” of temperature profile), we

need at least 200 grid blocks, or grid blocks less than 1cm. For other kinetic models

such as the three reaction model, we have observed a similar grid-size requirement.

One important observation is that coarsening the grid typically leads to a greater peak

temperature and a lower front propagation speed. When the front sweeps through

a unit reservoir volume, the total amount of hydrocarbons burned increases with

grid-block size. This observation of excessive coke formation and slower front speed

on coarse grids is also well documented in the literature [22, 47, 61, 112]. In many

engineering applications, the reaction kinetics parameters are changed or tuned to

overcome this effect in a large field-scale grid block [47].

6.4.2 The Cause of Grid Size Effects

When the combustion front approaches a grid block, part of the oil inside the block

pyrolyzes and generates solid coke. This coke is the main portion of the fuel for later

exothermic oxidation reactions. The rest of the oil moves out of this cell by either

viscosity reduction or by distillation effects. This is a physically continuous process.


Figure 6.15: Grid-size effects in 1D combustion tube simulation. We require at least200 grid blocks to achieve convergent solution in this 0.8m long combustion tubesimulation.

When we heat up the oil inside a block, the oil cracks. In the mean time, the oil also

flows out of the cell. If we use a larger simulation grid block, the block has larger

heat capacity, which generates relatively slower heating rate. This makes the oil takes

longer to pyrolyse/crack. The competing mechanism between pyrolysis/coking and

oil flowing out of the reaction cell does not scale with grid block sizes. This is caused

by the default instantaneous mixing or averaging assumption and discrete nature in

first order finite volume reservoir simulation Fig. 6.14. Consequently, less fraction of

oil moves to the next cell and more fraction of solid fuel is generated as a numerical

artifact. As a result, the front speed slows down. Eventually, the excessive amount

of fuel or coke also introduces higher peak temperatures.

6.4.3 Why not Adaptive Mesh Refinement?

Cartesian Adaptive Mesh Refinement (AMR) [84] was previously proposed for kinetics

based in-situ combustion simulation. The main challenge for using AMR for field-

scale ISC simulation is that it is practically very difficult to implement because of

the dramatic grid size contrast between areas near the front and away from it. The

typical grid block size for full field reservoir simulation is on the order of 10m. We have

observed < 1cm grid size requirement for kinetics based ISC simulations. In practice,


it is difficult to generate a high quality AMR grid when we have a narrow combustion

front of large spatial extent and large shrinkage requirements of the grid size. A multi-

level AMR grid may be necessarily needed for such dramatic changes of grid sizes.

Another practical concern is the choice of time step sizes. When implementing AMR

grid for kinetics based ISC simulation, the time step size is still restricted by the fast

chemical reactions right at the combustion zone. In current kinetics based simulation,

we observe time step size requirements in minutes. Therefore, if we intend to simulate

the field-scale ISC process for years, we will necessarily have many time steps and the

simulation becomes impractical to execute. Furthermore, the computational workload

per grid cell is not the same everywhere. In areas away from the combustion zone, the

workload is dramatically smaller than near the combustion front. When coarsening

the grid far from the front in AMR, the cost saving is relatively small. Because of

the intense workload in the near front region, the computational cost is still intense.

Actually, the implementation of a two-level dynamic gridding in commercial thermal

simulator is shown in [22, 95]. The speed-up gained (reported 2× to 3×) is still

limited in 3D realistic cases. This is the main reason that we instead seek the reaction

upscaling approach as shown previously.

6.4.4 The Need for Upscaling

The computationally affordable grid block sizes for full field models are typically tens

to hundreds of meters laterally. There are orders of magnitude differences between

the affordable and the desired grid block sizes for kinetics based in-situ combustion

simulation. The cost of kinetics based simulation hinders the application of numerical

simulation for field-scale simulations.

In-situ combustion is indeed a multi-scale multi-physics process bringing together

porous media flow, chemical reaction kinetics, and phase equilibria. First, the chem-

ical reactions happen very fast, in the time-scale of minutes or hours. The advection

of fluid flow, however, is a much slower process that has a typical time scale of years.

The Suplacu field ISC process has been more than 40 years [91]. Second, the chemical

reactions are localized inside the centimeter-sized reaction front, while the advective


flow takes place in the reservoir, typical length scale in kilometers. Therefore, both

the temporal and spatial scales of reaction kinetics and advection are very different in

the ISC process. An important dimensionless number that characterizes this differ-

ence is the Damkohler number (Da). It represents the ratio of the characteristic time

scale for the fluid flow to that of the kinetic reactions. In a lab-scale combustion tube

experiment, the Da number is usually of the order of 103. In field-scale ISC, the Da

number is typically around 107, because the advection time scale is in years rather

than hours. To alleviate this scaling problem, we must find grid-insensitive reaction

models. Pursuing such an upscaled reaction model is the goal of this work.


In traditional thermal reactive reservoir simulation, mass and energy balance equa-

tions are solved numerically on discretized reservoir grid blocks. The reaction terms

are calculated through Arrhenius kinetics using cell-averaged properties, such as aver-

aged temperature and reactant concentrations. The chemical reaction front is phys-

ically very narrow, typically a few inches thick. To capture accurately this front,

centimeter-sized grids are required that are orders of magnitude smaller than the af-

fordable grid block sizes for full field reservoir models. Therefore, the reaction kinetics

is a subgrid-scale phenomenon for field-scale simulation. The kinetics temporal and

spatial scales are significantly smaller than that of advection. It is extremely diffi-

cult to solve flow and kinetically determined reactions together simultaneously. We

need to upscale the reaction kinetics to the similar temporal/spatial scale of flow and

transport to solve this problem on the full field scale. This upscaling is explained in

detail in the next chapter.

Chapter 7

Upscaling for In-situ Combustion

Reactions

In this chapter, we focus on reaction upscaling of In-situ Combustion reactions. The

critical physics for the ISC process is shown in Fig. 7.1. When the reaction front

sweeps through the reservoir, a certain amount of oil becomes fuel and burns (usually

x = 5 ∼ 10% So). The rest of the mobile oil gets pushed further downstream

through either viscosity reduction or distillation. This process is also envisioned as

a bulldozing effect [47]: the reaction front burns some fuel and bulldozes everything

forward downstream. In order to simulate field-scale ISC, we need to capture this

process through upscaling of the reactions.

Figure 7.1: The critical physics for 1D ISC. When the reaction front sweeps throughthe reservoir, a certain amount of oil becomes fuel and burns (usually x = 5 ∼10% So). The rest of the mobile oil is displaced further downstream.

95

CHAPTER 7. UPSCALING FOR IN-SITU COMBUSTION REACTIONS 96

7.1 Non-Arrhenius Reaction Modeling

We know from previous analysis that the main cause for grid-size effects is the rela-

tively larger amount of fuel generated as grid block volume increases. We also know

for ISC, the reaction terms Qreaci in Eq. 3.18 need to be captured accurately in

particular.

Our main idea is to consider the equivalent reaction effects on a large field-scale

grid block. The most critical reaction effects should match the reaction effects on a

fine-scale model during a simulation time step from tn to tn+1. A certain amount of

oil becomes fuel and burns, consumes oxygen, generates light ends through pyrolysis,

heat and flue gas, and the rest of the oil flows further downstream. Because the

size of the reaction front is much smaller than the reservoir grid block, we do not

calculate the source and sink terms using cell averaged properties and Arrhenius

kinetics. Instead, we treat the reaction front as a sub-grid scale phenomenon and

calibrate the reaction effects through fine-scale simulations and lab experiments. We

only replace the reaction terms in the governing equations, with transport solved as

before. Equivalent reaction effects (fuel amounts) are calibrated here through 1D

fine-scale simulation as in Chapter 6, or collected experimentally using combustion

tube runs. Once we calibrate the equivalent reaction effects (fuel amount) from 1D

fine-scale simulation, we then infer reaction effects for a 3D ISC process accordingly.

A similar concept has been applied to idealized convection-reaction problems in [49].

Tracking the fuel amount concept is very similar to the classical Gates and Ramey

analytical model [44].

We follow these steps:

1. We first achieve a reasonable history match of the kinetic cell and combustion

tube using a chosen kinetics based model [10, 68].

2. We run fine-scale 1D simulations (typically 5 to 10 meters long), with various

uniform initial conditions and constant air-flux boundary conditions. As shown

in section 6.2, after the initial ignition the reaction front very quickly reach a

steady-state propagation with almost constant speed (usually within the first

0.1m). This implies that the front consumes a constant amount of fuel ”x” per


unit reservoir volume, with a constant oxygen utilization ratio. We define the

amount of fuel as the amount of hydrocarbon components that are oxidized

when the reaction front sweeps through. The fuel amount is collected in a

table over the range of conditions tested (usually the fuel amount is 5 ∼ 10%

So). This amount of fuel will be a certain value for each oleic hydrocarbon

component [C1burn, C2burn · · ·Cnburn]. According to the local conditions such

as saturations and pressure, we decide the fuel amount for each grid block in

the 3D reservoir, when the combustion front sweeps through. Calibration tests

for fuel amount and oxygen efficiency are performed for different initial and

boundary conditions, including different pressure, different initial saturation,

different air flux and petrophysical conditions. Petrophysical properties such

as type of porous media, mineralogy and clay content, also affect kinetics and

the ISC fuel amount that are currently not supported in Arrhenius kinetics

based commercial simulators. We are able to include these effects into our new

upscaled reaction modeling framework.

3. We assume that with enough initial heating and proper air flux, the combustion

ignites around the injection wells. When the reaction front starts propagating,

we calculate the reaction source and sink terms. During each time step, we

flag the cells that currently have combustion reactions. Inside each combusting

cell, we have a reasonable estimate of the quantity of the fuel based on tests

performed under step 2. We now perform equilibrium combustion reactions

according to the supplied air flux and the remaining fuel. If all fuel is consumed,

the cell is un-flagged. When the reaction front moves into a new cell, we flag

that cell as one of the combustion cells.

In this method, we only track how much fuel has been burned during a time step.

The equivalent reaction effects are used for source and sink term calculations on a large

field-scale grid block. This approach also removes the stiffness encountered in kinetics

based simulation. We can let the transport decides the time step sizes. Typically this

results in time step sizes ∆t = tn+1 − tn of the order of days. Thus reaction effects

and flow advection are effectively modeled at the same temporal and spatial scales.


Figure 7.2: ISC upscaling work flow.

As we discussed earlier in section 6.3, traditional kinetics based simulation is strongly

limited by the grid size requirement. This change makes the numerical simulation

much more efficient.

The complete upscaling work flow is shown in Fig. 7.2. First, we start the kinetic

cell experiments to understand the oxidation reactions for the crude. Combustion

tube experiments are also conducted to test the 1D combustion process. Then, we

perform numerical simulation to match the lab experiments. At this stage, we use

Arrhenius kinetics to model the reactions and history match lab experiments. We

use a large number of grid blocks with enough resolution to ensure convergence and

accuracy at lab scale. Based on this, we extract the most important parameter

for in-situ combustion design: the fuel amount ”x”. When the reaction front is

propagating through the reservoir, we directly calculate the reaction source and sink

terms according to this fuel amount ”x”. In this way, we achieve consistency when

scaling up the simulation from lab to field.


7.2 Implementation

7.2.1 Pseudo Code

The pseudo code implementation is shown here. The explanations for major variables

and major functions are given in Table 7.1 and Table 7.2. The main idea is to track

the remaining fuel amount for each component Ci in each reaction grid block. Instan-

taneous equilibrium oxidation reaction is assumed. The fuel is generated when the

reaction front moves into a virgin cell. When the fuel has been completely consumed,

the reaction front moves onto the next cell. We use special treatment for the reaction

cell oil phase flux to ensure complete sweep Sor = 0, i.e. all the oil flows out after the

front has completely swept through that cell.

Within each Newton-Rhapson Iterations: Loop (i,j,k)

If (Flag_reaction_cell(i,j,k)==TRUE)

CALL Calculate_Oil_Flux(Flux_O2(i,j,k),Fuel_amount(i,j,k),Ci(i,j,k))

CALL Calculate_Reac_Term (Flux_O2(i,j,k),Fuel_amount(i,j,k))

Else

If (Flux_O2(i,j,k)>0 AND So(i,j,k)>0)

Flag_reaction_cell(i,j,k)=TRUE

If (So(i,j,k)>So_burn)

Fuel_amount(i,j,k)=Ci_burn

Ci(i,j,k)=Ci(i,j,k)-Ci_burn

!Remaining oleic components eventually be swept out by ISC front

Else

Fuel_amount(i,j,k)=Ci(i,j,k)

Ci(i,j,k)=0

!If So(i,j,k)<So_burn, all the oil will be consumed as fuel

!Nothing flows out

End If

CALL Calculate_Oil_Flux(Flux_O2(i,j,k),Fuel_amount(i,j,k),Ci(i,j,k))

CALL Calculate_Reac_Term (Flux_O2(i,j,k),Fuel_amount(i,j,k))


End If

End If

End Loop

Function Calculate_Reac_Term(Flux_O2(i,j,k),Fuel_amount(i,j,k))

If (Oxygen Limited) !More fuel than Oxygen

Reac_Term=Flux_O2(i,j,k)*Stoichiometry

Fuel_amount(i,j,k)=Fuel_amount(i,j,k)-Flux_O2(i,j,k)*Stoichiometry

Else If (Fuel Limited) !More Oxygen than fuel

Reac_Term=Fuel_amount(i,j,k)*Stoichiometry Fuel_amount(i,j,k)=0

Flag_reaction_cell(i,j,k)=FALSE

End If

End Function

Function

Calculate_Oil_Flux(Flux_O2(i,j,k),Fuel_amount(i,j,k),Ci(i,j,k))

If (Oxygen Limited) !More fuel than oxygen

Temp_ratio=Flux_O2(i,j,k)*Stoichiometry/Fuel_amount(i,j,k)

!Ratio of fuel be consumed, over the total fuel initially in this cell

Flux_Oil=Ci(i,j,k)*Blksize*Temp_ratio

Else If (Fuel Limited) !More oxygen than fuel

Flux_Oil=Ci(i,j,k)*Blksize

End If

End Function

7.2.2 Implementation in Commercial Software

To test our upscaling methodology, we have preliminarily implemented our concept

in commercial reservoir simulators [37, 101] using a simple technique as follows.

All current commercial simulators are based on Arrhenius kinetics. Therefore,

we have developed some techniques to implement the reaction upscaling. All the test


Table 7.1: Explanation of major variables in the upscaling pseudo code.

Major Variables ExplanationFlag_reaction_cell Flag for cell (i,j,k) that are undergoing combustion reactions.Fuel_amount The remaining fuel in cell (i,j,k) for Nc oleic components.Flux_O2 Total amount of supplying O2 flux flowing into cell (i,j,k).So_burn The oil saturation that will be burned as fuel, if initial oil

saturation Soi>So_burn. If initial oil saturation Soi<So_burn,then all the Soi will be burned as fuel.

Ci_burn Amount of the i-th component that will be consumed as fuel,if Soi>So_burn.

Ci Amount of the i-th component currently in cell (i,j,k).

problems are small scale problems with uniform initial conditions, i.e. constant initial

saturations Soi, Swi, Sgi and pressure Pi. The original kinetic reaction model we use

is the three reaction model shown in Eq. 6.2. The corresponding upscaled model is

based on two pseudo kinetic reactions given by,

1. 1.0 Oil1 −→ (1.0− x) Oil2(inert) + x Coke (at the start of simulation)

2. Coke + O2 −→ CO + CO2 + H2O (during the simulation)

(7.1)

Oil2 is an inert component that does not engage in any reactions. Both reactions

have very large constant reaction rates (equilibrium reaction). The fuel amount is

pre-determined by the value x in the first reaction, that takes place at the beginning

of the simulation. In this way, we make combustion reactions insensitive to grid size.

Because we lay down the fuel at the start of the simulation, the relative permeability

is adjusted accordingly to account for this loss of oil saturation. This adjustment is

similar to the method described in [98]. A sample STARS input file for upscaled three

reaction model is provided in Appendix D. We compare the original kinetics based

simulation and the upscaled model results.

We have tested 1D, 2D, and 3D cases with varying grid sizes. We do not history

match any particular experiments or field history, but show the consistency of results


Table 7.2: Explanation of major functions in the upscaling pseudo code.

Major Functions ExplanationCalculate_Reac_Term Function to calculate the reaction source and sink terms

for cell (i,j,k). The reaction terms are calculated accordingto the supplying oxygen flux flowing into the cell Flux_O2,and the remaining fuel amount Fuel_amount(i,j,k).If (O2 flux > remaining fuel), fuel all consumed, (i,j,k) unflagged.If (O2 flux < remaining fuel), fuel=Flux_O2*Stoichiometry.

Calculate_Oil_Flux Special treatment for oil flux flowing out of a reaction cell:when the fuel all consumed, remaining oil in this reactioncell completely flows out.

between the fine-scale kinetics based reference and the coarse scale upscaled model.

The major inputs of the kinetics of the three reaction model are provided in Appendix

A. The reference kinetics based simulation cases are performed in small physical

sizes, because we are limited by the computational costs associated with the fine-

scale Arrhenius kinetics based simulations. In all test cases, we have a large grid-size

upscaling ratio to show the consistent results achieved by the upscaled model.

7.3 Upscaling Test Cases

7.3.1 1D Tests

The first example is a 2.1m long 1D combustion tube problem shown in Fig. 7.3.

We inject air from the top and produce from the bottom. To achieve an accurate

reference solution, we first use the kinetics reaction model Eq. 6.2 with N=600 grid

blocks. Then, we simulate the same problem with N=10 grid blocks using both the

kinetics based model and our upscaled reaction model. The fuel amount in this case

is calibrated as Sofuel = 8%.

As we can see, the N=10 upscaled model is capable of predicting a similar reaction

front velocity and peak temperature as the fine-scale reference. If we instead use the


Figure 7.3: 1D ISC upscaling tests: kinetic and upscaled model using different gridresolutions. The coarse scale upscaled model matches the fine-scale kinetics basedreference, while the coarse scale kinetic model fails.

Figure 7.4: 1D ISC upscaled model with different equivalent fuel amount Sofuel. Thefront location has a direct relationship with the fuel amount Sofuel.


original kinetic model and N=10 grid blocks, we fail to predict the correct front

location. We observe ”over coking” in this case, as explained in section 6.3. The

reaction front moves more slowly and we generate an excessive amount of coke in the

pyrolysis reaction. The excessive amount of coke also leads to greater heat generation.

This is consistent with the discussions in section 6.3.

We further show the results by making adjustments to the equivalent fuel amount

Sofuel value in Fig. 7.4. We show that by directly changing the Sofuel value in the

upscaled model we are able to adjust how fast the front propagates to match a certain

fine scale solution. There is a reciprocal relationship between fuel amount and front

speed, providing the same amount of injected air.

7.3.2 2D Tests

We test a 2D quarter five spot problem, Fig. 7.5. The size of the 2D problem is

0.72m× 0.72m with homogeneous permeability and porosity. We inject air from one

corner and produce from the other corner. We first run the simulation using Arrhenius

kinetics with 200×200 grid blocks. This is used as the fine-scale reference result. We

then use coarser grid blocks (20×20 and 5×5) and compare the results. We plot the

oxygen fraction in the gas phase to show the exact location of the ISC reaction fronts.

For grid resolutions 20×20 and 5×5, the kinetic simulation underestimates the front

speed (shown by the red dashed line). The coarse 5× 5 kinetic simulation introduces

so much averaging that we have an initial ignition problem. We have much oxygen

bypass initially because of the low temperature in the near well region. Furthermore,

the peak temperature is also greater because of the excessive amount of fuel (coke)

generated at later times in the larger grid blocks. On the other hand, the upscaled

model predicts front locations very similar to the fine-scale reference. This establishes

that a similar amount of fuel is combusted.

We also test a 2D problem with multiple wells, as shown in Fig. 7.6. The size of

the 2D domain is 0.72m × 0.36m with homogeneous permeability and porosity. We

inject from two air injectors on the left and produce from the right. Again, a fine-

scale kinetics based simulation (200 × 100 grid) is performed to achieve a reference


Figure 7.5: ISC upscaling in horizontal 2D 1/4 five spot case: O2 fraction in gas phasewith different grid resolution and both kinetic and upscaled model.

Figure 7.6: ISC upscaling in horizontal 2D multi well case: coarse grid upscaled modeland fine grid kinetic model.


Figure 7.7: ISC upscaling in horizontal 2D heterogeneous case: coarse grid upscaledmodel and fine grid kinetic model.

solution. We upscale the model to a 20× 10 grid and run the simulation again with

our upscaled model. In this case we show the oil saturation contour. The upscaled

model successfully predicts a similar front location (the red dashed line) and also the

size and shape of the oil bank.

We further test a simple 2D heterogeneous case, as shown in Fig. 7.7. We the

permeability field (60 × 20) show in Fig. 7.7 with physical size 1.08m × 0.36m. We

further downscale the 60 × 20 permeability field to 300 × 100 using first order flat

reconstruction for fine scale simulation. We inject air uniformly from the left side

and produce from the right side. The oxygen mole fraction in the gas phase is again

used to show the reaction front movement. We observe clearly that the reaction front

locations are consistent between fine-scale kinetic model (300× 100) and coarse-scale

upscaled model (60× 20).

A 2D vertical reservoir cross section case is also performed to test gravity effects,

as shown in Fig. 7.8. We inject air from a vertical injector on the left and produce


Figure 7.8: ISC upscaling in 2D vertical case: coarse grid upscaled model and fine-gridkinetic model.

from a vertical producer on the other side. Again we show the oxygen mole fraction

contour. Gravity override is clearly observed in this case. Reaction front movement is

consistent between the fine-scale kinetic model and the coarse scale upscaled model.

7.3.3 3D Tests

Our final test case is a 3D large field-scale forward in-situ combustion process. The

reservoir is homogeneous with the size 150m×63m×15m. We have a vertical air

injector on the left and a producer on the right, as shown in Fig. 7.9. We have

implemented two different grid resolution here, discretizing the reservoir by 10×11×5,

25× 11× 10 and 50× 21× 10 grid blocks. This is to show the consistency achieved

by the upscaled model with different grid resolutions. We have only implemented the

upscaled model in this case, because of the lack of computational power for achieving

a fine-scale kinetics based reference solution with centimeter sized grid. If we use

the 0.36cm× 0.36cm× 0.36cm grid as in kinetics based simulation, the total number

of grid blocks is 3.04 × 1012 in this case. The results are shown in Fig. 7.9. We

observe the combustion front moving towards the producer. For the grid resolution

25× 11× 10 and 50× 21× 10, gravity override is also clearly captured. When using


10×11×5 grid, the simulation has some problem in representing the gravity override

because of the coarse grid in the Z direction. The simulation results are consistent

with these three different grid sizes. This demonstration purpose example indicates

the potential capability of the upscaled reaction model for field-scale simulation.

We also compare the average time step sizes taken in this 3D field-scale case

(50 × 21 × 10 grid) in Fig. 7.9 with the previous small scale 1D kinetics based

combustion tube simulation in Fig. 7.3. In this 3D field-scale with upscaled reaction

model, we take average time steps of 0.25day. The average time step size in kinetics

based small scale simulation is 0.002day. In the kinetics based model, we need to

capture the fast transient chemical reactions in every small grid block with combustion

reactions. As a result, the time step size in the simulation is highly restricted. On

the other hand, the upscaled model captures the cumulative reaction effects over a

much longer time period and also a larger spatial scale. Reaction upscaling helps to

overcome the stiffness in the governing equations, that makes the simulation capable

to take larger time steps (in days) and simulate for reservoir scale time periods.

7.4 Discussion and Applications

7.4.1 Valid Range

Our upscaling methodology replaces the Arrhenius kinetics terms with upscaled terms

that have, at the scale of the coarse grid block, the equivalent overall effects on mass

and energy transport. Other than that, the governing equations are not changed. In

the preliminary test cases shown, the upscaled model gives similar results on coarse

grids as the original kinetics based model on fine grids. This is because the fuel

amount, determined by ”x” in Eq. 7.1, is carefully calibrated. This also ensures the

consistency in the reaction term calculations, when using different grid block sizes

on all levels of resolution. The methodology is also different than simple parameter

tweaking [28, 47, 53], that is dependent on the grid block sizes. Because the very

stiff kinetics have been removed from the governing equations, we are able to take

dramatically larger time steps. Accordingly, we believe that the efficiency of our


Figure 7.9: 3D field-scale case using the upscaled reaction model. Consistency isachieved between different grid resolutions.


upscaled method is much greater than traditional kinetics based simulation. We are

strongly encouraged by the accuracy obtained by our upscaling methodology.

Assumptions and restrictions for our ISC reaction upscaling methodology are laid

out next.

• In our upscaling approach, it is assumed that the combustion reaction has been

sustained in the HTO mode. In many field operations, failure to keep the

combustion in the HTO mode is the actual cause for many unsuccessful ISC

projects [85]. A further improvement that we suggest is the minimal air flux

approach. An operational window is defined for combustion to sustain in the

HTO mode [85]. This improvement is discussed in detail in the future direction

section 9.2.2.

• The upscaling approach is designed specifically for heavy oil combustion prob-

lems, where the reaction is sustained in the high temperature oxidation mode

with a very thin combustion zone. A light oil high pressure air injection (HPAI)

process operated in LTO mode could have wide extended reaction zone, instead

of the thin combustion front for heavy oil [46]. Combustion operated in LTO

mode is beyond the scope of the current work [42, 46, 79].

• We only consider the forward drive-type in-situ combustion process, in which

the different fluid phases flow almost co-currently. We currently do not consider

cases with counter-current combustion gravity drainage, because such cases do

not appear in practice at this time.

• We consider a single porosity reservoir. For a fractured dual porosity reservoir,

we need to understand the matrix-fracture heat and mass transfer. A specialized

upscaling technique is required, which is an ongoing research topic [39].

• The upscaling technique is for chemical reactions only. More research is needed

for sub-grid scale phenomena such as fluid fingering or unstable displacement

fronts. We discuss this in detail in the next section.

• We have either complete or partial oxygen utilization. In the case of partial

oxygen utilization, part of the oxygen bypasses the reaction front and is not


fully combusted. This may be due to local small-scale heterogeneity, that is

discussed in detail in the next section. We need to define an extra parameter:

oxygen utilization efficiency Eu(0 ∼ 1), that describes the fraction of oxygen

consumed in the reaction zone. This parameter is also calibrated through fine-

scale simulation or experiment.

7.4.2 Sub-grid Scale Heterogeneity and its Effects on Reac-

tion Upscaling

In most of the previous simulation studies, we implement small-scale problems with

moderately heterogeneous permeability fields (usually within 5 orders of magnitude).

In all these cases, the reaction front is topologically continuous. The domain is

divided into the burned zone and the unburned zone. Almost all the injected oxygen

is consumed at the thin reaction front, moving from injector towards the producer.

We do not observe oxygen or residual oil bypassing. One immediate question is what

if we have a more heterogeneous porous medium. Related questions arise. Is the

combustion front still topologically continuous and sweeping the whole area? Do we

get discontinuous fronts? Is there oil left behind? Do we have oxygen bypass? How

is fuel amount Sofuel obtained?

We perform tests of some sub-grid scale heterogeneous systems to test the oxygen

and residual oil bypassing and answer the above questions. The test cases are per-

formed using Arrhenius kinetics (three reaction model). We present a series of cases

here, that include white noise permeability, systems with square size low permeabil-

ity, layered systems, upper and lower layer of SPE 10 permeability [23]. The first

example is a 2D problem (0.072m × 0.54m) with uncorrelated white noise, 4 orders

of magnitude permeability contrast from 100mD to 1000D in Fig. 7.10. In this case,

we do not observe much oxygen bypass. The residual oil in the burned zone has been

fully swept forward by the continuous combustion front. Using material balance, we

calculate the equivalent fuel amount as Sofuel = 8.1%, which is very close to the fuel

amount of homogeneous case Sofuel = 8.2%.

The second example we tested is a 2D problem with a square low permeability


Figure 7.10: Permeability and oil saturation results for 2D case with uncorrelatedwhite noise permeability field. The combustion front propagates stably.

zone in the middle, shown in Fig. 7.11. The simulation results show three contin-

uous snapshots 1 to 3. The red dashed line shows the approximate location of the

high temperature front of the main combustion zone. Because of the strongly het-

erogeneous permeability field, some oil is initially bypassed and not fully combusted.

Eventually, by heat conduction and oxygen diffusion in the gas phase, the bypassed

oil is fully combusted after the main reaction front moves through. We also observe

oxygen bypassing the reaction front because of the strongly correlated heterogeneity.

Through the use of material balance, we further compare equivalent fuel amount to

the same 2D problem with 10D homogeneous permeability. For the 10D homoge-

neous case, the fuel amount is Sofuel = 8.2%. With the low permeability square, the

fuel amount becomes Sofuel = 9.0%. The fuel amount has been changed because of

the heterogeneity. When the combustion front moves in, the oil in the lower perme-

ability zone is more difficult to flow further downstream. This causes more oil in this

zone to coke, thus forming more fuel for later combustion reactions. Heterogeneity

causes oxygen bypassing and formation of a greater amount of coke relative to the

homogeneous case. We also calculate the cross sectional oxygen utilization efficiency

Eu = (Oxygen Consumed)(Oxygen Injected)

through material balance. For this low permeability square

case, the overall cumulative oxygen utilization efficiency is Eu = 0.85.

The third example is a 2D problem with its permeability field shown in Fig. 7.12.


Figure 7.11: 2D case with correlated heterogeneity (low perm in the middle). Weobserve some oxygen bypass and change in equivalent fuel amount.

We randomly generate low permeability zones in the 2D problem. We again combust

from left to the right. A small amount of oxygen bypass is detected. We calculate the

overall oxygen utilization efficiency and fuel amount as Eu = 0.90 and Sofuel = 9.7%,

respectively.

The fourth example we show is a 2D horizontal layered system in Fig. 7.13. We

implement two permeability contrasts K1/K2 = 2 and K1/K2 = 10. We observe

some oxygen bypass and residual oil bypass. After material balance calculations,

we find oxygen utilization efficiency as Eu = 0.81 for K1/K2 = 2 and Eu = 0.66

for K1/K2 = 10. The overall fuel amount for K1/K2 = 2 is Sofuel = 8.1% and

Sofuel = 8.7% for K1/K2 = 10.

The fifth example we show is a 2D problem with layer 1 permeability of SPE 10

[23] in Fig. 7.15. In this case, we do not observe residual oil bypass. We also observe

very little oxygen bypassing the main reaction front. We calculate the fuel amount

and oxygen utilization efficiency as Sofuel = 8.6% and Eu = 0.91.

The sixth example we show is a 2D problem with highly heterogeneous channel-

ized permeability (layer 51 of SPE 10 [23]) in Fig. 7.15. In this case, we have the

temperature front moving relatively stably because of heat conduction. Again, some


Figure 7.12: 2D case with correlated low permeability squares. We calculate oxygenefficiency Eu = 0.90 and fuel amount Sofuel = 9.7%.


Figure 7.13: 2D case with horizontal layered permeability. We observe oxygen bypassand change in equivalent fuel amount.


Figure 7.14: 2D case with heterogeneity (layer 1 of SPE 10 [23]). The oxygen efficiencyis Eu = 0.91 and fuel amount Sofuel = 8.6%.


Figure 7.15: 2D case with channelized heterogeneity (layer 51 of SPE 10 [23]). Theoxygen efficiency is Eu = 0.85 and fuel amount Sofuel = 13%.


Table 7.3: Summary of Sub-grid Scale Heterogeneity ISC Tests

Case Number and Description Eu Sofuel

1. White Noise 0.88 8.1%2. Low Permeability Middle Square 0.85 9.0%3. Low Permeability Squares 0.90 9.7%4a. Layered System (K1/K2 = 2) 0.81 8.1%4b. Layered System (K1/K2 = 10) 0.66 8.7%5. Layer 1 of SPE 10 [23] 0.91 8.6%6. Layer 51 of SPE 10 [23] 0.85 13%

oil is initially bypassed. We also observe oxygen bypassing the main reaction front.

We calculate the fuel amount and oxygen utilization efficiency as Sofuel = 13% and

Eu = 0.85.

In this section, we tested ISC front propagation in heterogeneous systems, cover-

ing a range of different permeability fields. A list of the summary of the test cases

are shown in Table 7.3. We have included some extreme cases here, such as the

channelized lower layer of the SPE 10. From the previous test studies, we see oxygen

utilization and residual oil bypassing events are case dependent. In cases with more

correlated heterogeneous permeability fields, we observe more different oxygen uti-

lization (oxygen bypass) and its equivalent fuel amount. We observe fuel amount and

oxygen bypass generally increases for more correlated heterogeneity. These numbers,

however, are still varying mostly within a small range, as shown in Table 7.3. The

fuel amount Sofuel varies from 8.1% to 13%, and oxygen utilization Eu from 0.66 to

0.91.

The different scales in measurements, geomodeling and reservoir simulation are

shown in Fig. 7.16. The typical geomodelling grid block size for a reservoir model is

on the order of 100ft×100ft×1ft. Between core/well log data and geo-cellular grid,

there is a ”missing scale”. This means the core/well log data is directly assigned to

the entire large geo-cellular grid in modern reservoir engineering. The scale of ISC

combustion front is actually inside this ”missing scale”. The simulation cases we have

shown here are small-scale test cases, typically less than 1m × 1m. We are actually


testing the potential impact of sub-grid scale heterogeneities in these cases. The

previous material balance calculations for Sofuel and Eu serve as a first step upscaling

methodology if ideally we are able obtain this detailed sub-grid scale heterogeneity.

For example, when we scale up the 2D domain in these tests to one geo-cellular

grid block in full field reservoir model, we calibrate the fuel amount and oxygen

utilization efficiency. In Fig. 7.16, we show the ”missing scale” from core/well log to

geomodelling grid (uncertainty in permeability, porosity, wetting, clay contents, etc.).

Considering the lack of sub-grid scale heterogeneity information through well log,

seismic and well tests in this ”missing scale”, we need to bridge the gap between the

core/well log and geo-cellular grid. We suggest performing uncertainty study based

on different realizations of the sub-grid scale heterogeneity. The simulations and the

sensitivities with respect to permeability distributions is a first step in understanding

the spread of the fuel amount and oxygen utilization. When history matching full field

reservoir models, the equivalent fuel amount and oxygen utilization coefficient are also

allowed to be perturbed. We hope this helps to represent heterogeneity in all scales.

This is very analogous to the concept of sand to shale net-to-gross ratio (NTG), or

the mixing parameter ω for sub-grid scale heterogeneity induced dispersion in Todd-

Longstaff miscible flooding formulation [108]. This enables us to account for these

sub-grid scale heterogeneities that cause oxygen bypass and change in fuel amount.

From geomodelling to reservoir simulation grid, we suggest the same classical flow

transport upscaling technique [20]. The fuel amount and oxygen utilization efficiency

are upscaled similarly as porosity φ or NTG.


In this work, we propose a new upscaling method for heavy-oil ISC reaction modeling.

We no longer calculate the reaction terms using Arrhenius kinetics and cell-averaged

values. We instead find and calibrate the equivalent reaction effects, i.e. the amount

of heat released, the volume of fuel burned and the reaction products generated. This

calibration is done through fine-scale numerical simulation or from lab combustion

tube experiments. Test cases have shown the consistency achieved between fine-scale


Figure 7.16: Different scales in measurements, geomodeling and reservoir simulation,lecture notes of ERE 241 Seismic Reservoir Characterization.

reference and coarse-scale upscaled model. This approach significantly improves the

computational speed for in-situ combustion simulation. It is also applicable for other

subsurface reactive flow modeling problems with fast chemical reactions and sharp

reaction fronts.

Chapter 8

Front Stability Study for In-situ

Combustion

8.1 1D Flow Regimes

Very limited work has analyzed front stability during ISC. This topic is important for

the design of ISC processes, because an unstable front and viscous fingering can lead

to premature front break through. If we do have viscous fingering due to unstable

frontal displacement, we probably also need to implement some upscaling approach

for the viscous fingers, similar to the pseudo relative permeability technique used in

miscible flooding [51]. We need to answer these questions as we move on to field-scale

ISC simulation, and after implementing the chemical reaction upscaling technique

mentioned earlier. In this chapter, we intend to explore this issue.

We start by studying the different flow regimes and their behaviors in a 1D ISC

process, and then move to 2D and 3D problems. We focus on the study of the scenario

of the ISC process in a conventional mobile heavy oil reservoir (in-situ oil viscosity

µoi ∼ 103cp), instead of highly viscous bitumen reservoirs (µoi ∼ 106cp). This is due to

the fact that most of the past successful ISC trials are within heavy-oil reservoirs, such

as in California and the Lloydminster region in Canada. Conventional ISC processes

with vertical wells in bitumen reservoirs have well-known injectivity problems, that

may require extremely small well spacing (< 1 acre) or extensive heating with cyclic

121

CHAPTER 8. FRONT STABILITY STUDY FOR IN-SITU COMBUSTION 122

steam stimulation, as discussed in reference [55, 48].

The conditions in field operations are different than the laboratory experiments

and also the corresponding combustion tube simulations. For example, most combus-

tion tube lab tests performed at Stanford University are at low initial oil saturation

Soi ∼ 0.25. Small pressure drop is observed through the combustion tube [66]. Field

operations usually involve much higher Soi, that may induce the potential flow re-

strictions (choking) problems, that has been noticed in many Canadian projects [85].

The predictability of flow restriction problems is essential to the design of the ISC

process, especially when deciding the well spacing and sizing air injection capacity.

We identify three different fronts in the typical 1D dry/normal wet forward com-

bustion process shown in Fig. 8.1: the combustion zone 1, the leading edge of the

steam plateau 2, and the leading edge of the oil bank 3. These three fronts also

divide the 1D process into several flow regimes, shown as A, B, and C in Fig. 8.1.

In the area A, we only have air and water flowing (wet combustion process). Across

combustion zone 1, we assume a constant fuel amount Sofuel, as in the classical Gates

and Ramey method [44]. This fuel consumes the supplied oxygen, generating steam

and non-condensable flue gas, that flows further down into the area B. At the leading

edge of the steam plateau 2, steam condenses, releasing latent heat and mobilizing

the viscous oil into the oil bank. Flue gas keeps moving through into the area C,

the cold oil bank. In area C, both oil, water and flue gas flows forward. The high

pressure drop is due to the fact that the gas saturation (fractional flow of gas) has

been reduced to a very small value.

These three fronts influence each other in a 3D forward combustion process. By

inspection, we identify the leading edge of steam plateau 2 as the most unstable front.

This is due to the high pressure gradient contrast between region B and C, shown in

Fig. 8.1, which is also analogous to the miscible flood front instability. This will be

the most critical front that instability will first occur.


Figure 8.1: Different fronts and flow regimes in 1D ISC process.

8.2 Contributors for Front Stability/Instability

We identify five contributors for front stability/instability: viscous force, condensa-

tion, heat conduction, coke plugging and gravity. We do not consider here chemical

reaction instability in this study. This is because we have Da ·Pe � 1 the product of

Damkohler number and Peclet number much larger than 1 (Da � 1 is because of the

fast chemical reaction kinetics in ISC, Pe � 1 is because ISC process is advection

dominated). Thus, the problem is flow transport limited [34]. Some of these factors

can be partially quantified by dimensionless numbers. Some of them are not easy to

be described merely by dimensionless numbers. They are discussed in detail later.

The five contributors to front stability/instability are discussed next.

• Viscous force: Viscous force is quantified by the mobility ratio M between the

displacing and displaced fluid. Complex multi-phase flow exists both upstream

and downstream of the steam plateau front. This complicates the mobility ratio

calculations. We suggest the pressure gradient contrast to be used instead. The

pressure gradient contrast between the steam plateau and the cold oil bank is,

∇PB

∇PC

=λtButB

λtCutC

, (8.1)

where λtB and λtC are the total phase mobilities in region B and C, and utB and

utC are the total phase velocities. The mobility contrast of the displacing (steam

and flue gas mixture) and displaced fluid (heavy oil) makes λtB � λtC . The


magnitude of total phase velocities utB and utC are close to each other. Thus

the overall pressure gradient contrast ∇PB/∇PC is � 1. This viscous force will

cause the front to be unstable. For example, in the mobile heavy oil ISC process

in section 6.2, the calculated pressure gradient contrast is ∇PB/∇PC ≈ 110.

• Steam condensation: In classical front stability analysis of steam flooding hot

water (no heat conduction included), the criterion is stated as,

(1 + 1−φ

φcv(Ts−Tr)

ρs(Hs−Hw)

)· µs(

1 + 1−φφ

cv(Ts−Tr)ρw(Hs−Hw)

)· µw

> 1, (8.2)

where cv is the volumetric heat capacity of rock, Ts and Tr are the steam sat-

uration temperature and initial temperature in the reservoir, ρs and ρw are the

densities of steam and water, Hs and Hw are the enthalpy of steam and water.

In this expression, the term (1−φ)cv provides extra stabilizing force. This is due

to the fact that steam condensation reduces the downstream magnitude of hot

water flux, thus reducing the pressure gradient contrast, also shown in Fig. 5.1.

The greater the rock porosity and heat capacity cv, the stronger the stabilizing

effect [50]. Condensation is intrinsic property of steam, thus no dimensionless

number derived here.

• Heat conduction: In the classical steam flooding hot water problem by Miller

[78], with the heat conduction included, the stability criterion becomes,

2π

λ

(Vsµs

ks

− Vwµw

kw

)+

(µs

ρsks

+µw

ρwkw

)·

Keγ (γr − γ)

(Hs −Hw)− cv,w

ρw(Ts − Tw) (γr − γ) λ

2π

> 0

(8.3)

where Ke is the effective thermal conductivity, and λ is the average wavelength

of a perturbation to the displacement front, shown in Fig. 8.2. Displacement

front stability improves with the added heat conduction, compared with the

previous analysis without conduction Eq. 8.2. The heat conduction stabilizing

effect is also dependent on the wave length of perturbation λ. The smaller

the λ, the more stable the front. This is also consistent with physical intuition,


Figure 8.2: Perturbation length λ at leading edge of steam plateau, from [50].

because heat conduction dissipates the heat of smaller wave length steam fingers.

The heat conduction stabilizing effect is partially characterized by the Nusselt

number Nu = F h·Lα

, where α is the heat conductivity, L is the characteristic

length of the problem and F h is the convective heat flux. The dimensionless

number, however, does not take into account the perturbation wave length λ in

Eq. 8.3.

• Coke plugging: Inside the ISC combustion zone, we have the pyrolysis reac-

tion that generates solid fuel coke that is deposited onto the porous media

surface. This deposited coke influences the porosity and permeability of the

porous medium. The coke reduces the permeability of the sand through this

geomechanical effect. Whenever an unstable front develops, this permeabil-

ity reduction improves the stability by creating flow resistance behind the fast

advancing part of the front. The permeability reduction is modeled in the com-

mercial thermal simulator CMG STARS through the option of solid component

blocking (*Blockage key word) [101]. The magnitude of permeability reduction

is highly dependent on the locations where solid coke is deposited. For exam-

ple, the deposition of coke around the pore throat and pore body will have very

different effects on pore blocking. Deposition at the pore throat causes much

more reduction in permeability than at the pore body. This is very similar to

the clay swelling problem in steam injection process [16]. In order to model the

pore blocking, we have implemented different blockage magnitude. We have


used the classical Kozeny-Carman correlation of porosity and permeability:

k = αφi

3

(1− φi)2 . (8.4)

We have also performed sensitivity on stronger blockage by coke deposition near

pore throats. Detailed sensitivity studies follow.

• Gravity: Consider the displacement of steam and flue gas mixture invading the

cold oil bank from top to bottom. The gravity difference between the steam

flue gas mixture and the liquid oil tends to stabilize the interface between them.

Whenever a steam finger is formed, the density difference slows down the finger

growth. To achieve an efficient gravity stable process, the injection rate (the

viscous force) should be controlled within a certain level and has been discussed

in detail in the gravity drainage section in [16]. In many field projects, we

tend to combust from up dip in the reservoir and move to down dip. This

injection configuration improves the sweep efficiency of the combustion front

[91]. For two phase oil and gas flow, gravity effect is usually described by the

gravity number Ng = (ρo−ρg)KφµovT

g, where K is the permeability, ρo and ρg are oil

and gas phase density, vT is the total phase velocity. Gravity number is one

parameter that characterizes the gravity stabilizing effect. In the ISC process,

factors such as temperature dependent viscosity, phase behavior and three phase

flow complicate the gravity number calculation and makes this dimensionless

number highly variable. It is not easy to estimate a single gravity number for a

particular problem.

8.3 Numerical Tests

8.3.1 Minimizing Numerical Errors

In this section, we show the numerical simulation we tested for ISC front stability.

The first work we performed is to probe the numerical errors (grid orientation effects

and temporal error in particular) in our simulations. Grid orientation effects are well


Figure 8.3: Trigger instability by changing the initial oil saturation at the boundary.

known as having strong influence on thermal and miscible flood EOR simulations

[60, 101]. We check the simulation results with different grid block sizes, different

numerical schemes (five point stencil and nine point stencil in CMG STARS [101]) and

different time step sizes. We draw conclusions about the behavior of the instability

if that behavior is very much the same across these variations. Because only then

we know it is the physical process that controls stability/instability. We perform

sensitivity studies to make sure the stable/unstable fronts we observe are not due to

numerical artifacts. We always trigger the instability at the same place by putting in

small perturbations in the initial condition (usually involving several grid blocks), as

shown in Fig. 8.3. We use a homogeneous permeability field so that we do not have

the problem of upscaling/downscaling when changing grid block sizes. We check here

whether the finger (instability) dimensions and growth rate are grid size, time step

size and numerical scheme (five point or nine point scheme) dependent.

We first show the simulation test case of a ISC process in 2D horizontal plane by

different grid block sizes and numerical schemes in Fig. 8.4. FIM time stepping is used

here. The time step sizes are selected automatically by the heuristics in commercial

simulator [101]. The size of the problem is 10m by 30m. We start the combustion

from the left side and propagate to the right side. The upscaled non-kinetic reaction

model is used here for this larger size problem. We use the same mobile heavy-oil

property as in section 6.2. The permeability is 10D and porosity is φ = 0.36. The

total simulation time is 7 days, which corresponds to an approximate front velocity of


about 2ft/day in this 2D problem. The instability is triggered initially at the center

of the left boundary by changing the initial oil saturation Soi, as shown in Fig. 8.3.

The simulation results are very dependent on the choice of grid sizes. At larger

grid block sizes, the results of five and nine point stencil diverge from each other.

When the grid block sizes are reduced to 0.222m×0.222m and 0.167m×0.167m, the

five and nine point stencil results start to converge.

This behavior is explained by the relative magnitude of numerical and physical

diffusion as follows. We start with an idealized model of a simple linear conservation

equation in 2D space,

ct + γcs = 0, (8.5)

where t is time, s is the coordinate along the velocity direction, and γ =√

u2 + v2 is

the magnitude of velocity (u and v are velocities along x and y directions). We show

the modified equation written in the flow aligned coordinate system (s, n) [60],

ct + γcs = γh (α1css + α2cnn + α3csn) , (8.6)

where h is the dimension of the grid size, α1, α2 and α3 are constants depending on

the numerical schemes used. The terms on the right hand side of the equation are

numerical diffusion terms. The numerical diffusion magnitude is proportional to the

grid size h. And the ratio between transverse cnn, longitudinal css, and cross term

csn diffusion are grid-size independent. This difference between numerical diffusion

terms css, cnn and csn are the cause of grid orientation effects. As seen in the work

by Kozdon [60], the grid orientation effects do not disappear as we refine the grid. In

adverse mobility ratio flooding problems, this is often seen as the five point stencil

tends to break through along the grid lines, while the nine point stencil tends to break

through in diagonal directions [101, 60].

In our thermal ISC process, we also have heat conduction effects, that can be

idealized as an extra diffusion term to this linear conservation equation. Now the

modified equation becomes,

ct + γcs = α (css + cnn) + γh (α1css + α2cnn + α3csn) , (8.7)


Figure 8.4: Test of numerical errors with triggered instability. The effect of grid sizeand 5-point versus 9-point scheme is illustrated.


Figure 8.5: Test of temporal numerical errors with automatic time-step selection andrestricted time step sizes.

where α is the constant for physical diffusion term. If we reduce the grid block size

h, the numerical diffusion term magnitude decreases. At a certain point, the physical

diffusion term dominates. At such refined grid resolution, we draw conclusions that

the stability/instability is caused by a physical process, not numerical artifacts. The

previous test case is for such purposes. The observations are consistent with the

analysis here. We have conducted similar tests for all the the simulation studies later

on to ensure control of the numerical errors within a reasonable range.

We also test the choice of time step sizes to probe the temporal errors in our

simulations. We use the same 10m by 30m 2D problem, with grid block size 0.222m×0.222m. The same instability is triggered in the middle of the left boundary. We

compare the base case with the case having highly restricted maximum time step

sizes. We show one test case here in Fig. 8.5. The average time step size of the

base case is 23min. The restricted maximum time step size is 2min. The simulation

results are very similar to each other in this case. Spatial discretization error is more

dominant than temporal error as in this case. Similar tests, however, are performed

for the following test cases to ensure the temporal numerical error is controlled within

a certain range.


8.3.2 Small Sub-grid Scale Front Stability

We first perform tests of 2D lab-scale simulations to study frontal stability. The

size of the 2D problem is 0.36m× 1.08m. We test dry forward combustion with the

same oil. Again we inject from the left side and produce from the right side. The

simulation time is 4 hours. This corresponds to an approximate front velocity of

6ft/day. We show the base case front propagation in Fig. 8.6. The permeability and

porosity field is homogeneous (K = 10D and φ = 0.36). We trigger the instability by

perturbing the initial condition on the left injection side, shown in Fig. 8.3. For all

the test cases we show, we have performed the previous test to ensure that physical

conduction/diffusion is dominating the numerical diffusion.

We have found at small scale that heat conduction plays an important role in

stabilizing the front. The original conductivity value we use is 2.58J/(cm ·min ·o C).

For simplicity we use a uniform conductivity value for the entire domain without

considering the fluid saturation effects. We demonstrate this by arbitrarily changing

the heat conductivity. We also show the temperature and oil saturation profiles with

50% conductivity (1.29J/(cm·min·oC)) and 30% conductivity (0.774J/(cm·min·oC))

in Fig. 8.6. As we decrease the heat conductivity values, unstable fronts appear. The

smaller the heat conductivity, the faster the steam finger becomes unstable and grows.

As seen from both the oil saturation and temperature profile, the instability starts

at the leading edge of the steam plateau where it is invading the oil bank. The

combustion zone follows right after the unstable steam plateau. This is consistent

with the previous analysis. We further calculate the heat Nusselt numbers Nu = F h·Lα

.

The heat Nusselt numbers are Nu = 4 for 100% heat conductivity case, Nu = 8 and

Nu = 13.3 for 50% and 30% heat conductivity cases, respectively. This means the

flow transport in this case is advection dominated.

We also repeat a simulation test with twice the front speed (air injection rate),

but keeping the original 100% heat conductivity. The calculated Nusselt number is

Nu = 13.3, the same as the 50% heat conductivity case. We show the oil saturation

and temperature profile, that corresponds to the same unstable front in Fig. 8.7.

This shows that the Nusselt number serves as one parameter characterizing the heat


Figure 8.6: Sensitivity to thermal conductivity. Heat conduction stabilizes the dis-placement front in small lab-scale tests. Reduced thermal conductivity cases showgreater instability.


Figure 8.7: Nusselt number characterizes the heat conduction stabilizing effect, com-pared to Fig. 8.6.

conduction stabilizing effect. Furthermore, the conduction stabilizing effect also de-

pends on the perturbation wavelength, shown in Eq. 8.3 by Miller [78]. Because of

the confined geometry here, this is currently not taken into account.

We test the pore blocking effects by introducing blockage in the simulation. The

first example we show is the same lab-scale case as before with reduced 50% heat

conduction (1.29J/(cm · min ·o C)). The blockage is adjusted to be similar to the

Kozeny-Carman correlation Eq. 8.4. We show the temperature, oil saturation and

also the blockage coefficient profile Fig. 8.8. The simulation time is 5 hours. The

maximum blockage coefficient (permeability reduction factor) achieved is around 1.5


Figure 8.8: Pore blocking with the Kozeny-Carman correlation. Very small changesare observed compared to base case.

to 2.0. The steam plateau finger growth is very close to the base case without any

blockage. Tests show that mild pore blocking has very limited effect on self-correcting

the front. This is also consistent with the elliptic nature of the pressure equation.

Mild permeability change in one place does not give significant change to the overall

pressure distribution, and thus will not strongly affect the flow transport.

We further test the scenario with even stronger blockage with exaggerated perme-

ability reductions in Fig. 8.9. We again show the temperature profile, oil saturation

and permeability reduction factor. In this case, the permeability reduction is as great

as a factor of 30. We find that blockage deters the finger growth and stabilizes the

front somewhat at these conditions. To conclude, we observe coke plugging stabilizing

the front when we have at least an order of magnitude reduction in the permeability.


Figure 8.9: Pore blocking with an exaggerated permeability reduction. The combus-tion front slows down only when implementing pore blocking of more than one orderof magnitude reduction in permeability.


8.3.3 Large Field-scale Front Stability

We further test the ISC front stability in larger field-scale cases. We use the upscaled

reaction model for all the tests here because of the large dimensions. We show the

first example in Fig. 8.10. It is a 2D problem with the size 13.2m × 26.4m, with

the grid block size 0.22m × 0.22m. The initial oil saturation is Soi = 0.75 and

the fuel amount in the upscaled model is set at Sofuel = 7.5%. The same mobile

heavy oil property is used in this case. Heat conductivity is set at a constant value

of 2.58J/(cm ·min ·o C). We again introduce perturbation at the left boundary by

changing initial saturations. Because we use the upscaled reaction model as described

in section 7.2.2, we also plot the remaining fuel amount to show the area that has been

burned. We simulate this problem for 21 days, with an approximate front speed of

1.5ft/day. We observe unstable displacement, as shown in Fig. 8.10. Heat conduction

is incapable of dissipating the energy of these large wavelength perturbations (in the

order of several meters).


In this chapter, we study the front stability behavior during the ISC process. We

first performed detailed analysis of a 1D ISC process with high initial oil saturation

mobile heavy oil. We identified three main fronts in the process: combustion zone,

leading edge of the steam plateau and leading edge of the oil bank. Among them,

we find the leading edge of the steam plateau as the most critical front because of

the large pressure gradient contrast. This is the place where instability first occurs.

Numerical test cases are performed for small lab-scale forward combustion problems.

We observe heat conduction plays an important role in stabilizing the front, because

of the small length scale of perturbations. Coke plugging (pore blockage) is found to

have very minor effects in stabilizing the front, unless permeability reductions of up to

one order of magnitude are introduced. We have not observed significant ”front self-

correction”, unless with such large permeability reductions [54]. We also test front

stability in larger field-scale cases. The upscaled reaction model is used for these


Figure 8.10: Unstable displacement in 2D larger scale ISC process. Heat conductionis incapable of dissipating the energy of large wavelength perturbations.


studies. We find unstable fronts in these simulations because the heat conduction is

incapable of dissipating large wavelength perturbations.

Chapter 9

Future Directions

9.1 Thermal Streamline Simulation

9.1.1 Non-advective Forces

The original idea of solving transport along the streamline direction makes streamline

simulation an ideal solution for strong advection dominated flow problems [7]. In ther-

mal EOR processes, often we have non-advective forces, such as heat conduction, heat

losses to over and under burden, gravity segregation, etc. The incorporation of these

non-advective effects, while at the same time retaining accuracy and improving simu-

lation speed is crucial to the success of thermal streamline simulation. In this thesis,

we show the usage of Glowinski θ-scheme operator splitting for solving non-advective

effects. We show the accuracy achieved and improved computational efficiency. The

current approach of θ-scheme operator splitting may not be the unique solution.

Other techniques such as putting some heat conduction along the streamlines may

also be applicable. We propose further study related to the efficient incorporation of

non-advective effects into the thermal streamline simulation framework.

9.1.2 Guidelines for Commercial Code Development

In this thesis, we have solved some of the numerical problems for applying the stream-

line method to thermal flood problems. Even though most of the test cases we show

139

CHAPTER 9. FUTURE DIRECTIONS 140

are in 2D space (horizontal and vertical), the same methodology is applicable to full

3D streamline simulator. Difficulty in 3D implementation, however, still exists.

It is recommended the 3D thermal streamline simulator to be developed on the

basis of an existing thermal reservoir simulator. In the 1D transport part of the

streamline global time step, we solve a series of full 1D problems. We can directly

utilize the existing thermal simulator to provide the 1D solution [29]. Multiple stream-

line solves are combined into one 1D problem. This is very similar to the work by

Crane et al., [29].

The implementation of complex well models is a potential challenge for thermal

streamline simulators. In streamline 1D transport, we usually specify a certain bound-

ary condition according to the physical properties in the well grid block, or certain

upstream location [71]. In thermal problems, we have complex multi-tubing string

wells with complicated flow and heat exchanges. Thermal problems such as steam

flooding also often have back flow problems. This means that the total phase velocity

is pointing backwards into the well grid block. We need further study for incorporat-

ing the detailed flows inside the well block into the streamline simulation framework.

Complex well models may be actually difficult to implement.

Specific issues such as mappings occurred inside the streamline global time step,

control of mass balance error, streamline tracing, selection of global time step sizes,

etc., are the common numerical issues for general streamline simulation. They have

been studied extensively in previous works by various parties [71, 72, 92, 58], which

are out of the scope of this work. Similar techniques are applicable here to thermal

streamline simulation.

9.1.3 Thermal Streamline Simulation for SAGD?

We have shown in this work the use of thermal streamline simulation for hot water

flood and steam flood problems. One interesting question here is can we also use

streamline simulation for SAGD simulation. The SAGD process is different than

conventional steam flooding in that we create a slow drainage process that fully takes

advantage of the gravity segregation [16]. Therefore, gravity may serve as the main


Figure 9.1: Total phase velocity vectors in 2D SAGD process. Convection cell/loopcurrent occur when tracing streamlines.

driving force in SAGD process, instead of viscous force.

We show one typical SAGD simulation example for bitumen reservoir in Fig. 9.1.

The simulation input file is attached in Appendix E. We show the temperature profile

and also plot the total phase velocity vectors in this 2D SAGD problem. As we can

see, the steam rises upwards inside the steam chamber. Once it reaches the cap rock

ceiling, it starts moving sideways. At the edge of the steam chamber, the steam

condenses into hot water. The hot water and oil flows downwards by the gravity

force. If we trace the total phase velocity in this SAGD problem, we will get flow

patterns similar as the convection cells or loop currents. We can have streamlines

start from one cell and eventually trace all the way back to the same cell. This will

require special numerical treatment for the boundary conditions in these streamline

loops, which is also noted in [7]. The numerical challenges described here remain to

be solved for applying streamline simulation to SAGD process.


9.2 In-situ Combustion

9.2.1 History Matching of Field-scale ISC process

History match of field-scale ISC process has been tried previously by different parties

for fields such as Suplacu [91] and West Heidelberg [63]. Most of these simulation

trials do not take into account the grid size effects for reaction kinetics modeling,

and more or less implement some heuristics to deal with the numerical artifact of

excessive coke formation. The simulation usually only involve a small number of

wells with relatively large grid blocks. A qualitative match has been achieved in

these early reservoir simulation studies.

The successful history match of a ISC field or part of the field is the subject of our

future work. Detailed laboratory measurements such as ramped temperature oxida-

tion kinetic cell and 1D combustion tube should be performed first to understand the

burning characteristics of the crude oil. After history matching the experiment, we

will upscale the kinetics based model to field-scale simulation. When history match-

ing the field production data, a first step Ramey and Gates analytical approach [44]

will be conducted to get the first order estimation. The Gates and Ramey method

and its variants provide the correlation of injected air volume, burned reservoir vol-

ume and the corresponding field production. They serve very similar roles as the

classical material balance approach used for modern reservoir engineering. A detailed

characterization of the reservoir will be followed. Eventually, we will try to match the

geologic model and tuning other fluid and rock properties to achieve the final match.

The proposed entire work flow from lab experiment to field-scale simulation has been

discussed by Bazargan et al., [10].

Through this history matching process, we have uncertainties associated within

each stage. In the lab-scale kinetics based simulation, we have uncertainty in the

reaction model and its parameters. The pseudo reactions are based on simplifica-

tion/lumping of the thousands of real chemical reactions. The parameters for each

pseudo reaction is based on matching the limited information in the RTO experiment

(mainly gas production data). When we scale up to non-kinetic upscaled reaction


model, we have uncertainty in the sub-grid scale heterogeneity. As we discussed pre-

viously, this affects the fuel amount and oxygen utilization efficiency. During the

upscaling from geo-cellular grid to simulation, we also have errors in flow transport

upscaling. Eventually, when simulating full field problem, we also have uncertainty

in the geologic realizations of the field. As we can see, uncertainty is carried forward

within each stage. Our ultimate goal is to provide eventually ranking for general

reservoir engineering purposes.

9.2.2 Calibrating the Operational Range for Sustaining ISC

Combustion

In field ISC operations,sufficient air must be supplied to maintain the propagation of

combustion front in the desired HTO mode. Otherwise unfavorable LTO reactions

consume oxygen and immobilize the oil. To improve our current upscaling approach

that assumes HTO combustion mode, we propose to use a minimal air flux approach

to define the operational window for sustaining ISC combustion. We check the air

flux behind the combustion front if it is enough to sustain the HTO mode. This min-

imal air flux can be calibrated through three different ways: engineering knowledge,

physical experiment, and numerical simulation. Various parties have proposed empir-

ically the minimal air flux or minimal combustion front velocity based on engineering

field knowledge [85]. For example, a minimal front front velocity of 0.125ft/day is

proposed for successful operation of ISC project [83]. Recently, a novel conical com-

bustion tube design was built as it enables continuous air flux reductions without

having to adjust the air injection rate in a combustion tube experiment [4]. The

conical design is expected to define experimentally the minimal air flux. Numerical

simulation has also been used to define operational window for ISC. For example, the

ignition/extinction regimes are defined as a function of both air injection rate and

magnitude of heat loss in the kinetic cell experiment [61]. A similar technique can

be used here to calibrate the ignition/extinction behavior of 1D ISC process once we

achieve a history match using the Arrhenius kinetics based simulation.


9.2.3 Grid Size Effects for Other Thermal or Non-Thermal

EOR Processes

The choice of simulation grid block sizes has always been an issue for reservoir sim-

ulation accuracy, especially in modeling EOR processes. This is due to the fact that

many EOR processes are often associated with large property changes within a small

distance. Capturing these property changes and their flow response usually requires

certain sized grid blocks. Most of the commercial simulators use first-order numerical

schemes, that calculate both the flux terms and source/sink terms in the governing

equations using cell averaged properties. This is the main cause for grid-size effects

in thermal or non-thermal EOR processes.

Compositional gas injection simulation is well known for its deteriorated accu-

racy when using large size grid blocks due to numerical dispersion effects [71]. Thus,

higher order numerical schemes such as TVD (Total Variation Diminishing) and ENO

(Essentially Non-Oscillatory) have been successfully applied for the accurate capture

of multiple fronts/contacts in multi-contact miscible flooding [71]. When simulat-

ing multi-dimensional first-contact miscible flooding, pseudo relative permeability

approach is also utilized to capture the sub-grid scale viscous fingering [51].

In this work, we propose the upscaling approach for capturing the sub-grid scale

chemical reaction kinetics in ISC process. For other thermal or non-thermal heavy

oil EOR processes, grid size effects have also been noticed, for example SAGD [22],

ESSAGD (Expansion Solvent Steam Assisted Gravity Drainage) [82] and VAPEX

(Vapor Extraction) processes [31, 95]. We test a simple SAGD process in a bitumen

reservoir with different grid block sizes, shown in Fig. 9.2 and Fig. 9.3 for both early

time and late time. The simulation input file is attached in Appendix F. We have

homogeneous reservoir and show half the plane of the steam chamber temperature

profile. We observe that we need 0.5m×0.5m size grid blocks to achieve the convergent

solution in this test case. This is a very demanding requirement for large 3D field-

scale simulation, especially when simulating multiple pads with many well pairs. This

phenomenon is due to the fact that the viscosity of bitumen is a strong function of

the local temperature (up to 106cp at reservoir temperature, and down to 10cp at


Figure 9.2: Grid size effects in 2D SAGD simulation (early time 360 days). Thesimulation achieves convergent solution when using grid size 0.5m × 0.5m in thiscase.

the steam chamber temperature). In classical Butler’s analytical approach, the oil

drainage rate is calculated through an exact integration from the interface of the steam

chamber into the cold bitumen zone [16]. However, in numerical simulation, the total

oil drainage rate is calculated by the oil fluxes inside each discretized grid block. The

temperature is represented as a step-wise function, instead of continual variation. The

flux terms are calculated using the cell averaged temperature and its corresponding

viscosity value. This is the main cause for the grid size problems in SAGD. We propose

further work to explore the improvements for SAGD simulation accuracy. Techniques

such as higher order numerical schemes (similar as in compositional gas injection),

or upscaling the drainage flux (finding the equivalent drainage rate, instead of using

cell averaged mobility terms) may be applied in this area.

9.2.4 How to Use In-situ Combustion as a Gravity Drainage

Process in Fractured Carbonate Reservoir?

Significant volumes of heavy oil resources reside in the fractured carbonate reservoirs

worldwide. For instance, large amount of bitumen resource exists in the ”carbonate

triangle” area in Alberta, Canada [16]. Some of these reservoirs may be good candi-

dates for thermal EOR. For example, currently a major steam injection and gravity

drainage project is underway in the Qarn Alam field in Oman [90]. The purpose of


Figure 9.3: Grid size effects in 2D SAGD simulation (late time 1100 days). Thesimulation achieves convergent solution when using grid size 0.5m × 0.5m in thiscase.

steam injection in this project is to promote the thermally assisted gas-oil gravity

drainage mechanism.

A considerable percentage of Mexican oil production is from naturally fractured

heavy oil carbonate reservoirs, such as the Cantarell field and the Ku-Maloob-Zaap

field in the Gulf of Mexico [2]. The field has been under extensive nitrogen injection

to promote the gas-oil gravity drainage in these thick reservoirs (more than 1000m

thick pay zone for Cantarell). Chemical EOR methods such as surfactant injection

are currently being evaluated as a possible option for Cantarell field [2]. The major

disadvantage of chemical EOR is its large cost of the injectant and the transportation

logistics in the offshore environment. Thermal EOR (steam injection or ISC) may be

a viable option for the Cantarell field and the Ku-Maloob-Zaap field. The reservoirs,

however are often deemed too deep for steam injection because of the high pressure

and heat losses through the injector well bore.

Is it possible to use in-situ combustion as a thermal EOR method for fractured

carbonate heavy oil reservoirs? Previous work by Fadaei et al., [39] has shown through

numerical simulation that oxygen diffusion plays an important role in the successful

propagation of the combustion front in fractured media. The measurement of oxy-

gen diffusion in carbonate rocks could be performed by the experiment described by

Stokka et al., [102]. Physical experiments have also been conducted to test the com-

bustion front propagation in fractured carbonate core [97]. The combustion front is


successfully sustained in this lab experiment. The authors conclude that it is possible

to perform ISC in the fractured carbonate.

Furthermore, gravity drainage is a promising recovery mechanism for fractured

carbonate reservoirs. Classical in-situ combustion processes, however, are often deemed

as a forward drive process, instead of a drainage process [85]. Steam injection, on

the other hand, has been applied both as a flood process (steam flood) and gravity

drainage process (SAGD) [16]. One interesting problem is whether or not can we also

use in-situ combustion as a gravity drainage process. As discussed before in Chapter

6, the fundamental difference between steam injection and ISC is that in steam in-

jection steam carries heat into the reservoir and condenses into hot water, while ISC

continuously generates heat and non-condensable flue gas through the combustion

reaction. One key question to address is where the generated flue gas will be vented.

In classical forward ISC process with vertical injectors and producers, the flue gas

passes through the oil bank and exists at the producer. In a gravity drainage process,

we wish to keep the fluid close to hydrostatic condition so that the gas and liquid

segregates and maximizes the drainage production.

In the early 1990s, a task force was assembled to test the idea of Combustion

Overhead Split Horizontal (COSH) [69]. In this process, the concept of offset venting

wells is introduced. Instead of moving concurrently with the oil towards the producer,

the flue gas moves horizontally towards the offset venting wells, while the oil drains

downward into the horizontal producer placed at the bottom of the reservoir. The

process, however, never got advanced to a field pilot. COSH does bring new insight

into the traditional thinking of ISC as a drive process. The concept of vent wells

may also serve as a potential solution to using ISC as a drainage process for fractured

carbonate reservoirs.

Inspired by the concept of venting wells in COSH process [69], we also propose

the use of ISC to generate thermal gravity drainage in thick fractured carbonate

reservoirs. The schematic of the process is shown in Fig. 9.4. In this process,

we have three sets of wells, air injectors, chimney (vent) wells and producer wells.

The air injectors and vent wells are perforated in the upper part of the reservoir at

the similar depth level, while the producers are placed at the bottom to efficiently


Figure 9.4: Concept of ISC assisted gas oil gravity drainage in fractured media. Ventwell is added to achieve hydrostatic gravity drainage condition in the reservoir.

drain the reservoir. We continuously inject air into the reservoir to propagate the

combustion front from the injector towards the venting well. The oxygen diffuses into

the matrix to react and combust with the crude. This generates heat, steam, hot

water and flue gas. Once warmed up, the carbonate matrix wettability is changed

to assist the oil to flow into the fracture. By the density difference, the flue gas

flows horizontally towards the vent well through the fracture network. The hot water

and oil flow downwards by gravity drainage. This process is analogous to the steam

injection based thermal assisted gas-oil gravity drainage process [90]. We perform

some preliminary numerical simulation, combusting a single block of fractured media.

The simulation input is the same as the three reaction model in Appendix B, with

the geometry and permeability field changed accordingly. The diffusivity of O2 in the

gas phase is set as 1m2/day. We show the results in Fig. 9.5. The the combustion

reaction is able to extend into the matrix of the fractured media. By the density

difference, most of the gas is produced on the upper right corner, while most of the

liquid is produced at the lower right corner.


Figure 9.5: Gravity stable ISC process for a single block of fractured media. Becauseof the gas diffusion, we are able to combust into the matrix. Most of the flue gas isproduced from upper right corner.

Chapter 10

Conclusions

This dissertation addressed the problem of efficient simulation of thermal enhanced

oil recovery processes and proposed two major solutions: the usage of streamline

simulation for thermal recovery problems and chemical reaction kinetics upscaling for

in-situ combustion processes.

For the thermal streamline simulation, we first introduce the extension of stream-

line simulation to hot water flood. We tested our streamline simulator and compared

the results with a commercial thermal simulator. The main conclusions are summa-

rized next.

• We show our streamline simulator is capable of handling a thermal hot water

flood problem. We successfully included the effects of temperature dependent

viscosity changes, compressibility, gravity and heat conduction.

• Thermal streamline simulation is able to handle temperature induced viscosity

changes. For the small compressibility effects encountered in hot water flood,

the dimensionless velocity approach is able to capture the volume changes dur-

ing the 1D transport. We also solve heat conduction and gravity segregation on

separate steps using operator splitting. Thermal streamline simulation achieves

comparable accuracy compared to finite volume simulation, with reduced com-

putational cost.

Steam flooding poses a larger challenge for the use of the streamline method,

150

CHAPTER 10. CONCLUSIONS 151

because of the large volume changes, coupled mass and energy transport and gravity

effects. We implemented several new techniques to tackle these issues. A research

streamline simulator is developed for the steam flooding problem. We tested our

streamline simulator on a series of test problems and again compared the results with

a commercial thermal simulator. Specific conclusions are summarized next.

• The 1D pressure update approach is a major improvement to streamline simula-

tion. It makes streamline simulation capable of handling large volume changes

due to condensation/vaporization during the 1D transport. By implementing

the Glowinski θ-scheme operator splitting, we are also able to handle successfully

the coupling between mass and energy transport when solving heat conduction

and gravity in the operator splitting step. For problems which are dominated

by non-advection forces such as the soaking period in cyclic steam stimulation,

we also successfully implement the preliminary SL/FV hybrid approach.

• For the cases investigated, we are able to retain solution accuracy compared to

finite volume simulation.

• We analyze the computational complexity of both streamline simulation and

finite volume simulation, and show the reduced computational cost. For the

studied case, we are able to decouple the large solving system (7200 grid blocks)

into a series of smaller problems (410 streamlines, each with 68 grid blocks on

average) to achieve speedup.

• We also show the potential applications of thermal streamline simulation as

a fast tool for optimization studies, history matching, flow visualization and

achieving the well connectivity information.

Traditional simulation technique for the in-situ combustion process is limited by

the requirement of using extremely small grid blocks to capture the centimeters-

sized combustion front. We analyze the cause of grid size effects and propose a new

method based on a non-Arrhenius reaction upscaling approach. We try to capture

the equivalent effects of the combustion reactions. The upscaled reaction model is


calibrated using fine-scale numerical simulation. We show the detailed procedures to

implement this methodology in a thermal reservoir simulator. The conclusions are

summarized next.

• Field scale ISC process usually has a large Damkohler number, which means

the chemical reaction kinetics is much faster than flow advection. An upscaled

equilibrium type reaction model is more appropriate for field scale ISC modeling.

• Various test cases have illustrated the consistency achieved in the upscaled

reaction model when scaling up the grid sizes in multidimensional ISC problems.

The reaction upscaling is not sensitive to grid block sizes. We do not need to

change reaction parameters when changing grid block sizes.

• We show this method significantly improves the computational efficiency of ISC

simulation by the capability of using larger grid blocks.

• The limitation and valid range of the reaction upscaling approach is summa-

rized. A minimal air flux criterion is suggested to model the extinction/transition

to LTO behavior.

• We further study the sub-grid scale heterogeneity effects on ISC reaction up-

scaling. We implement material balance approach to upscale the final reaction

effects to fuel amount and oxygen utilization efficiency. We find that sub-grid

scale heterogeneity induces changes in the fuel amount and oxygen utilization

efficiency. When history matching an ISC field, the sub-grid scale heterogeneity

needs to be taken into account.

Driving front stability in ISC process has been studied in the final part of this

thesis. We identified the different flow regimes in the 1D ISC process. Then we

analyzed the different contributors for ISC front stability based on analytical analysis.

Numerical test cases are presented to show the relative importance of these factors.

Specific conclusions drawn are summarized next.


• The ISC process involves multiple fronts. We show that the most critical front

for stability is the leading edge of the thermal front driving the cold oil bank.

This is the front that has the highest mobility/pressure gradient contrast.

• We found coke plugging has a relatively small effect on front stability. To achieve

stabilization, we need a very large permeability reduction (two orders of magni-

tude in the case studied). Heat conduction plays an important role in stabilizing

the front with small wavelength perturbations. With large scale perturbations,

however, unstable front and pre-mature break through of combustion front may

occur. We have not observed significant ”front self-correction” in our numerical

test cases.

Appendix A

Simulation Inputs

The simulation input parameters for all the test cases are listed here.

Table A.1: Reservoir properties for hot water flood

Reservoir dimension [m2] 500× 500Thickness [m] 1φ 0.4Rock thermal capacity [kJ/m3 ·K] 2300Rock thermal conductivity [kJ/(m · day ·K)] 302

154

APPENDIX A. SIMULATION INPUTS 155

Table A.2: Viscosity relationships for hot water flood

Viscosity Coefficient (µj = Aj exp(Bj/T )) Aj [cp] Bj [K]Water 0.5 0.0Oil 0.6541× 10−5 3969

Table A.3: Fluid parameters for incompressible hot water flooding

Injection well pressure Pinj [kPa] 24000Production well pressure Pprod [kPa] 14000Injection well water temperture Tinj [oC] 80Initial oil temperature Tinit [oC] 20Water heat conductivity [kJ/(m · day ·K)] 51.8Oil heat conductivity [kJ/(m · day ·K)] 51.8Water density [kg/m3] 1004.26Oil density [kg/m3] 981.76Water heat capacity [kJ/kg] 4.19Oil heat capacity [kJ/kg] 2.02

Table A.4: Coefficients of density calculations in compressible hot water flood

Coefficients of density calculations ρscj [kg/m3] cj [1/kPa] aj [1/K]

Water 998 1.0× 10−7 1.0× 10−4

Oil 972 1.0× 10−7 1.0× 10−4

Table A.5: Reservoir properties for steam flood

Reservoir Thickness [m] 30 Porosity φ at standard condition 0.35Rock compressibility [1/kPa] 1e-5 Rock thermal expansion [1/K] 1e-4Rock heat capacity [kJ/m3 ·K] 2000 Rock conductivity [kJ/(m · day ·K)] 200


Table A.6: Well control for steam flood

Injection well pressure Pinj [kPa] 6000 Production well liquid rate BHL [m3] 54Injection well temperature Tinj [oC] 276 Initial reservoir temperature Tinit [oC] 50Steam Quality fs 0.8 Initial reservoir temperature Tinit [oC] 50

Table A.7: Fluid parameters for steam flood

Water density at standard conditions [kg/m3] 998Oil density at standard conditions [kg/m3] 972Water and oil compressibility [1/kPa] 1.0e-7Water and oil thermal expansion [1/K] 1.0e-4Water vaporization enthalpy HV R [kJ/mol] 4820Water critical temperature Tc [K] 647.3Oil vaporization enthalpy HV R [kJ/mol] 8569Oil critical temperature Tc [K] 767Water heat capacity CPG1 [J/mol ·K] 32.2Water heat capacity CPG2 [J/mol ·K2] 1.92E-3Water heat capacity CPG3 [J/mol ·K3] 1.06E-5Water heat capacity CPG4 [J/mol ·K4] -3.60E-9Oil heat capacity CPG1 [J/mol ·K] -22.4Oil heat capacity CPG2 [J/mol ·K2] 1.94Oil heat capacity CPG3 [J/mol ·K3] -1.12E-3Oil heat capacity CPG4 [J/mol ·K4] 2.53E-7Oil molecule weight [g/mol] 282

Table A.8: Reservoir properties for cyclic steam stimulation

Reservoir Thickness [m] 30 Porosity φ at standard condition 0.35Rock compressibility [1/kPa] 5e-5 Rock thermal expansion [1/K] 1e-4Rock heat capacity [kJ/m3 ·K] 3000 Rock conductivity [kJ/(m · day ·K)] 300


Table A.9: Fluid properties in three reaction ISC model.

Components Oil CO CO2 N2 O2 Coke1 Coke2Molecular Weight 537.6 28 44 28 32 673.04 179.8Tcrit (oC) 1472 -140.3 31.1 -147.0 -118.6Pcrit (kPa) 1168 3496 7376 3394 5046CPG1 (J/gmol ·K) 1138.8 30.0 16.9 30.3 27.63CPG2 (J/gmol ·K) 0 -1.54e-3 0.1063 -2.572e-3 6.437e-3ρSTD

o (kg/cm3) 1.008e-3

Table A.10: Rock properties in ISC simulation.

Rock Heat Capacity (J/cm3 ·K) 1.206 + 0.00236 · T (T is temperature)Rock Heat Conductivity (J/cm ·min ·K) 2.5833Porosity 0.36Permeability (mD) 10

Table A.11: Kinetics parameters for three reaction ISC model.

Reactions Pyrolysis LTO HTOActivation Energy (kJ/gmol) 22.6 67.5 87.6Pre-exponential Factor (1/(kPa ·min)) 0.01 250 220Reaction Enthalpy (kJ/gmol) 1.60× 103 1.28× 104 4.85× 103

Table A.12: Reaction stoichiometry for three reaction ISC model.

Pyrolysis 1.0Oil + 4.24O2 → 1.0Coke1LTO 1.0Coke1 + 33.8O2 → 1.0Coke2 + 4.75CO + 23.77CO2 + 22.0H2OHTO 1.0Coke2 + 12.29O2 → 1.73CO + 8.56CO2 + 8.0H2O

Appendix B

STARS Input File for Three

Reaction Model

**----------------------INPUT-OUTPUT CONTROL----------------------------

*TITLE1 ’Simulation of Dry Combustion Tube Test’

*INUNIT *LAB

*OUTUNIT *LAB

*WPRN *GRID *TIME

*WPRN *ITER *TIME

*OUTPRN *GRID *PRES *SW *SO *SG *TEMP *Y *X *MASS *SOLCONC

*OUTPRN *WELL *WELLCOMP

*OUTPRN *ITER *BRIEF

*OUTSRF *WELL *MASS *COMPONENT *ALL

*OUTSRF *GRID *PRES *SW *SO *SG *TEMP *Y *X *MASS *SOLCONC *VISO *VISW

*INTERRUPT *STOP

**-------------------------RESERVOIR DESCRIPTION------------------------

*GRID *CART 1 302 1

*KDIR *DOWN

*DI *CON 6.2

*DJ *CON 0.36

158

APPENDIX B. STARS INPUT FILE FOR THREE REACTION MODEL 159

*DK *CON 6.2

*POR *CON 0.36

*PERMI *CON 10000

*PERMJ *CON 10000

*PERMK *CON 10000

*END-GRID

**------------------------OTHER RESERVOIR PROPERTIES--------------------

*CPOR 0

*CTPOR 0

*ROCKCP 1.20572 0.00236

*THCONR 2.5833

*THCONO 0.0

*THCONG 0.0

*THCONS 0.0

**--------------------------COMPONENT PROPERTIES------------------------

*MODEL 8 6 2

*COMPNAME ’WATER’ ’OIL’ ’CO’ ’CO2’ ’N2’ ’ O2’ ’COKE1’ ’COKE2’

*CMM 0 0.5376 0.028 0.044 0.028 0.032 0.67304 0.17976

*TCRIT 0 1472 -140.25 31.05 -146.95 -118.55

*PCRIT 0 1168 3496 7376 3394 5046

*KV1 0 0

*KV2 0 0

*KV3 0 0

*KV4 0 0

*KV5 0 0

*PRSR 101

*TEMR 15

*PSURF 101

*TSURF 15


*SURFLASH W O G G G G

** ’WATER’ ’OIL’ ’CO’ ’CO2’ ’N2’ ’ O2’ ’COKE1’ ’COKE2’

*CPG1 0 1138.8 29.987 16.864 30.288 27.627

*CPG2 0 0 -1.542e-3 0.1063 -2.572e-3 6.437e-3

*CPG3 0 0 0 0 0 0

*CPG4 0 0 0 0 0 0

*HVR 0 0

*EV 0 0

*MASSDEN 0 1.008E-3

*CP 0 0

*CT1 0 0

*CT2 0 0

*SOLID_DEN ’COKE1’ 0.0326265 0 0

*SOLID_CP ’COKE1’ 17.0 0

*SOLID_DEN ’COKE2’ 0.0088 0 0


*AVISC 0 1E-4

*BVISC 0 5087


*STOREAC 0.0 1.0 0.0 0.0 0.0 4.24 0.0 0.0

*STOPROD 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0

*FREQFAC 0.1 **1/kPa-MIN

*RENTH 1.597E6 ** J/gmol

*RPHASE 0 2 0 0 0 3 0 0

*RORDER 0 1 0 0 0 1 0 0

*EACT 45560 ** J/gmol

*O2PP ’O2’



*STOREAC 0.0 0.0 0.0 0.0 0.0 33.8 1.0 0.0

*STOPROD 22.0 0.0 4.75 23.77 0.0 0.0 0.0 1.0

*FREQFAC 250 **1/kPa-MIN


*RPHASE 0 0 0 0 0 3 4 0

*RORDER 0 0 0 0 0 1 1 0


*O2PP ’O2’


*STOREAC 0.0 0.0 0.0 0.0 0.0 12.29 0.0 1.0

*STOPROD 8.0 0.0 1.73 8.56 0.0 0.0 0.0 0.0

*FREQFAC 220 **1/kPa-MIN


*RPHASE 0 0 0 0 0 3 0 4

*RORDER 0 0 0 0 0 1 0 1


*O2PP ’O2’

**----------------------------------ROCK~FLUID DATA---------------------

*ROCKFLUID

*RPT 1 *LININTERP

** Sw Krw Krow Pcow

*SWT

0.068 0.0000 1.0000 0

0.15 0.0077 0.8229 0

0.2 0.0200 0.7232 0

0.218 0.0259 0.6889 0

0.25 0.0381 0.6300 0

0.3 0.0619 0.5432 0

0.35 0.0915 0.4629 0


0.4 0.1268 0.3889 0

0.45 0.1679 0.3214 0

0.5 0.2148 0.2604 0

0.55 0.2674 0.2057 0

0.6 0.3258 0.1575 0

0.65 0.3899 0.1157 0

0.7 0.4598 0.0804 0

0.75 0.5354 0.0514 0

0.8 0.6168 0.0289 0

0.85 0.7040 0.0129 0

0.9 0.7969 0.0032 0

0.932 0.8956 0.0000 0

1 1 0.0000 0

*SLT

** Sl Krg Krog Pcgo

0.068 0.8975 0 0

0.1368 0.6845 1E-3 0

0.2 0.6233 0.0077 0

0.25 0.5429 0.0200 0

0.268 0.5154 0.0259 0

0.3 0.4681 0.0381 0

0.35 0.3989 0.0619 0

0.4 0.3352 0.0915 0

0.45 0.277 0.1268 0

0.5 0.2244 0.1679 0

0.55 0.1773 0.2148 0

0.6 0.1357 0.2674 0

0.65 0.0997 0.3258 0

0.7 0.0693 0.3899 0

0.75 0.0443 0.4598 0

0.8 0.0249 0.5354 0


0.85 0.0111 0.6168 0

0.9 0.0028 0.7040 0

0.932 1E-3 0.7969 0

0.98 0 0.8956 0

1 0 1 0

**----------------------------------INITIAL CONDIRION--------------------

*INITIAL

*PRES *CON 780

*TEMP *CON 15

*SW *CON 0.068

*SO *CON 0.66

*MFRAC_WAT ’WATER’ *CON 1.00

*MFRAC_OIL ’OIL’ *CON 1.00

*MFRAC_GAS ’N2’ *CON 1.00

*MFRAC_GAS ’CO2’ *CON 0.00

*MFRAC_GAS ’CO’ *CON 0.00

*MFRAC_GAS ’O2’ *CON 0.00

**-----------------------------------NUMERICAL CONTROL---------------------

*NUMERICAL

*MAXSTEPS 100000

**DTMAX 0.1

*DTMIN 0.000001

**-----------------------------------RECURRENT DATA-------------------------

*RUN

*TIME 0

*DTWELL 0.01

*WELL ’INJECTOR’

*WELL ’PRODUCER’


**--------------NITROGENE INJECTION ---

*PRODUCER ’PRODUCER’

*OPERATE *BHP 780

*GEOMETRY *K 0.50 1 1 0

*PERF *TUBE-END ’PRODUCER’

1 302 1 1.00

*INJECTOR ’INJECTOR’

*TINJW 20

*PINJW 780

*INCOMP *GAS 0 0 0 0 1 0

*OPERATE *STG 3000

*GEOMETRY *K 0.50 1 1 0

*PERF *TUBE-END ’INJECTOR’

1 1 1 1.00

**----------------HEATERS ON!----------------

*TIME 18

*HEATR *IJK 1 1:30 1 320 **

*HEATR *IJK 1 1:25 1 260 **

*TIME 78

*INJECTOR ’INJECTOR’

*TINJW 20

*PINJW 780

*INCOMP *GAS 0 0 0 0 0.79 0.21

*OPERATE *STG 3000

*GEOMETRY *K 0.50 1 1 0

*PERF *TUBE-END ’INJECTOR’

1 1 1 1.00

**----------------HEATERS OFF!----------------

*HEATR *IJK 1 1:30 1 0

*HEATR *IJK 1 1:25 1 0

*TIME 138


*TIME 198

*TIME 258

*TIME 318

*TIME 378

*TIME 438

*STOP

Appendix C

Eclipse Input File for Three

Reaction Model

RUNSPEC

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

LAB

MESSAGES

8* 100000 /

DIMENS

1 1 602 /

COMPS

7 /

UNIFOUT

WATER

THERMAL

SOLID

HWELLS

WELLDIMS

1* 180 7* 180 /

REACTION

166

APPENDIX C. ECLIPSE INPUT FILE FOR THREE REACTION MODEL 167

3 /

START

22 JAN 2010 /

HEATDIMS

30 /

ROCKDIMS

1 52 2 /

ACTDIMS

2 50 132 3 /

GRID

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

-- GRID GEOMETRY

TOPS

0 /

DX

602*6.2 /

DY

602*6.2/

DZ

602*0.36 /

PORO

602*0.36 /

PERMX

602*10000.0 / mD

PERMY

602*10000.0 / mD

PERMZ

602*10000.0 / mD


-- Thermal keywords:for over/under burden ROCKPROP

1 15 0. 1. /

/

-- Thermal keyword: THCONR

602*155 /

-- Thermal keyword: HEATCR

602*1.20572 /

HEATCRT

602*0.00236 /

EDIT

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

PROPS

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

SWOF

--SW KRW KRO PCOW

0.068 0.0000 1.0000 0

0.15 0.0077 0.8229 0

0.2 0.0200 0.7232 0

0.218 0.0259 0.6889 0

0.25 0.0381 0.6300 0

0.3 0.0619 0.5432 0

0.35 0.0915 0.4629 0

0.4 0.1268 0.3889 0

0.45 0.1679 0.3214 0

0.5 0.2148 0.2604 0

0.55 0.2674 0.2057 0

0.6 0.3258 0.1575 0

0.65 0.3899 0.1157 0

0.7 0.4598 0.0804 0


0.75 0.5354 0.0514 0

0.8 0.6168 0.0289 0

0.85 0.7040 0.0129 0

0.9 0.7969 0.0032 0

0.932 0.8956 0.0000 0 /

SGOF

--SG KRG KROG PCOG

0 0 0.8956 0

0.068 1E-3 0.7969 0

0.1 0.0028 0.7040 0

0.15 0.0111 0.6168 0

0.2 0.0249 0.5354 0

0.25 0.0443 0.4598 0

0.3 0.0693 0.3899 0

0.35 0.0997 0.3258 0

0.4 0.1357 0.2674 0

0.45 0.1773 0.2148 0

0.5 0.2244 0.1679 0

0.55 0.277 0.1268 0

0.6 0.3352 0.0915 0

0.65 0.3989 0.0619 0

0.7 0.4681 0.0381 0

0.732 0.5154 0.0259 0

0.75 0.5429 0.0200 0

0.8 0.6233 0.0077 0

0.8632 0.6845 1E-3 0

0.932 0.8975 0 0 /

-- saturation endpoints in INIT file, FILLEPS -- Stone’s Rel Perm

Model 2 STONE2


EOS RK /

-- Component data

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

CNAMES

"OIL" "COKE1" "COKE2" "O2" "CO2" "N2" "CO" /

CVTYPE

DEAD SOLID SOLID GAS GAS GAS GAS /

CVTYPES

DEAD SOLID SOLID GAS GAS GAS GAS /

MW

537.6 673.04 179.76 32 44 28 28 /

PCRIT

11.68 1* 1* 50.46 73.76 33.94 34.96 /

TCRIT

1745 1* 1* 154.6 304.2 126.2 132.9 /

DREF

1.008 1* 1* 1* 1* 1* 1* /

PREF

1.0 1* 1* 1* 1* 1* 1* /

TREF

288.15 1* 1* 1* 1* 1* 1* /

CREF

0.0 1* 1* 1* 1* 1* 1* /

THERMEX1

0.0 1* 1* 1* 1* 1* 1* /

SDREF

1* 32.626 8.8 1* 1* 1* 1* /

SPECHA

2.1183 1* 1* 1* 1* 1* 1* /


SPECHB

0.0 1* 1* 1* 1* 1* 1* /

SPECHG

1* 1* 1* 0.8633 0.3833 1.082 1.0710 /

SPECHH

1* 1* 1* 2.0116e-4 0.0024 -9.186e-5 -5.507e-5 /

SPECHS

1* 0.0253 0.0946 1* 1* 1* 1* /

HEATVAPS

0.0 0.0 0.00 0.00 0.00 0.0 0.0 /

OILVISCT

5 1636262 0.01 0.01 0.01 0.01 0.01 0.01

15 229374 0.01 0.01 0.01 0.01 0.01 0.01

25 54485.9 0.01 0.01 0.01 0.01 0.01 0.01

30 30000 0.01 0.01 0.01 0.01 0.01 0.01

48 10600 0.01 0.01 0.01 0.01 0.01 0.01

65 3000 0.01 0.01 0.01 0.01 0.01 0.01

100 290 0.01 0.01 0.01 0.01 0.01 0.01

150 32 0.01 0.01 0.01 0.01 0.01 0.01

200 7.5 0.01 0.01 0.01 0.01 0.01 0.01

250 2.7 0.01 0.01 0.01 0.01 0.01 0.01

300 1.0 0.01 0.01 0.01 0.01 0.01 0.01

325 0.75 0.01 0.01 0.01 0.01 0.01 0.01

350 0.60 0.01 0.01 0.01 0.01 0.01 0.01

375 0.50 0.01 0.01 0.01 0.01 0.01 0.01

400 0.47 0.01 0.01 0.01 0.01 0.01 0.01

425 0.41 0.01 0.01 0.01 0.01 0.01 0.01

450 0.36 0.01 0.01 0.01 0.01 0.01 0.01

475 0.315 0.01 0.01 0.01 0.01 0.01 0.01

500 0.28 0.01 0.01 0.01 0.01 0.01 0.01


525 0.25 0.01 0.01 0.01 0.01 0.01 0.01

550 0.22 0.01 0.01 0.01 0.01 0.01 0.01

2000 0.12 0.01 0.01 0.01 0.01 0.01 0.01

/ GASVISCT

0 0.0136 0.0136 0.0136 0.0136 0.0136 0.0136 0.0136

137 0.0188 0.0188 0.0188 0.0188 0.0188 0.0188 0.0188

210 0.0216 0.0216 0.0216 0.0216 0.0216 0.0216 0.0216

615 0.0370 0.0370 0.0370 0.0370 0.0370 0.0370 0.0370

1000 0.0516 0.0516 0.0516 0.0516 0.0516 0.0516 0.0516

2000 0.0896 0.0896 0.0896 0.0896 0.0896 0.0896 0.0896

/

STCOND --Temp Pressure

15 1.0 /

-- Chemical reaction data

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

STOREAC -- "OIL" "COKE1" "COKE2" "O2" "CO2" "N2"

"CO" "Water"

1.0 0.0 0.0 4.24 0.0 0.0 0.0 0.0 / REACTION 1

0.0 1.0 0.0 33.8 0.0 0.0 0.0 0.0 / REACTION 2

0.0 0.0 1.0 12.29 0.0 0.0 0.0 0.0 / REACTION 3

STOPROD -- "OIL" "COKE1" "COKE2" "O2" "CO2" "N2"

"CO" "Water"

0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 / REACTION 1

0.0 0.0 1.0 0.0 23.77 0.0 4.75 22.0 / REACTION 2

0.0 0.0 0.0 0.0 8.56 0.0 1.73 8.0 / REACTION 3

REACRATE

60 1.5E6 1.32E6 /


REACACT

22560 67525 87575 /

REACENTH

1.597E6 1.28E7 4.85E6 /

REACPHA --

"OIL" "COKE1" "COKE2" "O2" "CO2" "N2" "CO" "Water"

OIL 1* 1* GPP 1* 1* 1* 1* / REACTION 1

1* SOLID 1* GPP 1* 1* 1* 1* / REACTION 2

1* 1* SOLID GPP 1* 1* 1* 1* / REACTION 3

REACCORD -- "OIL" "COKE1" "COKE2" "O2" "CO2" "N2" "CO""Water"

1 0 0 1 0 0 0 0 / REACTION 1

0 1 0 1 0 0 0 0 / REACTION 2

0 0 1 1 0 0 0 0 / REACTION 3

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

SOLUTION

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

TEMPI

602*15 / Initial Temperature C

PRESSURE

602*7.8 /

SOIL

602*0.26 /

SWAT

602*0.24 /

SSOLID

602*0 /

------specify initial oil composition in cells

XMF

602*1.0 602*0.0 602*0.0 602*0.0 602*0.0 602*0.0 602*0.0 602*0.0


/

---------specify initial gas composition in cells

YMF

602*0.0 602*0.0 602*0.0 602*0.0 602*0.0 602*1.0 602*0.0 602*0.0

/

SMF

602*0.0 602*0.0 602*0.0 602*0.0 602*0.0 602*0.0 602*0.0 602*0.0

/

RPTSOL

SOLVD TEMP PRESSURE SSOLID SOIL SGAS SWAT MLSC TEMP HOIL HSOL

XMF YMF /

RPTRST

PRESSURE SSOLID SOIL SGAS SWAT MLSC TEMP REAC HOIL HSOL XMF

YMF ZMF VOIL VGAS VWAT KRO KRW KRG PCOW PCOG FLOC1 FLOC2 FLOC4 FLOC5

FLOC6 FLOWAT FLOE DENW DENO DENG /

SUMMARY

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

WBHP

INJE PROD /

WTEMP INJE PROD /

-- Rates

FOPR FWPR FGPR FGIR FREAC /

-- Totals

FOPT FOIP FWPT FWIP FGPT FGIT FREAT /

PERFORMA

RUNSUM

SCHEDULE

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


RPTSCHED

FIP=1 /

-- Convergence & time step criteria

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

TSCRIT

CVCRIT

.005 10 4* .00005 0.05 / Need tight convergence criteria for small cells

TSCRIT

1.0E-8 1.0E-9 /

7* 10.0 / Need small time steps

WELSPECS INJE

1* 1 1 1* GAS /

PROD 1* 1 1 1* OIL /

/

COMPDAT --

WELL I J K1 K2

INJE 1 1 1 1 OPEN 1* 1E5 /

PROD 1 1 602 602 OPEN 1* 1E5 /

/

WCONPROD

PROD OPEN BHP 5* 7.8 /

/

WELLSTRE -- HEAVY LIGHT COKE O2 CO2 N2

AIR 0.0 0.0 0.0 0.21 0.0 0.79 /

NIT 0.0 0.0 0.0 0.00 0.0 1.00 /

/

WCONINJE -- Rate BHP

INJE GAS OPEN RATE 1.8e+5 1* /


/

WINJGAS

INJE STREAM NIT /

/

WINJTEMP -- T P

INJE 1* 20 7.8 /

/

TSTEP 1*0.3 /

HEATER

HEAT1 1 1 2 17400 1*/

HEAT1 1 1 3 17400 1*/

HEAT1 1 1 4 17400 1*/

HEAT1 1 1 5 17400 1*/

HEAT1 1 1 6 17400 1*/

HEAT1 1 1 7 17400 1*/

HEAT1 1 1 8 17400 1*/

HEAT1 1 1 9 17400 1*/

HEAT1 1 1 10 17400 1*/

HEAT1 1 1 11 17400 1*/

HEAT1 1 1 12 17400 1*/

HEAT1 1 1 13 17400 1*/

HEAT1 1 1 14 17400 1*/

HEAT1 1 1 15 17400 1*/

HEAT1 1 1 16 17400 1*/

HEAT1 1 1 17 17400 1*/

HEAT1 1 1 18 17400 1*/

HEAT1 1 1 19 17400 1*/

HEAT1 1 1 20 17400 1*/

HEAT1 1 1 21 17400 1*/

HEAT1 1 1 22 17400 1*/

HEAT1 1 1 23 17400 1*/


HEAT1 1 1 24 17400 1*/

HEAT1 1 1 25 17400 1*/

HEAT1 1 1 26 17400 1*/

HEAT1 1 1 27 9600 1*/

HEAT1 1 1 28 9600 1*/

HEAT1 1 1 29 9600 1*/

HEAT1 1 1 30 9600 1*/

HEAT1 1 1 31 9600 1*/

/

TSTEP

1*1 /

WCONINJE -- Rate BHP

INJE GAS OPEN RATE 1.8e+5 1* /

/

WINJGAS

INJE STREAM AIR /

/

WINJTEMP -- T P

INJE 1* 20 7.8 /

/

HEATER

HEAT1 1 1 2 0 1*/

HEAT1 1 1 3 0 1*/

HEAT1 1 1 4 0 1*/

HEAT1 1 1 5 0 1*/

HEAT1 1 1 6 0 1*/

HEAT1 1 1 7 0 1*/

HEAT1 1 1 80 1*/

HEAT1 1 1 9 0 1*/

HEAT1 1 1 10 0 1*/


HEAT1 1 1 11 0 1*/

HEAT1 1 1 12 0 1*/

HEAT1 1 1 13 0 1*/

HEAT1 1 1 14 0 1*/

HEAT1 1 1 15 0 1*/

HEAT1 1 1 16 0 1*/

HEAT1 1 1 17 0 1*/

HEAT1 1 1 18 0 1*/

HEAT1 1 1 19 0 1*/

HEAT1 1 1 20 0 1*/

HEAT1 1 1 21 0 1*/

HEAT1 1 1 22 0 1*/

HEAT1 1 1 23 0 1*/

HEAT1 1 1 24 0 1*/

HEAT1 1 1 25 0 1*/

HEAT1 1 1 26 0 1*/

HEAT1 1 1 27 0 1*/

HEAT1 1 1 28 0 1*/

HEAT1 1 1 29 0 1*/

HEAT1 1 1 30 0 1*/

HEAT1 1 1 31 0 1*/

/

TSTEP

8*1 /

END

Appendix D

STARS Input File for Upscaled

Three Reaction Model

In this input file, the fuel amount in this sample input file is set at Sofuel = 7.9%.

The oxygen utilization efficiency is set at Eu = 0.917.

**----------------------INPUT-OUTPUT CONTROL----------------------------

*TITLE1 ’Simulation of Dry Combustion Tube Test’

*INUNIT *LAB

*OUTUNIT *LAB

*WPRN *GRID *TIME

*WPRN *ITER *TIME

*OUTPRN *GRID *PRES *SW *SO *SG *TEMP *Y *X *MASS *SOLCONC

*OUTPRN *WELL *WELLCOMP

*OUTPRN *ITER *BRIEF

*OUTSRF *WELL *MASS *COMPONENT *ALL

*OUTSRF *GRID *PRES *SW *SO *SG *TEMP *Y *X *MASS *SOLCONC *VISO *VISW

*INTERRUPT *STOP

**-------------------------RESERVOIR DESCRIPTION------------------------

*GRID *CART 1 602 1

*KDIR *DOWN

179

APPENDIX D. STARS INPUT FILE FOR UPSCALED THREE REACTION MODEL180

*DI *CON 6.2

*DJ *CON 0.36

*DK *CON 6.2

*POR *CON 0.36

*PERMI *CON 10000

*PERMJ *CON 10000

*PERMK *CON 10000

*END-GRID

**------------------------OTHER RESERVOIR PROPERTIES--------------------

*CPOR 0

*CTPOR 0

*ROCKCP 1.20572 0.00236

*THCONR 2.5833

*THCONO 0.0

*THCONG 0.0

*THCONS 0.0

**--------------------------COMPONENT PROPERTIES------------------------

*MODEL 10 8 3

*COMPNAME ’WATER’ ’OIL1’ ’OIL2’ ’CO’ ’CO2’ ’N2’ ’O2’ ’O2_B’ ’COKE1’ ’COKE2’

*CMM 0 0.5376 0.5376 0.028 0.044 0.028 0.032 0.032 0.5376 0.17976

*TCRIT 0 1472 1472 -140.3 31.05 -146.95 -118.6 -118.6

*PCRIT 0 1168 1168 3496 7376 3394 5046 5046

*KV1 0 0 0

*KV2 0 0 0

*KV3 0 0 0

*KV4 0 0 0

*KV5 0 0 0

*PRSR 101

*TEMR 15


*PSURF 101

*TSURF 15

** ’WATER’ ’OIL1’ ’OIL2’ ’CO’ ’CO2’ ’N2’ ’O2’ ’O2_B’ ’COKE1’ ’COKE2’

*CPG1 0 1138.8 1138.8 29.987 16.864 30.288 27.627 27.627

*CPG2 0 0 0 -1.542e-3 0.1063 -2.572e-3 6.437e-3 6.437e-3

*CPG3 0 0 0 0 0 0 0 0

*CPG4 0 0 0 0 0 0 0 0

*HVR 0 0 0

*EV 0 0 0

*MASSDEN 0 1.008E-3 1.008E-3

*CP 0 0 0

*CT1 0 0 0

*CT2 0 0 0

*SOLID_DEN ’COKE1’ 1.008E-3 0 0


*SOLID_DEN ’COKE2’ 1.008E-3 0 0


*VISCTABLE

** Tempe Water Oil1 Oil2

5 0 1636262 1636262

15 0 229374 229374

25 0 54485.9 54485.9

30 0 30000 30000

48 0 10600 10600

65 0 3000 3000

100 0 290 290

150 0 32 32

200 0 7.5 7.5

250 0 2.7 2.7

300 0 1.0 1.0

325 0 0.75 0.75


350 0 0.60 0.60

375 0 0.50 0.50

400 0 0.47 0.47

425 0 0.41 0.41

450 0 0.36 0.36

475 0 0.315 0.315

500 0 0.28 0.28

525 0 0.25 0.25

550 0 0.22 0.22

2000 0 0.12 0.12


*STOREAC 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

*STOPROD 0.0 0.0 0.605 0.0 0.0 0.0 0.0 0.0 0.395 0.0

*FREQFAC 1E5 **1/kPa-MIN

*RENTH 0 ** J/gmol

*RPHASE 0 2 0 0 0 0 0 0 0 0

*RORDER 0 1 0 0 0 0 0 0 0 0

*EACT 0 ** J/gmol

*O2PP ’O2’


*STOREAC 0.0 0.0 0.0 0.0 0.0 0.0 38.04 0.0 1.0 0.0

*STOPROD 22.0 0.0 0.0 4.75 23.77 0.0 0.0 0.0 0.0 1.0



*RPHASE 0 0 0 0 0 0 3 0 4 0

*RORDER 0 0 0 0 0 0 1 0 1 0

*EACT 0 ** J/gmol

*O2PP ’O2’


*STOREAC 0.0 0.0 0.0 0.0 0.0 0.0 12.29 0.0 0.0 1.0


*STOPROD 8.0 0.0 0.0 1.73 8.56 0.0 0.0 0.0 0.0 0.0



*RPHASE 0 0 0 0 0 0 3 0 0 4

*RORDER 0 0 0 0 0 0 1 0 0 1

*EACT 0 ** J/gmol

*O2PP ’O2’

**----------------------------------ROCK~FLUID DATA---------------------

*ROCKFLUID

*RPT 1 *LININTERP

** Sw Krw Krow Pcow

*SWT

0.074447121 0 1 0

0.16422159 0.0077 0.8229 0

0.21896212 0.02 0.7232 0

0.23866871 0.0259 0.6889 0

0.273702649 0.0381 0.63 0

0.328443179 0.0619 0.5432 0

0.383183709 0.0915 0.4629 0

0.437924239 0.1268 0.3889 0

0.492664769 0.1679 0.3214 0

0.547405299 0.2148 0.2604 0

0.602145829 0.2674 0.2057 0

0.656886359 0.3258 0.1575 0

0.711626889 0.3899 0.1157 0

0.766367418 0.4598 0.0804 0

0.821107948 0.5354 0.0514 0

0.875848478 0.6168 0.0289 0

0.930589008 0.704 0.0129 0

0.985329538 0.7969 0.0032 0


1 0.8956 0 0

*SLT

** Sl Krg Krog Pcgo

** -- --- ---- ----

0.054959492 0.6845 1.00E-03 0

0.124151522 0.6233 0.0077 0

0.178892052 0.5429 0.02 0

0.198598642 0.5154 0.0259 0

0.233632582 0.4681 0.0381 0

0.288373111 0.3989 0.0619 0

0.343113641 0.3352 0.0915 0

0.397854171 0.277 0.1268 0

0.452594701 0.2244 0.1679 0

0.507335231 0.1773 0.2148 0

0.562075761 0.1357 0.2674 0

0.616816291 0.0997 0.3258 0

0.671556821 0.0693 0.3899 0

0.726297351 0.0443 0.4598 0

0.78103788 0.0249 0.5354 0

0.83577841 0.0111 0.6168 0

0.89051894 0.0028 0.704 0

0.925552879 1.00E-03 0.7969 0

1 0 0.8956 0

**----------------------------------INITIAL CONDITION--------------------

*INITIAL

*PRES *CON 780

*TEMP *CON 15

*SW *CON 0.24

*SO *CON 0.20

*MFRAC_WAT ’WATER’ *CON 1.00


*MFRAC_OIL ’OIL’ *CON 1.00

*MFRAC_GAS ’N2’ *CON 1.00

*MFRAC_GAS ’CO2’ *CON 0.00

*MFRAC_GAS ’CO’ *CON 0.00

*MFRAC_GAS ’O2’ *CON 0.00

**-----------------------------------NUMERICAL CONTROL---------------------

*NUMERICAL

*MAXSTEPS 100000

**DTMAX 0.1

*DTMIN 0.000001

**-----------------------------------RECURRENT DATA-------------------------

*RUN

*TIME 0

*DTWELL 0.01

*WELL ’MB-INJECTOR’

*WELL ’MB-PRODUCER’

*PRODUCER ’MB-PRODUCER’

*OPERATE *BHP 780

*GEOMETRY *K 0.50 1 1 0

*PERF *TUBE-END ’MB-PRODUCER’

1 1 602 1.00

*INJECTOR ’MB-INJECTOR’

*TINJW 20

*PINJW 780

*INCOMP *GAS 0 0 0 0 0 1 0 0

*OPERATE *STG 3000

*GEOMETRY *K 0.50 1 1 0

*PERF *TUBE-END ’MB-INJECTOR’

1 1 1 1.00


*TIME 18

*HEATR *IJK 1 1 1:30 320

*HEATR *IJK 1 1 1:25 260

*TIME 78

*INJECTOR ’MB-INJECTOR’

*TINJW 20

*PINJW 780

*INCOMP *GAS 0 0 0 0 0 0.79 0.1926 0.0174

*OPERATE *STG 3000

*GEOMETRY *K 0.50 1 1 0

*PERF *TUBE-END ’MB-INJECTOR’

1 1 1 1.00

*HEATR *IJK 1 1 1:30 0

*HEATR *IJK 1 1 1:25 0

*TIME 138

*TIME 198

*TIME 258

*TIME 318

*TIME 378

*TIME 438

*STOP

Appendix E

SAGD Input File

RESULTS SIMULATOR STARS

*interrupt *stop

*INUNIT *SI

*OUTPRN *WELL *ALL

*OUTPRN *GRID *NONE

*OUTPRN *ITER *NEWTON

**restart

*WPRN *GRID 200

*WPRN *ITER 200

*PRNTORIEN 2 0

*WPRN *SECTOR 100

*WSRF *SECTOR 100

*OUTSRF *SPECIAL

MATBAL WELL ’OIL’

MATBAL WELL ’WATER’

*OUTSRF *GRID *PRES *SW *SO *SG *TEMP *QUALBLK *CCHLOSS

*VISO *VISW *VISG *MOLDENO

*MASDENO *MASDENW *MASDENG *THCONDUCT *VPOROS *KRW *KRO *KRG

** ============== GRID AND RESERVOIR DEFINITION =================

*GRID *CART 45 1 30

187

APPENDIX E. SAGD INPUT FILE 188

*KDIR *DOWN

*DI *CON 1 **

*DJ *CON 30 **

*DK *CON 1 **

*DEPTH 1 1 1 1000

*POR *CON 0.35

*PERMI *CON 2000

*PERMJ *EQUALSI

*PERMK *EQUALSI

*NINEPOINT *IK

*END-GRID

*PRPOR 100

*ROCKTYPE 1

*CPOR 1e-5

*CTPOR 1e-4

*ROCKCP 1500E3

*THCONR 3E5

*thconw 0

*thcono 0

*thcong 0

*THTYPE *CON 1 ** Assigns the properties of the rocktype 1 to whole res.

** ============== FLUID DEFINITIONS =================================

MODEL 2 2 2 1

COMPNAME ’WATER’ ’OIL’

** -------- --------

CMM 0.018 0.282

PCRIT 22048 1115

TCRIT 374.15 493.85

KV1 1.1860E7 0.0

KV2 0.00000 0.000E+0

KV3 0.00000 0.000E+0


KV4 -3816.44 -4680.46

KV5 -227.02 -132.05

MASSDEN 0.998E3 0.972E3

CP 1E-7 1E-7

CT1 1E-4 1E-4

** T, deg C ’Water’ ’OIL’

*avisc 0.0047352 1E-8

*bvisc 1515.7 10000

** ’WATER’ ’OIL’

*CPG1 32.243 -22.383

*CPG2 1.924E-3 1.939

*CPG3 1.055E-5 -1.117E-3

*CPG4 -3.596E-9 2.528E-7

*HVR 4820 8569

*EV 0.38 0.38

PRSR 100 ** reference pressure, corresponding to the density

TEMR 20 ** reference temperature, corresponding to the density

PSURF 1E2 ** pressure at surface, for reporting well rates, etc.

TSURF 20 ** temperature at surface, for reporting well rates, etc.

** ============== ROCK-FLUID PROPERTIES ======================

rockfluid

swt ** Water-oil relative permeabilities

** Sw Krw Krow

** ---- -------- --------

0.00 0 1

0.05 0.0025 0.9025

0.10 0.01 0.81

0.15 0.0225 0.7225

0.20 0.04 0.64

0.25 0.0625 0.5625


0.30 0.09 0.49

0.35 0.1225 0.4225

0.40 0.16 0.36

0.45 0.2025 0.3025

0.50 0.25 0.25

0.55 0.3025 0.2025

0.60 0.36 0.16

0.65 0.4225 0.1225

0.70 0.49 0.09

0.75 0.5625 0.0625

0.80 0.64 0.04

0.85 0.7225 0.0225

0.90 0.81 0.01

0.95 0.9025 0.0025

1.00 1.0000 0.0000

slt **NOSWC ** Liquid-gas relative permeabilities

** Sl Krg Krog

** ---- -------- --------

0.00 1 0

0.05 0.9025 0.0025

0.10 0.81 0.01

0.15 0.7225 0.0225

0.20 0.64 0.04

0.25 0.5625 0.0625

0.30 0.49 0.09

0.35 0.4225 0.1225

0.40 0.36 0.16

0.45 0.3025 0.2025

0.50 0.25 0.25

0.55 0.2025 0.3025

0.60 0.16 0.36


0.65 0.1225 0.4225

0.70 0.09 0.49

0.75 0.0625 0.5625

0.80 0.04 0.64

0.85 0.0225 0.7225

0.90 0.01 0.81

0.95 0.0025 0.9025

1.00 0.0000 1.0000

** ============== INITIAL CONDITIONS ======================

*INITIAL

*PRES *CON 6000

*SW *CON 0.0

*SO *CON 1.0

*SG *CON 0.0

*TEMP *CON 55

*mfrac_oil ’OIL’ *CON 1.0

*mfrac_gas ’WATER’ *CON 1.0

*mfrac_wat ’WATER’ *CON 1.0

** ============== NUMERICAL CONTROL ======================

*NUMERICAL

*DTMAX 2

*NEWTONCYC 15

*NORM *PRESS 100 *SATUR 0.1 *TEMP 10 *Y 1 *X 1

*MINPRES 0.01

*CONVERGE *PRESS 0.1 *SATUR 0.1 *TEMP 0.1

*MATBALTOL 0.00001

*PIVOT *ON ** Pivoting stabilization

*RANGECHECK OFF

*RUN

** ============== RECURRENT DATA ========================================

*DATE 2002 01 01


*DTWELL 1

*WELL ’Injector1’

*WELL ’Producer1’

*INJECTOR ’Injector1’

**--------------’WATER’ ’C2 toC14’

*INCOMP WATER 1.0 0.0

*TINJW 275.86** degree C

*QUAL 1.0

*OPERATE *BHP 6000 **IWELLBORE 1 *PWELLBORE 1

**---------- rad geofac wfrac skin

*GEOMETRY *K 2.0 0.249 1.0 0.0

*PERF GEO ’Injector1’ ** i j k wi.

2 1 23 1.0

*PRODUCER ’Producer1’

*OPERATE *BHP 6000 **IWELLBORE 1 *PWELLBORE 1

**---------- rad geofac wfrac skin

GEOMETRY *K 2.0 0.249 1.0 0.0

*PERF GEO ’Producer1’ ** i j k wi.

2 1 29 1.0

*TIME 100

*TIME 200

*TIME 300

*TIME 400

*TIME 500

*TIME 600

*STOP

Appendix F

SAGD Input File for Grid Size

Effects Study

RESULTS SIMULATOR STARS

*INTERRUPT *STOP

** ============== INPUT/OUTPUT CONTROL ======================

dim mdiclu 200000

*INUNIT *SI

*OUTUNIT *SI

*WRST *TIME

**RESTART 185

**REWIND XXXX

*OUTPRN *GRID *PRES *SW *SO *SG *TEMP *OBHLOSS *VISO

*OUTPRN *WELL *ALL

*OUTSRF *GRID *PRES *SW *SO *SG *TEMP *VISO

*WPRN *GRID *TIME

*PRNTORIEN 2 0

*RANGECHECK *OFF

wprn iter 1

outprn iter *newton

**restart 154 **186

193

APPENDIX F. SAGD INPUT FILE FOR GRID SIZE EFFECTS STUDY 194

** ========= DEFINITION OF FUNDAMENTAL CARTESIAN GRID ========

*GRID *VARI 50 3 47

*KDIR *DOWN

*DI *IVAR 50*1

*DJ *JVAR 3*100

*DK *KVAR 34*1 1 1 8*1 3*1

*DTOP 50*734.4 50*739.6 50*744.0

*WELLBORE 0.0365

*LAMINAR

wellinfo

*CIRCWELL 0.0875 50 3 36 0

*RANGE 50 1:3 36

*WELLBORE 0.0365

*LAMINAR

*CIRCWELL 0.0875 50 3 46 0

*RANGE 50 1:3 46

*POR *IJK

1:50 1 1:20 0.25368

1:50 1 21:47 0.253509

1:50 2 1:20 0.250318

1:50 2 21:47 0.280487

1:50 3 1:20 0.232101

1:50 3 21:47 0.301246

*PERMI *CON 1468

*PERMJ *EQUALSI * 1.00

*PERMK *EQUALSI * 0.3828

*END-GRID

*PRPOR 7600

*ROCKTYPE 1

*CPOR 1.8E-5

*ROCKCP 1.5E6


*THCONR 1.5E5

*THCONW 1.5E5

*THCONO 1.5E5

*THCONG 1.5E5

*HLOSSPROP *OVERBUR 1.0E6 8.0E4

*UNDERBUR 1.0E6 8.0E4

*ROCKTYPE 2 copy 1

*ROCKCP 2.35E6

*THCONR 1.5E5

*THCONW 1.5E5

*THCONO 1.5E5

*THCONG 1.5E5

**cpor 0.0

THTYPE CON 1 ** RESERVOIR

THTYPE WELLBORE 50 1:3 36 CON 2 **TOP WELLBORE

THTYPE WELLBORE 50 1:3 46 CON 2 **BOTTOM WELLBORE

** ============== FLUID DEFINITIONS ======================

*MODEL 2 2 2 ** Components are water and dead oil. Most water

** properties are defaulted (=0). Dead oil K values

** are zero, and no gas properties are needed.

*COMPNAME ’WATER’ ’OIL’

** ------- --------

CMM 0.018 0.45 ** kg/gmol

MOLDEN 0 2222.22 ** gmol/m3

CP 0 7.25E-7 ** oil cr

CT1 0 6.84E-4 **3.8e-4

PCRIT 0 0 **217.6

TCRIT 0 0 **374.15

CPG1 0 942.0 **2.09

AVG 0 0 **1.13e-5

BVG 0 0 ** 1.075


*VISCTABLE

** TEMP

47 0 122807.

60. 0 35541.

80. 0 6751.

100 0 1659.

140 0 186.

180 0 39.

260 0 6.

300 0 3.3

900 0 3.3

** REFERENCE CONDITIONS

*PRSR 101.3

*TEMR 20.0

*PSURF 101.3

*TSURF 20.0

** ============== ROCK-FLUID PROPERTIES ======================

*ROCKFLUID

*RPT 1

*SWT smoothend quad

** Sw Krw Krow

0.35 0.00000 1.00000

0.40 0.00638 0.73551

0.43 0.01290 0.59032

0.46 0.02080 0.45618

0.49 0.02987 0.33409

0.52 0.03997 0.22540

0.55 0.05100 0.13201

0.58 0.06290 0.05702

0.61 0.07560 0.00713

0.62 0.08000 0.00000


**1.00 0.14000 0.00000

*SLT

** Sl Krg Krog

0.73 1.000 0.000

0.76 0.838 0.037

0.79 0.686 0.105

0.82 0.544 0.192

0.85 0.414 0.296

0.88 0.296 0.414

0.91 0.192 0.544

0.94 0.105 0.686

0.97 0.037 0.838

1.00 0.000 1.000

** ROCK 1

*KRTEMTAB *SWR *SORW *SGR *SORG *KRWRO *KROCW *KRGCW

48 0.35 0.40 0.00 0.38 0.08 1.0 1.0

200 0.40 0.28 0.00 0.25 0.12 1.0 1.0

*RPT 2 **--- WELLBORE ---

*SWT

** Sw Krw Krow

0.0 0.00 1.00

1.00 1.00 0.00

*SLT

** Sl Krg Krog

0.0 1.00 0.00

1.00 0.00 1.00

** ============== INITIAL CONDITIONS ======================

*INITIAL

*VERTICAL *DEPTH_AVE

*REFPRES 7600.0

*REFDEPTH 760.0


**PRES *CON 7600.

*TEMP *CON 48.

** ============== NUMERICAL CONTROL ======================

*NUMERICAL

***AIM *OFF

*DTMAX 15.0

*ITERMAX 60 **12

*SDEGREE 2

*SORDER rcmrb ***REDBLACK

*NORM *PRESS 400 *TEMP 50

*SATUR .3 *Y .3 *X .2

*NEWTONCYC 17

**RANGECHECK *OFF

**PVTOSCMAX 4

*upstream klevel

**converge totres normal

** ============== RECURRENT DATA ======================

*RUN

*DATE 1996 01 01

*DTWELL 1.0e-1

*WELL 1 ’UPCASI’ *FRAC 0.5

*WELL 2 ’UPTUB’ *FRAC 0.5

*WELL 3 ’LWCASI’ *FRAC 0.5

*WELL 4 ’LWTUB’ *FRAC 0.5

*******************

** PREHEAT PHASE

*******************

*PRODUCER ’UPCASI’ ** UPCAS

*OPERATE *MIN *BHP 11700

*GEOMETRY *J 0.0875 0.249 1 0 ** 7" liner

*PERF *GEO ’UPCASI’


50 1 36 / 2 1 1 1.0

*INJECTOR *MOBWEIGHT ’UPTUB’ ** UPTUB

*TINJW 324.6

*QUAL 0.5


*OPERATE *MAX *STW 200

*GEOMETRY *J 0.0875 0.249 1 0 ** 7" liner

*PERF *GEO ’UPTUB’

50 1 36 / 1 1 1 1.0

*PRODUCER ’LWCASI’ ** LWCASI


*GEOMETRY *J 0.0875 0.249 1 0 ** 7" liner

*PERF *GEO ’LWCASI’

50 1 46 / 2 1 1 1.0

*INJECTOR *MOBWEIGHT ’LWTUB’ ** LWTUB


*TINJW 324.6

*QUAL 0.5


*OPERATE *MAX *BHP 11650

**STOP

*GEOMETRY *J 0.0875 0.249 1 0 ** 7" liner

*PERF *GEO ’LWTUB’

50 1 46 / 1 1 1 1.0

*DATE 1996 02 01

*DATE 1996 03 01

*DATE 1996 04 01

*DTWELL 1.0e-4

*******************

** HIGH PRESSURE SAGD

*******************


*SHUTIN ’UPCASI’

*SHUTIN ’LWTUB’

*INJECTOR *MOBWEIGHT ’UPTUB’

** UPTUB


*TINJW 324.6

*QUAL 0.9



*PRODUCER ’LWCASI’

*OPERATE *MAX *STL 50

*OPERATE *MAX *STEAM 1 cont

*WELL 5 ’LWTUBP’ *FRAC 0.5

*PRODUCER ’LWTUBP’

** LWTUBP




*NEXTSEG

*GEOMETRY *J 0.0875 0.249 1 0 ** 7" liner

*PERF *GEO ’LWTUBP’

50 1 46 / 1 1 1 1.0

*DATE 1996 05 25

*DTWELL 1.0e-2

*******************

** DEPRESSURIZATION PHASE

*******************






*PRODUCER ’LWCASI’


*SHUTIN




** LWTUBP


*NEXTSEG



*DATE 1996 07 15

*DTWELL 1.0e-3

*******************

** SAGD PHASE

*******************


*TINJW 270

*QUAL 0.9








*DATE 1996 09 30






*MONITOR *MIN *STO 5

*STOP

transient ’LWTUBP’ on

*DATE 1996 12 31

*DATE 1997 09 30





*MONITOR *MIN *STO 5

*STOP

*DATE 1997 12 31

*DATE 1998 12 31

*DATE 1999 12 31

*DATE 2000 12 31

*DATE 2001 12 31

*DATE 2002 12 31

*DATE 2003 12 31

*DATE 2004 12 31

*DATE 2005 12 31

*DATE 2006 12 31

*DATE 2007 12 31

*STOP

Bibliography

[1] 3DSL. 3DSL Version 2010 Software Manual. StreamSim Technologies, Inc.,

2010.

[2] M. Abbaszadeh, F.R. Garza, C. Yuan, and G.A. Pope. Single-well simulation

study of foam eor in gas-cap oil of the naturally-fractured cantarell field. In

SPE129867 presented at the SPE Improved Oil Recovery Symposium. Society

of Petroleum Engineers, April 2010.

[3] I.Y. Akkutlu and Y.C. Yortsos. The dynamics of in-situ combustion fronts in

porous media. Combustion and Flame, 134:229–247, 2003.

[4] A. Alamatsaz, R.G. Moore, S.A. Mehta, and M.G. Ursenbach. Experimental

investigation of in situ combustion at low air fluxes. In SPE144517 presented

at the SPE Western North American Region Meeting. Society of Petroleum

Engineers, 2011.

[5] M.E. Armento and C.A. Miller. Stability of moving combustion fronts in porous

media. SPE Journal, 17(6):423–430, December 1977.

[6] K. Aziz and A. Settari. Petroleum Reservoir Simulation. Elsevier Applied

Science Publishers, London, 1979.

[7] Roderick Panko Batycky. A Three-dimensional Two-phase Field Scale Stream-

line Simulator. PhD thesis, Stanford University, 1997.

[8] R.P. Batycky, M.J. Blunt, and M.R. Thiele. A 3d field-scale streamline-based

reservoir simulator. SPE Reservoir Engineering, 12(4):246–254, November 1997.

203

BIBLIOGRAPHY 204

[9] R.P. Batycky, M. Forster, M.R. Thiele, and K. Stuben. Parallelization of a

commercial streamline simulator and performance on practical models. SPE

Reservoir Evaluation & Engineering, 13(3):383–390, June 2010.

[10] M. Bazargan, B. Chen, M. Cinar, G. Glatz, A. Lapene, Z. Zhu, L. Castanier,

M. Gerritsen, and A.R. Kovscek. A combined experimental and simulation

workflow to improve predictability of in situ combustion. In SPE144599 pre-

sented at the SPE Western Regional Meeting. Society of Petroleum Engineers,

2011.

[11] J.D.M. Belgrave, R.G. Moore, M.G. Ursenbach, and D.W. Bennion. A compre-

hensive approach to in-situ combustion modeling. SPE Advanced Technology

Series, 1(1):37–58, April 1993.

[12] V.T. Beraldo, M.J. Blunt, and D.J. Schiozer. Compressible streamline-based

simulation with changes in oil composition. SPE Reservoir Evaluation & En-

gineering, 12(6):963–973, December 2009.

[13] I. Berre, H.K. Dahle, K.H. Karlsen, and H.F. Nordhaug. A streamline front

tracking method for two- and three-phase flow including capillary forces. Con-

temporary Mathematics, 295:49–61, 2002.

[14] F. Bratvedt, T. Gimse, and C. Tegnander. Streamline computations for porous

media flow including gravity. Transport in Porous Media, 25(1):63–78, October

1996.

[15] J. Burger, P. Sourieau, and M. Combarnous. Thermal Methods of Oil Recovery.

Paris, France, 1985.

[16] R.M. Butler. Thermal Recovery of Oil and Bitumen. 1991.

[17] M.L. Byl, R.G. Moore, and M.G. Ursenbach. Field observations of in situ

combustion in a waterflooded reservoir in the kinsella field. Journal of Canadian

Petroleum Technology, 32(7), July 1993.

BIBLIOGRAPHY 205

[18] Hui Cao. Deveopment of Techniques for General Purpose Simulators. PhD

thesis, Stanford University, 2002.

[19] Qing Chen. Assessing and Improving Steam-Assisted Gravity Drainage: Reser-

voir Heterogeneities, Hydraulic Fractures, and Mobility Control Foams. PhD

thesis, Stanford University, 2009.

[20] Yuguang Chen. Upscaling and Subgrid Modeling of Flow and Transport in Het-

erogeneous Reservoirs. PhD thesis, Stanford University, 2005.

[21] H. Cheng, I. Osako, A. Datta-Gupta, and M. King. A rigorous compress-

ible streamline formulation for two- and three-phase black-oil simulation. SPE

Reservoir Evaluation & Engineering, 11(4):407–417, December 2006.

[22] J.R. Christensen, G. Darche, B. Dechelette, H. Ma, and P.H. Sammon. Ap-

plications of dynamic gridding to thermal simulations. In SPE86969 presented

at the SPE International Thermal Operations and Heavy Oil Symposium and

Western Regional Meeting. Society of Petroleum Engineers, 2004.

[23] M.A. Christie, Heriot-Watt U., and M.J. Blunt. Tenth spe comparative solution

project: A comparison of upscaling techniques. SPE Reservoir Engineering,

4(4):308–317, August 2001.

[24] M. Cinar, L.M. Castanier, and A.R. Kovscek. Isoconversional kinetic analysis

of the combustion of heavy hydrocarbons. In SPE120995 presented at the SPE

Western Regional Meeting. Society of Petroleum Engineers, 2009.

[25] Murat Cinar. Kinetics of Crude-oil Combustion in Porous Media Interpreted

Using Isoconversional Methods. PhD thesis, Stanford University, 2011.

[26] R. Coates, S. Lorimer, and J. Ivory. Experimental and numerical simulations

of a novel top down in-situ combustion process. In SPE30295 presented at the

SPE International Heavy Oil Symposium. Society of Petroleum Engineers, 1995.

[27] K.H. Coats. In-situ combustion model. SPE Journal, 20(6):533–554, December

1980.

BIBLIOGRAPHY 206

[28] K.H. Coats. Some observations on field-scale simulation of the in-situ combus-

tion process. In SPE12247 presented at the SPE Reservoir Simulation Sympo-

sium. Society of Petroleum Engineers, 1983.

[29] M. Crane, F. Bratvedt, K. Bratvedt, and R. Olufsen. A fully compositional

streamline simulator. In SPE63156 presented at the SPE Annual Technical

Conference and Exhibition. Society of Petroleum Engineers, 2000.

[30] R.B. Crookston, W.E. Culham, and W.H. Chen. A numerical simulation model

for thermal recovery processes. SPE Journal, 19(1):37–58, February 1979.

[31] S. Das. Diffusion and dispersion in the simulation of vapex process. In SPE97924

presented at the SPE/PS-CIM/CHOA International Thermal Operations and

Heavy Oil Symposium. Society of Petroleum Engineers, 2005.

[32] R.J. Davies. The thin flame technique for in-situ combustion simulation. In

SPE19647 presented at the SPE Annual Technical Conference and Exhibitio.

Society of Petroleum Engineers, 1989.

[33] B. Dechelette, O. Heugas, G. Quenault, J. Bothua, and J.R. Christensen. Air

injection-improved determination of the reaction scheme with ramped temper-

ature experiment and numerical simulation. Journal of Canadian Petroleum

Technology, 45(1), January 2006.

[34] I. Diabira, L.M. Castanier, and A.R. Kovscek. Porosity and permeability evo-

lution accompanying hot fluid injection into diatomite. Petroleum Science and

Technology, 19:1167–1185, 2001.

[35] Birol Dindoruk. Analytical Theory of Multiphase Multicomponent Displacement

in Porous Media. PhD thesis, Stanford University, 1992.

[36] G.D. Donato and M.J. Blunt. Streamline-based dual-porosity simulation of

reactive transport and flow in fractured reservoirs. Water Resources Research,

40(W04203), 2004.

[37] Eclipse. Eclipse Version 2009 Software Manual. Schulumberger Ltd., 2009.

BIBLIOGRAPHY 207

[38] N.R. Edmunds. A model of the steam drag effect in oil sands. Journal of

Canadian Petroleum Technology, 23(5), Sep 1984.

[39] H. Fadaei, G. Debenest, A.M. Kamp, M. Quintard, and G. Renard. How the

in-situ combustion process works in a fractured system: 2d core- and block-scale

simulation. SPE Reservoir Evaluation & Engineering, 13(1):118–130, February

2010.

[40] Mohammad Reza Fassihi. Analysis of Fuel Oxidation in In-Situ Combustion

Oil Recovery. PhD thesis, Stanford University, 1981.

[41] M.R. Fassihi and T.H. Gillham. The use of air injection to improve the double

displacement processes. In SPE26374 presented at the SPE Annual Technical

Conference and Exhibition. Society of Petroleum Engineers, 1993.

[42] M.R. Fassihi, D.V. Yannimaras, and V.K. Kumar. Estimation of recovery factor

in light-oil air-injection projects. SPE Reservoir Engineering, 12(3):173–178,

August 1997.

[43] N.P. Freitag and D.R. Exelby. A sara-based model for simulating the pyrolysis

reactions that occur in high-temperature eor processes. Journal of Canadian

Petroleum Technology, 45(3), March 2006.

[44] C.F. Gates and H.J. Ramey. A method for engineering in-situ combustion oil

recovery projects. Journal of Petroleum Technology, 32(2):285–294, February

1980.

[45] M.G. Gerritsen, K. Jessen, B.T. Mallison, and J. Lambers. A fully adaptive

streamline framework for the challenging simulation of gas injection processes.

In SPE97270 presented at the SPE Annual Technical Conference and Exhibition.

Society of Petroleum Engineers, 2005.

[46] M. Greaves, S.R. Ren, R.R. Rathbone, T. Fishlock, and R. Ireland. Improved

residual light oil recovery by air injection (lto process). Journal of Canadian

Petroleum Technology, 39(1), January 2000.

BIBLIOGRAPHY 208

[47] D. Gutierrez, F. Skoreyko, R.G. Moore, S.A. Mehta, and M.G. Ursenbach. The

challenge of predicting field performance of air injection projects based on lab-

oratory and numerical modelling. Journal of Canadian Petroleum Technology,

48(4):23–33, April 2009.

[48] R.J. Hallam and J.K. Donnelly. Pressure-up blowdown combustion: A chan-

neled reservoir recovery process. SPE Advanced Technology Series, 1(1):153–

158, April 1993.

[49] H. Hassanzadeh, J. Abedi, and M. Pooladi-Darvish. Equivalent rate constant for

numerical simulation of linear convection-diffusion-reaction. Journal of Cana-

dian Petroleum Technology, 49(4):51–57, April 2010.

[50] B.T. Hoffman and A.R. Kovscek. Displacement front stability of steam in-

jection into high porosity diatomite rock. Journal of Petroleum Science and

Engineering, 46(4):253–266, April 2005.

[51] M.H. Hui, D. Zhou, X.H. Wen, and L.J. Durlofsky. Development and application

of a new technique for upscaling miscible processes. SPE Reservoir Evaluation

& Engineering, 8(3):189–195, June 2005.

[52] M.K. Hwang. An in-situ combustion process simulator with a moving-front

representation. Society of Petroleum Engineers Journal, 22(2):271–279, April

1982.

[53] Y. Ito and A.K. Chow. A field scale in-situ combustion simulator with chan-

neling considerations. SPE Reservoir Engineeringg, 3(2):419–430, May 1988.

[54] S. Javad, P. Oskouei, R.G. Moore, B. Maini, and S.A. Mehta. Front self-

correction for in-situ combustion. Journal of Canadian Petroleum Technology,

50(3):43–56, March 2011.

[55] G.R. Jenkins and J.W. Kirkpatrick. Twenty years’ operation of an in-situ

combustion project. Journal of Canadian Petroleum Technology, 18(1), Jan-

Mar 1975.

BIBLIOGRAPHY 209

[56] K. Jessen, M.G. Gerritsen, and B.T. Mallison. High-resolution prediction of

enhanced condensate recovery processes. SPE Journal, 13(2):257–266, June

2008.

[57] K. Jessen and F.M. Orr Jr. Gravity segregation and compositional streamline

simulation. In SPE89448 presented at the SPE/DOE Symposium on Improved

Oil Recovery. Society of Petroleum Engineers, 2004.

[58] E.A. Jimenez, A. Datta-Gupta, and M.J. King. Full-field streamline tracing in

complex faulted systems with non-neighbor connections. SPE Journal, 15(1):7–

17, March 2010.

[59] C.S. Kabir, N.I. Al-Khayat, and M.K. Choudhary. Lessons learned from en-

ergy models: Iraq’s south rumaila case study. SPE Reservoir Evaluation &

Engineering, 11(4):759–767, August 2008.

[60] Jeremy E. Kozdon. Numerical Methods with Reduced Grid Dependency for

Enhanced Oil Recovery. PhD thesis, Stanford University, 2009.

[61] Morten Rode Kristensen. Development of Models and Algorithms for the Study

of Reactive Porous Media Processes. PhD thesis, Technical University of Den-

mark, 2008.

[62] Archana Kumar. Hybrid simulation of the eulerian method and the streamline

method: Investigating the role of cross-product in cross-over time between the

two methods. Master’s thesis, Stanford University, 2010.

[63] M. Kumar. A cross-sectional simulation of west heidelberg in-situ combustion

project. SPE Reservoir Engineering, 6(1):46–54, February 1991.

[64] M. Kumar and A.M. Garon. A method for predicting oxygen and air fireflooding

performance. SPE Reservoir Evaluation & Engineering, 6(2):245–252, May

1991.

BIBLIOGRAPHY 210

[65] M.A. Langley. In-situ combustion experiences, husky oil operation. In Technical

Meeting/Petroleum Conference Of The South Saskatchewan Section. Petroleum

Society of Canada, 1985.

[66] A. Lapene, L.M. Castanier abd G. Debenest, M. Quintard, A.M. Kamp, and

B. Corre. Effects of water on kinetics of wet in-situ combustion. In SPE93243

presented at the SPE Western Regional Meeting. Society of Petroleum Engi-

neers, 2009.

[67] A. Lapene, D.V. Nichitae, G. Debenesta, and M. Quintard. Three-phase free-

water flash calculations using a new modified rachfordcrice equation. Fluid

Phase Equilibria, 297(1):121–128, October 2010.

[68] Alexandre Lapene. Experimental and numerical study of in-situ combustion of

heavy oil. PhD thesis, Institut de Mecanique des Fluides de Toulouse, 2010.

[69] E.C. Lau and K.E. Kisman. Attractive control features of the combustion

override split - production horizontal well process in a heavy oil reservoir. In

Annual Technical Meeting, Calgary, Alberta. Petroleum Society of Canada, Jun

1994.

[70] H. Lof, M. Gerritsen, and Marco Thiele. Parallel streamline simulation. In

SPE113543 presented at the Europec/EAGE Conference and Exhibition. Society

of Petroleum Engineers, 2008.

[71] Bradley Thomas Mallison. Streamline-Based Simulation of Two-phase, Multi-

component Flow in Porous Media. PhD thesis, Stanford University, 2004.

[72] B.T. Mallison, M.G. Gerritsen, and S.F. Matringe. Improved mappings for

streamline-based simulation. SPE Journal, 11(3):294–302, September 2006.

[73] E.J. Manrique, V.E. Muci, and M.E. Gurfinkel. Eor field experiences in carbon-

ate reservoirs in the united states. SPE Reservoir Evaluation & Engineering,

10(6):667–686, December 2007.

BIBLIOGRAPHY 211

[74] D.M. Marjerrison and M.R. Fassihi. A procedure for scaling heavy-oil combus-

tion tube results to a field model. In SPE24175 presented at the SPE/DOE

Enhanced Oil Recovery Symposium. Society of Petroleum Engineers, 1992.

[75] D.M. Marjerrison and M.R. Fassihi. Performance of morgan pressure cycling

in-situ combustion project. In SPE27793 presented at the SPE/DOE Improved

Oil Recovery Symposium. Society of Petroleum Engineers, 1994.

[76] J.C. Martin and R.E. Wegner. Numerical solution of multiphase, two-

dimensional incompressible flow using streamtube relationships. Society of

Petroleum Engineers Journal, 19(5):313–323, October 1979.

[77] M.L. Michelsen and J.M. Mollerup. Thermodynamic Models: Fundamentals

and Computational Aspects. Tie-Line Publications, Holte, Denmark, 2004.

[78] C.A. Miller. Stability of moving surfaces in fluid systems with heat and mass

transport iii. stability of displacement fronts in porous media. AIChE Journal,

21(3):474–479, May 1975.

[79] A.R. Montes, D. Gutierrez, R.G. Moore, S.A. Mehta, and M.G. Ursenbach. Is

high-pressure air injection (hpai) simply a flue-gas flood? Journal of Canadian

Petroleum Technology, 49(2):56–63, February 2010.

[80] R.G. Moore, C.J. Laureshen, M.G. Ursenbach, S.A. Mehta, and J.D.M. Bel-

grave. A canadian perspective on in situ combustion. Journal of Canadian

Petroleum Technology, 38(13), 1999.

[81] R.J. Morgan. Can horizontal wells inject new life into an old combustion pilot?

eyehill thermal project update. In Technical Meeting/Petroleum Conference Of

The South Saskatchewan Section. Petroleum Society of Canada, 1983.

[82] T.N. Nasr, G. Beaulieu, H. Golbeck, and G. Heck. Novel expanding solvent-sagd

process es-sagd. Journal of Canadian Petroleum Technology, 42(1), January

2003.

BIBLIOGRAPHY 212

[83] T.W. Nelson and J.S. McNeil. How to engineer an in-situ combustion project.

The Oil and Gas Journal, page 58, June 1961.

[84] J. Nilsson, M. Gerritsen, and R. Younis. A novel adaptive anisotropic grid

framework for efficient reservoir simulation. In SPE93243 presented at the SPE

Reservoir Simulation Symposium. Society of Petroleum Engineers, 2005.

[85] J. Nodwell, R.G. Moore, M.G. Ursenbach, C.J. Laureshen, and S.A. Mehta.

Economic considerations for the design of in-situ combustion projects. Journal

of Canadian Petroleum Technology, 39(8), August 2000.

[86] Olufolake Ogunbanwo. Uncertainty assessment on in-situ combustion simula-

tions using experimental design. Master’s thesis, Stanford University, 2011.

[87] I. Osako, M. Kumar, V. Hoang, and G. Balasubramanian. Evaluation of stream-

line simulation application to heavy oil waterflood. In SPE122922 presented at

the Latin American and Caribbean Petroleum Engineering Conference. Society

of Petroleum Engineers, 2009.

[88] Ichiro Osako. A Rigorous Compressible Streamline Formulation for Black Oil

and Compositional Simulation. PhD thesis, Texas A&M University, 2006.

[89] U. Pasarai and N. Arihara. Application of streamline method to hot waterflood-

ing simulation for heavy-oil recovery. In SPE93149 presented at the SPE Asia

Pacific Oil and Gas Conference and Exhibition. Society of Petroleum Engineers,

2005.

[90] R. Penney, R. Moosa, G. Shahin, F. Hadhrami, A. Kok, G. Engen, O. van

Ravesteijn, K. Rawnsley, and B. Kharusi. Steam injection in fractured car-

bonate reservoirs: Starting a new trend in eor. In IPTC10727 presented at

the International Petroleum Technology Conference. Society of Petroleum En-

gineers, 2005.

[91] H.J-M. Petit, P.L. Thiez, and P. Lemonnier. History matching of a heavy-

oil combustion pilot in romania. In SPE20249 presented at the SPE/DOE

Enhanced Oil Recovery Symposium. Society of Petroleum Engineers, 1990.

BIBLIOGRAPHY 213

[92] D.W. Pollock. Semianalytical computation of path lines for finite-diffference

models. Ground Water, 26(6):743–750, 1988.

[93] M. Prats. Thermal Recovery. 1986.

[94] S. Roychaudhury, N.S. Rao, S.K. Sinha, S. Sur, K.K. Gupta, A.V. Sapkal, A.K.

Jain, and J.S. Saluja. Extension of in-situ combustion process from pilot to semi-

commercial stage in heavy oil field of balol. In SPE37547 presented at Inter-

national Thermal Operations and Heavy Oil Symposium. Society of Petroleum

Engineers, 1997.

[95] P.H. Sammon. Dynamic grid refinement and amalgamation for compositional

simulation. In SPE79683 presented at the SPE Reservoir Simulation Sympo-

sium. Society of Petroleum Engineers, 2003.

[96] P.S. Sarathi. In-Situ Combustion Handbook C Principles and Practices. Na-

tional Petroleum Technology Office, U.S.D.O.E., Tulsa, OK, 1998.

[97] W.M. Schulte and A.S. de Vries. In-situ combustion in naturally fractured

heavy oil reservoirs. SPE Journal, 25(1):67–77, February 1985.

[98] J.P. Seidle and L.E. Arri. Use of conventional reservoir models for coalbed

methane simulation. In SPE21599 presented at the CIM/SPE International

Technical Meeting. Society of Petroleum Engineers, 1990.

[99] C.J. Seto, K. Jessen, and F.M. Orr Jr. Compositional streamline simulation

of field scale condensate vaporization by gas injection. In SPE79690 presented

at the SPE Reservoir Simulation Symposium. Society of Petroleum Engineers,

2003.

[100] C.J. Seto, K. Jessen, and F.M. Orr Jr. Using analytical solutions in composi-

tional streamline simulation of a field-scale co2-injection project in a conden-

sate reservoir. SPE Reservoir Evaluation & Engineering, 10(4):393–405, August

2007.

BIBLIOGRAPHY 214

[101] STARS. STARS Version 2009 User’s Guide. Computer Modeling Group Ltd.,

Calgary, Canada, 2009.

[102] S. Stokka, A. Oesthus, and J. Frangeul. Evaluation of air injection as an ior

method for the giant ekofisk chalk field. In SPE97481 presented at the SPE

International Improved Oil Recovery Conference in Asia Pacific. Society of

Petroleum Engineers, 2005.

[103] T.W. Stone and J.S. Nolen. Practical and robust isenthalpic/isothermal flashes

for thermal fluids. In SPE118893 presented at the SPE Reservoir Simulation

Symposium. Society of Petroleum Engineers, 2009.

[104] M. Thiele, R.P. Batycky, S. Pollitzer, and T. Clemens. Polymer-flood modeling

using streamlines. SPE Reservoir Evaluation & Engineering, 13(2):313–322,

April 2010.

[105] M.R. Thiele. Streamline simulation. In 8th International Forum on Reservoir

Simulation, 2005.

[106] M.R. Thiele and R.P. Batycky. Using streamline-derived injection efficiencies for

improved waterflood management. SPE Reservoir Evaluation & Engineering,

9(2):187–196, April 2006.

[107] M.R. Thiele, R.P. Batycky, and M.J. Blunt. A streamline-based 3d field-scale

compositional reservoir simulator. In SPE38889 presented at the SPE Annual

Technical Conference and Exhibition. Society of Petroleum Engineers, 1997.

[108] M.R. Todd and W.J Longstaff. The development, testing and application of a

numerical simulator for predicting miscible flood performance. Trans. AIME,

pages 874–882, 1972.

[109] S. Turek, L. Rivkind, J. Hron, and R. Glowinski. Numerical analysis of a new

time-stepping θ-scheme for incompressible flow simulations. 2005.

BIBLIOGRAPHY 215

[110] M.G. Ursenbach, R.G. Moore, and S.A. Mehta. Air injection in heavy oil reser-

voirs - a process whose time has come (again). Journal of Canadian Petroleum

Technology, 49(1):48–54, January 2010.

[111] F.K. Veedu, M. Delshad, and G.A. Pope. Scaleup methodology for chemical

flooding. In SPE135543 presented at the SPE Annual Technical Conference and

Exhibition. Society of Petroleum Engineers, 2010.

[112] G. K. Youngren. Development and application of an in-situ combustion reservoir

simulator. SPE Journal, 20(1), February 1980.

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