implicit coupling framework for multi-physics reservoir

IMPLICIT COUPLING FRAMEWORK

FOR MULTI-PHYSICS RESERVOIR SIMULATION

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

Ruslan Rin

October 2017

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

This dissertation is online at: http://purl.stanford.edu/xz567nc2513

© 2017 by Ruslan Rin. 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/xz567nc2513

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.

Hamdi Tchelepi, Primary Adviser


Pavel Tomin, Co-Adviser


Louis Durlofsky

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for 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

The growth in the complexity of the subsurface flow processes of interest and the contin-

uous increase in computational power have created the need for simulation frameworks

that can be extended to handle multi-physics problems. Having a generalized framework

reduces the development cost of incorporating additional physical mechanisms and explor-

ing new numerical formulations and solution algorithms. In addition, a general simulation

framework is necessary as we the transition from the existing paradigm, in which a small

number of specific process models are coupled in a very limited number of predefined

combinations, to a new paradigm, where many different nonlinear processes can be cou-

pled in arbitrary permutations. The flexibility of such a framework allows us to investigate

new nonlinear coupling configurations across the different physics processes and improve

our understanding of the nature of the interactions. Such a flexible simulation framework

will allow us to improve the robustness and computational efficiency of modeling coupled

systems by finding new configurations and more efficient solvers for a given physics pro-

cess. The goal of this work is the development and implementation of a general sequential

reservoir simulation framework for multi-physics problems.

The limitations associated with existing ‘general-purpose’ platforms include: (1) Lim-

ited coupling strategies for a predetermined small set of physical processes. (2) The ab-

sence of accurate and consistent comparisons between the different coupling strategies.

(3) Highly specialized linear solvers for a predefined set of physical processes. (4) Lim-

ited paths for the extension to new physics and additional complex features. In contrast to

the existing platforms, the new framework can accommodate any coupling strategy across

the different nonlinear physics modules. This flexibility is achieved using a ‘sub-problem’

tree structure and a global status table. These components are integrated with the subsets

iv

through an automatic differentiation layer (library). The exploration of different coupling

strategies facilitates the study of new multi-physics preconditioners. The modular design

with transparent subset-based interfaces greatly improves code extensibility. In addition,

we developed a unique ‘mapping operators’ concept that enables advanced coupling strate-

gies, including (parallel) domain-decomposition and multiscale approaches. We have de-

signed, developed, implemented, and tested a new GENeral Implicit Coupling framework

for multi-physics problems (GENIC) that allows for rapid prototyping and customization.

This GENIC framework is the next generation of the Automatic-Differentiation General-

Purpose Research Simulator (AD-GPRS) at Stanford. GENIC represent a flexible and effi-

cient ‘computational’ laboratory for advanced reservoir simulation that allows researchers

to try their ideas rapidly and specialize the preferred strategies efficiently.

We demonstrate the power and flexibility of the GENIC framework in modeling com-

plex multi-physics problems using several nonlinear problems that involve several mecha-

nisms of flow and transport, and we show that the GENIC framework allows for investigat-

ing the full range of possible coupling strategies. We explore the use of existing and novel

sequential coupling schemes for flow-thermal problems (i.e., multi-component multiphase

flow and transport of mass and energy). We investigate the convergence properties of three

different schemes: constant pressure, constant density, and a hybrid scheme where con-

stant pressure is used for single-phase blocks, and constant density is used for two-phase

blocks. The hybrid approach performed the best out of the three schemes tested. This

hybrid method was able to avoid the divergent, or slow converging, behavior in the single-

phase (constant density) and the two-phase (constant pressure) regions. Next, we present a

new approach for coupling flow and thermal displacement processes – modified sequential

fully implicit (m-SFI) method. We demonstrate m-SFI method for several challenging ex-

amples where standard sequential approaches fail; specifically, we show that m-SFI takes

a similar number of time steps as the full implicit method (FIM). In cases where accurate

front prediction is possible and the two-phase region is limited, such as strictly injection or

production problems, the m-SFI approach outperforms FIM. However, when most of the

reservoir has two phases, the overall performance of the m-SFI method is more expensive

than FIM due to the increased cost of the sequential iterations.

We then demonstrate the applicability of the GENIC framework for flow-mechanics

v

problems using three examples: (1) Mandel’s problem; (2) Dean’s problem; (3) an extended

SPE 10 problem. We tested two existing ‘coupled properties’: fixed-stress and fixed-strain.

The results are consistent with published results, where the fixed-strain approach is only

conditionally stable and the fixed-stress is unconditionally stable. In addition, we compare

the performance of the sequential coupling strategy with FIM for the extended SPE 10

problem. For moderate and weak coupling, the sequential approach performs better than

FIM. However, strongly coupled problems where both physics are highly nonlinear poses

remains an open challenge for sequential schemes. We conclude with a solution for a

flow-thermal-mechanics problem using three different coupling strategies: fully coupled,

sequentially coupled, and nested sequentially coupled. This is the first example - to our

knowledge - where both fully coupled and sequentially coupled approaches are applied

to this three-physics problem. When the mechanics problem is linear, and the coupling

strength is weak, the sequential coupling strategy performs faster than FIM; however, as

the coupling strength involving mechanics increases, sequential coupling strategies become

less effective.

vi

Acknowledgments

I would like to thank my advisor, Hamdi Tchelepi, for his help and constant encouragement.

Hamdi has provided me with latitude and freedom to explore ideas and direct my research.

He has taught me how to work independently and think creatively and has helped me to

grow as a researcher. More importantly, Hamdi has supported me during a hard time in

Stanford when I took an academic leave of absence, and he encouraged me to come back

to pursue my Ph.D. degree. Throughout the entire process, he has found a way to foster my

growth and development through insightful research advice, kind words, and in a multitude

of other ways, and I am deeply grateful for his generosity and unconditional support.

My second advisor, Pavel Tomin, played a key role in both my professional and personal

development. I am most grateful for the strong professional relationship and friendship we

were able to build. I thank Pavel for spending countless hours working together on the

foundation of the new research platform. His hard work and passion for research have had

a profound impact on my motivation to do the best research possible. Pavel contributed to

virtually every single aspect of this work, and it would have been impossible to accomplish

it without his advice.

I would like to thank Denis Voskov for his help and support during my M.S. research,

and serving as co-adviser in the early stages of my Ph.D. program. Denis greatly con-

tributed to my professional development and choice of research area.

Many thanks to Professor Lou Durlofsky and Professor Khalid Aziz for reading my dis-

sertation and providing valuable comments and suggestions. I also thank Professor Biondo

L. Biondi for being the chair during my defense. I would like to acknowledge Timur

Garipov for valuable discussions on geomechanics, Yang Wong for collaboration in the

development of thermal physics, Kirill Terekhov for his contribution in the idea of subsets,

vii

and Eiko Rutherford, Joanna Sun and Sandy Costa for providing administrative assistance

during my stay in the ERE department. Special thanks to Michael Connolly for providing

extremely valuable feedback on my dissertation.

My wonderful time at Stanford was enriched by colleagues and friends at Stanford,

including Sergey Klevtsov, Karine Levonyan, Yongduk Shin, Sumeet Trehan, Waqas Ali,

Rustem Zaydullin, Jincong He, Motonao Imai, Da Huo, Mehrdad Shirangi, Elnur Aliyev,

Amir Salehi, Nicola Castelletto, Oleg Volkov, Jason Chiang, Yimin Liu, and Olga Fuks.

Most importantly, I want to thank my parents, Nuriya and Mansur, for their support and

constant encouragement, and my wife, Yuliya, for her love, our son, and endless support.

viii

Contents

Abstract iv

Acknowledgments vii

1 Introduction 11.1 Multi-Physics Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Coupling Methods for Multi-Physics Problems . . . . . . . . . . . . . . . 8

1.3 Current Progress in Multi-Physics Simulator Development . . . . . . . . . 12

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 General Implicit Coupling Framework (GENIC) 162.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Multi-Component Transport . . . . . . . . . . . . . . . . . . . . . 20

2.1.2 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.3 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.4 Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Coupling Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Framework Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 Sub-Problem Tree Structure and Global Status Table . . . . . . . . 36

2.3.3 Coupling Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.4 Coupled Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.5 Mapping Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.6 Solving Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 47

ix

3 Numerical Applications 533.1 Flow and Thermal Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1.1 Example 1. Avdonin’s Problem . . . . . . . . . . . . . . . . . . . 56

3.1.2 Example 2. Injection Flash Front . . . . . . . . . . . . . . . . . . 57

3.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2 Modified SFI for Flow-Thermal Problems . . . . . . . . . . . . . . . . . . 66

3.2.1 Example 3. 3D m-SFI Case . . . . . . . . . . . . . . . . . . . . . 67

3.2.2 Example 4. m-SFI SPE 10 . . . . . . . . . . . . . . . . . . . . . . 72

3.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3 Flow and Mechanics Coupling . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3.1 Example 5. Mandel Problem . . . . . . . . . . . . . . . . . . . . . 80

3.3.2 Example 6. Depletion of a Box-Shaped Reservoir (Deans’ Problem) 83

3.3.3 Example 7. Water Flooding Problem . . . . . . . . . . . . . . . . . 86

3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.4 Coupled Flow, Thermal, and Mechanics . . . . . . . . . . . . . . . . . . . 93

3.4.1 Example 8. Cold Water Injection . . . . . . . . . . . . . . . . . . . 93

3.4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4 Future Extensions of GENIC 984.1 Automatic Coupled Properties . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Multi-Scale Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Conclusions and Future Work 1075.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . 109

Nomenclature 111

Bibliography 115

A Programming Model of AD-GPRS 133A.1 The Structure of AD-GPRS . . . . . . . . . . . . . . . . . . . . . . . . . . 134

x

A.2 Flow Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

B Two-Phase Prototype 143

xi

List of Tables

2.1 Example of physics modules (sub-problems), their hypothetical subset prop-

erties and corresponding unknowns. . . . . . . . . . . . . . . . . . . . . . 30

2.2 Example of a variable set for three-phase black-oil problem expanded by

inactive status. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 Parameters for Avdonin’s problem. . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Parameters for Injection Flash Front problem. . . . . . . . . . . . . . . . . 59

3.3 Single-phase injection flash front. Comparison between sequential cou-

pling schemes, such as Constant Pressure (CP), Constant Density (CD),

Hybrid (H) and fully coupled method (FIM). . . . . . . . . . . . . . . . . . 62

3.4 Two-phase injection flash front. Comparison between sequential coupling

schemes, such as Constant Density (CD), Hybrid (H) and fully coupled

method (FIM). Note: Constant Pressure does not converge for two-phase

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Parameters for 3D model for m-SFI study. . . . . . . . . . . . . . . . . . . 68

3.6 m-SFI 3D case for 10× 20× 18 blocks. Comparison between different

coupling schemes: FIM, SEQ, m-SFI and MSFI-N. . . . . . . . . . . . . . 71

3.7 m-SFI 3D case for low grid resolution 4×5×6 (L) and finer grid resolution

20× 40× 36 (H). Comparison between different coupling schemes: FIM,

SEQ, m-SFI and MSFI-N. . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.8 Parameters for SPE 10 model for m-SFI study. . . . . . . . . . . . . . . . . 73

3.9 m-SFI SPE 10 upscaled case with top four layers for 30× 110× 2 grid.

Comparison between different coupling schemes: FIM, m-SFI and MSFI-N. 74

xii

3.10 m-SFI SPE 10 case for very coarse grid resolution 15×55×1 (L) and orig-

inal grid resolution (top four layers) 60×220×4 (H). Comparison between

different coupling schemes: FIM, m-SFI and MSFI-N. . . . . . . . . . . . 76

3.11 Rock and fluid properties used for Mandel’s problem. . . . . . . . . . . . . 81

3.12 Mandel’s problem. Convergence between fixed-strain and fixed-stress meth-

ods, and FIM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.13 Rock properties for Dean’s model. . . . . . . . . . . . . . . . . . . . . . . 84

3.14 Number of cells and total degrees of freedom (DOF) for the simulation cases. 87

3.15 Normalized simulation time for Water Flooding Problem. . . . . . . . . . . 89

3.16 Main parameters for Cold Water Injection problem. . . . . . . . . . . . . . 94

3.17 Cold Water Injection case. Comparison between FIM and SEQ schemes . . 95

A.1 Sub-problems, subsets and properties. . . . . . . . . . . . . . . . . . . . . 138

xiii

List of Figures

2.1 Key principles of the GENIC framework. . . . . . . . . . . . . . . . . . . 19

2.2 Schematic representation of fully implicit coupling. . . . . . . . . . . . . . 26

2.3 Schematic representation of mixed implicit coupling. . . . . . . . . . . . . 26

2.4 Example of possible coupling strategies for a flow-thermal-mechanics prob-

lem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Example of a flow sub-problem made up of two gridblock with subsets and

corresponding unknowns. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Example of mixing formulations between flow and thermal sub-problems. . 33

2.7 Activation logic of subsets for different space discretization schemes for a

flow sub-problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.8 Example of flow-thermal-mechanics problem. . . . . . . . . . . . . . . . . 34

2.9 Subset applications for domain decomposition and multiscale. . . . . . . . 35

2.10 Example of extended fixed-stress split strategy for a flow-thermal-mechanics

problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.11 Subset and subproblem activation state . . . . . . . . . . . . . . . . . . . . 37

2.12 Subset activation logic for different subsets. . . . . . . . . . . . . . . . . . 38

2.13 Example 1 for sub-problem tree and corresponding global status table. . . . 40



2.16 Example of a coupled property. . . . . . . . . . . . . . . . . . . . . . . . . 43

2.17 Modified sequential implicit (m-SFI) approach for a flow-thermal problem. 45

2.18 Schematic representation of m-SFI approach. . . . . . . . . . . . . . . . . 46

2.19 Extension of global status table concept for mapping operators. . . . . . . . 47

xiv

2.20 Example of global status table for m-SFI coupling strategy. . . . . . . . . . 47

3.1 Schematic representation of Avdonin’s problem (uniform linear spacing). . 57

3.2 Avdonin’s problem. Comparison of the analytical and numerical solutions

of the temperature distribution and the temperature evolution. . . . . . . . . 58

3.3 Avdonin’s problem. Sequentially implicit method convergence profile after

the first time step ∆t = 0.001 days. . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Schematic representation of Injection Flash Front problem (uniform radial

spacing). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Single-phase injection flash front. Pressure and temperature distribution

after 100 days (50 grid cells). . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Single-phase injection flash front. Pressure and temperature profile at x =

167 m (50 grid cells). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.7 Grid refinement for single-phase radial one-dimensional injection scenario.

Pressure and temperature distribution after 100 days. . . . . . . . . . . . . 61

3.8 Grid refinement for single-phase radial one-dimensional injection scenario.

Pressure and temperature profile at x = 167 m. . . . . . . . . . . . . . . . . 61

3.9 Two-phase radial one-dimensional injection scenario. Pressure and tem-

perature distribution after 100 days (50 grid cells). . . . . . . . . . . . . . . 62

3.10 Two-phase radial one-dimensional injection scenario. Pressure and tem-

perature profile at x = 167 m (50 grid cells). . . . . . . . . . . . . . . . . . 63

3.11 Grid refinement for two-phase radial one-dimensional injection scenario.

Pressure and temperature distribution after 100 days. . . . . . . . . . . . . 63

3.12 Grid refinement for two-phase radial one-dimensional injection scenario.

Pressure and temperature profile at x = 167 m. . . . . . . . . . . . . . . . . 64

3.13 Schematic of the three-dimensional model. . . . . . . . . . . . . . . . . . . 68

3.14 3D m-SFI case. Pressure and temperature profile for 20× 40× 36 grid at

production well block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.15 3D m-SFI case. Pressure (a) and temperature (b) distribution for 20×40×36 grid after 10 years of production at 16th layer. . . . . . . . . . . . . . . 70

xv

3.16 3D m-SFI case. Relative error between FIM and MSFI-N approach for

pressure (a) and temperature (b) for 20×40×36 grid after 10 years at 16th

layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.17 Grid refinement for 3D m-SFI case. Pressure and temperature profile at

production well block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.18 Schematic of the SPE 10 setup: permeability (top) and porosity (bottom)

maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.19 SPE 10 m-SFI case. Pressure, temperature, and water saturation profile for

60×220×4 grid at central block of top layer. . . . . . . . . . . . . . . . . 75

3.20 SPE 10 m-SFI case. Pressure (left) and temperature (right) distribution for

60×220×4 grid after 100 days at top layer. . . . . . . . . . . . . . . . . . 75

3.21 SPE 10 m-SFI case. Relative error between FIM and MSFI-N approach

for pressure (left) and temperature (right) distribution for 60×220×4 grid

after 100 days at top layer. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.22 Schematic representation of different operator-splitting strategies for flow-

mechanics problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.23 Sketch of the setup for Mandel’s problem. . . . . . . . . . . . . . . . . . . 80

3.24 Mandel’s problem. Comparison of the analytical and the numerical solu-

tions for displacement and pressure. . . . . . . . . . . . . . . . . . . . . . 83

3.25 Mandel’s problem. Sequentially implicit method convergence profile after

first time step ∆t = 0.001 days. . . . . . . . . . . . . . . . . . . . . . . . . 84

3.26 Dean’s model geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.27 Dean’s problem solution comparison. . . . . . . . . . . . . . . . . . . . . 86

3.28 Water Flooding Problem. Properties distribution. . . . . . . . . . . . . . . 87

3.29 Water Flooding Problem. Comparison of FIM (solid color) and SEQ (dashed)

simulation time for single-phase setup. . . . . . . . . . . . . . . . . . . . . 89


simulation time for two-phase setup. . . . . . . . . . . . . . . . . . . . . . 90


simulation time for the two-phase plastic setup. . . . . . . . . . . . . . . . 91

3.32 Cold Water Injection problem description. . . . . . . . . . . . . . . . . . . 93

xvi

3.33 Cold Water Injection. Sequentially implicit method convergence profile

after first time step ∆t = 0.001 days. . . . . . . . . . . . . . . . . . . . . . 95

3.34 Saturation and temperature distributions in the middle of the reservoir at the

end of simulation for FIM approach (left) and difference in them between

FIM and SEQ simulations (right). . . . . . . . . . . . . . . . . . . . . . . 96

3.35 Oil production and water injection rates. FIM and SEQ approaches produce

nearly identical results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1 Schematic representation of the solution for sequential mechanics-thermal

coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 Example of update for coupled properties. . . . . . . . . . . . . . . . . . . 99

4.3 A dependency graph for a flow-thermal-mechanics problem. . . . . . . . . 100

4.4 Scheme of the interaction between simulator and the ADETL library in the

GENIC framework. Both mappers and subsets are part of ADETL. . . . . . 101

4.5 Differences between regular and parallel mapping operators. . . . . . . . . 102

4.6 Parallel implementation through parallel mapping operators for a flow-

thermal problem for n computation nodes. . . . . . . . . . . . . . . . . . . 103

4.7 Schematic representation of multi-scale sub-problems and corresponding

sub-domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.8 Interaction between multi-scale sub-problems and their coupled properties

- basis functions ϕi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

A.1 Overall structure of the entire simulator. . . . . . . . . . . . . . . . . . . . 134

A.2 Structure of SubProblemTree class. . . . . . . . . . . . . . . . . . . . . . . 135

A.3 Structure of SubProblem class. . . . . . . . . . . . . . . . . . . . . . . . . 136

A.4 Class diagram for subsets. . . . . . . . . . . . . . . . . . . . . . . . . . . 139

A.5 Class diagram for formulations. . . . . . . . . . . . . . . . . . . . . . . . 140

A.6 Class diagram for LinearSystemBase. . . . . . . . . . . . . . . . . . . . . 141

B.1 Class diagram for subproblems and their interaction with subsets in Two-

Phase Prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

B.2 Class diagram for subsets in Two-Phase Prototype. . . . . . . . . . . . . . 144

xvii

Chapter 1

Introduction

Reservoir simulation is a primary tool for planning and managing subsurface flow processes

including oil/gas recovery, subsurface water resources, and sequestration of carbon diox-

ide. Reservoir simulation is the art and science of developing numerical techniques to solve

the equations that describe mass and heat flow in subsurface porous geologic formations

and implementing these numerical methods in computer software (Aziz and Settari, 1979).

In the last three decades, there has been a strong interest across academia, industry, and

the national laboratories to develop ‘general-purpose’ computational tools and frameworks

to simulate various dynamic processes, and Reservoir Simulation technology is a prime

example of such efforts. In the context of Reservoir Simulation, ‘general-purpose’ refers to

the ability to apply a software built using a unified computer code capable of simulating dif-

ferent subsurface flow processes (e.g., water flooding, steam injection, miscible/CO2 gas

injection, chemical flooding). Developing a general-purpose reservoir simulator is chal-

lenging because the requirements on robustness and efficiency are extreme. That is, the

reservoir simulator is expected to model different recovery processes of practical interest,

while maintaining the highest standards of robustness and efficiency for the particular pro-

cess being modeled at a given time. In a research setting, such as the Reservoir Simulation

Research group at Stanford (SUPRI-B), an additional critical requirement is for the sim-

ulation framework to be modular and extensible; that is, the framework must serve as an

effective research platform whereby multiple researchers work on various aspects includ-

ing adding new physics modeling capabilities, developing advanced numerical methods

1

CHAPTER 1. INTRODUCTION 2

and solution schemes, and using the simulator as an integral part of investigating and un-

derstanding the complex interactions associated with subsurface flow processes.

In the SUPRI-B group, significant efforts have been invested into developing a flexible

research simulation platform. The efforts date back to the 1980s. Aziz and Wong (1989)

developed a mathematical framework that accommodates different formulations to describe

thermal-compositional processes at the so-called ‘Darcy’ scale. Wong (1988) implemented

and compared the most important black-oil and compositional formulations that were avail-

able in the literature at the time in a unified simulation code base. The work by Wong and

Aziz provided important insights about the choice of variables and different formulations,

especially for compositional problems. Their effort was a significant step toward realizing

a general-purpose reservoir simulator. Several efforts led by Aziz and his students were

invested. Important examples include the graduate work of Verma (1996), Nogaret (1996),

and Byer (2000).

The first truly general-purpose research reservoir simulator was GPRS, which was de-

veloped from scratch by Hui Cao as a part of his Ph.D. work (Cao, 2002). Since then,

GPRS has evolved quite significantly; nevertheless, the connection back to Cao’s General-

Purpose Research Simulator (GPRS) is strong. At the time, the primary objective of devel-

oping the GPRS simulator was to allow for flexible nonlinear compositional formulations

(e.g., natural-variables and mass/molar variables), so that researchers can compare existing

methods and propose new and improved models. GPRS is distinguished from the previ-

ous efforts in its extensible modular design and object-oriented computer code written in

C++. The code design of GPRS reflected the physical components associated with oil-field

development. Each field was split into several spatial domains, and each domain includes

one reservoir and all of the wells that are in that reservoir. In addition, each domain also

had an ‘equation selector’ (module to select a subset of the governing equations) and a

linear solver. In GPRS, the Jacobian matrices were calculated separately for the reservoir

part and for the wells part. The equation selector was used to recast the Jacobian using the

desired variables. In addition, one was able to specify different levels of implicitness in

the treatment of the variables (Cao, 2002). The first version of GPRS included black-oil

and isothermal compositional models, unstructured grids and Multi-Point Flux Approxi-

mation (MPFA) discretization schemes. Cao proposed a general formulation approach in


order to test and validate different choices for the variable sets used to describe composi-

tional flow. He integrated different formulations into the simulation framework, including

IMPES (implicit pressure and explicit saturations and mole fractions), IMPSAT (implicit

pressure and saturations and explicit mole fractions), AIM (Adaptive Implicit Method), and

FIM (Fully Implicit Method). Note that GPRS had a connection-based design that allowed

for the use of structured and unstructured grids (Lim, 1995).

Jiang (2007) made significant contributions to GPRS in terms of architectural design,

extensibility, computational efficiency, and new advanced well modeling capabilities. He

designed and implemented a new framework, in which the facilities model was treated

as a separate component and promoted to the same level as the reservoir component.

The framework enabled the combination of any available facilities object (e.g., standard

wells, multi-segment wells, and well groups). Jiang proposed a general discrete wellbore

model that accommodates complex (surface) network topology with general branching,

loops, and multiple exits. Jiang extended the GPRS linear solver capability with novel

Multi-Level Block Sparse (MLBS) data storage of the matrices. One of the lasting con-

tributions of Jiang is the advanced multi-stage linear solution framework, including the

two-stage CPR (Constrained Pressure Residual) preconditioning approach, which was ex-

tended to support general wellbore models. His advanced linear solvers with multi-stage

preconditioners made it possible to simulate complex compositional displacements using

highly detailed models with unstructured grids and complex multi-segment wells (Jiang and

Tchelepi, 2009). Several students and researchers in SUPRI-B expanded the capabilities of

GPRS. Notable efforts include reactive transport Fan (2010), phase-equilibrium computa-

tions Pan and Tchelepi (2010, 2011), and tie-line based compositional simulation Voskov

and Tchelepi (2009a,b).

The next highly significant development was due to Younis (Younis, 2011). He de-

veloped and implemented the Automatically Differentiable Expression Templates Library

(ADETL). ADETL provides generic representations of variables and discretized expres-

sions on a graph and automatically computes the sparse analytical Jacobian. Unlike existing

AD packages at the time (Shiriaev and Griewank, 1996; Griewank et al., 1996; Bendtsen

and Stauning, 1996; Bischof et al., 2008), the computational performance of ADETL was

demonstrated to be reasonably on par with hand-coded differentiation routines. In addition,


ADETL provides the capability to define a ‘variables’ formulation and performs variable

switching, ordering, and alignment behind the scenes. Using the library, large-scale numer-

ical simulations can be performed rapidly by simply writing the discrete residual equations

without the need for hand differentiation and hand-crafted performance tuning. The Ph.D.

work of Younis led to the first automatic differentiation platform for GPRS, which we call

AD-GPRS (Younis, 2011).

Zhou (2012) made significant contributions to the development of AD-GPRS. He im-

proved the efficiency of the ADETL library with high-performance memory allocation and

data storage, block-sparse operators, and interfaces for variable switching and ordering

(Zhou, 2009). In addition, Zhou integrated advanced well modeling capabilities, the MLBS

data structure, and the BILU and CPR preconditioners into AD-GPRS. Zhou (2012) also

developed a unified code-base for both Two-Point Flux Approximation (TPFA) and MPFA

in space, and for any combination of FIM, AIM, IMPES, and IMPSAT in time. This code

unification designed and implemented by Zhou (Zhou, 2012) differentiates AD-GPRS from

GPRS where each functionality was often implemented as a new ‘code branch’. His con-

tributions allowed for a smooth transition from GPRS to AD-GPRS as the default platform

for reservoir simulation research at Stanford.

AD-GPRS was initially developed with focus on isothermal compositional displace-

ment processes with advanced capabilities, such as a variety of nonlinear formulations

(Voskov and Tchelepi, 2012), multi-stage linear solvers with powerful preconditioners

Zhou et al. (2013), complex multi-segment wells (Zhou, 2012), optimization packages

(Voskov and Volkov, 2013), nonlinear finite-volume approximations (Terekhov et al., 2017),

and flexible restart capabilities (Iskhakov, 2013). There has been significant growth in the

complexity of simulation models of practical interest, such as unconventional gas produc-

tion, thermal Enhanced Oil Recovery (EOR), enhanced geothermal systems, and Steam

Assisted Gravity Drainage (SAGD). These problems are highly nonlinear with complex

interactions of a wide range of physical phenomena that operate at different spatial and

temporal scales. In the last few years, AD-GPRS has been extended to support thermal-

compositional, EOS-based multiphase flow and transport (Zaydullin et al., 2014), chemical

reactions (Farshidi, 2016), geothermal simulation (Wong et al., 2017), and coupled flow

and geomechanics (Garipov et al., 2015, 2016a).


AD-GPRS has evolved into a powerful computational platform for modeling nonlinear

multi-component multiphase flow processes in heterogeneous reservoirs. Nevertheless, the

addition of new physical mechanisms into the AD-GPRS framework has proved to be very

challenging. For example, the incorporation of the geomechanics module required a full

understanding of a wide range of unrelated capabilities in the simulator, including details

of the Jacobian-matrix construction for the flow and thermal parts. Moreover, the exten-

sion of a particular physics module to additional constitutive relations and formulations

had become quite complex despite using the automatic-differentiation layer. This made the

further extension to new physics quite difficult and created the need for a general simula-

tion framework applicable to different and new physical processes and numerical methods,

including multiscale formulations (Jenny et al., 2003, 2006; Zhou, 2010). The availability

of a generalized framework can decrease the development cost of incorporating additional

physical mechanics and exploring new formulations and solution methods. In addition, a

general simulation framework would aid in the transition from the existing paradigm, in

which a small number of process models are coupled together in a limited number of pre-

defined combinations, to the new paradigm, where many different processes are coupled

in arbitrary permutations. The flexibility of such a framework allows us to investigate new

nonlinear coupling configurations across the different physical processes and improve our

understanding of the coupled nature of the processes. This would enable us to improve the

performance and stability of coupled systems by finding new coupling solution schemes

and more efficient nonlinear and linear solvers for a given physics. The goal of this work is

the development and implementation of such a multi-physics reservoir simulation research

platform.

1.1 Multi-Physics Problems

In reservoir engineering, many subsurface processes of interest are associated with En-

hanced Oil Recovery (EOR). Examples include gas injection, chemical flooding, and ther-

mal recovery methods. These displacement processes are characterized by complex inter-

actions across the multiple components that make up the fluid-rock system. The interac-

tions often lead to the presence of multiple fluid phases with inter-phase mass transfer as


a function of space and time. The multi-component transport and the multiphase flow are

coupled in a highly nonlinear manner. Moreover, the changes in the pressure and temper-

ature distributions inside the subsurface formation often lead to changes in the mechanical

stresses, which impact the solid structure of the porous medium and affect the distribu-

tion of fluids. The complex interactions among different physical mechanisms pose strict

requirements on the numerical methods and solution strategies. Together with numerical

stability, nonlinear convergence, and truncation errors, the solution strategy should account

for different types of coupling; this demands a robust multi-physics simulator.

One example of a complex subsurface processes is Steam Assisted Gravity Drainage

(SAGD) – a widely used recovery technique for heavy oil and bitumen reservoirs (Butler,

1985; Li and Chalaturnyk, 2007). The process involves multi-physics phenomena, such

as fluid thermal expansion, viscosity reduction, rock thermal dilation, and complex phase

behavior. Typical SAGD operations involve a pair of horizontal wells located one a few me-

ters above the other (Butler, 1985). Steam is injected into the upper well to form a steam

chamber and to mobilize the reservoir oil. The mobilized oil drains to the production well

that lies below. Significant mechanical stresses associated with this process increase the

risk of fracturing the reservoir or the cap-rock. In the case of hydrate-bearing sediments,

when methane is produced from the hydrate reservoirs, flow is tightly coupled to the me-

chanics because the changes of pressure and temperature can have a significant impact on

the effective stresses in the porous medium (Rutqvist and Moridis, 2009; Moridis et al.,

2013). Moreover, changes in the stress and temperature fields influence hydrate formation

or dissociation. This can cause lead to wellbore instabilities or possible methane emission

to the surface due to geological failure (Kim et al., 2011).

In environmental engineering, land subsidence due to the exploitation of subsurface re-

sources, its damage to surface infrastructure, and its impact on our environment have been

the subject of extensive studies in the subsurface modeling of fluid flow and geomechanics

(White and Morton, 1997; Hermansen et al., 1997; Hansen et al., 1995; Settari et al., 2008).

Compaction of unconsolidated aquifer systems that accompanies excessive groundwater

pumping is by far the single largest cause of subsidence in the United States (Gai, 2004).

The increasing development of land and water resources threatens to exacerbate the existing

subsidence problem while initiating new ones. Surface subsidence may also be induced by


hydrocarbon withdrawal from weak formations and chalk reservoirs. While reservoir com-

paction itself has been widely recognized as an additional driving mechanism for increasing

oil and gas recovery, its side effects are undesirable. The most obvious one is surface, or

sea-floor settlement, which may create environmental problems and cause damage to oil-

field structures and seabed pipelines (Gai, 2004; Chaussard et al., 2013). One well-known

case is the excessive groundwater removal that has caused Venice to start sinking (Lubick,

2002). The same problem has occurred in other cities, such as Tokyo, Shanghai, Houston,

and New Orleans (Nicholls, 1995; Chaussard et al., 2013), where the simulation of ground-

water usage enabled effective environmental regulation to reduce groundwater use. The

effective treatments for these problems can be accurately assessed with flow-mechanics

simulations.

In nuclear waste management, despite more than thirty years of active research and

development (Ames and Rai, 1978; Abrajano and Bates, 1987; Wronkiewicz et al., 1996;

Gaucher et al., 2004), there is no operating geologic repository for highly radioactive waste.

This is due to the long times (thousands of years) for which predictions need to be per-

formed, and where the overall scenario cannot be verified entirely, as it is deduced from

many individual observations made by different laboratories or in the field. These issues

can be resolved by use of numerical simulations which allow modeling over a wide range

of parameters and longer time scale. In nuclear waste disposal, radioactive materials gen-

erate heat, which can perturb an initial stable condition, causing fluid flow or deformation.

Modeling of this process requires the use of a coupled flow-thermal-mechanics and chem-

ical reactions simulator (Rutqvist and Tsang, 2003; Grambow et al., 2014). Additionally,

there is great interest in the development of ceramics materials that can withstand radiation

damage over millennia without losing their properties; the ceramic crystalline structure can

be found from numerical simulations when a large number of scales can be considered (De-

vanathan et al., 2010). Recent studies (Devanathan et al., 2010; Verma et al., 2015) reveal a

demand for additional scales (quantum and atomic) and physics (quantum mechanical cal-

culations, classical molecular dynamics simulations, kinetic Monte Carlo and phase field

methods) for a correct prediction. This further increased interest in a multi-physics simu-

lator.


In earthquake engineering, there is building legislation that attempts to ensure that struc-

tures would suffer no damage in moderate events and would not collapse in major events.

However, this legislation does not guarantee ‘earthquake-proof’ buildings. In addition to

damage caused by severe ground shaking, damage to structures during earthquakes can

be due to the soil response (such as ground cracking, liquefaction, lateral spreading, and

landslides) and to the effects of adjacent structures, either through direct contact or through

structure-soil-structure interaction (Elnashai and Di Sarno, 2008). Numerical simulation

of soil-structure allows for the design of structures that meet specific performance criteria

set by the structure’s owners and regulators. This improves the prediction of the structural

response and enables the design of earthquake-resistant buildings. For most soil responses,

a mechanical simulator is enough (McKenna, 2011). However, in the case of liquefaction

(a state in which fluid pressure in a porous medium becomes greater than the forces hold-

ing the solid together, that makes solid behave like fluid), a complex flow-elasto-plastic

simulator is needed (Carey et al., 2016).

Each of the above-mentioned fields has its set of simulators; however, it is still challeng-

ing to extend a software for additional physics within the same field or employ similar ideas

with slightly different governing equations towards another field. We believe it is impor-

tant to develop a flexible multi-physics simulation platform capable of accurate modeling

of complex subsurface processes.

1.2 Coupling Methods for Multi-Physics Problems

In the reservoir engineering community, the interest in different coupling strategies emerged

at early stages of numerical simulations. While the fully coupled approach – the Fully Im-

plicit Method (FIM) - ensures unconditionally stable numerical solutions for well-posed

problems and is a straightforward method of solving systems of equations, it is computa-

tionally expensive and cannot effectively handle problems of practical interest when the

computational resources are limited (Aziz and Settari, 1979). This has led to the develop-

ment of sequential methods for coupled flow (pressure) and transport (saturation) problems.

The IMplicit Pressure, Explicit Saturation (IMPES) was developed by Sheldon et al.

(1959), Stone and Garder (1961), and Danilov et al. (1968). The main idea in IMPES is to


split the coupled system into a pressure equation, which is solved implicitly in time, and

saturation equations that are advanced using explicit time approximation. Because of the

explicit treatment of saturation, IMPES is only conditionally stable and the time step is

limited by a CFL-condition (Coats, 2000, 2003). IMPES is often considered as a special

form of the FIM method with only one implicit variable, since it can be obtained from

FIM by applying algebraic reduction (Aziz et al., 2005). While IMPES is computationally

efficient for a single time step, the total cost of an IMPES simulation can easily exceed that

of FIM given the prohibitively small time steps that are required in real-field applications.

Nevertheless, IMPES remains a popular method in the community to solve large-scale

problems, and it is still used today.

The IMplicit Pressure and SATuration(s) (IMPSAT) sequential method was developed

to overcome the time step limitations of the IMPES method by incorporating implicit treat-

ment of saturations without solving simultaneously for pressure and saturations. IMPSAT

was proposed by MacDonald and Coats (1970) and later reported by Spillete et al. (1973),

Coats et al. (1974) and Coats (1976). The method consists of two steps. The first step is

to obtain an implicit pressure solution in exactly the same way as for the IMPES method;

however, the saturations are updated by solving the transport problems with implicit time

stepping (Watts, 1986). IMPSAT is an operator splitting method, in which the flow problem

(pressure field) and the transport problem (saturations) are decoupled and solved sequen-

tially. However, for compressible flow, mass is generally not conserved for one of the fluid

phases. As a result, the mass-balance errors are proportional to the areal variation of ρn/ρw

(Aziz and Settari, 1979; Coats et al., 1974) and can be significant for large time steps.

With growing computational power, FIM became the main approach for solving cou-

pled multiphase flow and transport problems and the development of sequential methods

for such problems slowed down significantly. The Sequential Fully Implicit was developed

and integrated with the Multiscale Finite-Volume method (MsFV) Jenny et al. (2006). In

the SFI scheme, each time step consists of an outer loop and two inner loops. The first

inner loop deals with the pressure; the second inner loop deals with the saturations; the

outer loop deals with the iterating on all the variables. If only a single outer-loop iteration

is performed, we obtain the IMPSAT method. If a single iteration of both the outer and

inner Newton loops is performed, we obtain the IMPES method (Jenny et al., 2005). The


SFI method for MsFV was tested for large-scale (millions of grid blocks) multiphase flow

problems in highly heterogeneous domains, and the coupling scheme has been shown to be

robust and computationally efficient, even for large time steps. However, the method re-

quires a multiscale framework and has not been used extensively in traditional (i.e., single

scale) compositional simulators.

While FIM is a popular coupling method for the pressure-transport problem, the situa-

tion is different for flow-mechanics problems. The coupling between the flow and mechan-

ics codes has been the subject of intense activity in the last decade. The coupling schemes

can be split into two broad categories (Settari and Walters, 2001): iteratively coupled and

fully coupled. Similar to the pressure-transport problem, FIM for flow-mechanics ensures

unconditionally stable numerical solutions for well-posed problems (Jha and Juanes, 2007).

However, the fully coupled approach is computationally expensive and requires the devel-

opment of a unified flow-geomechanics simulation framework (Hu et al., 2013; Yang et al.,

2014). In addition, the simulation domain for the mechanical problem is usually larger

than the domain for the flow problem, since it includes the overburden, base, and sides.

Consequently, the computational cost of the mechanical problem at each simulation time

step can be quite high. Another disadvantage of the fully coupled approach consists of the

fact that the resulting linear system associated with the FIM Jacobian has a mixed character

of near-elliptic and hyperbolic components. The fully coupled approach for flow and me-

chanics requires complex preconditioning procedures for the efficient solution of the linear

system (White and Borja, 2011; White et al., 2016; Castelletto et al., 2015; Klevtsov et al.,

2016).

The development and application of sequential-implicit solution schemes are motivated

by a simple idea to avoid the solution of a complex full system with mixed character. In

sequential coupling, the system of governing equations is divided into several physical sub-

problems. Each subproblem can be solved by a different method, which includes dedicated

nonlinear and linear solvers. In modern reservoir simulation, different types of sequential

coupling between the flow and geomechanics are used.

Initially, the coupled solution of flow and geomechanics were implemented in explicitly

coupled fashion (Park, 1983; Zienkiewicz et al., 1988), where each physical process was

solved separately with a partial update of the solution between different processes. In this


approach, the convergence of the full system of the governing equations often was not

checked. While this method can be efficient and accurate in some situations, it does not

guarantee the correctness of the solution. Later, a loosely coupled scheme was proposed

for the solution of flow and geomechanics problems (Bevillon and Masson, 2000; Settari

and Walters, 2001). This approach is similar to the explicitly coupled scheme with an

occasional solution of the coupling between different physics. The loosely coupled scheme

may achieve a very good performance (Minkoff et al., 2003b; Samier et al., 2008) with a

reasonable match to the results of a fully coupled solution; however, it is not stable and it

is not clear when the full system should be resolved.

Several sequential-implicit approaches have been proposed (Prevost, 1997; Settari and

Walters, 2001; Thomas et al., 2003; Tran et al., 2004; Gai, 2004; Wheeler and Gai, 2007).

Sequential-implicit formulations have several advantages over FIM, such as the use of sep-

arate modules for each physical process, use of specialized linear solvers and precondi-

tioners for different physics components, and separate treatment of the different compu-

tational domains (Mainguy and Longuemare, 2002). Moreover, for flow-mechanics prob-

lems, such an approach can provide accurate and unconditionally stable solutions (Armero

and Simo, 1992; Kim et al., 2013; Mikelic and Wheeler, 2013). Due to an increased in-

terest in the Reservoir Simulation community, several implementations of the sequential-

implicit method for a flow-thermal-geomechanics simulation have been developed. How-

ever, existing reservoir simulators capable of solving challenging problems are limited to

a specific coupling strategy, namely, an iterative sequential approach. Examples include:

the Subsurface Transport Over Multiple Phases (STOMP) (White and Oostrom, 2006),

TOUGHT+FLAC (Rutqvist, 2011), IPARS-JAS3D (Minkoff et al., 2003a), and Open-

GeoSys (Kolditz et al., 2012). However, without reference solutions of the fully coupled

physics, it is often not clear whether the obtained solutions are accurate, let alone having the

detailed assessment of the computational performance. Issues related to the accuracy of the

computed solutions can be addressed when a fully coupled multi-physics simulator is used;

examples include DuMuX (Flemisch et al., 2011) and MRST (Lie, 2014; Keilegavlen and

Nordbotten, 2017). Unfortunately, published results are reported for small-scale problems

and for relatively simple multi-physics interactions. To our knowledge, none of the avail-

able simulators are capable of running both fully-implicit and sequential-implicit methods


within the same code platform for challenging nonlinear problems, such as SAGD prob-

lems. There is also a lack of systematic studies of the robustness and efficiency of different

nonlinear coupling strategies. An available comparison (Markert et al., 2010) performed

in PANDAS (Ehlers and Ellsiepen, 1998) is limited to simplified flow-mechanics prob-

lems without nonlinear effects. Moreover, this study employs a finite-element methodol-

ogy, while large-scale compositional simulations are heavily reliant on conservative finite-

volume approximations. This motivates the development of a framework that allows for

accurate comparison between coupling strategies.

1.3 Current Progress in Multi-Physics Simulator Devel-

opment

We can categorize numerical simulators into three groups. The first group consists of

highly specialized, fast, parallel, state-of-the-art simulators for a specific set of problems

where no more than two physics modules are solved simultaneously. Examples include

compositional-thermal and flow-mechanics. All commercial simulators belong to this group.

The second group is a set of fully-coupled multi-physics simulators. These simulators are

either non-extendable for additional physics, or have limited scalability. There are ongoing

efforts to overcome these limitations by developing efficient multi-physics precondition-

ers, such as in the case for compositional-mechanics (White and Borja, 2011; White et al.,

2016; Castelletto et al., 2015; Klevtsov et al., 2016). The third group is represented by

sequential iterative multi-physics simulators.The sequential coupling is often rigid, does

not permit extension to new physics, and the order of the involved physics modules and

coupling terms are fixed.

In summary, available multi-physics simulators have several issues. First, they exhibit

poor scalability for fully coupled approach caused by a limited number of efficient multi-

physics preconditioners. Second, the design of coupling strategies is severely limited due

to the extreme complexity of simulator development. Third, to the best our knowledge,

an accurate and consistent comparison of different coupling strategies for multi-physics

problems has not been performed. Finally, available simulators in reservoir engineering are


not extendable for additional physics and processes, discretization domains, and coupling

terms.

This work is not the first attempt to create a flexible multi-physics simulator. Devel-

oping process-rich simulators with great fidelity is a common challenge across many dis-

ciplines. In most cases, the focus is on the richness of the modeled processes and not on

providing wide flexibility in the nonlinear coupling strategy across the different physics

modules. Some of the strategies used to manage complexity in a multi-physics simula-

tor include: data managers (Larson et al., 2005; Slattery et al., 2013), dependency graphs

(Berzins et al., 2011), and process couplers (Redler et al., 2010; Craig et al., 2012). The

Data Manager is a software that uses database solutions to enable users to manage struc-

tures of simulation and process data across the complete product lifecycle. Data managers

are common in simulations that involve multiple domains, such as atmosphere-ocean-land

climate models (Redler et al., 2010), or in engineering applications with solid and fluid

domains (Larson et al., 2005). Data managers help achieve solutions of process-rich sim-

ulation, but they do not focus on the scalability of multi-physics problems. Dependency

graphs is a system representing dependencies of several objects to each other and started

getting attention in extreme scale computing, where it is important to identify the correct

order of the final complex system and identify concurrency in the processes (Kale et al.,

2007; Agullo et al., 2009; Notz et al., 2012). Dependency graphs are used in combination

with process couplers - a software that links together and mediates interactions between

models (physics modules). Couplers are abstractions with different implementations; no

standard reference architecture for couplers has emerged. The functions typically associ-

ated with couplers include managing data transfer between two, or more, models (often in

parallel), interpolating field data when the grid resolution, or discretization, differs between

these models, ensuring conservation of physical quantities across the coupling boundary,

accumulation and averaging of physical quantities (e.g., when the timesteps are different

in different modules), and coordinating execution of the constituent models (Dunlap et al.,

2013). Examples of frameworks based on process couplers include the Community Earth

System Models - CESM (Kay et al., 2015), CPL7 (Craig, 2010), and OASIS (Redler et al.,

2010). These types of efforts result in clean, modular code; however, the coupling is limited

to sequential schemes and is allowed only between different sub-domains.


The recent work of Coon et al. (2016) is a conceptual model for managing complex-

ity in multi-physics codes using a process tree, a dependency graph, and a data manager.

These ideas were applied to develop the Arctic Terrestrial Simulator, which uses the process

coupler approach to manage the coupling among the process models present in the same

sub-domain. In addition, dependency graphs and the associated evaluation process are used

extensively to manage and organize shared data requirements among different process rep-

resentations. While the ATS framework is an example of a very flexible platform, there

are some possible disadvantages. The first is that it requires the developer to cede some

control and change their approach to software development through a significant learning

curve. The developer is forced to implement small blocks of code with significant code

redundancy. Additionally, there are performance concerns. The overhead of traversing,

maintaining, and evaluating dependency graphs can entail indirection and therefore wall-

clock time and memory requirements to the simulation. Finally, implementation of both

fully coupled and sequentially coupled approaches requires significant additional efforts.

Here, we developed a modular multi-layer architecture for the multi-physics frame-

work that can employ any coupling strategy across the modules. This level of flexibility is

achieved using a ‘sub-problem’ tree structure, a global status table, and the integration of

these concepts with subsets through an automatic differentiation layer. The modular design

with transparent subset-based interfaces greatly improves code extensibility. In addition,

we have employed a unique mapping operators concept, which enables advanced coupling

strategies, including domain-decomposition and multiscale approaches. We have designed,

developed, implemented and tested a novel general implicit coupling framework for multi-

physics problems (GENIC) that allows for rapid customization. This GENIC framework

constitutes the next generation of the Automatic-Differentiation General-Purpose Research

Simulator (AD-GPRS), and it represents a flexible and efficient reservoir simulation re-

search laboratory in which students and researchers can easily test their ideas.

1.4 Outline

The dissertation proceeds as follows. In Chapter 2, we present the governing equations for

flow, thermal, mechanics, and wells, and the different coupling strategies between them.


Then, we describe the design of the framework based on three key principles: modular

design, extension to ‘new’ physics, and seamless coupling. We elaborate on how to add

new functionality for to existing physics models, how to add new physics, how to control

domain discretization and formulations, how to specify coupled properties and coupling

strategies. Finally, we describe the general algorithm and mapping operations for advanced

sequential coupling strategies.

In Chapter 3, we demonstrate numerical applications of the framework using several

multi-physics problems with different coupling strategies. For each physics coupling, we

apply different coupling strategies and splitting operators. We demonstrate examples with

two (flow-thermal, flow-mechanics) and three (flow-thermal-mechanics) physics. We show

that the best choice of the coupling strategy depends on the coupling strength across the

physics models, which depends strongly on where a particular problem lies in the multi-

physics parameter space. Finally, we demonstrate the flexibility of the framework by trying

several new coupling methods for highly nonlinear flow-thermal problems.

In Chapter 4, we describe how we can further extend the GENIC framework to solve

large-scale multi-physics problems using automatic coupled properties, parallel domain

decomposition, and multiscale formulations. These new capabilities can be developed in

alignment with the key principles of the framework and preserve its flexibility and effi-

ciency. We conclude this thesis in Chapter 5 with a summary and suggestions for future

research.

Chapter 2

General Implicit Coupling Framework(GENIC)

The primary design objective of the General Implicit Coupling Framework (GENIC) reservoir-

simulation framework is the creation of a platform where different coupling strategies for

the simulation of complex physical processes can be designed and implemented with great

flexibility. For a given set of coupled nonlinear processes, such a capability makes it pos-

sible to find the best solution strategy in terms of robustness and computational efficiency.

The GENIC platform is based on three key principles. The first principle is modular de-

sign. Each module represents a particular physics process that can be described by a set of

conservation equations and associated constitutive relations. We are interested in dynamic

processes that are nonlinear and tightly coupled to other physical processes. Isothermal

compositional displacement processes are one example; compositional simulation involves

multi-component transport across multiple flowing fluid phases with interphase mass trans-

fer. Another important problem space is thermal Enhanced Oil Recovery (EOR), of which

steam injection is a primary example. In thermal EOR processes, one must keep track of

energy transfer in addition to mass transfer. A growing area of interest is poromechanics

whereby the interactions of the fluid with the solid skeleton must be accounted for, and

a momentum balance for the solid-fluid aggregate is required.

The GENIC design allows for testing each ‘physics module’ independently, or in com-

bination with other physics modules. For example, compositional-thermal, thermal-mechanics,

16

CHAPTER 2. GENERAL IMPLICIT COUPLING FRAMEWORK (GENIC) 17

thermal-reactions, or compositional-thermal-mechanics type of a multi-physics problem

can be solved. Here, compositional refers to the coupled mass conservation equations of

multiple species; thermal refers to the energy balance; mechanics refers to the Biot for-

mulation of poromechanics; reactions refers to the accounting for chemical reactions (both

equilibrium and non-equilibrium) in the species mass conservation equations. Importantly,

only the specified physics modules are involved in constructing the nonlinear solution strat-

egy in GENIC, and the details of the construction are applied at Run-Time.

Since each module is independent of the other ones, the requirements for a researcher

to contribute to GENIC are focused on knowledge of the relevant physics. This is expected

to widen the user base and allow individuals to contribute to the module of his/her exper-

tise, if they have adequate programming skills. Modularity, or component-based design,

is particularly important for modeling nonlinear multi-physics. Modularity facilitates de-

coupling of multi-physics problems, but it is really critical for methodical management of

solution methods for coupled nonlinear problems. Modularity and extensibility go hand in

hand. Additionally, the modular structure reduces memory and processing consumption by

optimizing the required resources for large-scale problems of practical interest.

The second key principle is physics extension. The GENIC framework enables a gen-

eral representation of a given physics with flexible discretization in both space and time, the

choice and ordering of variables for each physics. GENIC also allows for associating dif-

ferent physical regions with different physics. For example, the compositional module can

be discretized in space using finite-volume methods, such as Two-Point Flux Approxima-

tion (Aziz and Settari, 1979), Multi-Point Flux Approximation (Aavatsmark et al., 1998,

2010; Edwards and Rogers, 1998), Nonlinear Finite Volume Approximation (Le Potier,

2005; Nikitin et al., 2014; Terekhov et al., 2017) or Mixed-Finite Element Methods (Ma-

tringe, 2008). For time discretization, the Fully Implicit Method (FIM), IMPES (Sheldon

et al., 1959; Stone and Garder, 1961; Danilov et al., 1968; Coats, 2000) or Adaptive Implicit

Method (Thomas and Thurnau, 1983; Russell, 1989; Cao, 2002; De Loubens et al., 2009;

Maes et al., 2013). GENIC provides wide flexibility in choosing the primary variables and

the ordering of equations and unknowns; these include the natural-variables (Coats, 1980),

molar variables (Young and Stephenson, 1983; Fussell and Fussell, 1979; Chien et al.,

1985) or the tie-simplex space (Zaydullin et al., 2013). This flexibility is available for every


physics module. Because the modules are independent of each other, different combina-

tions of discretization and choice of variable sets are allowed. For example, one can solve

a compositional-thermal problem with a choice of pressure p, phase saturations Sp, molar

fractions xcp and temperature T . Another choice can be pressure p, overall composition zc

and molar enthalpy hp. Similar flexibility is simultaneously allowed for discretization in

time and space. For example, one can apply FIM and Nonlinear Finite Volume Approxi-

mation for time and space discretization of a compositional module, and at the same time,

discretize the thermal module with AIM and Two-Point Flux Approximation. In addition,

physics extension is aligned with modular design: each extension is implemented as a sepa-

rated sub-module and can be activated and deactivated independently. There is no resource

allocation for non-active sub-modules - this enables the extension of capabilities without

the performance drop.

The last key principle is seamless coupling. The platform provides seamless coupling

between the physics modules without resorting to conditional branches and intermediate in-

terfaces. GENIC treats terms that are coupled across multiple physics modules efficiently.

This principle enables a choice of coupling strategy for a given problem on the fly. For

example, the same problem can be run with FIM, Sequential Fully Implicit (SEQ) (Jenny

et al., 2006) or hybrid (FIM-SEQ) solution methods by using an input parameter for the

coupling logic (solution scheme). The framework design is not limited to the specific num-

ber or ordering of the physics modules; therefore, wide flexibility is achieved to investigate

coupling strategies in order to find the most effective strategy for the given multi-physics

problem. For three or more physics modules, nested FIM-SEQ constructions are allowed.

This level of coupling is not available in other multi-physics simulators. In fact, existing

simulators are usually limited by the number of the physics modules and how they can be

sequenced. Building a framework that allows for ’seamless coupling’ is one of the key

developments in this work.

Figure 2.1 shows the interaction between the key principles. Modularity brings clarity

and simplicity to the software development; physics extensions provide simulator capabili-

ties, and seamless coupling maintains flexibility of the framework. The different principles

are synergistic, and combining them effectively has been one of the primary challenges in

developing GENIC. For example, the implementation of physics extension as a sub-module


Figure 2.1: Key principles of the GENIC framework.

enables efficient resources management, when no resources are allocated for unused capa-

bilities. The seamless coupling of different physics modules is the key to multi-physics in-

tegration: without a modular design, the platform would not be extensible to new physics,

while without seamless integration only one coupling method (FIM or SEQ) would be al-

lowed. Finally, the choice of coupling strategy requires various coupled properties such as

fixed-stress or fixed-strain for compositional-mechanics coupling. Such flexibility is pro-

vided by seamless coupling of coupled properties implemented with physics extensions

principle.

This chapter is organized as follows. First, we give a brief description of the governing

equations and the different coupling strategies. Then, we describe in detail the design of

the framework based on the key principles.

2.1 Mathematical Formulation

The GENIC framework is applicable for an arbitrary number of physics modules as long as

we can specify the physics with a set of governing partial differential equations, unknowns,

and computational domain. Presently, the framework is capable of solving compositional,

thermal, mechanical, and chemical processes with the presence of wells. For brevity, we

limit the description here to thermal multi-component flow in deformable subsurface for-

mations with wells. The full set of governing equations that describes these processes


includes the conservation of mass, energy, and momentum, and well constraints. The ap-

propriate constitutive relations and boundary conditions are assumed to be fully specified.

2.1.1 Multi-Component Transport

The mass balance equations for nc components that can exist in np fluid phases can be

written as:

RF ≡∂

∂t

(φ

np

∑j=1

xi jρ jS j

)+∇ ·

(np

∑j=1

xi jρ jvvv j

)+

np

∑j=1

xi jρ jq j = 0, i = 1..nc, (2.1)

where i is the index of a component, j the index of a phase, φ the porosity, xi j the phase

compositions, ρ j the phase density, S j the phase saturation, q j the source term. Phase

velocity, vvv j, is given by the Darcy’s law

v j =−kkr j

µ j

(∇p j−ρ jg

), (2.2)

where kkk is the absolute permeability, kr j the relative permeability of the phase, µ j the

viscosity of the phase, and g is the gravity.

These equations are supplemented with linear constraints for the phase compositions

and the phase saturations:

nc

∑i=1

xi j = 1, ∀ j = 1..np, (2.3)

np

∑j=1

S j = 1, (2.4)

and the equality condition for fugacities (instantaneous thermodynamic phase equilibrium

is assumed):

fi j(

p,T,xi j)− fik (p,T,xik) = 0, ∀ j 6= k = 1..np, i = 1..nc. (2.5)

Here, we employ the natural-variables formulation (Coats, 1980; Voskov and Tchelepi,


2012), where the unknowns are the pressure, p, phase saturations S j, and phase composi-

tions, xi j.

2.1.2 Conservation of Energy

We assume thermal equilibrium between the rock and the fluids. It follows that the total-

energy balance equation can be expressed as:

RT ≡∂

∂t

(φ

np

∑j=1

u jρ jS j +(1−φ)ur

)+∇ ·

(np

∑j=1

h jρ jvvv j

)−∇ · (κ∇T )+

np

∑j=1

h jρ jq j = 0,

(2.6)

where κ is the thermal-conductivity coefficient, u j the phase internal energy, ur the rock

internal energy, and h j the phase enthalpy. The unknown related to the energy equations

is the temperature T . All the thermal properties are evaluated based on p, T , and xi j using

correlations and a cubic Equation of State (EoS). The details of the thermal-compositional

implementation are given elsewhere (Zaydullin et al., 2014).

2.1.3 Momentum Balance

The mechanical deformation equations are formulated in terms of total stresses, bulk strains,

pore pressure, and temperature. They are written as follows (Nikolaevskij, 1990; Jaeger

et al., 2009):

RM ≡ ∇ ·σ+ρggg = 0, (2.7)

where ρ = ρs(1−φ)+ρ f φ is the overall-mass density, ρs is the density of the solid skeleton

and ρ f is the average fluid density. The total-stress tensor, σ, contains the contributions of

the fluid and solid-skeleton (Zienkiewicz et al., 1999):

σ = Cεe−bP, (2.8)

where εe is the second-order elasticity strain tensor, b is the second-order tensor of Biot

coefficients, C is the forth-order elasticity moduli tensor. The fluid pressure is defined as

P =np

∑j=1

S j p j.


We follow the ‘small deformation’ theory and assume that the total strain tensor, ε,

can be decomposed into the three components representing elastic, plastic, and thermal

contributions:

ε = εe + ε

p + εθ, (2.9)

so the elastic strain can be expressed as

εe = ε− ε

p− εθ. (2.10)

We also assume that the dilation induced by the temperature is isotropic (Ottosen and

Ristinmaa, 2005), i.e.,

εθ = αT 1 (2.11)

with α being the skeleton linear thermal expansion/dilation coefficient. Then, combining

Eq. 2.9 and Eq. 2.8, we obtain:

σ = C · (ε− εp)−aT −bP, (2.12)

where we define the thermal expansion tensor, a = αC1. In the following, we also assume

that the Biot’s tensor is isotropic, i.e. b = b1, where b is the Biot’s coefficient.

Eq. 2.7 and Eq. 2.12 together with the total-strain definition, ε =12(∇uuu+∇

T uuu)

(uuu is

the displacement vector), provide a framework for modeling the deformation behavior of

natural saturated porous formations. A material model needs to be added to complete the

description. We use the Drucker–Prager (Drucker and Prager, 1952) model to compute the

stress-strain response in Eq. 2.12. To determine the plastic behavior, the classical associated

Drucker–Prager model employs the following yield surface F (Borja, 2013):

F =√

J2 +B · I1 +A, (2.13)

where I1 and J2 are volumetric and deviatoric invariants of the effective stress, and A and

B are material parameters. We assume that the Drucker–Prager yield surface circumscribes


the Mohr-Coulomb yield surface, and consequently express A, B as follows:

A =6 c cosψ√

3(3− sinψ), B =

2 sinψ√

3(3− sinψ), (2.14)

where ψ is the internal friction angle and c is the cohesion. This model primarily captures

plastic dilation, and the size of the yield surface is controlled by the internal state vari-

ables, such as softening and hardening parameters. In this work, we employ an isotropic

softening/hardening rule:

c = Hvεpv −Ht T , (2.15)

where Hv and Ht are two material parameters, εpv = tr(εp) is the volumetric plastic strain,

and dot defines an increment. The first term on the right hand side of Eq. 2.15 allows

the yield surface to grow with plastic volumetric compaction (hardening) and shrink with

plastic dilation (softening), and the second terms allows shrinking of the yield surface due

to temperature increase.

Following Coussy (2004) we introduce the porosity as the ratio of the volume of the

connected porous space to the total volume. The relationship between the porosity and

stress/strain is assumed to be:

φ−φp = φo +

(b−φo)(1−b)Kd

(P−Po)+b(εv− εpv − εv,0)−αφ (T −To) . (2.16)

Here, Kd is the local drained bulk modulus, εv = tr(ε) is the volumetric total strain, and αφ

is the volumetric thermal dilation coefficient related to the porosity. The reference porosity,

pressure, and volumetric strain are denoted as φo, Po, and εv,o, respectively. Assuming that

the irreversible porosity is defined as φp = bεpv , we obtain:

φ = φo +(b−φo)(1−b)

Kd(P−Po)+b(εv− εv,o)−αφ (T −To) . (2.17)

We use a widely used model for the permeability calculation (David et al., 1994):

k = k0

(φ

φ0

)m

. (2.18)


The permeability exponent, m, defines how permeability deviates from a reference value ko.

2.1.4 Wells

A general multi-segment well is represented by a set of conservation equations written for

each segment. The mass balance equation for each segment is:

RW,F ≡∂

∂t

np

∑j=1

ρ jα jxi j−∂

∂z

np

∑j=1

ρ jVs, jxi j +np

∑j=1

ρ jxi jq j = 0, i = 1..nc (2.19)

where α j is the in-situ phase fraction (holdup) of phase j, Vs, j is the superficial phase ve-

locity (flux over the surface area), q j is the inflow of phase j to the well, z is the coordinate

along the well, and ∑npj=1 ρ jxi jq j represents the mass source/sink through the wellbore.

The energy balance equation for each segment is:

RW,T ≡∂

∂t

np

∑j=1

ρ jα j

(u j +

V 2in, j

2

)+

∂

∂z

np

∑j=1

ρ jα jVin, j

(h j +

V 2in, j

2

)

−gcosθ

np

∑j=1

ρ jα jV 2in, j +qloss−

np

∑j=1

ρ jh jq j = 0 (2.20)

where Vin, j is the interstitial velocity of phase j (Vin, j =Vs, j/α j), gcosθ is the gravitational

component along the well, in which θ is the inclination angle of the segment from vertical,

qloss is the heat loss from the wellbore fluid to surroundings, and it is calculated using the

following relation:

qloss =−2πrwuto(T w−T ) (2.21)

where uto is overall heat transfer coefficient, rw is the wellbore radius and T w is the tem-

perature of wellbore fluids. We close the system of equations Eq. 2.19–2.21 with linear

constraints for xi j and α j, Eq. 2.3, and thermodynamic phase equilibrium equations for

components, Eq. 2.5. The details of the multi-segment well implementation and modeling

of phase fractions inside the wellbore are given elsewhere (Zhou, 2012).


2.2 Coupling Strategies

We define the set of a given physics as ℘. Next, we introduce the following notation to

describe a particular coupling strategy Ψ:

Ψ = ϒA [A1,A2, ...,An] , (2.22)

where ϒA is a type of coupling such as FIM or SEQ, i.e. ϒA ∈ FIM, SEQ, and Ai

either represents a particular physics module (Ai ∈℘), or refers to another coupled set

Ai = ϒB [B1, ...,Bk], where B j is defined similarly to Ai. Such formulation enables a nested

coupling strategy. For example, solving fully coupled physics A1 and A2 followed by se-

quentially coupled physics A3, can be written as Ψ = SEQ[FIM[A1,A2], A3]. Notice that

ϒ = FIM is invariant with respect to the order of arguments; on the other hand, ϒ = SEQ is

sensitive to the order of the arguments.

Here, we assume ℘= FLOW, THERMAL, MECHANICS, where FLOW, THER-

MAL, MECHANICS represent multi-component flow (Eq. 2.1–2.5), thermal (Eq. 2.6),

and mechanics (Eq. 2.7–2.18). For a given ℘ we introduce a vector of unknowns, x =

(xF ,xT ,xM), where xF are the flow unknowns, xT are the thermal unknowns, and xM are the

mechanics unknowns. The corresponding residuals are RF(xF ,xT ,xM), RT (xF ,xT ,xM), and

RM(xF ,xT ,xM). Notice that there is no assumption about which set of residuals depends

on which unknowns in the general case. So, some residuals may depend only on a subset

of the unknowns. In the case of a compositional-thermal-mechanics problem, the residu-

als for each physics module – RF ,RT ,RM – are assumed to depend on all the unknowns –

xF ,xT ,xM.

The two commonly used coupling strategies for the compositional-thermal-mechanics

multi-physics problem available in the literature are briefly outlined below.

Option 1: Fully implicit coupling: solve all the equations simultaneously (Figure 2.2):Rn+1

F(xn+1

F ,xn+1T ,xn+1

M ,xn)= 0,

Rn+1T(xn+1

F ,xn+1T ,xn+1

M ,xn)= 0,

Rn+1M(xn+1

F ,xn+1T ,xn+1

M ,xn)= 0.

(2.23)


Figure 2.2: Fully implicit coupling:Ψ = FIM[FLOW, THERMAL, MECHANICS].

Figure 2.3: Fully coupled flow and energy-balance,followed by sequentially coupled mechanics:Ψ = SEQ[FIM[FLOW,THERMAL], MECHANICS].

Using Newton’s method leads to the following system:∂RF/∂xF ∂RF/∂xT ∂RF/∂xM

∂RT/∂xF ∂RT/∂xT ∂RT/∂xM

∂RM/∂xF ∂RM/∂xT ∂RM/∂xM

δxF

δxT

δxM

=−

RF

RT

RM

. (2.24)

Option 2: Fully coupled flow and energy-balance, followed by sequentially coupled

mechanics (Figure 2.3):

Step 1: Solve Rn+1F

(xn+1

F ,xn+1T ,xn+1/2

M ,xn)= 0,

Rn+1T

(xn+1

F ,xn+1T ,xn+1/2

M ,xn)= 0.

(2.25)

or in the linearized matrix form[∂RF/∂xF ∂RF/∂xT

∂RT/∂xF ∂RT/∂xT

][δxF

δxT

]=−

[RF

RT

]. (2.26)

Step 2: Solve

Rn+1M (xn+1

F ,xn+1T ,xn+1

M ,xn) = 0. (2.27)

matrix form [∂RM/∂xM

][δxM

]=−

[RM

]. (2.28)

We repeat Steps 1 and 2 until global convergence is achieved.


(a) (b) (c)

Figure 2.4: Example of other possible coupling strategies for a flow-thermal-mechanics problem:(a) Ψ = SEQ[SEQ[FLOW,MECHANICS],THERMAL] (b) Ψ = SEQ[FLOW, SEQ[MECHANICS,THERMAL]] (c) Ψ =SEQ[FLOW, MECHANICS, THERMAL]. Dashed box represents nested sequentially coupled blocks.

There are 13 different coupling strategies for problems involving multiphase flow, ther-

mal effects, and mechanical deformation (Figure 2.4 depicts some of them):

FIM[FLOW, THERMAL, MECHANICS]

SEQ[FIM[FLOW, THERMAL], MECHANICS] SEQ[FLOW, FIM[THERMAL, MECHANICS]] SEQ[MECHANICS, FIM[FLOW, THERMAL]]

SEQ[FIM[MECHANICS, FLOW], THERMAL] SEQ[THERMAL, FIM[FLOW, MECHANICS]] SEQ[FIM[THERMAL, MECHANICS], FLOW]

SEQ[FLOW, THERMAL, MECHANICS] SEQ[FLOW, MECHCANICS, THERMAL] SEQ[THERMAL, FLOW, MECHANICS]

SEQ[THERMAL, MECHANICS, FLOW] SEQ[MECHANICS, FLOW, THERMAL] SEQ[MECHANICS, THERMAL, FLOW]

For problems with four physics modules (e.g., flow, thermal, mechanics, reactions), the

number of possible coupling strategies is 49. This would require implementing a large

number of different simulators, which is simply not feasible. Although not all coupling

strategies are potentially effective, numerical analysis of nonlinear solution schemes is

often limited to simplified physics (Kim et al., 2013; Kim, 2015). In modeling coupled

nonlinear dynamics problems, it is often the case that one has to experiment with different

schemes to figure out what works best for which problem class. The objective here is to

build an advanced reservoir-simulation framework that allows for implementing, testing,

and comparing different coupling strategies in a unified manner.


2.3 Framework Design

We employ a modular design by splitting a given multi-physics problem into sub-problems.

Each sub-problem represents a particular physical process, e.g., flow, thermal, reactions,

mechanics, wells. A sub-problem contains a set of subsets, where we define unknowns

and properties. We employ physics extension principle by defining the governing equa-

tions, computational domain, spatial discretization and nonlinear formulation for each sub-

problem. As a result, the framework can be viewed as a collection of large modules (sub-

problems) and their sub-modules (subsets). We arrange these pieces together to solve a

particular multi-physics problem. For example, the choice of subsets for a compositional-

thermal problem with Two-Point Flux Approximation, natural-variables formulations, and

FIM is different from a set of subsets for a compositional-mechanics problem with Nonlin-

ear Finite Volume approximation, molar-variables formulation, and AIM.

Flexibility in coupling the different sub-problems is of prime importance. The objective

is to provide wide flexibility in the number of different physics modules and how they are

coupled. The choice of the coupling strategy is a part of the simulation input. To allow

this level of flexibility, two important questions have to be answered. First: how to solve

each sub-problem for a particular coupling strategy? Second: how to stitch all the sub-

problems together to produce a global numerical solution for a given timestep? To deal

with the first question, we devise a tree strategy for sub-problems. The second question

is handled using a ‘global status table’. In the following section, we outline the basis of

sub-problems organization, and we elaborate the details of the sub-problem tree and the

global status table. Then, we describe the strategy for accommodating complex coupling

terms and properties.

2.3.1 Subsets

The concept of subsets lies at the core of the GENIC framework. A subset is an abstract

computational domain with a defined set of unknowns and properties. A particular com-

bination of subsets describes a sub-problem. For example, the flow physics module may

have a cell-based subset with unknowns, such as pressure p, phase saturations S j and prop-

erties, such as relative permeability kr j and phase viscosity µ j. In addition, we may define


(a) (b) (c)

Figure 2.5: Example of a flow sub-problem of two blocks with following hypothetical subsets and corre-sponding unknowns: (a) cell-based (pressure p and phase saturation Sp), (b) face-based (phase velocity vp)and (c) boundary-based unknowns (boundary flux b).

c l a s s F l o w C e l l S u b s e t : p u b l i c SubSet F l o w C e l l S u b s e t ( n c e l l s , nphase , ncomp , . . . )ADD SCALAR( p r e s ) / / p r e s s u r eADD VECTOR( s a t , nphase ) / / phase s a t u r a t i o nADD MATRIX( xcp , nphase , ncomp ) / / molar f r a c t i o n o f component c i n phase p. . . / / any o t h e r p r o p e r t y

c l a s s F lowFaceSubse t : p u b l i c SubSet

FlowFaceSubse t ( n f a c e s , nphase , . . . )ADD VECTOR( p h a s e f l u x , nphase ) / / molar phase f l u x. . . / / any o t h e r p r o p e r t y

c l a s s T h e r m a l C e l l S u b s e t : p u b l i c SubSet

T h e r m a l C e l l S u b s e t ( n c e l l s , nphase , . . . )ADD SCALAR( temp ) / / t e m p e r a t u r eADD VECTOR( e n t h a l p y , nphase ) / / phase e n t h a l p y. . . / / any o t h e r p r o p e r t y

Listing 2.1: Simplified code example: cell- and face-based subsets for flow, and cell-based subset for thermalsub-problems. Number of blocks is different for each subset (number of cells for cell-based, and number offaces/connections for face-based subsets).

a connection-based subset with unknowns defined on the interfaces of a control-volume

(e.g. phase velocities v j) and properties defined on boundaries as shown in Figure 2.5. The

geomechanics module employs vertex-based displacements, uuu, and cell-based properties,

such as the fluid pressure p, volumetric strain εv, and volumetric stress σv. In Listing 2.1

we define three subsets: FlowCellSubset, FlowFaceSubset and ThermalCellSubset for cell

and face-based compositional properties and cell-based thermal properties. Notice that we

do not establish pairing between subsets and sub-problems because the same subset can be


Table 2.1: Example of physics modules (sub-problems), their hypothetical subsets properties and correspond-ing unknowns. We can extend capabilities of existing sub-problems by adding additional subsets or extendingexisting ones.

Sub-problem Subset Unknowns and properties

cells p, S j, xi j, µ, ρ, ...FLOW faces v j

boundaries boundary conditions

cells T, h j, u j, htot , ...THERMAL faces ve

jboundaries boundary conditions

vertices uuucells p

MECHANICS faces traction valuesintegration points ε, σ, C ...

boundaries boundary conditions

used by different sub-problems as needed.

Table 2.1 represents three physics modules: Flow, Thermal, and Mechanics with their

hypothetical subsets. Each subset has a list of related properties. We can extend this exam-

ple with additional physics modules, subsets and unknowns. Listing 2.2 demonstrates the

relationship between a sub-problem and its subsets. Notice that we can reuse subsets from

other sub-problems. This is important for well sub-problems, or advanced applications of

the framework such as multi-scale formulations (Jenny et al., 2003; Lunati and Jenny, 2006;

Hajibeygi et al., 2008), or domain decomposition (Killough and Wheeler, 1987; Wheeler

et al., 1999).

We define status as a specific choice and ordering of the problem unknowns. The status

change may involve dynamic variable switching, which is used in various formulations of

multiphase flow models when phases appear or disappear (Voskov and Tchelepi, 2012).

Moreover, it is often the case that the number of unknowns is different for each status,

which implies that the size of the residual vector and the Jacobian matrix can change.

The Automatically Differentiable Expression Templates Library (ADETL) (Younis, 2011;


c l a s s FlowProblem : p u b l i c SubProblem FlowProblem ( . . . )c r e a t e S u b s e t s ( ) ;F l o w C e l l S u b s e t * f l o w c e l l s u b s e t ;F lowFaceSubse t * f l o w f a c e s u b s e t ;. . .

FlowProblem : : c r e a t e S u b s e t s ( ) / / c r e a t e each s u b s e tf l o w c e l l s u b s e t = new F l o w C e l l S u b s e t ( n c e l l s , nphase , ncomps , . . . ) ;f l o w f a c e s u b s e t = new FlowFaceSubse t ( n f a c e s , nphase ) ;/ / add s u b s e t s i n t o t h e g l o b a l J a c o b i a na d d S u b s e t ( f l o w c e l l s u b s e t ) ;a d d S u b s e t ( f l o w f a c e s u b s e t ) ;

Listing 2.2: Simplified code example: flow sub-problem consists of cell and face subsets.

Zhou, 2012) provides a capability to define a status table (a set of statuses with a corre-

sponding order of unknowns) and automatically rearranges the column structure and enu-

meration of the variables when the status changes. A variable formulation defines a choice

of a status table. We apply different formulations (i.e. status tables) of the variables for

each subset independently. The resulting mixed formulation for a multi-physics problem is

formed using the subsets of those individual formulations through a combination of related

status tables. For example, Figure 2.6 depicts two formulations for flow (natural and molar)

and two formulations for thermal (temperature and enthalpy) problems. Instead of defining

a new formulation for a coupled compositional-thermal problem, a mixed formulation is au-

tomatically employed based on applied formulations for each involved physics. This leads

to four possible mixed formulations. The automatic construction of mixed formulation is

of great importance since the number of formulations for a coupled multi-physics problems

grows very quickly with the number of formulations for each physics. Listing 2.3 demon-

strates related code where each formulation for each sub-problem is enabled independently.

We first define formulations for each physics. In the presented case, a compositional formu-

lation such as natural or molar is the implementation of InterfaceFlowFormulation class,

while thermal formulation is the implementation of InterfaceThermalFormulation. Then,

we independently assign formulations for each physics.

Since each subset is introduced independently from other subsets, we may enable or


c l a s s I n t e r f a c e F l o w F o r m u l a t i o n : p u b l i c F o r m u l a t i o n I n t e r f a c e F l o w F o r m u l a t i o n ( . . . )/ / used s u b s e t sF l o w C e l l S u b s e t * f l o w c e l l s u b s e t ;F lowFaceSubse t * f l o w f a c e s u b s e t ;c o n s t T h e r m a l C e l l S u b s e t * t h e r m c e l l s u b s e t ; / / used on ly t o a c c e s s t e m p e r a t u r e. . .

c l a s s I n t e r f a c e T h e r m a l F o r m u l a t i o n : p u b l i c F o r m u l a t i o n

I n t e r f a c e T h e r m a l F o r m u l a t i o n ( . . . )T h e r m a l C e l l S u b s e t * t h e r m c e l l s u b s e t ; / / no need f o r f low s u b s e t s. . .

S i m u l a t o r : : s e t F o r m u l a t i o n s ( . . . )

/ / p i c k any i m p l e m e n t a t i o n o f I n t e r f a c e F l o w F o r m u l a t i o n/ / N a t u r a l / Molar F o r m u l a t i o n s a r e i m p l e m e n t a t i o n o f I n t e r f a c e F l o w F o r m u l a t i o npFlowProblem−>s e t F o r m u l a t i o n ( new N a t u r a l F o r m u l a t i o n ) ;

/ / p i c k any i m p l e m e n t a t i o n o f I n t e r f a c e T h e r m a l F o r m u l a t i o n/ / Thermal / E n t h a l p y F o r m u l a t i o n a r e i m p l e m e n t a t i o n o f I n t e r f a c e T h e r m a l F o r m u l a t i o npThermalProblem−>s e t F o r m u l a t i o n ( new T h e r m a l F o r m u l a t i o n ) :

Listing 2.3: Simplified code example: each sub-problem has its own formulation (mixed Natural and Thermalformulation).

c l a s s F V D i s c r e t i z a t i o n F o r m u l a t i o n : p u b l i c F o r m u l a t i o n F V D i s c r e t i z a t i o n F o r m u l a t i o n ( . . . )F l o w C e l l S u b s e t * f l o w c e l l s u b s e t ; / / used s u b s e t s. . .

c l a s s M F E D i s c r e t i z a t i o n F o r m u l a t i o n : p u b l i c F o r m u l a t i o n

M F E D i s c r e t i z a t i o n F o r m u l a t i o n ( . . . )F l o w V e r t i c e s S u b s e t * f l o w v e r t i c e s s u b s e t ; / / used s u b s e t s. . .

c l a s s FlowProblem : p u b l i c SubProblem

FlowProblem ( . . . )/ / used s u b s e t s , a l l r e q u i r e d s u b s e t s f o r bo th d i s c r e t i z a t i o n a r e p r e s e n t/ / n o t e t h a t f a c e / b o u n d a r i e s s u b s e t s a r e n o t shown f o r b r e v i t yF l o w C e l l S u b s e t * f l o w c e l l s u b s e t ;F l o w V e r t i c e s S u b s e t * f l o w v e r t i c e s s u b s e t ;. . .

S i m u l a t o r : : s e t F o r m u l a t i o n s ( . . . )

i f ( f i n i t eVo lumeMethod )/ / f l o w v e r t i c e s s u b s e t i s d i s a b l e d and n e v e r a l l o c a t e d f o r FVpFlowProblem−>s e t F o r m u l a t i o n ( F V D i s c r e t i z a t i o n F o r m u l a t i o n ) ;

e l s e i f ( mixedF in i t eE lemen tMe thod )/ / f l o w c e l l s u b s e t i s d i s a b l e d and n e v e r a l l o c a t e d f o r MFEpFlowProblem−>s e t F o r m u l a t i o n ( M F E D i s c r e t i z a t i o n F o r m u l a t i o n ) ;

Listing 2.4: Simplified code example: FV and MFE space discretization for a flow sub-problem. Each classbelow contains face and boundaries subset which are not shown for brevity.


Sub-problem Formulations Unknowns

FLOW natural p, Sp, xcpFLOW molar p, zc

THERMAL temperature TTHERMAL enthalpy hp

(a)

Mixed Formulations Unknowns

natural and temperature p, Sp, xcp, Tnatural and enthalpy p, Sp, xcp, hp

molar and temperature p, zc, Tmolar and enthalpy p, zc, hp

(b)

Figure 2.6: Example of mixing natural and molar formulations for a flow sub-problem, and temperatureand enthalpy formulations for a thermal sub-problem: (a) list of sub-problem and formulations, (b) possiblemixed formulations with unknowns.

Flow Subsets FV MFE

cell-based X –face-based X Xboundaries X X

vertices-based – X

Figure 2.7: Activation logic of subsets for different space discretization schemes for a flow sub-problem.

disable any subset for a given problem throughout a simulation run, or at the initializa-

tion stage. When a subset is disabled, it is removed at the Jacobian level and corre-

sponding memory is freed. We can create an alternative sub-problem package without

changing its core functionality by enabling/disabling related subsets. For example, we

might introduce a Mixed Finite Element (MFE) formulation for the flow sub-problem

by adding additional subsets defined on the vertices (Figure 2.7). The residuals of both

Finite Volume (FV) and MFE methods contain similar terms (accumulation, fluxes, and

source/sink) and vary only in the subset choice, and how these terms are calculated. List-

ing 2.4 demonstrates the reuse of subsets using formulations FV DiscretizationFormulation

and MFE DiscretizationFormulation for FV and MFE space discretization. Sub-problem

FlowProblem contains all subsets from both formulations, however, only part of them are


Figure 2.8: Example of flow-thermal-mechanics problem. SM , SF , ST , STiso represent all subsets related tomechanics, flow, thermal and iso-thermal sub-problems correspondingly. Mechanics sub-problem includesover- and under-burden and, therefore, is larger than a reservoir. The reservoir domain SF has two parts:thermal ST and iso-thermal STiso .

used at any moment. The selected subsets depend on the chosen formulation.

The overall flexibility of the framework depends strongly on how subsets are designed

and implemented. Subsets can be identical in terms of the variable sets, yet a subset can

be applied to different domains. At the same time, different subsets can belong to different

parts of the overall problem. For example, we may have a small domain for the flow

sub-problem and a large domain for the mechanics sub-problem which is often the case

for problems with geomechanical effects. Figure 2.8 shows an example of a flow-thermal-

mechanics problem, where SM, SF , ST , STiso represent all subsets related to mechanics, flow,

thermal and iso-thermal sub-problems, respectively. Each of those subsets has different

domain size: mechanics subsets SM include over- and under-burden and, therefore, cover

a larger domain than the reservoir. The reservoir domain SF has two parts: thermal ST

and iso-thermal STiso . Listing 2.5 is a code example of how different size of subsets is

supported in the framework. Notice that mapping of properties between different physics

requires explicit management. For example, both cells A and B are the first elements for

the mechanics SM and the reservoir SF domains correspondingly. However, the cell B

has another index in the domain SM. Such mapping between physics defined on different

domains should be provided on the input stage.

In addition, for low-order transport equations, we can separate the flux part from the

accumulation part (Maes et al., 2016) using connection-based subsets. Similarly, we can

decouple other terms for any sub-problem. The GENIC framework supports all of the


s i z e t nSM = 1000 ; / / number o f b l o c k s i n mechan ics domains i z e t nSF = 200 ; / / number o f b l o c k s i n r e s e r v o i r domain ( f low )s i z e t nST = 160 ; / / number o f b l o c k s i n t h e r m a l domains i z e t nSTiso = nSF−nST ; / / number o f b l o c k s i n i s o−t h e r m a l domainpMechanicsProblme−>m e c h c e l l s u b s e t = new MechCe l lSubse t (nSM, . . . )pFlowProblem−>f l o w c e l l s u b s e t = new F l o w C e l l S u b s e t ( nSF , . . . ) ;pThermalProblem−>t h e r m a l c e l l s u b s e t = new T h e r m a l C e l l S u b s e t ( nST , . . . ) ;pThermalProblem−> i s o t h e r m a l c e l l s u b s e t = new I s o t h e r m a l C e l l S u b s e t ( nSTiso , . . . ) ;. . . / / d e f i n e e x p l i c i t mapping

Listing 2.5: Simplified code example: different size domain for each sub-problem.

(a) (b)

Figure 2.9: (a) Domain decomposition example: S1, ..S9 represents identical flow-thermal subsets related todifferent parts of a global simulation domain, shaded cells represent additional boundaries subsets b1, ..b9 (b)Multiscale example: a primary coarse grid (solid thick lines) and the dual coarse grid (dashed lines) formsubsets Sc and Sd correspondingly; the dual grid naturally defines a partition of the fine-mesh into internal(white cells), edge (shaded cells), and node points (grey cells) of the dual grid - subsets S f , Se and Sn.

above-mentioned capabilities. We elaborate on the details of coupling terms in Section

2.3.3.

Additionally, subsets may be used for domain decomposition and multi-scale approaches.

We employ flow-thermal subsets S1, ..Sk (where k is a number of domains) to solve a prob-

lem using a domain decomposition, where for each domain we specify a related boundaries

subset (see Figure 2.9a). Alternatively, subsets of properties defined on a fine grid, dual and

coarse grids allow for multiscale formulations and solution schemes (Figure 2.9b). While

GENIC framework helps with a structure to solve these problems, additional code modifi-

cations are required to fully support these capabilities. These modifications are discussed

in detail in Chapter 4.


(a) (b)

Figure 2.10: Example of extended fixed-stress split strategy for a flow-thermal-mechanics problem: (a) aschematic representation of split and (b) the corresponding tree structure. Note that ’Linear solver A’ isapplied for both FLOW and THERMAL sub-problems, while ’Linear solver B’ is applied only for ME-CHANICS sub-problem.

2.3.2 Sub-Problem Tree Structure and Global Status Table

A tree structure provides a general representation of a given coupling scheme defined by

Eq. 2.22. Each inner node in the tree defines a coupling strategy – ϒ (FIM or SEQ) – that

is applied to the child nodes. Each terminal node represents a particular physics problem

with a dedicated linear solver applied to a single or multi-physics block. Figure 2.10 ex-

plains this concept for a flow-thermal-mechanics problem. We have one head node with

the global coupling strategy Ψ = SEQ[FIM[FLOW,THERMAL], MECHANICS], one internal

node for coupling between two physics – Ψ2 = FIM[FLOW,THERMAL], and two terminal

nodes for flow-thermal and mechanics physics.

We extend the status table concept to allow for flexible coupling. First, we mark all

possible statuses within a status table with non-fixed unknowns as active. Second, we add

an auxiliary inactive status when all unknowns are fixed (i.e. derivatives are not formed).

Consequently, a subset can be active, or inactive, at any given moment. Table 2.2 shows an

example of a variable set for black-oil model with a corresponding Jacobian block structure.

We define a sub-problem as active when any of its enabled subsets is active and inactive

otherwise (Figure 2.11). An active sub-problem forms a corresponding part of the Jacobian

matrix. Similarly, for a sub-problem in the inactive state, its Jacobian part is not formed.

Notice that auxiliary subsets used by a sub-problem with only defined properties do not

form derivatives once related unknowns are in the inactive state. Figure 2.12 shows activa-


Table 2.2: Example of a variable set for three-phase black-oil problem expanded by inactive status. Noderivatives are formed for inactive subset status.

Local Subset Status p So Sg RS Jacobian Block Structure Global State Status

O, W 0 1 – 2 J3x3 activeO, G 0 1 2 – J3x3 active∅ – – – – ∅ inactive

Figure 2.11: Subset and subproblem activation state. Sub-problem is inactive only when all enabled subsetsS1 and S2 are inactive, disabled subsets S3 and S4 do not affect the active status of its sub-problem.

tion logic of subsets for cell-based subset SC and face-based subset SF . Subset SC defines

both unknowns XC and properties YC, while subset SF is an auxiliary subset and contains

only properties YF , which are a function of unknowns from subset SC. Both active sub-

sets form a full Jacobian matrix. When subset SC is active only related part of Jacobian is

formed. However, when SF is the only active subset - Jacobian is empty because properties

YF depend on unknowns XC which are fixed.

/ / G l ob a l S t a t u s Tab le c o n s t r u c t e d from SEQ[ FIM [FLOW,THERMAL] , MECHANICS] p l u s FIM/ / s t a t u s FLOW THERMAL MECHANICS/ / [ 0 ] t r u e t r u e f a l s e/ / [ 1 ] f a l s e f a l s e t r u e/ / [ 2 ] t r u e t r u e t r u edo ub l e d t = 0 . 1 ; / / c u r r e n t t i m e s t e p s i z e i n days/ / one i m p l i c i t s e q u e n t i a l i t e r a t i o n f o r e x t e n d e d f i x e d−s t r e s s s p l i tsubProb lems . s e t G l o b a l S t a t u s ( 0 ) ; / / a c t i v a t e FLOW & THERMALsubProb lems . s o l v e ( d t ) ; / / s o l v e FIM [FLOW, THERMAL]subProb lems . s e t G l o b a l S t a t u s ( 1 ) ; / / a c t i v a t e MECHANICSsubProb lems . s o l v e ( d t ) ; / / s o l v e MECHANICS on ly/ / f u l l y c o u p l e d a p p r o a c hsubProb lems . s e t G l o b a l S t a t u s ( 2 ) ; / / a c t i v e a l l sub−prob lems t o g e t h e rsubProb lems . s o l v e ( d t ) ; / / s o l v e FIM [FLOW, THERMAL, MECHANICS]

Listing 2.6: Simplified code example: sub-problems activation depends on the global status. Note there is aunified interface for different coupling strategies.


Subsets Unknowns Properties

cell-based SC XC YCface-based SF – YF

(a)

Active Subset

SC SC and SF SF

J =

[∂RC∂XC

∂RC∂YF

= 00 ∂RF

∂YF= 0

]= ∂RC

∂XCJ =

[∂RC∂XC

∂RC∂YF

0 ∂RF∂YF

]J =

[∂RC∂XC

= 0 ∂RC∂YF

= 00 ∂RF

∂YF= 0

]= ∅

(b)

Figure 2.12: Subset activation logic for different subsets. On top (a) - definition of subsets and relatedunknowns X , and properties Y . Face-based subset SF contains only properties YF (for example, phase velocityvp) without unknowns. At the bottom (b) - activation logic of subsets affects the Jacobian matrix. Both activesubsets form full Jacobian matrix. When subset SC is active only related part of Jacobian is formed. However,when SF is only active subset - Jacobian is empty, because properties YF depend on unknowns XC which arein non-active state (i.e. fixed and do not have derivatives), as a result ∂RF

∂YF= 0.

A given coupling scheme defines a set of states when a related physics sub-problem

needs to be solved, or not (see Eq. 2.23 – 2.28). These states form the global status ta-

ble, which controls the activation and deactivation logic of the physics modules. When the

fully implicit strategy is applied, all the physics modules are active, and a full Jacobian

matrix is formed. However, if one of the physics modules is inactive, no derivatives will be

constructed for this sub-problem. The combination of the sub-problem tree structure and

the global status table provides several unique improvements over existing multi-physics

simulators. First, the desired coupling strategy can be chosen simply by activating and

deactivating specific sub-problems. Second, such approach facilitates an efficient code de-

sign: no specific property storage, conditional branches or interfaces are required to couple

physics modules that use a sequential strategy; as well as it avoids an excessive copying of

data between the physical modules. Finally, we can test hybrid schemes: one can employ

a fully coupled strategy for strongly coupled domains of the problem and apply a sequen-

tially coupled strategy for the rest (see, e.g., Tomin and Lunati (2015) for a discussion).

Listing 2.6 explicitly shows the usage of a global status table activation logic for solving

two sub-problems together or separately.


Figure 2.13, 2.14 and 2.15 represent several examples of different coupling strategies

applied for a four-physics problem, namely, flow, thermal, mechanics and chemical reac-

tions. For each example, we specify the associated tree structure and the corresponding

global status table. Each terminal node has an assigned global status with a particular lin-

ear solver. Figure 2.15 emphasizes that the fully implicit coupling strategy is just another

case of the global status table with a single state when all sub-problems are active. Unlike

a subset status table where the order of unknowns is flexible (for details refer to Younis

(2011)), the order of sub-problems in the global status table is fixed once the correspond-

ing tree is constructed. This guarantees efficient memory allocation for the linear solver

and proper update of coupling terms. The framework generates a consistent global status

table from the sub-problem tree. There is no need to control it manually unless the non-

standard strategy is tested. One example of such a strategy is a hybrid scheme where we

mix fully coupled and sequentially coupled strategies. Listing 2.7 demonstrates an example

of a flow-thermal-mechanics problem that is solved in the fully implicit manner for the first

100 days, followed by extended fixed-stress split for the rest of the simulation. Details of

the solution algorithm for a general multi-physics problem and convergence are outlined in

Section 2.3.6.

2.3.3 Coupling Terms

This section describes the mechanism to support splitting of a sub-problem to treat its terms

independently, or to describe the coupling terms of the different physics. For example,

wells are coupled with flow and thermal physics by a source/sink term in the corresponding

equations. We can rewrite the compositional and energy equations (Eq. 2.1–Eq. 2.6) as

follows: RF ≡ gF,i + fWF,i = 0, i = 1..nc,

RT ≡ gT + fWT,i = 0,

(2.29)

where fWF,i = ∑

npj=1 xi jρ jq j is the well contribution for a multi-component transport, gF,i

and gT are Eq. 2.1 and Eq. 2.6 without source/sink input, and fWT,i = ∑

npj=1 h jρ jq j is the

well contribution for a thermal part. Each well in the framework is modeled as a separate


do ub l e tmax = 300 ; / / t o t a l s i m u l a t i o n t ime i n daysdo ub l e t f i m = 100 ; / / FIM s i m u l a t i o n t ime i n daysdo ub l e t = 0 , d t = 0 . 1 ; / / c u r r e n t t ime and t i m e s t e p/ / FIM s t r a t e g y a p p l i e d f i r s tsubProb lems . s e t G l o b a l S t a t u s ( 2 ) ;w h i l e t < t f i m

subProb lems . s o l v e ( d t ) ; / / s o l v e FIM [FLOW, THERMAL, MECHANICS]t = t + d t ; / / u p d a t e t ime

/ / e x t e n d e d f i x e d s t r e s s s p l i t a p p l i e d l a t e rw h i l e t < tmax

/ / check c o n v e r g e n c e o f r e l a t e d prob lemsw h i l e ( n o t conve rged ( subProb lems ) )

subProb lems . s e t G l o b a l S t a t u s ( 0 ) ;subProb lems . s o l v e ( d t ) ; / / s o l v e FLOW & THERMALsubProb lems . s e t G l o b a l S t a t u s ( 1 ) ;subProb lems . s o l v e ( d t ) ; / / s o l v e MECHANICS

t = t + d t ; / / u p d a t e t ime

Listing 2.7: Simplified code example demonstrates a hybrid scheme where a flow-thermal-mechanicsproblem is solved in fully implicit manner for first tfim = 100 days, followed by extended fixed-stress splitfor the rest of the simulation (tmax = 300 days). Global status table is taken from Listing 2.6, timestep isfixed for simplicity.

(a)

Global Status FLOW MECHANICS THERMAL REACTIONS

status 1 X – – –status 2 – X – –status 3 – – X –status 4 – – – X

(b)

Figure 2.13: Example 1: (a) sub-problem tree and (b) corresponding global status table. The coupling strat-egy is Ψ = SEQ[FLOW, SEQ[MECHANICS, THERMAL], REACTIONS], where the global sequential approach isused to couple all the sub-problems together (ϒA = SEQ), and the mechanics and thermal sub-problems aresolved sequentially (ϒB = SEQ).


(a)


status 1 X – – –status 2 – X X –status 3 – – – X

(b)

Figure 2.14: Example 2: (a) sub-problem tree and (b) corresponding global status table. The coupling strategyis Ψ = SEQ[FLOW, FIM[MECHANICS, THERMAL], REACTIONS], where the global sequential approach isused to couple all the sub-problems together (ϒA = SEQ), but the geomechanics and thermal sub-problemsare solved in a fully implicit manner (ϒB = FIM). Status 1 activates FLOW, status 2 activates THERMALand MECHANICS simultaneously, and status 3 activates REACTIONS.

sub-problem with corresponding subsets. For example, a simple well has subsets for flow

and thermal perforations, while a more complex multi-segment well has additional subsets

for its segments. However, a well sub-problem is different from flow, thermal or mechan-

ics sub-problems. While each sub-problem might be coupled with other sub-problems

via properties (temperature T in the flow equations for EoS parameters, or pressure p as

an input for mechanics equations), a well sub-problem has additional terms to other sub-

problems such fWF,i for flow and fW

T,i for thermal. These additional terms should appear both

for active and inactive well sub-problems.

For example, Eq. 2.29 can be solved using different coupling strategies such as Ψ1 =

FIM[FLOW,THERMAL, WELLS], Ψ2 = SEQ[FIM[FLOW,WELLS], THERMAL] or Ψ3 =SEQ [FLOW,

FIM[THERMAL,WELLS]]. A fully-coupled strategy Ψ1 is solved when all sub-problems are

active, while THERMAL and FLOW sub-problems are solved with inactive wells for Ψ2 and

Ψ3 correspondingly. Well related terms fWF,i and fW

T,i should appear in flow and thermal

equations both in active and inactive states (well phase inflow q is fixed for the inactive


(a)


status 1 X X X X

(b)

Figure 2.15: Example 3: (a) sub-problem tree and (b) corresponding global status table. The fully couplingstrategy is Ψ = FIM[FLOW, MECHANICS, THERMAL, REACTIONS].

state). Additionally, we may employ a coupling strategy Ψ4 = SEQ[FIM[FLOW,WELLS],

FIM[THERMAL,WELLS]], where sub-problems WELLS are always active. The last strategy is

allowed in the framework since there are no restrictions of uniqueness in a construction of

a global status table.

We extend the framework by adding coupling terms. The logic of an active and inactive

state for additional coupling terms is identical to the activation logic of subset and is enabled

automatically via subsets. However, the appearance of additional terms should be specified

for each sub-problem manually. Thus, we construct the residuals of sub-problems in two

steps. First, we build residual for each active sub-problem, then we add related coupling

terms for all active sub-problems. Using the coupling term mechanism, we can decouple

a sub-problem s into series of sub-problems si (i= 1...ns), where each si is solved with fixed

terms from other sub-problems. For example, we may solve a flux and an accumulation

term in a sequential manner for a flow sub-problem by splitting it into two sub-problems

with one additional term each. This splitting is of practical interest since it can improve

convergence for some problems as reported by Maes et al. (2016).

2.3.4 Coupled Properties

In this section, we describe how to treat properties that are shared by different physics

equations and modules. For example, porosity φ (see Eq. 2.17) is a function of cell pressure


Figure 2.16: Example of a coupled property. Here we have three sub-problems MECHANICS, FLOW andTHERMAL with corresponding mech cell, flow cell and therm cell subsets. Solid lines between sub-problems and subsets indicate that properties defined in these subsets can be changed by a correspondingsub-problem, while dashed lines indicate that these properties are fixed for a related sub-problem. There aretwo coupled properties introduced in this system. Porosity is a function of volumetric stress σv (mech cellsubset) and pressure p (flow cell subset), while mobility depends on pressure p and composition xcp fromflow cell subset, and on temperature T from therm cell subset. Notice flow cell subset coupled withmechanics sub-problem via porosity.

p (flow subset), volumetric strain εv, and mean stress σv (mechanics subsets), while the

phase mobility λ j is a function of pressure p, phase saturation S j, compositions xi j (flow

subsets), and temperature T or enthalpy h (thermal subset). Figure 2.16 demonstrates the

relationship between subsets, sub-problems and coupled properties, such as mobility and

porosity. While in this example both coupled properties depend on two subsets there is no

such limitation in the framework. A coupled property can be a function of multiple subsets

and multiple sub-problems. We can present a coupled property as a sub-problem without

its own subsets and governing equations and with only one defined property belonging to

other subsets.

Both coupled properties and regular (non-coupled) properties relate to subsets; how-

ever, there are several major differences between them:

(i) A property is a function of several subsets of the same problem, while a coupled

property is a function of several subsets from different sub-problems. We link subsets

of other sub-problems to the coupled property at the construction stage. This is similar

to a sub-problem when it uses other subsets without modifying them.

(ii) All subsets and all related properties of the same sub-problem are activated and de-

activated simultaneously. When a sub-problem is an inactive state, corresponding

property is fixed and does not require updates. For a coupled property that does not


hold: a property is a combination of active and inactive subsets and updates every

time when any related active subset is updated. For example, for both coupling strate-

gies Ψ1 = FIM[FLOW, MECHANICS] and Ψ2 = SEQ[FLOW,MECHANICS] porosity φ is

always calculated and updated. In Ψ2, porosity φ is a function of fixed mechanics

variables (volumetric stress σv) and non-fixed flow variables (pressure p) when solv-

ing a flow sub-problem. Similarly, when solving a mechanics sub-problem porosity

is a function of fixed flow properties and non-fixed mechanics unknowns.

(iii) Whenever we apply a global status switch, we calculate all coupled properties first,

followed by computation of the properties that are associated with active sub-problems.

This guarantees the correctness of derivatives produced by automatic differentiation.

In addition, we can update related coupled properties when the corresponding subset

is updated.

(iv) The order of the sub-problems and their properties is invariant. However, the order

of coupled properties depends on the sequential evaluation of corresponding func-

tions used for this property. For example, in thermal-mechanics problems, we may

have two coupled properties: porosity φ and enthalpy h, where φ is a function of h.

While the order of physics (thermal followed by mechanics, or mechanics followed

by thermal) is not critical, the order of coupled properties is strictly defined by the

corresponding relations, thus, h should be calculated prior to φ. The flexibility of

order of coupled properties can be greatly extended using a concept of dependency

properties graph described in Chapter 4.

Listing 2.8 demonstrates a coupled property (porosity φ) structure and usage in the

framework. Any other coupled property can be defined in a similar fashion. Since a coupled

property relates to subsets and not to sub-problems, the relationship towards a sub-problem

is established through a connection between a sub-problem and its subsets.

2.3.5 Mapping Operators

While the presented framework is flexible in terms of physics and coupling properties,

sometimes it is still limited by the stability of coupling strategies. For flow and transport


c l a s s P o r o s i t y : p u b l i c C o u p l e d P r o p e r t y P o r o s i t y ( . . . )l i n k S u b s e t s ( . . . )F l o w C e l l S u b s e t * f l o w c e l l s u b s e t ; / / p o r o s i t y i s a p a r t o f t h i s s u b s e tc o n s t M e c h a n i c s C e l l S u b s e t * m e c h c e l l s u b s e t ; / / l i n k e d f i x e d s u b s e t. . .

P o r o s i t y : : compute ( i n t i b l o c k )

f l o w c e l l s u b s e t −>p o r o s i t y [ i b l o c k ] = func ( f l o w c e l l s u b s e t , m e c h c e l l s u b s e t ) ;

Listing 2.8: Simplified code example for defining porosity as a coupled property, subsets defined bylinkSubsets() method.

Figure 2.17: Modified sequential implicit (m-SFI) approach for a flow-thermal problem. Color annotation:yellow – active domain, gray – inactive domain. We iteratively solve full domain for FLOW together witha strongly coupled domain for THERMAL, followed by solving THERMAL in the rest.

splitting in compositional flow simulation, one way to improve the convergence properties

of a sequential coupling scheme is to apply a modified sequential implicit (m-SFI) algo-

rithm (Moncorge et al., 2017). The m-SFI approach separates the strongly coupled domain

where it solves fully coupled flow and transport, and the weakly coupled domain where

it solves sequentially flow and transport. Using this coupling strategy we take advantage

of the sequential approach in the regions, taking the advantage of the stability of the fully

coupled approach in the rest. In this work, we extend the m-SFI approach for general

multi-physics problems. Figure 2.17 demonstrates the m-SFI approach for a flow-thermal

problem, where we have a strong coupling between flow and thermal in the two-phase

regions around wells and weak coupling in the rest of the reservoir.


Figure 2.18: Schematic representation of m-SFI approach. We apply coupling strategy Ψ = SEQ[FIM[FLOW,THERMAL<M1>], THERMAL<M2>], where M1 and M2 are mapping operators for strongly and weakly coupledphysics correspondingly.

We extend the flexibility of the framework with mapping operators. A mapping oper-

ator specifies the active domain and can dynamically change through a simulation run. It

has several practical applications such as m-SFI, domain decomposition, and multi-scale.

Figure 2.17 is schematically represented by Figure 2.18. We specify two mapping oper-

ators M1 and M2, where M1 defines a strongly coupled domain, and M2 relates to a weakly

coupled part of the reservoir. The criteria for strong coupling may include the number

of phases, CFL number, and distance to a well. Using the mapping operator concept

we can write the m-SFI approach for flow-thermal problem in a general form as Ψ =

SEQ[FIM[FLOW, THERMAL<M1>], THERMAL<M2>], where <> relates to an applied map-

ping operator for a given sub-problem. Similarly, we can write m-SFI for other physics

by applying another mapping operators for different physics, and changing the order as

needed.

In subsection 2.3.1 we introduced one approach to handle domain decomposition by

specifying different subsets for each subdomain. While it has a lot of flexibility in terms

of the interaction between subsets, this approach is difficult for multi-physics domain de-

composition extension. An alternative approach is to specify different mapping operators

for each subset. For example, we can specify domain-decomposition approach for a flow

problem as coupling strategy Ψ = SEQ[FLOW<DD-1>,...,FLOW<DD-N>], where DD-K is

a mapping operator to activate K’s sub-domain.

The GENIC framework supports mapping operators through the extension of the global

status table, where we replace a global status from the activation state (active or inactive)

by a corresponding mapping operator. For example, Ψ1 = SEQ[FLOW,THERMAL] can be


Global Status FLOW THERMAL

status 1 X –status 2 – X

(a)


status 1 FULL NONEstatus 2 NONE FULL

(b)

Figure 2.19: Extension of global status table concept: a) Ψ1 = SEQ[FLOW,THERMAL] and b) Ψ2 =SEQ[FLOW<FULL>,THERMAL<FULL>]. Coupling strategy Ψ2 is Ψ1 written with mapping operators. Here,mapping operator <FULL> specifies a whole applied domain, and <NONE> denotes an inactive domain.


status 1 FULL M1status 2 NONE M2

Figure 2.20: Example of global status table for m-SFI coupling strategy Ψ = SEQ[FIM[FLOW,THERMAL<M1>], THERMAL<M2>]

rewritten explicitly with mapping operators as Ψ2 = SEQ[FLOW<FULL>, THERMAL<FULL>]

(see Figure 2.19). Here, mapping operator <FULL> specifies a whole applied domain, and

<NONE> denotes an inactive domain. Figure 2.20 shows a global status table for m-SFI

coupling strategy Ψ = SEQ[FIM[FLOW, THERMAL<M1>], THERMAL<M2>]. For optimal ef-

ficiency, the mapping operator implementation is a part of the ADETL library. Notice that

each mapping operator can be updated after each Newton iteration.

Listing 2.9 demonstrates an example of the M1 mapping operator where full physics are

required. Mapper M1 is an implementation of interface class Mapper where we only rede-

fine updateMapping() function. We add cells with strong coupling based on CFL criteria

and number of phases into the vector of active cells – m actives. Any mapping operator

can be defined in a similar fashion and will be automatically enabled in the framework.

2.3.6 Solving Algorithm

Typically, different algorithms are used with different coupling strategies (Zienkiewicz

et al., 1988; Settari and Walters, 2001; Kim et al., 2013; Castelletto et al., 2015). How-

ever, because every solution scheme for a multi-physics problem in the GENIC framework

is represented by a tree structure and a global status table, we can apply a general algorithm

for any coupling strategy for which a tree structure can be constructed. We start solving


c l a s s Mapper M1 : p u b l i c Mapperp u b l i c :

Mapper MSFI T1 ( . . . )

v i r t u a l vo id updateMapping ( )

m a c t i v e s . c l e a r ( ) ;f o r ( s i z e t i b = 0 ; i b < t h e r m c e l l s u b s e t −>num blocks ( ) ; ++ i b )

/ / two−phase , w e l l o r c f l c r i t e r i a f o r s t r o n g c o u p l i n gi f ( i s t w o p h a s e ( i b ) | | i s w e l l ( i b ) | | c f l ( i b ) > m i n c f l )

m a c t i v e s . p u s h b a c k ( i b ) ;u p d a t e i s a c t i v e ( ) ;

;p r i v a t e :

/ / used s u b s e t sF l o w C e l l S u b s e t * f l o w c e l l s u b s e t ;T h e r m a l C e l l S u b s e t * t h e r m c e l l s u b s e t ;

;

Listing 2.9: Simplified code example for defining active cells for flow-thermal m-SFI approach.

our system at the head node, which always has at least one child node, by applying the

SOLVE function outlined in Algorithm 1. Note that the SOLVE function is recursively

applied to each node of the tree. In Step 2 of the algorithm, we check the terminal condi-

tion, i.e., this node has no children nodes. Each terminal node has a corresponding status

in the global status table. We activate the global status at Step 3 (see Algorithm 2 for de-

tails) and on Step 4 we solve active sub-problems using a dedicated solver and return the

convergence state. Steps 5–9 describe the convergence logic of a local node; it is similar

to a global convergence check of a node (Step 18–27) with some key differences. First,

the global convergence is often achieved after several sequential iterations (as outlined in

Step 29), while the local convergence takes only one sequential iteration. Second, global

convergence is achieved when a residual norm is below a threshold prior to solving a re-

lated linearized system of equations, i.e. at the first Newton iteration (Step 22). Local

convergence, instead, is achieved when the residual norm is below the threshold at the last

Newton iteration (Step 7).


Algorithm 1: Node solve function pseudocode. The function is called recursively in the sub-problems treepassing algorithm. Here node is the sub-problem tree node, node.children gives an ordered set of correspond-ing children elements, k is a number of sequential iterations between node.children tree nodes, and kmax isa maximum number of such iterations.

1: function SOLVE(node)2: if node is terminal then3: activate global status . see Algorithm 2 for details

4: solve active sub-problems SA using a dedicated linear solver5: for all s ∈ SA do . check local convergence

6: if ||Rs||> εs then . residual norm at the last iteration

7: return node is NOT converged

8: end for9: update variables related to SA

10: return node is converged11: else12: set k = 013: do14: for all i ∈ node.children do15: if SOLVE(node.children[i]) is not converged then16: return node is NOT converged . stop in case of convergence problem

17: end for18: set conv = true . check global convergence

19: for all i ∈ node.children do20: find active sub-problems SA related to node i21: for all s ∈ SA do22: if ||R 1

s ||> εs then . residual norm at the first iteration

23: conv = f alse

24: end for25: end for26: if conv is true then27: return node is converged

28: set k = k+129: while k < kmax


31: end function


We apply a sequential iterative strategy for the children nodes when they exist, i.e., the

node is not a terminal node. We solve each child node independently using a recursive

procedure SOLVE (Step 15). If any node does not converge, we cannot solve the time step

for the coupled problem and a time step cut is required (Step 16). If the node converges,

then we check a global convergence criterion (Step 18–27). We repeat Steps 12-28 until

we reach the global convergence for a given tree node or exceed the maximum allowed a

number of sequential iterations kmax for this node.

Algorithm 2 explains the logic of switching to the new global status. For each sub-

problem, we set its local status to be consistent with the current global status (Step 5).

At Step 7 we assemble a set of active sub-problems SA and apply corresponding mapping

operators. We need to find coupled terms C related to active sub-problems SA. First, we find

which coupled terms Cg are related to the global status (Step 3). Then for each sub-problem,

we add intersection of its coupled terms and Cg into set C (Step 9). Note that we carry out

this procedure for all sub-problems – both active and inactive. For example, an inactive

sub-problem might have coupled terms which appear in equations for active sub-problems;

failing to add these coupling terms will lead to inconsistent derivatives. Once a set of

active sub-problems SA and related coupled terms C are defined, we reorder unknowns

using ADETL capabilities and update coupled terms followed by sub-problems (Step 11–

13).

Algorithm 1 provides a general approach to solve a multi-physics problem using a given

coupling strategy. It can be shown that it is identical to available algorithms (Settari and

Walters, 2001; Garipov et al., 2016b) when applied for two or three physics problems. Al-

gorithm 3 and Algorithm 4 are algorithms to solve fully coupled Ψ1 = FIM[FLOW, MECHANICS]

and sequentially coupled Ψ2 = SEQ[FLOW, MECHANICS] strategies for a flow-mechanics

problem correspondingly (adapted from (Garipov et al., 2016b)). The coupling strategy

Ψ1 is represented by a sub-problem tree with one terminal node FIM[FLOW, MECHANICS].

It can be seen that Steps 2–10 from Algorithm 1 coincide with Algorithm 3. The cou-

pling strategy Ψ2 is represented by a sub-problem tree with two terminal nodes (FLOW and

MECHANICS) and with the one parent node (SEQ[FLOW, MECHANICS]).

Algorithm 5 is Algorithm 1 applied to Ψ2 with two coupled terms: volumetric stress

σv and pressure p. The right side of Algorithm 5 explains Steps 4–9 which are identical to


Algorithm 2: Global status activation pseudocode. The procedure should be called prior to solving anysystem of equations to guarantee correct derivatives. S - set of all sub-problems available from sub-problemtree, gstatus - boolean array of active status for each sub-problem in S (a corresponding line from the globalstatus table).

1: procedure ACTIVATE GLOBAL STATUS(gstatus)2: set SA = ∅, C = ∅ . SA – set of active sub-problems, C – set of coupled problems3: find set of coupled terms Cg related to SA

4: for each sub-problem s ∈ S do5: status(s) = gstatus(s) . update local sub-problem status6: if gstatus(s) is active then7: SA← s8: find set of couple terms Cs related to s9: C←Cg∩Cs

10: end for11: restructure Jacobian12: update coupled terms C13: update variables related to SA

14: end procedure

Algorithm 3: Fully implicit solution strategy for a flow-mechanics problem (adapted from Garipov et al.(2016b)), xn represents all unknowns on the previous time step.

1: F = (xF ,xn),M = (xM,xn)2: do3: calculate residual and Jacobian R (F,M), J (F,M)4: solve system of linear equations5: update RF and RM

6: calculate error norm ‖R ‖7: while (‖R ‖> ε)

Steps 5–17 from Algorithm 4. The convergence condition of Algorithm 5 (Step 10) is iden-

tical to the convergence condition from Algorithm 4 (Step 11 and Step 18) because residual

norms are taken at the first Newton iteration prior to local convergence. The comparison of

these strategies makes clear that Algorithm 1 is indeed a general algorithm and consistent

with algorithms available from the literature for two and three physics problems. Similar

observations can be made for flow-thermal (van Odyck et al., 2009; Trangenstein, 1989) or

thermal-mechanics (Aboustit et al., 1985; Armero and Simo, 1992) problems.


Algorithm 4: Sequential implicit solution strategy for a flow-mechanics problem (adapted from Garipovet al. (2016b)).

1: F = (xF ,xn),M = (xM,xn)2: set sequential iteration k = 03: set Fk = F and Mk = M4: do5: update volumetric stress σk+1

v6: do7: calculate residual and Jacobian RF(Fk+1,Mk) JF(Fk+1,Mk)8: solve system of linear equations9: update Fk+1

10: while (RF > εF )11: calculate error norm for mechanics RM(Fk+1,Mk)

12: use fixed Fk+1

13: do14: calculate residual and Jacobian RM(Fk+1,Mk+1) JM(Fk+1,Mk+1)15: solve system of linear equations16: update Mk+1

17: while (RM > εM)18: calculate error norm for flow RF(Fk+1,Mk+1)19: set k = k+120: while (‖RF‖> εF ,‖RF‖> εF)

Algorithm 5: Example of application of Algorithm 1 to a specific coupling strategy Ψ =SEQ[FLOW,MECHANICS]. On the left – reduced algorithm, on the right – some specific details of Steps 4-9. The produced algorithm is identical to Algorithm 4.

1: F = (xF ,xn),M = (xM,xn)2: set k = 03: do4: activate global status FLOW5: solve FLOW sub-problem6: update variables related to FLOW7: activate global status MECHANICS8: solve MECHANICS sub-problem9: update variables related to MECHANICS

10: if ||R 1F ||> εF , ||R 1

M||> εM then11: return node is converged12: set k = k+113: while k < kmax


Steps 4-6 in details:4.1: Fk+1 - active, Mk - inactive (fixed)4.2: SA = Fk+1,C = σk+1

v (fixed stress)5.1: calculate RF(Fk+1,Mk), JF(Fk+1,Mk)5.2: solve system of linear equations6: update Fk+1

Steps 7-9 in details:7.1: Mk+1 - active, Fk+1 - inactive (fixed)7.2: SA = Mk+1,C = pk+1 (fixed pressure)8.1: calculate RM(Fk+1,Mk+1)8.2: calculate JM(Fk+1,Mk+1)8.3: solve system of linear equations9: update Mk+1

Chapter 3

Numerical Applications

The objective of this chapter is to demonstrate the power and flexibility of the GENIC

framework in modeling complex multi-physics problems. We consider a series of nonlinear

problems that involve several mechanisms of flow and transport, and we show that the

GENIC framework allows for investigating the full range of possible coupling strategies.

The chapter is organized as follows. First, we apply the framework to solve a flow-thermal

problem (i.e., multi-component multiphase flow and transport of mass and energy). Next,

we explore existing (constant pressure) and novel coupled properties, such as a constant

density and a hybrid scheme for flow-thermal sequential coupling strategies. We investigate

the convergence properties of three different sequential coupling schemes.

Then, we demonstrate the convergence improvement by extending the modified Se-

quential Fully Implicit (m-SFI) method proposed recently by Moncorge et al. (2017) for

isothermal compositional problems. Following that, we consider coupled flow and me-

chanics problems and demonstrate how the proper choice of the coupling strategy depends

on the coupling strength between the mass and momentum balance equations. Finally, we

apply the GENIC framework to a flow-thermal-mechanics problem for which little guid-

ance is available on which coupling strategy to employ.

53

CHAPTER 3. NUMERICAL APPLICATIONS 54

3.1 Flow and Thermal Coupling

Numerical modeling of multiphase flow with thermal effects (flow-thermal) is quite chal-

lenging. The mass conservation equations are highly nonlinear and tightly coupled, and

the overall energy conservation is also nonlinear with a wide range of coupling strength

to the mass balances both in space and in time. There is a limited investigation of se-

quential methods to model flow-thermal problems (Trangenstein, 1989; van Odyck et al.,

2009). The work of van Odyck et al. (2009) is an extension of a compositional solver

for non-isothermal multi-component two-phase flow and transport of mass and energy de-

veloped in Trangenstein (1989). It was shown that the system of equations can be split

into a parabolic pressure equation and a hyperbolic system of conservation laws for the

component density and the enthalpy. The authors successfully demonstrated a sequential

method, where pressure equation is solved implicitly, followed by an explicitly solved ther-

mal equation. Unfortunately, available studies have two limitations, which are addressed

here. First, due to the explicitness of enthalpy, all the numerical tests were conducted for

a CFL < 1. This CFL is relatively low in comparison to the CFL size used for problems

of practical interest. We couple flow and thermal sub-problems using a sequential method,

which does not have such a strict time step limitation. As a result, we were able to reach

CFL ≈ 100. Second, there is no comparison between FIM and published sequential meth-

ods for a compositional-thermal problem in terms of convergence and performance. We

compare both methods using the flexibility provided by the GENIC framework. The result

of this preliminary study was first published in Wong et al. (2017).

For a coupled flow-thermal problem we introduce a vector of unknowns, x = (xF ,xT ),

where xF are the flow unknowns, and xT are the thermal unknowns. The corresponding

residuals are RF(xF ,xT ) and RT (xF ,xT ). We use a pressure-enthalpy formulation (Wong

et al., 2016), and solve the following system of equations for FIM coupling strategy (xn is

all unknowns on the previous time step):Rn+1F(xn+1

F ,xn+1T ,xn)= 0,

Rn+1T(xn+1

F ,xn+1T ,xn)= 0.

(3.1)

For the implicit sequential coupling strategy SEQ, the flow and thermal equations are


solved separately, and coupled through carefully chosen coupled properties. We use cou-

pled properties cF and cT for flow and thermal equations correspondingly. We model se-

quential algorithm for the flow-thermal problem using Algorithm 1 applied for a problem

with two physics modules as following:

Step 1. Solve the flow sub-problem:

Rn+1F (xn+1

F , xT (δcn+1F = 0)) = 0 (3.2)

This involves solving the flow equation with a variable δcF = 0. We assume that the vari-

able cF is fixed while we solve the flow equation. The choice of variable for cF affects

the convergence and numerical stability of the coupled system. Here, we take xF ≡ p and

cF ≡ h – the fluid enthalpy of the system.

Once we have the solution xn+1F , we use that as the initial guess to the solution of the

thermal equation.

Step 2. Solve the thermal sub-problem

Rn+1T (xF(δcn+1

T = 0), xn+1T ) = 0 (3.3)

This involves solving the thermal equation using δcT = 0. Similar to cF , the variable cT

is held constant when solving the thermal equation. We repeat Steps 1 and 2 until global

convergence is achieved.

We apply three different strategies for the coupled property cT :

1. First scheme is with constant pressure, where we assume that the pressure is constant

(cT ≡ p) in the thermal residual equation. We can rewrite Eq. 3.3 as follows:

Rn+1T (xF(δpn+1 = 0), xn+1

T ) = 0

This is the same as solving the equation with ∂p/∂h = 0.

2. Second scheme is with constant density, where we assume each block has a constant

total density (cT ≡ ρ). We first compute the density ρ(p,h) after Step 1 (Eq. 3.2),


and solve Step 2 (Eq. 3.3) as follows:

Rn+1T (xF(δρ(p,h)n+1 = 0), xn+1

T ) = 0

We calculate ∂p∂h using the inverse function:

∂p∂h

(δρ = 0) =∂F∂h

(∂F∂p

)−1

where F(p,h) = p−ρ(p,h).

3. Third solution strategy is a combination of constant-pressure and constant-density

formulations. We refer to this formulation as the hybrid formulation. This mixed

formulation depends on the phase state of the block. For this formulation, we chose

a constant-pressure ‘coupled property’ for the single-phase blocks and constant-

density for two-phase blocks. We define the phase-state of the block by comparing

the enthalpy h with the saturated enthalpies of the block hw(p) and hs(p). More

formally, we define the coupled property as:

cT ≡

ρ, hw(p)≤ h≤ hs(p)

p, otherwise

3.1.1 Example 1. Avdonin’s Problem

We consider a geothermal problem (Avdonin, 1964; Wong, 2015) for which an analytical

solution is available. This test case involves a one-dimensional linear steady-state flow and

unsteady heat transport of single-phase liquid water. Cold water at T = 423 K is injected

with a flow rate of q = 10 m3/day into a reservoir with water at T = 473 K. The boundary

condition imposed is a pressure of 50 ·105 Pa at an outer radius of 1000 m (see Figure 3.1).

The model properties are shown in Table 3.1. The analytical solution is given by Avdonin

(1964):

u(χ,τ) =χ

(πτ)1/2

∫ 1

0exp

[−(

sγ(τ)1/2− χ

2s(τ)1/2

)2]

dss2 , (3.4)


Figure 3.1: Schematic representation of Avdonin’s problem (uniform linear spacing).

where χ = 2x/b, τ = 4Kmt/cmρmb2, γ = Qcmρw/4Km, Q is the injection rate, and b is the reservoir

thickness.

Figure 3.2 shows the temperature distribution at t = 1000 days and the temperature evo-

lution at x = 25 m, using fully implicit Ψ1 = FIM[FLOW, THERMAL], sequentially implicit

SEQ Ψ2 = SEQ[FLOW, THERMAL] coupling strategies, and an analytical solution. We use

constant pressure cT ≡ p as a coupled property for the sequential approach. The resulting

solutions match within the desired tolerance for all applied coupling strategies. The se-

quential approach normally requires 3–7 iterations for a tolerance ε = 10−6 to accurately

capture the analytical solution (Figure 3.3). We note that the error decreases monotonically

with iteration.

3.1.2 Example 2. Injection Flash Front

This numerical model is a one-dimensional, radial model. Cold water at T = 355 K is

injected at a constant pressure pin j = 90 bars into a reservoir with water at T = 523.15 K

(Wong et al., 2017). We consider two scenarios: single-phase and two-phase. The single-

phase case involves injection in the radial model, where the fluid remains as a single-phase

liquid water with a fixed outer boundary reservoir pressure at pres = 45 bars. The two-phase

Table 3.1: Parameters for Avdonin’s problem.

Reservoir length, m 1000Reservoir thickness, m 100Rock permeability, mD 1000Rock density, kg/m3 2500Rock specific heat capacity, J/(g·K) 1.0Porosity 0.2Spatial grid with uniform 20 m grid spacing


Figure 3.2: Avdonin’s problem. Comparison of the analytical and numerical solutions of the temperaturedistribution at t = 1000 days (left), and the temperature evolution at location x = 25 (right).

Figure 3.3: Avdonin’s problem. Sequentially implicit method convergence profile after the first time step ∆t= 0.001 days.

case involves the propagation of the injected cold-water as a sharp front with an initial wa-

ter saturation Sw = 0.1 and a fixed outer boundary reservoir pressure at pres = 10 bars (see

Figure 3.4). The model properties are shown in Table 3.2. We compare the fully implicit

Ψ1 = FIM[FLOW, THERMAL], and the sequential implicit SEQ Ψ2 = SEQ[FLOW, THERMAL]

coupling strategies. We study three different coupled properties for Ψ2: constant pressure,


Figure 3.4: Schematic representation of Injection Flash Front problem (uniform radial spacing).

Table 3.2: Parameters for Injection Flash Front problem.

Reservoir length, m 1000Rock permeability, mD 500Rock density, kg/m3 2500Rock thermal conductivity, kJ/(m·day·K) 1728Porosity 0.2Number of radial blocks 10, 50, 150Spatial grid (radial) uniform grid spacing

constant density, and hybrid schemes. We compare the time step size in terms of the maxi-

mum CFL number, which is defined as:

CFLmax =∆tQV φ

(3.5)

where ∆t is the maximum time step, Q is the injection rate, V φ is pore volume of the block.

Single-Phase scenario. Figure 3.5 shows the pressure and temperature distribution after

t = 100 days. The injection front is located around x = 200 m. The temperature front is

smeared due to the difference in dynamics between flow and thermal effects, and also due

to numerical dispersion. All of the tested coupled properties yield very similar results.

There is a small difference due to varying time step for the constant density case, which

is not stable for the single-phase scenario. Figure 3.6 shows the pressure and temperature

profiles at x = 167 m. The pressure reaches its maximum value (around 80 bars) when the

injection front reaches x = 167 m (around 10 days) and subsides after that. The pressure

increases slightly after 90 days, and this is caused by a decrease in temperature. We conduct

a refinement study using 10, 50, and 150 grid blocks. Figure 3.7 and Figure 3.8 show

convergence to the true solution.

Table 3.3 summarizes a comparison between sequential coupling schemes with three

different coupled properties and the fully coupled approach. In the single-phase case, the


Figure 3.5: Single-phase injection flash front. Pressure and temperature distribution after 100 days (50 gridcells).

Figure 3.6: Single-phase injection flash front. Pressure and temperature profile at x = 167 m (50 grid cells).

constant pressure and hybrid schemes achieve nearly identical nonlinear performance. This

is because the entire domain remains in single-phase water through the entire simulated

period; that makes pressure a good choice as a coupled property for all the grid blocks.

We can reach maximum time step for constant pressure, hybrid and fully implicit method

without time step cuts. The constant pressure takes 4-5 sequential iterations to decrease

the residual to ε = 10−6. For the constant density case, the maximum time step was about


Figure 3.7: Grid refinement for single-phase radial one-dimensional injection scenario. Pressure and temper-ature distribution after 100 days.

Figure 3.8: Grid refinement for single-phase radial one-dimensional injection scenario. Pressure and temper-ature profile at x = 167 m.

an order of magnitude less than the other three schemes; moreover, the constant-density

formulation suffers from a large number of wasted iterations (i.e. makes a lot of time step

cuts).

Two-Phase scenario. Figure 3.9 shows the pressure and temperature distribution after

t = 100 days. The injection front is located around x = 350 m. For this scenario, the

constant pressure does not converge regardless of the chosen time step.


Table 3.3: Single-phase injection flash front. Comparison between sequential coupling schemes, such asConstant Pressure (CP), Constant Density (CD), Hybrid (H) and fully coupled method (FIM).

Number of blocks

10 50 150

CP CD H FIM CP CD H FIM CP CD H FIM

number of timesteps 37 599 37 37 73 6432 73 73 120 17633 120 120total Newton iterations 199 12199 199 58 549 73751 549 137 952 185733 952 234

total seq iterations 133 18716 133 0 340 176742 340 0 581 478988 581 0timesteps wasted 0 586 0 0 0 6424 0 0 0 17625 0 0

wasted Newton iterations 0 23408 0 0 0 256784 0 0 0 702690 0 0maximum CFL 2.16 0.27 2.16 2.16 29.57 0.91 29.57 29.53 99.64 6.90 99.64 99.64

Figure 3.9: Two-phase radial one-dimensional injection scenario. Pressure and temperature distribution after100 days (50 grid cells).

Other tested coupled properties produce results that are quite similar to the fully implicit

approach. Figure 3.10 shows the pressure and temperature profile at x = 167 m. Here

we observe numerous fluctuations in the pressure profile; such behavior poses significant

challenges for the sequential scheme. We conduct a refinement study using 10, 50, and 150

grid blocks. Figure 3.11 and Figure 3.12 show convergence to the true solution.

Table 3.4 summarizes a comparison between sequential coupling schemes using two

coupled properties with the fully coupled approach. In all the cases studied here, the hybrid

scheme outperforms the constant-density formulation allowing for larger time steps. More-

over, the hybrid scheme requires less sequential iterations (9-10 in average versus 16-20 for

constant density) for a given time step. Overall, the hybrid scheme scales much better than

the constant density scheme. Nevertheless, the hybrid scheme is not efficient enough to


Figure 3.10: Two-phase radial one-dimensional injection scenario. Pressure and temperature profile at x =167 m (50 grid cells).

Figure 3.11: Grid refinement for two-phase radial one-dimensional injection scenario. Pressure and temper-ature distribution after 100 days.

replace the fully implicit approach. While the hybrid scheme produces less wasted time

steps for 10 grid blocks, it scales non-linearly with grid size (36 wasted time steps for 10

blocks and 3,121 for 150 blocks), while FIM shows excellent scalability.


Figure 3.12: Grid refinement for two-phase radial one-dimensional injection scenario. Pressure and temper-ature profile at x = 167 m.

Table 3.4: Two-phase injection flash front. Comparison between sequential coupling schemes, such as Con-stant Density (CD), Hybrid (H) and fully coupled method (FIM). Note: Constant Pressure does not convergefor two-phase case.

Number of blocks

10 50 150

CD H FIM CD H FIM CD H FIM

number of timesteps 325 58 71 1835 474 306 5978 3141 743total Newton iterations 6071 625 271 28956 8338 1179 90632 55309 3080

total seq iterations 9436 331 0 52103 9272 0 169631 85631 0timesteps wasted 314 36 51 1825 452 280 5969 3121 699maximum CFL 1.13 3.50 3.45 9.49 28.53 49.69 20.76 80.92 99.14

3.1.3 Summary

We demonstrated the applicability of the GENIC framework for flow-thermal problems.

First, we validated our implementation for Avdonin’s problem which has an analytical so-

lution. Next, we explored existing (constant pressure) and novel coupled properties, such

as a constant density and a hybrid scheme for flow-thermal sequential coupling strategies.

We investigated the convergence properties for three different sequential coupling schemes.

We found that a hybrid method where the ‘coupled property’ was pressure for single-phase

blocks and density for two-phase blocks performed the best. This hybrid method was able


to avoid convergence problems in the single-phase (constant density) or two-phase (con-

stant pressure) regions. More numerical tests also showed that the fully implicit method

performed significantly better than any sequential scheme we tried. This preliminary study

might be extended with mathematical analysis of the different coupling schemes for flow-

thermal problems. Further mathematical analysis could provide insight as to how to design

better nonlinear solvers for the fully implicit solution and test them with the GENIC frame-

work.

As we reported in Wong et al. (2017), existing sequential schemes for flow-thermal

problems have convergence issues for more complex scenarios. For example, for a two-

dimensional heterogeneous reservoir, the hybrid approach is 50-100 times slower than fully

implicit approach. We developed modified SFI for flow and thermal physics to overcome

this challenge and will present its results in the next section.


3.2 Modified SFI for Flow-Thermal Problems

We demonstrate the use of the GENIC framework for studying the robustness and efficiency

of different coupling strategies for thermal-compositional simulation. We focus on the

modified SFI (Moncorge et al., 2017), which was designed as a stable splitting approach for

the flow problem (i.e., coupled multi-component multiphase flow with mass transfer). Here,

we applied a variant of the m-SFI approach for flow and thermal physics with promising

results.

In the GENIC framework, the m-SFI approach is assembled by using the mapping

operator introduced in Section 2.3.5. We solve the flow-thermal problem at hand with the

following coupling strategy: Ψ = SEQ[FIM[FLOW, THERMAL<M1>], THERMAL<M2>]. We

define M1 as a region with strong coupling effects, which satisfies one of the following

conditions for each grid block: two or three phase region, connected to a well perforation,

CFL >CFLtol . Similarly, we define M2 as a domain with weak flow-thermal coupling such

that it meets one of the following criteria: single-phase, CFL <CFLtol . For m-SFI we use

constant pressure as a coupled property given its efficiency for a single phase region.

The fully coupled system can be written asRn+1F(xn+1

F ,xn+1T ,xn)= 0,

Rn+1T(xn+1

F ,xn+1T ,xn)= 0.

(3.6)

and in linearized matrix form, we can write[∂RF/∂xF ∂RF/∂xT

∂RT/∂xF ∂RT/∂xT

][δxF

δxT

]=−

[RF

RT

]. (3.7)

To solve the problem, we split the thermal equation RT into two equations RT1 and RT2

written for domains M1 and M2 correspondingly with unknowns xT1 and xT2 . Then, we solve

the flow-thermal problem in two steps:

Step 1: Solve full flow system with a strongly coupled thermal domain M1 and fixed


unknowns xT2 . [∂RF/∂xF ∂RF/∂xT1

∂RT1/∂xF ∂RT1/∂xT1

][δxF

δxT1

]=−

[RF

RT1

](3.8)

Step 2: Solve the thermal equation for the weakly coupled domain M2 with fixed xF :[∂RT2/∂xT2

][δxT2

]=−

[RT2

]. (3.9)

We repeat Step 1 and Step 2 until we reach convergence for RF , RT1 , and RT2 .

3.2.1 Example 3. 3D m-SFI Case

A three-dimensional problem based on a reservoir model from Stanford Geothermal Pro-

gram (1980) is used here. The reservoir model consists of a single-phase liquid water

with a two-phase zone of immobile steam sandwiched between hot and cold water regions.

Production is from a well that is completed below the two-phase zone. The parameters

were chosen such that boiling in the well occurs after a certain period of production. Al-

though the parameters are relatively homogeneous, it was used as a prototype for field-wide

reservoir development studies. Figure 3.13 depicts a schematic representation of the layer

configuration. This problem has both three-dimensional flow with phase transitions and

two-phase flow including gravity drainage. The production well was completed in the

corner block and perforates layers 16-20. The water rate at 500 m3/day was used as the

well control. The simulation time was 10 years. A full discussion on the problem can be

found in Stanford Geothermal Program (1980). The model parameters are summarized in

Table 3.5.

We compare several coupling strategies. First, we apply the fully implicit coupling

strategy Ψ1 = FIM[FLOW, THERMAL]. The second approach is the sequential coupling strat-

egy Ψ2 = SEQ[FLOW, THERMAL] with the hybrid coupling property discussed previously

(Section 3.1). Hybrid coupling approach is effective for single and two-phase regions in

1D or 2D cases; however, coupling strategy Ψ2 has limited stability and poor performance

for heterogeneous two or three-dimensional cases. Third, we apply m-SFI approach with

a coupling strategy Ψ3 = SEQ[FIM[ FLOW, THERMAL<M1>], THERMAL<M2>] over a range

of CFLtol parameters (0.1, 1, 10). Fourth, we change M1 domain in m-SFI approach and, in


Table 3.5: Parameters for 3D model for m-SFI study.

Reservoir dimensions, m 4000×5000×1800Reference porosity in each layer [0.2, 0.25, 0.25, 0.25, 0.2, 0.2]Layer permeability in x and y directions, md [100, 200, 200, 200, 100, 100]Layer permeability in z direction [2, 50, 50, 50, 2, 2]Layer gas saturation [0, 0.15, 0, 0, 0, 0]Depth at layer top, m [150, 450, 750, 1050, 1350, 1650]Temperature at layer top, K [433.15, 553.15, ..., 553.15]Pressure at layer top, bar [40, 64, 88, 112, 136, 160]

Figure 3.13: Schematic of the three-dimensional model.

addition to CFL condition equals to 1.0, we add one layer of neighboring cells for strongly

coupled domain M1 (referred as MSFI-N). We use a direct linear solver PARDISO for nu-

merical simulations.

Figure 3.14 shows the pressure and temperature profiles for a 20×40×36 grid at pro-

duction well block. Figure 3.15 represents the pressure and temperature distributions at

16th layer after 10 years of production. As a consequence of production, pressure drops in

the wellblock, so that horizontal and vertical flow towards the wellblock is initiated. Down-

ward flow from the two-phase zone gives rise to boiling and an increase in the vapor satura-

tion. As the pressure decline spreads to the margins of the field, water recharge is initiated.


Figure 3.14: 3D m-SFI case. Pressure and temperature profile for 20×40×36 grid at production well block.

One consequence of this is the occurrence of several phase transitions to single-phase con-

ditions in the two-phase layer. Thus, large pressure drops occur in the wellblock, as well

as in the adjacent grid blocks, causing several phase transitions to two-phase conditions

and subsequent boiling. Transition period is accompanied by a decline in temperatures

and pressures, as well as a buildup of vapor saturation. Steam-water counterflow occurs

as steam rises from the shallow two-phase layer, whereas water flows downward towards

the production well. Figure 3.16 depicts relative error map between FIM and MSFI-N

approach where the difference is less than 0.01%.

Table 3.6 summarizes a comparison between the sequential coupling scheme with the

hybrid coupled property, m-SFI, MSFI-N, and the fully coupled approach for 10×20×18

blocks. The SEQ scheme, even with hybrid splitting, is unstable and is not a suitable

candidate to solve this problem. We omit the results for the SEQ scheme for other grid

resolutions where we observe similar poor performance. All the presented m-SFI schemes

show better convergence than FIM, have fewer time steps that fail to converge (4 for FIM vs

1 for m-SFI), and converge to the desired tolerance (as shown on Figure 3.16). We observe

that tighter CFL tolerance criterion decreases the number of Newton iterations (from 324

for CFLtol = 10 to 306 for CFLtol = 0.1). However, the convergence improvement is rel-

atively small due to an inability of CFL criterion alone to accurately predict for two-phase

region. MSFI-N approach employs more aggressive strategy with additional neighboring


(a) (b)

Figure 3.15: 3D m-SFI case. Pressure (a) and temperature (b) distribution for 20×40×36 grid after 10 yearsof production at 16th layer.

(a) (b)

Figure 3.16: 3D m-SFI case. Relative error between FIM and MSFI-N approach for pressure (a) and temper-ature (b) for 20×40×36 grid after 10 years at 16th layer.

cells. As a result, two-phase region is predicted more accurately, and we observe the drop

in the number of Newton iterations (from 306 for m-SFI to 213 for MSFI-N) with the slight

increase in the coverage of M1 domain. For this grid resolution, m-SFI schemes simulation

takes 3% more CPU time than FIM.

Figure 3.17 demonstrates the convergence of the pressure and temperature with grid

refinement. The accuracy of the solution increases for a more refined grid. This is due to


Table 3.6: m-SFI 3D case for 10× 20× 18 blocks. Comparison between different coupling schemes: FIM,SEQ, m-SFI and MSFI-N.

m-SFI

FIM SEQ CFLtol = 0.1 CFLtol = 1 CFLtol = 10 MSFI-N

number of timesteps 40 215 39 39 39 39total Newton iterations 136 5050 306 308 324 213

total sequential iterations 0 6681 134 136 137 105number of timesteps wasted 4 201 1 1 1 1

% of blocks in M1 100 0 27.89 19.25 19.06 30.30maximum CFL 5.343 2.707 5.342 5.342 5.342 5.342

the complex nature of steam-water propagation in the system. Table 3.7 summarizes the

performance comparison between the applied coupling strategies for a low grid resolution

(4× 5× 6) and a finer grid resolution (20× 40× 36). In our example, a low grid resolu-

tion does not allow us to capture all the complex physics; as a result, all m-SFI coupling

strategies perform relatively well compared with FIM. Once we increase our grid resolu-

tion, more phase changes occur, and m-SFI strategies based on a CFL criterion struggle to

converge when a phase change occurs, leading to numerous time step cuts. MSFI-N does

not have this issue due to an additional layer of neighboring cells. As a result, the MSFI-N

approach is stable, and it performs as well as FIM on a finer grid. Moreover, the MSFI-

N simulation is about 10% faster than FIM. Unfortunately, the GENIC framework linear

solver capabilities are not fully developed, and more advanced linear solvers are needed to

conclude if MSFI-N can outperform FIM on complex large-scale problems.

Table 3.7: m-SFI 3D case for low grid resolution 4× 5× 6 (L) and finer grid resolution 20× 40× 36 (H).Comparison between different coupling schemes: FIM, SEQ, m-SFI and MSFI-N.

m-SFI

FIM CFLtol = 0.1 CFLtol = 1 CFLtol = 10 MSFI-N

L H L H L H L H L H

number of timesteps 19 162 19 584 19 592 19 592 19 162total Newton iterations 28 454 67 1982 67 2004 67 2016 41 585

total sequential iterations - - 49 1901 49 1918 49 1946 40 470number of timesteps wasted 0 1 0 433 0 436 0 436 0 1

% of blocks in M1 100 100 19.85 16.83 17.50 15.80 17.50 15.21 50.83 24.39maximum CFL 0.51 4.07 0.51 4.07 0.51 4.07 0.5 4.07 0.51 4.07


Figure 3.17: Grid refinement for 3D m-SFI case. Pressure and temperature profile at production well block.

3.2.2 Example 4. m-SFI SPE 10

Our second example for investigating m-SFI based coupling strategies is the SPE 10 test

problem (Christie and Blunt, 2001). The original data set, including relative permeabil-

ity curves, porosity and permeability distributions, and fluid properties, was used in this

example. The simulation domain Ω = 365.8×670.6×51.8 m is discretized using a regu-

lar Cartesian grid into 60×220×85 cells. The top 35 layers represent a formation with a

permeability field that has the Gaussian distribution, and the bottom 50 layers are character-

ized by a channelized permeability field with high-contrast features. We limit our domain

by using only the top four layers from the original SPE 10 problem. The initial reservoir

pressure is 50 bar filled with steam at temperature T = 523.26 K. We have two water in-

jectors on the left and right side of the field, injecting at T = 433.15 K and p = 90 bar,

and two producers at the inner part of the reservoir at p = 30 bar. All wells operate under

BHP constraint. Figure 3.18 represents porosity, permeability and well configuration. The

model parameters are summarized in Table 3.8.

We performed three levels of grid refinement: fine scale (60× 220× 4), coarse scale

(30× 110× 2), and very coarse scale (15× 55× 1). The upscaling of the properties was

performed using a simple volume-averaging approach. This leads to a slightly different

solution between the different grid resolutions. The resulting problem is challenging even


Figure 3.18: Schematic of the SPE 10 setup: permeability (top) and porosity (bottom) maps.

Table 3.8: Parameters for SPE 10 model for m-SFI study.

Reservoir dimensions, m 365.8×670.6×51.8Initial reservoir pressure, bar 50Initial reservoir temperature, K 523.26BHP, bar 30 for producers, 90 for injectorInjection temperature, K 433.15Relative phase permeability kr j = S2

jRock thermal expansion, 1/K 2 ·10−5

Rock heat capacity, kJ/(kg·K) 2000.0Rock thermal conductivity, kJ/(m·day·K) 150.0Oil thermal conductivity, kJ/(m·day·K) 11.5Gas thermal conductivity, kJ/(m·day·K) 3.586Water thermal conductivity, kJ/(m·day·K) 53.5

for the FIM method, and it struggles to converge with more aggressive time step and well

constraints.

We employ similar coupling strategies to those used in Example 3. First, we apply the

fully implicit coupling strategy Ψ1 = FIM[FLOW, THERMAL]. The second approach is the

sequential coupling strategy Ψ2 = SEQ[FLOW, THERMAL] with a hybrid coupling property


discussed previously (Section 3.1). Here, hybrid coupling approach proved to be unstable,

and failed to converge. Third, we apply m-SFI approach with a coupling strategy Ψ3 =

SEQ[FIM[FLOW, THERMAL<M1>], THERMAL<M2>] with CFLtol equal to 0.1. We observed

that larger CFL parameters lead to poor convergence and instability. Fourth, we employ

MSFI-N approach combined with CFL criteria, where we took CFLtol equal to 0.1 and 1.

Table 3.9 summarizes comparison between discussed coupling strategies for resolution

30×110×2 blocks. Here, we observe that m-SFI with CFLtol = 0.1 outperforms MSFI-N

with CFLtol = 1, and takes half the time of the last approach. This observation is different

from Example 3, where MSFI-N gave the best performance even when small CFL criterion

was used for m-SFI method. In our SPE 10 example, we do not only have a complicated

phase-behavior of steam and liquid of the previous example, but also injection and produc-

tion fronts. Therefore, MSFI-N cannot accurately predict the single-phase region which

leads to the poor stability and convergence we observe for the sequential flow-thermal

method in two-phase region. In this scenario, smaller CFL tolerance allows for better con-

trol over single-phase region change. Nevertheless, we can further improve convergence by

combining tight CFL tolerance and neighboring cells: MSFI-N with CFLtol = 0.1 takes 2.6

times less run time. Unfortunately, all presented m-SFI methods perform slower than FIM.

There are several reasons why. First, inaccurate prediction of single-phase region. As we

have seen CFL condition cannot accurately predict phase change. Second, a significantly

larger M1 domain which increased from 25% in the previous case to 70% in SPE 10. This

alone makes every sequential iteration almost as expensive as FIM approach.

Figure 3.19 shows pressure, temperature, and water saturation for 30×110×2 grid at

central block of top layer. All coupling strategies are in good agreement. The small tem-

perature variation caused by mixing cold injected water with water from reservoir region;

Table 3.9: m-SFI SPE 10 upscaled case with top four layers for 30× 110× 2 grid. Comparison betweendifferent coupling schemes: FIM, m-SFI and MSFI-N.

FIM m-SFI (CFLtol = 0.1) MSFI-N (CFLtol = 1) MSFI-N (CFLtol = 0.1)

number of timesteps 53 75 111 53total Newton iterations 179 1869 3974 793

total sequential iterations 0 797 1599 377number of timesteps wasted 9 37 81 8

% of blocks in M1 100 71.75 74.81 67.42normalized run time 1 8.3 17.43 3.15

maximum CFL 44.128 44.238 42.037 44.175


Figure 3.19: SPE 10 m-SFI case. Pressure, temperature, and water saturation profile for 60×220×4 grid atcentral block of top layer.

Figure 3.20: SPE 10 m-SFI case. Pressure (left) and temperature (right) distribution for 60× 220× 4 gridafter 100 days at top layer.

corresponding water saturation shows two-phase region at this block. Figure 3.20 repre-

sents pressure and temperature distribution for the same grid at the top layer after 100 days.

Here, the right part of the reservoir has a higher permeability and leads to a larger injection

region. Meanwhile, pressure and temperature map are not identical since steam rises from

the bottom layers to top.

Figure 3.21 depicts relative error between FIM and m-SFI which is within 10−3% for

pressure and less than 0.5% for temperature. Notice, errors are larger on the border of

two-phase and single-phase regions.

Table 3.10 summarizes performance comparison between applied coupling strategies


Figure 3.21: SPE 10 m-SFI case. Relative error between FIM and MSFI-N approach for pressure (left) andtemperature (right) distribution for 60×220×4 grid after 100 days at top layer.

Table 3.10: m-SFI SPE 10 case for very coarse grid resolution 15×55×1 (L) and original grid resolution (topfour layers) 60× 220× 4 (H). Comparison between different coupling schemes: FIM, m-SFI and MSFI-N.Normalized run time is normalized towards simulation time of FIM approach for each grid resolution.

FIM m-SFI (CFLtol = 0.1) MSFI-N (CFLtol = 1) MSFI-N (CFLtol = 0.1)

L H L H L H L H

number of timesteps 49 114 49 126 49 177 49 115total Newton iterations 77 443 708 4077 1061 8203 708 3412

total sequential iterations 0 0 371 1509 476 2969 371 1323number of timesteps wasted 0 19 0 40 0 106 0 21

% of blocks in M1 100 100 55.18 70.62 22.13 68.63 55.18 74.21normalized run time 1 1 4.92 4.83 6.69 10.21 4.91 4.15

maximum CFL 59.36 62.35 59.36 62.20 59.36 62.69 59.36 62.35

for fine (60× 220× 4) and very coarse grid (15× 55× 1) resolutions. In our example, a

low grid resolution does not allow us to capture all complex physics. As a result, all m-

SFI coupling strategies perform relatively well compared to FIM. Once we increase our

grid resolution, we observe a similar trend as with a coarse grid resolution (30× 55× 2)

in Table 3.9. MSFI-N with CFLtol = 0.1 slightly outperforms m-SFI with CFLtol = 0.1,

where MSFI-N with CFLtol = 1 performs much worse than on two coarser examples. For

presented example, m-SFI approaches are able to converge to the desired tolerance with

good convergence properties. However, given large percentage in M1 domain (over 60%),

m-SFI approach does not outperform FIM approach.


3.2.3 Summary

We presented a new approach for coupling flow and thermal physics. We demonstrated

m-SFI method on several challenging examples, where standard sequential approach fails,

while m-SFI takes a similar number of time steps as FIM approach. In cases where accurate

front prediction is possible and the two-phase region is limited, such as strictly injection

or production problems, m-SFI approach outperforms FIM. However, when the two-phase

process appears in most of the reservoir, the overall performance of the m-SFI method

is more expensive than FIM due to the increased cost of sequential iterations. The perfor-

mance of m-SFI can be improved by considering another coupled property for flow-thermal

which does not suffer in two-phase region. Additionally, m-SFI can be used as a precondi-

tioner for challenging flow-thermal problems.


3.3 Flow and Mechanics Coupling

Flow and mechanics coupling has been extensively studied in the last two decades (Prevost,

1997; Settari and Walters, 2001; Tran et al., 2004; Wheeler and Gai, 2007; Gai, 2004). To

solve the coupled flow and geomechanics problems, four different operator-splitting strate-

gies have been proposed, namely, fixed-stress, fixed-strain, drained, and undrained splits.

Figure 3.22 is a schematic representation of the operator-splitting strategies. The details

of these sequential solution schemes and numerical analysis of their stability and conver-

gence properties can be found in Kim (2010). Kim et al. (2011) and Mikelic and Wheeler

(2013) showed that both fixed-stress and undrained split are unconditionally stable, and

that fixed-stress converges faster than undrained split.

In this section, we demonstrate the applicability of the GENIC framework for flow-

mechanics problems. For a coupled flow-mechanics problem we introduce a vector of

unknowns, x = (xF ,xM), where xF are the flow unknowns, and xM are the mechanics un-

knowns. The corresponding residuals are RF(xF ,xM) and RM(xF ,xM). We solve following

system of equations for FIM coupling strategy (xn is all unknowns on the previous time

step): Rn+1F(xn+1

F ,xn+1M ,xn)= 0,

Rn+1M(xn+1

F ,xn+1M ,xn)= 0.

(3.10)

For the implicit sequential coupling strategy SEQ, the flow and mechanics equations are

Figure 3.22: Schematic representation of different operator-splitting strategies for flow-mechanics problems.(adapted from Kim et al. (2011).


solved separately, and coupled through carefully chosen coupled properties. We use cou-

pled properties cF and cM for flow and mechanics equations correspondingly. We model

sequential algorithm for the flow-mechanics problem using Algorithm 1 applied for a prob-

lem with two physics modules as following:

Step 1. Solve flow sub-problem:

Rn+1F (xn+1

F , xM(δcn+1F = 0)) = 0 (3.11)

This involves solving the flow equation with a variable δcF = 0. We assume that the vari-

able cF is fixed while we solve the flow equation. Here, we take cF ≡ εv (fixed-strain) or

cF ≡ σv (fixed-stress). Once we have the solution xn+1F , we use that as the initial guess to

the solution of the mechanics equation.

Step 2. Solve mechanics sub-problem

Rn+1M (xF(δcn+1

M = 0), xn+1M ) = 0 (3.12)

This involves solving the mechanics equation with a variable δcM = 0. Similar to the cF ,

the variable cM is constant when solving the mechanics equation. For both the fixed-stress

and fixed-strain coupled properties, cF ≡ p. We repeat Step 1 and Step 2 until the global

convergence is achieved.

Remark. Kim (2010) defines operator-splitting strategy as solving two subproblems

(flow and mechanics) sequentially. In this work, a coupling strategy defines a coupling

order of sub-problems (Eq. 2.22), while the exact coupling operator is defined through a

coupled property (see section 2.3.4). Both fixed-stress and fixed-strain operator-splitting

can be represented by the same coupling strategy Ψ = SEQ[FLOW, MECHANICS] with two

different coupled properties: volumetric strain or stress. Here, fixed-stress method is a

short notation for a sequential coupling strategy Ψ with stress as a coupled property. We

define fixed-strain method similarly.


3.3.1 Example 5. Mandel Problem

We consider the two-dimensional Mandel consolidation problem (Mandel, 1953). Man-

del’s problem consists of a homogeneous and isotropic poroelastic slab sandwiched be-

tween two rigid, frictionless, impermeable plates. Both plates are loaded by a compressive

vertical force as shown in Figure 3.23. The slab is infinitely long in the z-direction, with

a cross-section measuring 2a× 2b. The characteristic length to be used in the estimate of

the characteristic time is a. The side surfaces (x =±a) are stress-free, drained, and are kept

at ambient pressure. The force has a magnitude 2F (units of force per unit length in the

z-direction) and is suddenly applied at t = 0, producing both an instantaneous overpressure

p0 and linear displacement uuu0 =[ux,0,uy,0

]T.

We consider a computational domain Ω = (0,a)× (0,b) = (0,100m) × (0,100m), the

geometry represents a quarter of the original Mandel’s problem (see Figure 3.23). The

domain is fully saturated with compressible single-phase fluid, and the initial fluid pressure

is p|Ω = 44.17×105 Pa.

We impose no-flow boundary conditions everywhere, except on the right boundary

where a constant pressure is enforced, p|x=a = 0. We set zero horizontal displacement on

the left boundary (ux|x=0 = 0), zero vertical displacement on the bottom boundary (uz|z=0 =

y

x2b

2a

2F

2F

x

y

(0,0)

(a,b)

Figure 3.23: Sketch of the setup for Mandel’s problem: (left) physical problem, and (right) numerical dis-cretization. Taking advantage of symmetry of geometry and boundary conditions only the gray region corre-sponding to one-quarter of the domain is discretized. Figure is taken from Castelletto et al. (2015).


Table 3.11: Rock and fluid properties used for Mandel’s problem.

Reference porosity 0.375Young’s modulus, Pa 109

Biot’s constant 1.0Poisson ratio 0.25Undrained Poisson ratio 0.47Permeability, mD 1Fluid viscosity, Pa · s 9.81×10−5

Fluid compressibility, Pa−1 4.4×10−10

Reference fluid density, kg/m3 1000

0), and uniform loading conditions on the top boundary (F |z=b = 100×105 Pa). The verti-

cal displacement on the upper boundary is constrained by the condition, uz|∀x=constant.

The undrained Poisson ratio is used to initialize the initial displacement field, and the

drained Poisson ratio is used for the transient simulation. The model properties are shown

in Table 3.11.

The initial conditions are as follows (Abousleiman et al., 1996):

p0(xxx,0) =13a

B(1+νu)F xxx ∈Ω (3.13a)

ux,0(xxx,0) =Fνu

2Gxa

xxx ∈Ω (3.13b)

uy,0(xxx,0) =−F (1−νu)

2Gya

xxx ∈Ω (3.13c)

with B= bM/(K +b2M

)Skempton’s coefficient, νu = [3ν+bB(1−2ν)]/ [3−bB(1−2ν)]

the undrained Poisson ratio. The analytical solutions for p, ux, and uy read (Abousleiman


et al., 1996):

p(x,y, t) =2p0

∞

∑n=1

sinαn

αn− sinαncosαn×(

cosαnxa− cosαn

)exp(−

α2nc f ta2

)(3.14a)

ux (x,y, t) =

[Fν

2Ga− Fνu

Ga

∞

∑n=1

sinαncosαn

αn− sinαncosαn× exp

(−

α2nc f ta2

)]x+

+FG

∞

∑n=1

cosαn

αn− sinαncosαnsin

αnxa× exp

(−

α2nc f ta2

)(3.14b)

uy (x,y, t) =

[−F (1−ν)

2Ga+

F (1−νu)

Ga

∞

∑n=1

sinαncosαn

αn− sinαncosαn× exp

(−

α2nc f ta2

)]y (3.14c)

with αn the positive roots that satisfy the characteristic equation: tan αn =1−ν

νu−ναn.

We employ two coupled properties for sequential coupling strategy Ψ = SEQ[FLOW,

MECHANICS]: fixed-stress and fixed-strain. Table 3.12 summarizes a comparison between

sequential coupling schemes with two different coupled properties and the fully coupled ap-

proach. Fixed-strain method is conditionally stable and can solve Mandel’s problem only

with tiny time steps and loose tolerance (10−3); it diverges for a tighter tolerance. Both

FIM and fixed-stress are unconditionally stable and converge to the desired tolerance. Fig-

ure 3.24 shows the displacement and pressure evolution. Fixed-stress, FIM, and analytical

solution are identical, while fixed-strain converges to a different solution.

Both the FIM and fixed-stress approaches capture the transient solution behavior ac-

curately, and the sequential approach requires 3–20 iterations with tolerance ε = 10−6

(Figure 3.25) which is consistent with published results (Castelletto et al., 2015). The

fixed-strain method is incapable of resolving the solution to the desired tolerance, this is

consistent with the published results (Kim, 2010).


Table 3.12: Mandel’s problem. Convergence between fixed-strain and fixed-stress methods, and FIM. Note,fixed-strain method is conditionally stable and can solve Mandel’s problem only with tiny time steps andloose tolerance (10−3).

fixed-strain fixed-stress FIM

tolerance 10−3 10−3 10−4 10−6 10−3 10−4 10−6

number of timesteps 2,625 86 86 87 86 87 87total Newton iterations 26,501 196 218 1179 21 54 99

total seq iterations 14,723 174 196 688 - - -timesteps wasted 1,860 0 0 0 0 0 0maximum CFL 0.001 0.038 0.038 0.033 0.009 0.004 0.004

Figure 3.24: Mandel’s problem. Comparison of the analytical and the numerical solutions for displacementat location (x = 0, z = 100) (left), and pressure at location (x = 2.5, z = 0) (right). Fixed-strain methodconverges to a different solution.

3.3.2 Example 6. Depletion of a Box-Shaped Reservoir (Deans’ Prob-lem)

Next, we demonstrate the simulation of a test problem presented by Dean et al. (2006).

This example demonstrates the applicability of GENIC to solve a problem where differ-

ent physics modules are applied in different spatial domains. Here, the reservoir, where

both flow and mechanics interact, is fully enclosed within a larger domain that includes

the over- and under-burden formations. Mechanical deformation is accounted for in the

large domain. The simulation region is a box Ω = 18897.6 m × 9448.8 m × 3429.0 m and


Figure 3.25: Mandel’s problem. Sequentially implicit method convergence profile after first time step ∆t =0.001 days. Fixed-strain method cannot decrease error less than 10−3, while fixed-stress can reduce it to thedesired tolerance.

contains three layers: top layer (0 m – 3048 m), middle layer (3048 m – 3124.2 m), and

bottom layer (3124.2 m – 3429.0 m). The pay zone of the reservoir, Ωr = 6705.6 m ×3352.8 m × 76.2 m, is located in the middle layer and has the same thickness as the

middle layer and is surrounded by a stiff non-pay impermeable medium. Figure 3.26

shows geometrical configuration of the model, and Table 3.13 provides the rock proper-

ties. The pay and non-pay zones are fully saturated with a single-phase liquid which has

a density of 1000 kg/m3 at 105 Pa, the compressibility of 3 · 10−10 Pa−1, and the viscos-

ity of 10−3 Pa · s. The initial fluid pressure is 105 Pa at the surface with the hydrostatic

gradient of 0.98852022 ·105 Pa/m; the initial vertical stress is 0 Pa at the surface with the

vertical stress gradient of 0.9869 ·105 Pa/m, and initial horizontal stresses are equal to the

Table 3.13: Rock properties for Dean’s model.

Property Reservoir Nonpay zonesPorosity, % 25 25Horizontal permeability, mD 100 0Vertical permeability, mD 10 0Young’s modulus, Pa 0.0689476 ·109 6.89476 ·109

Biot’s constant 1 1Poisson ratio 0.25 0.25Rock density, kg/m3 2700 2700


18897.6

6705.6

9448.8

3352.8

Well

y

x

(x,y) plane

z

x

0

7949.365 m3/d

-3048.0

-3124.2

-3429.0

(x,z) plane

Figure 3.26: Dean’s model geometry.

half of the vertical stress. The bottom and side boundaries have the zero normal displace-

ment constraint. The wellbore is located at the center of the domain, has the radius of

0.0762 m, and fully penetrates the reservoir. The well operates at a constant production

rate of 7949.365 m3/day for 4000 days.

We employ a fixed-stress coupled property for sequential implicit coupling strategy.

We plot the pressure behavior at the top boundary of the reservoir (top right corner of the

dark box in Figure 3.26), and subsidence at the top of the reservoir and at the surface. The

simulation results have been compared against the published solution (Dean et al., 2006),

and show a good match (see Figure 3.27). Due to the Mandel-Crayer effect, the pressure in-

creases in the early period, reaches a maximum value, and then decreases monotonically as

shown in Figure 3.27a. Our numerical implementation reproduces Dean’s results both for

the fully-implicit and the fixed-stress sequentially implicit strategies. A small discrepancy

(less than 0.1%) appears due to presumable differences in the initialization procedure.


(a) (b)

Figure 3.27: Dean’s problem solution comparison: a) pressure at cell (6,11,6), b) displacement at point (0,0)m (top curve), and point (0, 3048) m (bottom curve). Solid line represents reference solution (Dean et al.,2006), and symbols represent fully-implicit and sequential-implicit methods respectively.

3.3.3 Example 7. Water Flooding Problem

The previous subsection results show that the sequentially implicit approach is able to

converge to the desired tolerance, yet requires performing coupling iterations between sub-

problems. A question then arises: how expensive are these iterations? The main purpose

of this subsection is to compare the performance of the fully-implicit and the sequentially

implicit methods.

To demonstrate the applicability of the proposed framework to solve problems of prac-

tical interest, we consider a modified SPE 10 test problem (Christie and Blunt, 2001). The

original data set, including relative permeability curves, porosity and permeability distri-

butions, fluid properties, and wells controls, was used in this study. The simulation domain

Ω = 365.8×670.6×51.8 m is discretized using a regular Cartesian grid into 60×220×85

cells. The top 35 layers represent a formation with a permeability field that has the Gaus-

sian distribution, and the bottom 50 layers characterized by a channelized permeability

field with high-contrast features. A set of coarsened models, which contains from 24,960

to 399,360 cells, was generated (see Table 3.14). The upscaling of properties was per-

formed using a simple volume-averaging approach.

The original SPE 10 test case is a two-phase flow problem and does not involve geome-

chanics. Our extended model is supplemented by boundary conditions and parameters for


Table 3.14: Number of cells and total degrees of freedom (DOF) for the simulation cases.

Case Name 25k 50k 100k 200k 400kNumber of cells 24,960 49,920 99,840 199,680 399 360

DOF 134,667 267,735 522,555 1,037,235 2,068,395

a pore-mechanical problem. We set zero normal displacement on the side boundaries, zero

vertical displacement on the bottom boundary, and uniform loading conditions on the top

boundary, Ftop = 900 ·105 Pa.

Figure 3.28: Water Flooding Problem. The Young’s modulus distribution (units are 1010 Pa). Reservoirdomain Ω = [0,250]× [0,500]× [−3650,−3800] m. The top part is characterized by smoothly varying rockproperties, whereas the bottom part has channelized distributions of the properties with long-correlation andhigh-contrast features.

We take the Biot’s coefficient equal to 1.0 and the Poisson ratio equal to 0.2. Three

different settings with different coupling strengths are used to study the performance of the

sequential scheme. A detailed description of the coupling strength for multiphase problem

can be found in Kim et al. (2013) and Castelletto et al. (2015). The base case uses the

original rock compressibility value, cr = 1.45 ·10−10 Pa−1, to estimate the average Young’s

modulus, Eav = 3(1−2ν)/cr. Then, Young’s modulus values are distributed on the grid

using the following porosity correlation:

E1.0(xxx) = Eav · (a ·φ(xxx)+b), (3.15)

where φ is the initial porosity, and parameters a and b are chosen to obtain min(E1.0) =

E1.0 (max(φ)) = 0.2Eav and max(E1.0) = E1.0 (min(φ)) = 5Eav. Figure 3.28 illustrates the


distribution of the Young modulus. The second and third cases use softer and stiffer moduli:

E0.1 = 0.1 ·E1.0 and E10.0 = 10 ·E1.0. The former setup corresponds to a stronger coupling

and the later setup corresponds to a weaker coupling (as compared with the base case). This

set of models allow us to investigate the performance of sequential scheme for 15 different

scenarios overall.

Usually, the most time-consuming part of the simulation is the solution of the linear sys-

tem. As has been mentioned above, the FIM approach for coupled geomechanical problems

requires a sophisticated linear solver development (White et al., 2016), and the choice of

efficient solvers is limited. In the following comparison, we employ our in-house iterative

linear solver (Klevtsov et al., 2016), which applies a multi-stage preconditioning strategy

combining the generalized Constrained Pressure Residual approach (Zhou et al., 2013) with

the fixed-stress block-partitioned algorithm (White et al., 2016). The performance of the

solver is validated against the widely used SAMG (Algebraic Multigrid Methods for Sys-

tems) software (Gries et al., 2014). We made a comparison for a set of problems of different

resolutions (∼ 104−106 degrees of freedom) and found that the difference in solution time

between the two solvers is comparable, and does not change overall conclusions. This

justifies using the in-house linear solver for FIM simulations. For the sequential scheme,

we employ the Constrained Pressure Residual (CPR) preconditioned GMRES solver (Zhou

et al., 2013) to solve the flow problem, and the SAMG-preconditioned GMRES solver to

solve the geomechanical problem. We use the same convergence tolerance values for FIM

and SEQ approaches.

In addition to the effect of coupling strength between flow and mechanics, we also

investigate how the nonlinearities related to each physics module affect the overall perfor-

mance. The simulation results are summarized in Table 3.15. In the first setup, to mimic

a single-phase poromechanics behavior, we set the initial water saturation equal to one,

Sw = 1.0. In this case, for the sequential scheme, flow problem converges in one nonlinear

iteration because the water compressibility is small, cw = 4.35 ·10−10 Pa. Since in this case,

the mechanical equation is linear and quasi-static, convergence is achieved in one nonlin-

ear iteration. It has been shown before (Castelletto et al., 2015) that the sequential scheme

requires larger numbers of iterations when the coupling is stronger. Consequently, the case

E10.0 requires less coupling iterations and has lower computational cost (Figure 3.29). The


Figure 3.29: Water Flooding Problem. Comparison of FIM (solid color) and SEQ (dashed) simulation timefor single-phase setup. Notice the logarithmic scale for the vertical axis.

Table 3.15: Normalized simulation time for Water Flooding Problem. Corresponding FIM simulation time forthe smallest problem (marked in bold) is used to normalize simulation time the single-phase setup (Sw = 1.0)and the two-phase setup (Sw = 0.2).

E0.1 E1.0 E10.0Case FIM SEQ FIM SEQ FIM SEQSw = 1.0, elasticity25k 1.39 2.17 1.00 1.00 0.94 0.8150k 3.08 4.79 2.31 2.15 2.18 1.71

100k 6.34 10.22 4.89 4.56 4.55 3.44200k 16.11 28.53 10.98 10.13 9.25 7.25400k 34.56 52.22 27.10 25.25 24.17 18.38Sw = 0.2, elasticity25k 1.33 1.35 1.00 0.57 0.98 0.4250k 2.99 2.95 2.46 1.25 2.41 0.94

100k 7.74 6.40 6.01 2.72 5.38 2.09200k 20.93 15.89 15.55 5.88 13.16 5.05400k 59.21 33.73 39.48 15.82 34.09 12.62Sw = 0.2, plasticity25k 1.53 2.55 1.10 1.13 1.07 0.8150k 3.34 5.47 2.54 2.44 2.58 1.77

100k 8.21 12.56 6.13 5.25 5.61 3.68

sequential scheme outperforms the fully-implicit scheme for E10.0. At the same time, it

performs comparably well for E1.0, and slower for E0.1 because the fully-implicit scheme


is less sensitive to the change of coupling strength.

Next, we solve two-phase problems with initial water saturation Sw = 0.2. Remember

that the porous matrix still has linear-elastic deformation characteristics, while the flow is

highly nonlinear because of the saturation transport. However, the fully-implicit approach

treats the overall system as nonlinear and multiple Newton steps are required for conver-

gence. Thus, the cost of the fully-implicit scheme increases due to the increased number

of nonlinear and linear iterations performed for the full system. On the other hand, the

sequential scheme requires approximately the same number of coupling iterations as in

the single-phase setup. In this solution scheme, multiple nonlinear iterations are required

only for the smaller flow and transport problem, while the mechanical problem still needs

only one nonlinear iteration. As a result, the sequential scheme is comparable or faster

even in the case of strong coupling strength and becomes a very attractive solution strategy

(Figure 3.30).

Figure 3.30: Water Flooding Problem. Comparison of FIM (solid color) and SEQ (dashed) simulation timefor two-phase setup. Notice the logarithmic scale for the vertical axis.

Finally, we consider the two-phase flow problem but include plastic deformation (Drucker

and Prager, 1952). Now both sets of equations are nonlinear. The fully-implicit approach

requires almost the same number of nonlinear iterations as before and consequently, its

performance differs only slightly from the elastic case. However, the performance of the


Figure 3.31: Water Flooding Problem. Comparison of FIM (solid color) and SEQ (dashed) simulation timefor the two-phase plastic setup. Light part of the bars correspond to the simulation time increase betweenelastic and plastic models.

sequential scheme suffers a lot since two nested nonlinear loops are performed for each

coupling step. This is especially evident in the case when the coupling is strong and the

sequential approach demands the higher number of coupling iterations. For the moder-

ate and weak coupling, the sequential approach still performs better than the fully-implicit

approach (Figure 3.31).

3.3.4 Summary

We demonstrated the applicability of the GENIC framework for coupled nonlinear flow-

mechanics problems. First, we validated our implementation on Mandel’s problem against

the analytical solution. We tested two existing coupled properties: fixed-stress and fixed-

strain. The results are consistent with published results (Castelletto et al., 2015; Kim,

2010), where fixed-strain is only conditionally stable, and fixed-stress is unconditionally

stable. Next, we validated our framework on Dean’s problem against the available nu-

merical solution. Dean’s problem demonstrated the framework’s ability to model physics

on different domains. Finally, we compared the performance of the sequential coupling


strategy against the fully implicit approach in the extended SPE 10 problem. For the mod-

erate and weak coupling, the sequential approach performs better than the fully-implicit

approach, while it is not efficient for strongly coupled problems where both physics are

highly nonlinear.


3.4 Coupled Flow, Thermal, and Mechanics

3.4.1 Example 8. Cold Water Injection

Here, we consider problems whereby the interactions involve three physics modules, namely,

thermal, compositional, and mechanics. We demonstrate the applicability of the proposed

framework to solve problems of practical interest (Rin et al., 2017). A modified version of

the SPE 9 model (Killough, 1995) is considered (Fig. 3.32). We extend the description from

three-phase black-oil to a 5-component system (H2S, C1, C5, C13, H2O). The K-values flash

algorithm (Rannou et al., 2013; Zaydullin, 2015) is used for the phase equilibrium calcula-

tions. There are 25 producers and one injector as in the original SPE9 model. All the wells

operate with a BHP control. The model parameters are summarized in Table 3.16.

Here, our starting point and reference is the fully coupled solution obtained using

Ψ1 = FIM[FLOW,THERMAL, MECHANICS]. (3.16)

For the sequential simulations, we apply the fixed-stress splitting (Kim, 2015). In the

first setup, the flow and thermal problems are solved together, followed by the mechanical

overburden −→SPE9 reservoir −→

underburden −→

Figure 3.32: Cold water injection problem. The domain includes the overburden, under-burden, and thereservoir. The reservoir geometry, porosity, permeability, well locations are from the SPE 9 problem. Thepermeability field is shown and black blocks represent the well locations.


Table 3.16: Main parameters for Cold Water Injection problem. See Killough (1995) for full specificationsof the reservoir properties.

Grid dimensions 24×25×24Grid block size in X and Y directions, m 91.44Reservoir depth range, m 2735.14 – 3231.52Reservoir dip in X direction, deg 10Reservoir layers thickness, m 6.1, 4.6, 7.9, 4.6, 4.9, 4.3, 2.4,

2.4, 5.5, 3.7, 5.8, 5.5, 6.1, 15.2, 30.5Young’s modulus, Pa 2 ·109

Biot’s constant 1.0Poisson’s ratio 0.25Undrained Poisson’s ratio 0.47Rock density, kg/m3 2250.0Rock thermal expansion, 1/K 2 ·10−5

Rock heat capacity, kJ/(kg·K) 2000.0Rock thermal conductivity, kJ/(m·day·K) 150.0Oil thermal conductivity, kJ/(m·day·K) 11.5Gas thermal conductivity, kJ/(m·day·K) 3.586Water thermal conductivity, kJ/(m·day·K) 53.5Initial reservoir overall composition 0.049, 0.18, 0.6, 0.157, 0.014Initial reservoir temperature, K 343.0BHP, Pa 107 for producers, 4 ·107 for injectorInjection temperature, K 323.0Relative phase permeability kr j = S2

j

deformation step:

Ψ2 = SEQ[FIM[FLOW, THERMAL], MECHANICS]. (3.17)

The method requires 3–10 iterations to converge for a tolerance ε= 10−4 (Figure 3.33). The

SEQ simulation results agree with the FIM solution. Figure 3.34 shows the final saturation

and temperature distributions together with the differences between FIM and SEQ results.

The calculated production and injection rates are also nearly identical (Figure 3.35).

Next, we run the sequential algorithm in a nested way: the inner sequential loop is

between the flow and thermal problems; the outer loop is between the flow-thermal block

and geomechanics:

Ψ3 = SEQ[SEQ[FLOW, THERMAL], MECHANICS]. (3.18)


Figure 3.33: Cold Water Injection. Sequentially implicit method convergence profile after first time step ∆t= 0.001 days.

The scheme is able to reproduce the fully coupled solution; however, it experiences sub-

stantial stability problems. Specifically, the coupling scheme does not converge for the

same time steps as the FIM method (the average time step is about two times smaller). The

investigation of stable coupling schemes for multiphase thermal problems is the subject of

future research.

Table 3.17 summarizes a comparison between fully implicit, sequentially implicit, and

nested sequential-implicit strategies. The fully implicit and sequentially implicit methods

show stable convergence behavior without time step cuts, and take the same time step size,

while the nested sequentially coupled scheme is less stable due to instabilities in resolving

the coupling between the flow and thermal physics modules. The SEQ scheme converges

in 1-2 sequential iterations per time step. While the SEQ scheme takes more total Newton

Table 3.17: Cold Water Injection case. Comparison between FIM and SEQ schemes

FIM (Ψ1) SEQ (Ψ2) Ψ3

number of timesteps 255 255 1155total Newton iterations 773 1170 12281

total sequential iterations 0 565 4120number of timesteps wasted 0 0 108

maximum CFL 3.51 3.51 0.96

normalized total time 1.0 0.32 2.69linear factorization time, % 91 90 90

linear solution time, % 5 4 4


Figure 3.34: Saturation and temperature distributions in the middle of the reservoir at the end of simulationfor FIM approach (left) and difference in them between FIM and SEQ simulations (right).

Figure 3.35: Oil production and water injection rates. FIM and SEQ approaches produce nearly identicalresults.


iterations than FIM (1170 vs 773), it requires fewer Newton iterations per physics (585 vs

773) taking advantage of weak coupling between the physics modules. As a result, SEQ

scheme takes 32% of the FIM simulation time, and is much more efficient for this problem.

Moreover, both linear factorization and solution time account for a smaller ratio of the total

time compared with the FIM scheme (94% vs 96%). This is due to the smaller system size,

and less memory overhead for the SEQ scheme.

3.4.2 Summary

We study flow-thermal-mechanics problems using three different coupling strategies: fully

coupled, sequentially coupled, and nested sequentially coupled. This is the first example

available in the literature where both fully coupled and sequentially coupled approaches are

applied to this type of three-physics problem. We found that when the mechanics problem

is linear, and the coupling strength is weak, the sequential coupling strategy performs faster

than FIM; however, as the coupling strength involving mechanics increases, sequential

coupling strategies become less effective.

Chapter 4

Future Extensions of GENIC

In this chapter, we elaborate on the details of possible framework extensions, such as auto-

matic coupled properties, parallelization, and multiscale formulations.

4.1 Automatic Coupled Properties

There are several limitations of the current design of Coupled Properties (CP), which is ex-

pected to become critical for problems with a large number of coupled properties. First, the

order of CP calculations requires careful arrangement. For example, in thermal-mechanics

problems we may have two coupled properties: porosity φ and enthalpy h, where φ is a

function of h. While we may solve thermal-mechanics and mechanics-thermal problems

without code modification, the order of the CP is fixed. Here, we must calculate h before

φ, otherwise both of them would use inconsistent parameters; therefore, in this case, only

one order for the coupled properties guarantees the desired sequence of operations. This

limitation is illustrated in Figure 4.1, where we demonstrate CP’s order dependency for Ψ

= SEQ[MECHANICS, THERMAL] coupling.

Second, all CPs are updated whenever a sub-problem is solved to avoid the use of non-

updated CPs when another sub-problem is solved. This guarantees the correctness of the

overall solution; however, it might become costly if CP is expensive to calculate when it

is not used. This limitation is based on the previous restriction: proper update needs to

update all relevant CPs. Figure 4.2 demonstrates an example of a sequentially coupled

98

CHAPTER 4. FUTURE EXTENSIONS OF GENIC 99

Figure 4.1: Schematic representation of the solution for Ψ = SEQ[MECHANICS,THERMAL]. In case 1, weupdate porosity prior to enthalpy. As a result, porosity φ(h) uses old enthalpy leading to a wrong value ofporosity. In case 2, we update enthalpy followed by porosity to account dependency between them.

Figure 4.2: Update of CPs for Ψ = SEQ[MECHANICS,THERMAL]: a) current design - all properties are updated,b) naive design: only related properties are updated ignoring a dependency between CPs, c) correct design -minimal CP update.

mechanics-thermal problem, where the mechanics sub-problem depends on porosity φ(h),

and the thermal sub-problem uses enthalpy h. It shows three designs: (a) full, (b) naive,

and (c) correct. While a naive design results in a minimal update, it ignores the dependency

between coupled properties causing the wrong solution of the mechanics sub-problem. The

correct design (c) produces minimal possible CP updates and accounts for dependency

between φ and h.

A possible way to resolve these framework limitations is by using a dependency graph,

which is a directed, acyclic graph describing the functional relationship of each CP in

the system of equations (see Figure 4.3). Leaf (terminal) nodes in the dependency are

unknowns and properties stored in corresponding sub-problem’s subsets (shown in blue).


Figure 4.3: A dependency graph for a flow-thermal-mechanics problem. The unknowns and properties areshown in blue, and coupled properties are shown in green. Arrows shows the relationship between coupledproperties and unknowns.

All other nodes are coupled properties (shown in green), and directed edges represent the

relationships between them. Based on the dependency graph, we can reconstruct a proper

order of CP independently of initialization. Moreover, the order automatically changes to

align with used properties and physics. From the example in Figure 4.3, we have three

coupled properties: mobility µ, enthalpy h, and porosity φ(h). Using a dependency graph it

is easy to find the right order: calculate mobility µ and enthalpy h independently from each

other, followed by porosity φ update.

Additionally, the dependencies between coupled properties allow the recalculation of

the CPs only if their dependencies have changed, resulting in automatic coupled properties

evaluation. As a consequence, dependency graphs allow for a more flexible and efficient

framework for a large number of coupled properties when identification of their correct

order may be challenging.

4.2 Parallelization

GENIC framework parallelization is implemented via shared memory multi-processing

through OpenMP. However, there are serious limitations to solve large-scale problems of

practical interest caused by the mostly serial implementation of the ADETL library. We can


Figure 4.4: Scheme of the interaction between simulator and the ADETL library in the GENIC framework.Both mappers and subsets are part of ADETL.

achieve much better scalability by utilizing a distributed memory system, such as MPI. We

proceed as follows. First, we explain the current issues with the parallelization of GENIC,

and we elaborate on how we can resolve them in distributed memory architecture.

Figure 4.4 depicts the GENIC framework in terms of the interaction between the ADETL

library, and a simplified representation of the simulator. All the properties are stored in AD

vectors, which contain both values and derivatives for each unknown. Subsets provide

efficient and unified access to AD vectors (and related data) through the use of mappers

and a status table. Finally, the main set organizes logic within subsets and performs up-

dates whenever a change of global status is needed causing a global renumeration. As we

explained in detail in Chapter 2, GENIC framework heavily relies on a subset concept to

provide seamless physics integration and straightforward physics extension. Subsets are

designed as a part of ADETL to minimize memory and CPU overheads. The paralleliza-

tion of GENIC is impossible without parallel access to data and subsets. Consequently,

parallelization of ADETL is a required step towards the development of a scalable parallel

multi-physics simulator with a flexibility provided by the framework. In addition, solver

part should also be modified to account for the distributed memory system.

One way to provide such modifications is through the use of Parallel Mapping Oper-

ators (PMO). Similar to the mapping operators introduced in Chapter 2, PMO defines a


(a) (b)

Figure 4.5: Differences between regular and parallel mapping operators. On the left (a): both M1,M2 belongto the same computational node. On the right (b): P1 and P2 might belong to different computational nodes,and P1 = Plocal

1 ∪Pghost1 . Note: dashed line separates two local parts of P1 and P2.

sub-domain related to the subset. However, it has several differences as shown in Fig-

ure 4.5. First, PMO has two parts: local (sub-domain itself) and ghost, which represents

cells from neighboring nodes. Ghost part minimizes communications betweens nodes and

performs synchronization. Second, PMOs are distributed among different nodes (along

with the memory allocated for related parts of AD vectors) and can be solved in paral-

lel. Finally, each PMO has a fixed domain with memory allocation in initialization stage

(regular mapping operator allows a dynamic domain change).

We employ PMO by replacing a sequential representation of the coupling strategy with

a parallel. For example, for a flow-thermal problem we substitute a sequential coupling

strategy Ψ1 = SEQ[FLOW, THERMAL]with Ψ2 = SEQ[FIM[FLOW<P-i>], FIM [THERMAL<P-i>]],

where P-i is short notation for P1,...,Pn, and each P-i is parallel mapping operator.

Notice, the number n of used PMOs are computed internally and based on partitioning

algorithm on initialization stage. For Ψ2 we employ an extended status table as shown

in Figure 4.6. This strategy preserves a flexibility of the framework in terms of coupling

strategies and mapping operators.

Next, we need to solve a sub-problem with PMO’s in parallel. This requires small

modification in Algorithm 1, where we split line 4 (solve active sub-problems using a

linear solver) in two steps. First, for each PMO we compute its ghost part by synchronizing

with neighboring nodes. This step is skipped if the ghost part is empty (as might be needed

for multi-scale). Second, we solve each sub-problem with applied PMO in parallel, and

other sub-problems in serial. Since we have parallel access to ADETL there is no need to

merge data from different parts of the sub-problems. Notice, Algorithm 1 transforms to its


Global Status F T

status 1 X –status 2 – X

(a)

Global Status F1 F2 ... Fn T1 T2 ... Tn

status 1 P1 P2 ... Pn – – ... –status 2 – – ... – P1 P2 ... Pn

(b)

Figure 4.6: Parallel implementation through parallel mapping operators for a flow-thermal problem for ncomputation nodes. For brevity, we use F for FLOW, and T for THERMAL. On the left (a), serial couplingstrategy. On the right (b), similar yet parallel coupling strategy, where P1, ...,Pn are PMOs, and F1, ...,Fn andT1, ...,Tn are related parts of flow and thermal domains correspondingly.

original when no parallel mapping operators are used.

We briefly described the GENIC framework modifications for distributed memory envi-

ronment. There are other challenges in implementation, such as an efficient parallel access

to AD Vector through subsets, the optimization of global status change routine in the main

set (which restructures a global Jacobian), and the updates of parallel coupled properties.

In addition, hiding this new level of complexity on the general level is another challenge.

4.3 Multi-Scale Framework

Multi-scale methods in general and particularly the Multi-scale Finite Volume (MsFV)

method represent an active area of research in the context of reservoir simulation (Jenny

et al., 2003; Lunati and Jenny, 2006; Hajibeygi et al., 2008; Zhou, 2010; Cusini et al., 2015;

Myner, 2016; Kozlova et al., 2015). Recently the MsFV approach was extended for multi-

physics applications (Tomin and Lunati, 2013, 2015, 2016), where pore-scale and Darcy-

scale physics were combined in a general conservation-based local-global framework with

flexible spatiotemporal adaptive strategies. Nevertheless, to the best of author’s knowl-

edge, available commercial solutions (DeBaun et al., 2005; Cusini et al., 2015; Myner,

2016; Kozlova et al., 2015) and research prototypes, are limited in terms of physics to

flow-transport problems at Darcy scale; a flow-thermal-mechanics multi-scale simulator

does not yet exist. This limitation can be resolved by extending the GENIC framework

to support multi-scale methods. The flexibility of GENIC allows coupling multi-scale and

standard methods within the same simulation. In addition, presented ideas could be ap-

plied for smart upscaling approaches including adaptive local-global upscaling (Chen and


Durlofsky, 2006; Li and Durlofsky, 2016).

Algorithm 6 presents the basic steps of the MsFV approach (see, e.g., Tomin and Lunati

(2015) for details). First a grid partitioning (coarse and dual grids) has to be provided

(step 1). Next, interpolators that include basis (and correction functions) are computed

locally on the dual grid cells (step 2) and a global problem is constructed (step 3). After the

global problem is solved, boundary conditions for the local sub-problems are constructed

(step 4) and the full system of equations is solved locally on the coarse grid cells (step 5).

Steps 2–5 are repeated till the end of the simulation.

Algorithm 6: Basic steps of the MsFV algorithm.

1: generate MsFV grids (dual and coarse)2: compute basis functions3: construct and solve coarse system4: construct boundary condition for coarse cells by prolongating the coarse solution back to the

fine scale5: solve local fine systems with prescribed boundary conditions

Based on Algorithm 6, we treat each scale in MsFV as a separate sub-problem. We

denote the coarse system as FLOW COARSE, dual grid computations as FLOW BASIS, and the

local fine systems as FLOW. We introduce parallel mapping operators DUAL and FINE for dual

and fine systems correspondingly. Since no global synchronization is required between ap-

plied PMO, ghost part of each PMO is empty. Figure 4.7 shows a schematic representation

of the multi-scale sub-problems and the corresponding parallel mapping operators.

We couple the fine and coarse scales with basis functions computed on the dual grid.

Thereby, the basis functions can be considered as coupled properties. The interaction be-

tween them and sub-problems is shown on Figure 4.8. Finally, we write the MsFV algo-

rithm in GENIC framework as

Ψ = SEQ[FIM[FLOW BASIS<DUAL-i>], FLOW COARSE, FIM[FLOW<FINE-i>]]

It must be noted that FLOW<FINE-i> differs from a standard flow problem by using basis

functions to reconstruct required quantities (boundary conditions). When basis functions

are not computed (i.e. in other non-MsFV methods), standard computations are called.

Therefore, the MsFV approach requires only the two new sub-problems (FLOW BASIS and


Figure 4.7: Schematic representation of multi-scale sub-problems and corresponding sub-domains on 10×10 grid. On the left (a) - four flow sub-problem on FINE-isub-domain (shown in blue), in the center (b) - onecoarse sub-problem, and on the right (c) - dual grid with 9 sub-domains (DUAL-i) for computation of basisfunctions (shown in green).

Figure 4.8: Interaction between multi-scale sub-problems and their coupled properties - basis functions ϕi.

FLOW COARSE), construction of MsFV grids at initialization stage, and one coupled property

(basis functions). There is no need to modify any general algorithm for MsFV methods.

We can couple multi-scale and standard methods within one coupling strategy by adding

MsFV logic into it. For example, we might employ three coupling strategies:

Ψ1 = SEQ[SEQ[FIM[FLOW BASIS<DUAL-i>], FLOW COARSE, FIM[FLOW<FINE-i>], THERMAL],

Ψ2 = SEQ[FLOW, SEQ[FIM[THERMAL BASIS<DUAL-i>], THERMAL COARSE, FIM[THERMAL <FINE-i>]]],

Ψ3 = SEQ[FIM[FLOW BASIS<DUAL-i>, THERMAL BASIS<DUAL-i>], FIM[FLOW COARSE, THERMAL COARSE], ...

FIM[FLOW<FINE-i>,THERMAL[FLOW<FINE-i>]]].

In Ψ1, we couple an MsFV flow problem followed by a thermal problem. In Ψ2, we

couple a flow problem with an MsFV thermal problem. Finally, in Ψ3, we solve a flow-

thermal problem using MsFV for both physics.

The flexibility in coupling MsFV and standard methods within the same framework

can identify new methods of coupling multi-physics within MsFV framework, or test new


multi-scale preconditioners. In addition, both MsFV and non-MsFV methods co-exist

within the same framework – this enables an accurate comparison between methods.

Chapter 5

Conclusions and Future Work

In this work, we developed a unique general implicit coupling framework for multi-physics

problems (GENIC). The framework is implemented as the next generation of the Automatic-

Differentiation General-Purpose Research Simulator (AD-GPRS) and represents a flexible

and efficient reservoir simulation research laboratory in which students and researchers can

easily practice their ideas. We applied GENIC to solve challenging two- and three-physics

problems with novel coupled properties and coupling strategies. Here, we summarize the

findings from this work followed by recommendations for future research efforts.

5.1 Conclusions

• We designed and implemented a multi-physics framework to address the challenges

in current multi-physics simulator development such as a code extensibility to new

physics and capabilities, and flexibility in the choice of coupling strategies for vari-

ous physical processes. The exploration of different coupling strategies facilitates the

study of new multi-physics preconditioners. The newly developed framework sup-

ports any coupling strategy, while the modular design with transparent subset-based

interfaces greatly improves code extensibility to new physics and complex features.

This level of flexibility is achieved with a sub-problem tree structure, global status

table, and its integration with subsets through an automatic differentiation layer. In

addition, we employed a unique mapping operators concept, which enables advanced

107

CHAPTER 5. CONCLUSIONS AND FUTURE WORK 108

coupling strategies including (parallel) domain-decomposition and multi-scale ap-

proaches. As a result, we have designed, developed, implemented and tested a novel

general implicit coupling framework for multi-physics problems (GENIC) that al-

lows rapid customizations.

• We explored existing (constant pressure) and novel coupled properties, such as a

constant density and a hybrid scheme for flow-thermal sequential coupling strategies.

We investigated the convergence properties for three different sequential coupling

schemes. We found that a hybrid method where the ‘coupled property’ was pressure

for single-phase blocks and density for two-phase blocks performed the best. This

hybrid method was able to avoid convergence problems in the single-phase (constant

density) or two-phase (constant pressure) regions.

• We presented a new approach for coupling flow and thermal physics – modified Se-

quentially Fully Implicit (m-SFI) method. We demonstrated its performance for sev-

eral challenging examples, where the standard sequential approach fails, while m-

SFI takes a similar number of time steps as the fully coupled approach (FIM). In

cases where accurate front prediction is possible and the two-phase region is limited,

such as strictly injection or production problems, m-SFI approach outperforms FIM.

However, when the two-phase process appears in most of the reservoir, the overall

performance of the m-SFI method is slower than FIM due to the increased cost of

sequential iterations.

• We demonstrated the applicability of the GENIC framework for coupled nonlinear

flow-mechanics problems. First, we validated our implementation on Mandel’s prob-

lem (Mandel, 1953) against the analytical solution. We tested two existing coupled

properties: fixed-stress and fixed-strain. The results are consistent with published

works (Kim, 2010; Castelletto et al., 2015), where the fixed-strain is only condition-

ally stable, and the fixed-stress is unconditionally stable. Next, we validated our

framework on Dean’s problem (Dean et al., 2006) against the available numerical so-

lution. Dean’s example demonstrated the applicability of GENIC to solve a problem

where different physics modules are applied in different spatial domains. Finally,

we compared the performance of the sequential coupling strategy against the fully


implicit approach for an extended SPE 10 problem. For the moderate and weak

coupling, the sequential approach performs better than the fully-implicit approach,

while it is not efficient for strongly coupled problems where both physics are highly

nonlinear.

• We investigated flow-thermal-mechanics problems using three different coupling strate-

gies: fully coupled, sequentially coupled, and nested sequentially coupled. This is

the first example available in the literature where both fully coupled and sequentially

coupled approaches are applied and analyzed to this type of three-physics problem.

We found that when the mechanics problem is linear, and the coupling strength is

weak, the sequential coupling strategy performs faster than FIM; however, as the

coupling strength increases, sequential strategies become less effective.

5.2 Recommendations for Future Work

In addition to automatic coupled properties, multi-scale and parallelization discussed in

Chapter 4, future research efforts in the areas addressed in this work may include the fol-

lowing:

• Extend the framework to new physics and new coupled properties. For example, the

next research objective might include a study of multi-physics processes with chem-

ical reactions and electrical resistivity phenomena. Both physics are already im-

plemented in the framework, however, the choice of coupled properties poses some

challenges for a sequential scheme.

• Apply an adaptive coupling strategy for a given problem. In this work, we showed

that the choice of optimal (in terms of performance) coupling strategy depends on the

coupling strength between physics. In addition, the overall performance and conver-

gence are affected by the choice of coupled property, coupling terms, and advanced

coupling strategies. Presently, the framework allows switching of a coupling strategy

between timesteps and should be extended to support various coupled properties and

coupling terms for different timesteps. We suggest applying machine learning for


a large set of realistic problems to establish reliable criteria for adaptively changing

between coupling strategies.

• Develop a technique to generate an optimized specialized simulator based on the

GENIC framework. The GENIC framework is a very flexible platform, however,

this flexibility comes with some performance and memory overhead. It is possible to

automatically create a low-level optimized code based on the used physics, coupled

properties, and coupling strategy. This challenging task can be accomplished with

the advancement of automatic code generation in computer science.

• Extend the framework to support different time discretization for different sub-problems.

It might be beneficial to use smaller timesteps for faster processes such as chemical

reactions or wellbore modeling.

• Apply the framework to study sequential coupling for a facility network which has

significant convergence issues in the simulation of large oil fields with hundreds and

thousands of wells even with the fully implicit method. Often facility network shows

slow convergence comparing to a reservoir compositional part. We might improve

convergence of such problems by solving them in a sequential fashion.

• Use the mapping operator concept from the GENIC framework to construct a better

preconditioner for coupled systems. For example, instead of solving FIM[FLOW,THERMAL],

we might solve SEQ[THERMAL<M1>, FIM[FLOW,THERMAL]]. The later coupling strat-

egy mimics a preconditioner for a thermal part of the coupled flow-thermal problem

and allows rapid exploration in this area which traditionally requires major changes

in linear solver part.

• Incorporate new linear solvers and preconditioners to solve large-scale problems.

• Optimize the GENIC framework in terms of memory consumption and performance

(through efficient cache usage).

Nomenclature

α Skeleton linear thermal expansion/dilation coefficient

α j Holdup, or in-situ phase fraction of phase j

αφ Volumetric thermal dilation coefficient related to the porosity

uuu Displacement vector

vvv j Velocity of phase j

εpv Volumetric plastic strain

εv,o Reference volumetric strain

εv Volumetric total strain

κ Thermal-conductivity coefficient

C Fourth order elasticity moduli tensor

F Yield surface

µ j Viscosity of phase j

q j Inflow of phase j to the well

φ Porosity

φp Irreversible porosity

φo Reference porosity

111


Ψ Coupling strategy

ψ Internal friction angle

ρ Overal-mass density

ρ f Average fluid density

ρ j Mass density of phase j

ρmj Molar density of phase j

ρs Density of solid skeleton

σ Total stress tensor

εe Second order elasticity strain tensor

b Second order tensor of Biot’s coefficients

a Thermal expansion tensor

θ Inclination angle of the segment from vertical

℘ Set of given physics

A Cross-sectional area of an multi-segment well segment

A, B Material parameters in Drucker–Prager model

c Cohesion

cF ,cT ,cM Coupled property for flow, thermal, and mechanics

d Distance between cell centers

fi, j Fugacity of component i in phase j

h j Enthalpy of phase j

I1 Volumetric invariants of the effective stress


J2 Deviatoric invariants of the effective stress

k Absolute permeability

Kd Local drainage bulk modulus

kr j Relative permeability of phase j

m Permeability exponent

nc Number of components

np Number of phases

P Average fluid pressure

p Pressure

Pc Capillary pressure

p j Pressure of phase j

Po Reference pressure

qloss Head loss from wellbore fluid to surroundings

rw Wellbore radius

RF ,RT ,RM Flow, thermal, and mechanics residuals

S j Saturation of phase j

T Temperature

T w Temperature of wellbore fluid

u j Internal energy of phase j

ur Rock internal-energy

uto Overall heat transfer coefficient


Vin, j Interstitial velocity of phase j

Vs, j Superficial phase velocity of phase j

xF ,xT ,xM Flow, thermal, and mechanics unknowns

xi, j Mass fraction of component i in phase j

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Appendix A

Programming Model of AD-GPRS

In this appendix, the programming model of AD-GPRS is described. The primary de-

sign objective of the GENIC framework is to create a platform where different coupling

strategies for the simulation of complex multi-physical processes can be employed with

a great flexibility. We adopt a modular object-oriented design. The extensibility is achieved

through the usage of a subset concept presented in Chapter 2. All of the code is written in

standard C++11. Presently, AD-GPRS code contains over 94k lines and over 400 files.

The GENIC AD-GPRS is very different from the previous version of AD-GPRS which

was built as a compositional simulator and later extended to some multi-physics functional-

ity. Instead, GENIC AD-GPRS was developed as multi-physics simulator from the begin-

ning. Many classes are heavily refactored or completely rewritten to meet a general multi-

physics paradigm. For example, the functionality of allStatus, totalBlocks, and full set is

fully incorporated into subsets, and many redundant structures are removed. Moreover, the

core classes of the previous version such as Reservoir and Facilities are evolved into Flow-

Problem/ThermalProblem and Well, which are on top level abstracted by the general class

of SubProblem. Another two new key classes to mention are Subset and CoupledProper-

ties. This entails significant changes in the whole simulator. The logic of input-output,

the configuration of parameters, convergence criteria, and discretization have been also re-

vised. We highly recommend to study in detail Two-Phase Prototype (see Appendix B)

prior to any code development in AD-GPRS.

133

APPENDIX A. PROGRAMMING MODEL OF AD-GPRS 134

A.1 The Structure of AD-GPRS

This section is organized similarly to Appendix A in Zhou (2012). The structure of AD-

GPRS is discussed level by level, from top to bottom, followed by a flow sequence of the

simulation framework. Here we use italic shape to denote class names, slanted shape to

denote variable names, and boldface to denote function names.

We explain the structure of AD-GPRS with multi-layer class diagrams. Figure A.1

shows the overall structure of the entire simulator. Simulator is the topmost class that is

instantiated in the main function and is an entry point for the whole simulation. We identify

four groups of class-members of Simulator:

1. The first group is the core of the simulator, and it contains adX, globalStatusTable,

subProblemTree, and vSubProblems. The global variable set (adX ) stores all the

simulation variables with both values and gradients, as well as the global backup

data, local status tables, and mapping operators. The global status table (globalSta-

tusTable) specifies which sub-problems are active based on mapping operators and

is used to define the sub-problems calculation order. The sub-problem tree (sub-

ProblemTree) represents tree structure of a given coupling strategy based on the

COUPLING keyword. Note that subProblemTree is accessible indirectly via simData.

The vector of sub-problems (vSubProblems) is a list of all unique sub-problems for

Figure A.1: Overall structure of the entire simulator.


a given coupling.

2. The second group represents additional members needed for coupling such as vCou-

plingSubProblems and the list of applied coupled properties (vCoupledProperties).

The set of sub-problems with coupling terms stored in vCouplingSubProblems. This

includes wells and any other sub-problem which contributes to other physics despite

being non-active at the moment. The first and second group complete the core of the

GENIC framework as discussed in Chapter 2.

3. The third group specifies the subset of sub-problems (vActiveSubProblems) which

are active for a given sub-problem tree node (see Algorithm 1 for details), corre-

sponding linear solver (pActiveLinearSystem), Jacobian, and residual. The later two

represented by ADVector residual (see ADETL manual for details).

4. The last group is responsible for input-output (pIO ) and data infrastructure (sim-

Data). simData enables change of parameters with time such as well controls, time

stepping and some parameters for physics. The input-output class IO is general for

every subset – newly introduced sub-problems and subsets will automatically appear

in the generated output.

Figure A.2 shows the structure of SubProblemTree and its relationship with the Sim-

ulator. SubProblemTree builds the global status table which is used by Simulator. The

elements of sub-problem tree represented by SubProblemNode. Each sub-problem node

contains the list of children nodes (children) again of SubProblemNode type, and this list

Figure A.2: Structure of SubProblemTree class.


is empty when the corresponding node is a terminal node. Each terminal node has a cor-

responding global status in the global status table (globalStatus), current and maximum

number of iterations for a given node for SEQ coupling (niter and niterMax). In addi-

tion, SubProblemNode stores the list of given problems in the node (problems), their linear

solver (pLinearSystem), and linear solver parameters (linearSolverParams).

Sub-problem is another core concept of GENIC AD-GPRS. We represent any physical

process as a sub-problem which is implemented as a derived class of the abstract Sub-

Problem class. Figure A.3 shows the structure of SubProblem class. Every sub-problem

has name and type; the type is used to properly link the subsets between different sub-

problems (function bindSubsets) and to apply a linear solver. We can add a new subset

to a given sub-problem using the function addSubset. It allocates memory in the global

variable set (adX ), and adds the pointer to the new subset into vSubsets. Notice, while

every sub-problem calls the constructor for each subset, its destructor is called in the de-

structor of the global variable set (adX ); therefore, calling the destruction of a subset in the

sub-problem would lead to a memory corruption. We write discretized governing equations

into the global residual (residual ) with the function addToResidual. The convergence is

checked with an object of type ResidualChecker, pResidualChecker is the pointer to that

object. All possible mappers for a given sub-problem are identified at the input stage and

stored in the corresponding vector of mappers mMappers. The set setMappedSubsets con-

tains the list of subsets affected by mappers. The change of global status leads to the change

of the applied mapper (changeStatusFromMapper) which is specified via the parameter

mapperStatus.

Figure A.3: Structure of SubProblem class.


Presently, AD-GPRS supports the following sub-problems:

• FlowProblem – compositional flow and transport,

• ThermalProblem – energy balance,

• GeomechProblem – momentum balance (geomechanics),

• Well – abstract parent class for standard (StandardWell) and multi-segment (MSWell)

wells,

• ReactionsProblem – chemical reactions (currently only kinetic reactions),

• TimeOfFlightProblem – time of flight equation,

• TracerProblem – tracer transport equation,

• ResistivityProblem – Poisson equation for electric potential field.

The governing equations for FlowProblem, ThermalProblem, GeomechProblem, MSWell,

and StandardWell are given in Chapter 2. The governing equations for other sub-problems

are given in the AD-GPRS keywords manual.

Table A.1 represents owned and binded subsets for each sub-problem. For example,

binded subsets for StandardWell and MSWell classes are identical to the parent class Well.

Both WellFlowSubsetPerf and WellFlowSubsetSC are of type WellFlowSubset and repre-

sent properties and unknowns defined at perforations and standard conditions. Similarly,

WellThermalSubsetPerf and WellThermalSubsetSC are of type WellThermalSubset. Seg-

ments subsets for MSWells – SegFlowSubset and SegThermalSubset are of types Flow-

CellSubset and ThermalCellSubset correspondingly.

Figure A.4 shows the class diagram for all available subsets. The topmost class AD Inde-

pendentSubSet defining all methods needed for a subset belongs to ADETL library. Deriva-

tive classes specify properties and unknowns; no additional methods are required to utilize

the subset. The details of subset methods and members are given in the ADETL manual.

Example of the subset creation and usage can be found in Appendix A.2.

The domain of active elements for a given subset is defined through class Mapper (part

of ADETL). The activation logic of class Mapper is different from ACTNUM (active cells)


Table A.1: Sub-problems, subsets and properties.

Sub-problem Owned subsets Binded subsets

FlowProblem FlowCellSubset ThermCellSubsetFlowConnectionSubset ReactionsSubset

ThermalProblem ThermCellSubset FlowCellSubsetFlowConnectionSubset

GeomechProblem GeomechSubset FlowCellSubsetThermalCellSubset

WellFlowSubsetPerf FlowCellSubsetWell WellFlowSubsetSC ThermalCellSubset

WellThermalSubsetPerf TOFSubsetWellThermalSubsetSC TracerSubset

StandardWell:WellBhpSubset

MSWell:SegFlowSubset

SegThermSubset

ReactionsProblem ReactionsSubset FlowCellSubsetThermalCellSubset

TimeOfFlightProblem TOFSubset FlowCellSubsetFlowConnectionSubset

TracerProblem TracerSubset FlowCellSubsetFlowConnectionSubset

FlowCellSubsetResistivityProblem ResistivityCellSubset ThermalCellSubset

TracerSubset

concept. The later disables inactive cells prior to a simulation (and all relevant memory

allocations); therefore, these cells can no longer be activated later on. In addition, ACTNUM

is limited to reservoir cells only. Mapper allows activation and deactivation at any moment

(time step or Newton iteration) and can be applied to any subset. The new mapper can

be derived from the base class Mapper, and only requires a redefinition of the function

updateMapping, which defines which cells are active and which are not.

We specify the choice and order of unknowns for each sub-problem by using the Formu-

lation class. The usage of the formulation is specific to the type of the sub-problem, unlike


Figure A.4: Class diagram for subsets.

the previous version of AD-GPRS where all formulations are derived from the base class

NonlinearFormulation. Such class design provides a higher level of flexibility, improves

the code readability and reduces the memory overhead. The usage of the formulation is not

mandatory for a subproblem, and not all sub-problems have a corresponding formulation.

For a sub-problem that requires a formulation, there is a corresponding base class Subprob-

lemNameFormulation, and a specific formulation for this sub-problem is derived from the

base class. For example, class FlowProblem has the base formulation class FlowFormula-

tion. Both natural (NaturalFormulation), and molar (MolarFormulation) formulations are

derived classes from the FlowFormulation class. By constructing new inherited classes of

a corresponding sub-problem base formulation with the same interfaces but possible differ-

ent realizations, we are able to introduce new formulations. Figure A.5 is a class diagram

for all available formulations. Notice, that class WellProblem does not have a formulation;

instead, WellProblem uses the formulation of FlowProblem. The modeling of a wellbore

without reservoir would require its own formulation.

The formulations of different sub-problems vary greatly. In general, all formulations


Figure A.5: Class diagram for formulations.

have three common functions:

• constructor: initialization of parameters and subsets,

• buildStatusTable: create the table of local statuses, for example, a flow-related for-

mulations it is a table of existing phases associated with each possible phase status,

• bindSubsets: link subsets used by the formulation.

In addition, each sub-problem’s formulation defines a set of additional functions. For ex-

ample, flow formulation contains member functions such as enforceConstraints (enforce

certain local constraints), changeStatus (check the phase status and change it if necessary),

calcFluidPropertiesInCell (compute all fluid properties), makeSecondaryTerm (com-

pute secondary terms such as fugacity), and makeAccumulation (compute accumulation

term).

The linear system of equations in class Simulator is of type LinearSystemBase. The

major functions of LinearSystemBase are

• setup: extraction of Jacobian matrix from the AD-residual vector,

• solve: solve linear system,

• getSolution: return the pointer to the linear solution).


Figure A.6: Class diagram for LinearSystemBase.

The structure of LinearSystemBase class in shown on Figure A.6. Presently, LinearSys-

temBase has only one derived class – CSRLinearSystem which works with the Compressed

Sparse Row (CSR) matrix format. Note that the BlockLinearSystem (available in the pre-

vious version of AD-GPRS), which works with a much more complicated matrix format,

is currently not supported due to limited extensibility for multi-physics problems. The in-

tegration of the block matrix storage and manipulation is the subject of future research and

development.

Each CSRLinearSystem contains corresponding CSR matrix (CSR Matrix), linear solver

(LibrarySolver), and its parameters (ParametersForLibrarySolvers). Generally, any solver

working with CSR format can be supported. Presently, LibrarySolver includes direct linear

solvers such as PARDISO and SuperLU, and iterative solvers such as HYPRE, INMOST,

and SAMG.

A.2 Flow Sequence

The essential steps of a forward simulation are as follows:

1. Read the input (problem definition) from the given simulation deck.

2. Build the sub-problem tree and the global status table based on the input coupling

strategy.


3. Initialization (allocate memory, create objects, link subsets, and apply initial condi-

tions).

4. While there is still a time interval that has not been simulated, do the following:

(a) Update the simulation parameters and corresponding data for the current input

time interval if it is not the first one.

(b) While the current input time interval has not been finished, do the following:

i. Calculate the timestep size.

ii. Initialize the timestep: if the nonlinear solution succeeded in the last time-

step, use this converged solution as the initial guess and save independent

variables into a backup set; otherwise, restore from the backup set and

compute all dependent variables.

iii. Call the recursive function solveNode that solves the coupled multi-physics

problem starting from the head of the sub-problem tree (see the details of

this function in Algorithm 1).

iv. If solveNode returned that the solution is accepted, report statistics for

the converged timestep and dump the solution if necessary (e.g., at a re-

port step); otherwise, update the statistics for wasted Newton and timestep

iterations.

v. Go to step 4b for the next timestep.

(c) Go to step 4 for the next input time interval.

5. Report the overall statistics and perform post-processing if necessary.

6. Deallocate everything and end the simulation.

Appendix B

Two-Phase Prototype

In addition to multi-physics AD-GPRS GENIC presented in this work, we developed a Two-

Phase Prototype. The Prototype is based on the key principles of the GENIC framework

and serves as an introductory step to the multi-physics AD-GPRS GENIC simulator. It

supports two different models: a water-oil (dead-oil) system with gravity and an oil-gas

system without gravity. Both are isothermal problems with single-perforation wells. The

data for physical properties and the reservoir was taken from Stanford Reservoir Simu-

lation class (ERE 223). While the Prototype has the simplified physics, it can be poten-

tially extended to handle multi-physics problems. The usage of the two key components

of GENIC framework, namely sub-problems and subsets, is demonstrated in the Prototype,

while the full framework with the sub-problem tree, global status table, coupling terms,

and properties can be found in the full AD-GPRS GENIC. Only one coupling strategy

Ψ = FIM[FLOW,WELLS] is available in the Prototype. The entire Prototype source code is

located in the master branch of AD-GPRS simulator in ExtraModules/ProtoTwoPhase

directory.

There are four types of sub-problems in the Prototype:

• FlowProblem for dead-oil system,

• FlowGasProblem for oil-gas flow and transport,

• WellProblem for a well,

143

APPENDIX B. TWO-PHASE PROTOTYPE 144

Figure B.1: Class diagram for subproblems and their interaction with subsets in Two-Phase Prototype.

Figure B.2: Class diagram for subsets in Two-Phase Prototype.

• WellGasProblem specifically for gas producers.

For simplicity, we specify dead-oil and oil-gas formulations directly in FlowProblem and

FlowGasProblem classes correspondingly. We use the three subsets:

• FlowCellSubset for unknowns and properties defined at reservoir cell centers

• FlowFaceSubset for phase fluxes,

• WellCellSubset for bottom hole pressure and rates values for each well.

Figure B.1 represents the class diagram for subproblems and their interactions with

corresponding subsets, while Figure B.2 depicts the class diagram for subsets.

In the following, we go through the building block for a simple flow and transport

simulator. For brevity, we limit our description to dead-oil formulation only.


First, we introduce subsets and its accessors classes FlowVarsT and WellVarsT:/ / Flow c e l l s u b s e t which c o n t a i n s a l l t h e v a r i a b l e s r e l e v a n t t o f low problem

c l a s s F l o w C e l l S u b s e t : p u b l i c a d e t l : : AD Independen tSubSe t

p u b l i c :

/ / c o n s t r u c t o r : number o f b locks , p h a s e s and components

F l o w C e l l S u b s e t ( c o n s t s i z e t y p e num blocks , c o n s t s i z e t np , c o n s t s i z e t nc ,

c o n s t Subse tType s t y p e = FLOW CELL)

: a d e t l : : AD Independen tSubSe t ( num blocks , b u i l d S t a t u s T a b l e ( np , nc ) , s t y p e ) ,

np ( np ) , nc ( nc )

ADD SCALAR( p r e s ) / / p r e s s u r e [ 1 ]

ADD VECTOR( s a t ) / / phase s a t u r a t i o n [ np ]

ADD VECTOR( m o b i l i t y ) / / phase m o b i l i t y [ np ]

ADD VECTOR( m o b i l i t y b ) / / phase m o b i l i t y wi th f o r m a t i o n f a c t o r [ np ]

ADD VECTOR( b f v f ) / / f o r m a t i o n f a c t o r [ np ]

ADD VECTOR( v i s c o s i t y ) / / phase v i s c o s i t i e s [ np ]

p r o t e c t e d :

s i z e t np ; / / number o f p h a s e s

s i z e t nc ; / / number o f components

vo id s e t O f f s e t s ( c o n s t s i z e t np , c o n s t s i z e t nc ) ;

a d e t l : : S t a t u s T a b l e b u i l d S t a t u s T a b l e ( c o n s t s i z e t np , c o n s t s i z e t nc ) ;

;

/ / Flow c o n n e c t i o n l i s t s u b s e t c o n t a i n s phase f l u x e s r e l e v a n t t o c o n n e c t i o n l i s t

c l a s s FlowConnSubset : p u b l i c a d e t l : : AD Independen tSubSe t

p u b l i c :

FlowConnSubset ( c o n s t s i z e t y p e num blocks , c o n s t s i z e t np ,

c o n s t Subse tType s t y p e = FLOW CONN)

: a d e t l : : AD Independen tSubSe t ( num blocks , b u i l d S t a t u s T a b l e ( np ) , s t y p e ) , np ( np )

ADD VECTOR( p h a s e f l u x ) / / phase f l u x e s [ np ]

p r o t e c t e d :


vo id s e t O f f s e t s ( c o n s t s i z e t np ) ;

a d e t l : : S t a t u s T a b l e b u i l d S t a t u s T a b l e ( c o n s t s i z e t np ) ;

;

/ / Well c e l l s u b s e t c o n t a i n s p r e s s u r e and r a t e p r o p e r t i e s f o r w e l l s

c l a s s W e l l C e l l S u b s e t : p u b l i c a d e t l : : AD Independen tSubSe t

p u b l i c :

/ / n b l o c k s = 1 ( s i n g l e p e r f o r a t i o n w e l l )

W e l l C e l l S u b s e t ( c o n s t s i z e t np , c o n s t Subse tType s t y p e = WELL CELL)

: a d e t l : : AD Independen tSubSe t ( 1 , b u i l d S t a t u s T a b l e ( np ) , s t y p e ) , np ( np )


ADD SCALAR( p r e s ) / / p r e s s u r e [ 1 ]

ADD VECTOR( r a t e ) / / phase r a t e s [ np ]

p r o t e c t e d :


vo id s e t O f f s e t s ( c o n s t s i z e t np ) ;

a d e t l : : S t a t u s T a b l e b u i l d S t a t u s T a b l e ( c o n s t s i z e t np ) ;

;

For this simplified physics each subset has only one active status. Below is the ex-

ample of status table for FlowCellSubset, where pressure and water phase saturation are

unknowns:a d e t l : : S t a t u s T a b l e F l o w C e l l S u b s e t : : b u i l d S t a t u s T a b l e ( c o n s t s i z e t np , c o n s t s i z e t nc )

s e t d e f a u l t s ( ) ;

s e t O f f s e t s ( np , nc ) ;

s i z e t n s t a t s = 1 ; / / s i n g l e phase s t a t u s

v e c t o r<v e c t o r<i n t> > s t a t u s T a b l e ( n s t a t s , v e c t o r<i n t >( n v a r s ( ) , −1) ) ;

s t a t u s T a b l e [ 0 ] [ p r e s O f f s e t ( ) ] = 0 ; / / p r e s s u r e − f i r s t unknown

s t a t u s T a b l e [ 0 ] [ s a t O f f s e t ( ) + PHASE WATER] = 1 ; / / Sw − second unknown

r e t u r n s t a t u s T a b l e ;

Next, we proceed with the FlowProblem implementation. Notice that there are two

ways to access properties and unknowns from subsets: directly (FlowCellSubset->pres(ib))

or through accessors (FlowVars.pres()). The latter is more convenient in most circum-

stances. The listing of the FlowProblem functions is below.FlowProblem : : FlowProblem ( R e s i d u a l& r e s i d u a l , AD Independen tSe t& adX )

: SubProblem ( r e s i d u a l , adX )

m name = ” f low ” , m type = FLOW PROBLEM; / / subprob lem name and t y p e

/ / add f low c e l l s and f low c o n n e c t i o n l i s t s

f l o w c e l l s u b s e t = new F l o w C e l l S u b s e t ( sd . n b l o c k s ( ) , 2 , 2 ) ; / / np = nc = 2

a d d S u b s e t ( f l o w c e l l s u b s e t , ”FLOW CELL” ) ;

f l o w c o n n s u b s e t = new FlowConnSubset ( sd . conn . s i z e ( ) , 2 ) ;

a d d S u b s e t ( f l o w c o n n s u b s e t , ”FLOW CONN” ) ;

vo id FlowProblem : : i n i t i a l i z e S u b P r o b l e m ( )

/ / f o r a l l t h e f low c e l l b locks , i n i t i a l i z e i n d e p e n d e n t unknowns : Sw and p

f o r ( s i z e t i b = 0 ; i b < f l o w c e l l s u b s e t −>num blocks ( ) ; ++ i b )

f l o w c e l l s u b s e t −>s a t ( ib , PHASE WATER) = 1 . − sd . S o i l i n i t [ i b ] ;


f l o w c e l l s u b s e t −>p r e s ( i b ) = sd . P i n i t [ i b ] ;

c o m p u t e A l l P r o p e r t i e s ( ) ; / / compute o t h e r p r o p e r t i e s from unknowns

vo id FlowProblem : : addToRes idua l ( c o n s t d ou b l e d t )

/ / add a c c u m u l a t i o n t e r m s f o r each b l o c k


c o n s t d ou b l e c o e f = −sd . volume [ i b ] * sd . BETA / sd . DT; / / c o n s t a n t c o e f f i c i e n t

FlowVars v a r s ( f l o w c e l l s u b s e t , i b ) ;

FlowVarsDbl v a r s o l d ( f l o w c e l l s u b s e t , X N , i b ) ;

R e s i d u a l I t e r a t o r r e s = r e s i d u a l ( f l o w c e l l s u b s e t , i b ) ;

r e s . v a l [ 0 ] += c o e f * sd . poro [ i b ] * ( v a r s . b f v f ( PHASE OIL ) * v a r s . s a t ( PHASE OIL )

− v a r s o l d . b f v f ( PHASE OIL ) * v a r s o l d . s a t ( PHASE OIL ) ) ;

r e s . v a l [ 1 ] += c o e f * sd . poro [ i b ] * ( v a r s . b f v f (PHASE WATER) * v a r s . s a t (PHASE WATER)

− v a r s o l d . b f v f (PHASE WATER) * v a r s o l d . s a t (PHASE WATER) ) ;

/ / add f l u x t e r m s f o r each b l o c k

f o r ( s i z e t i c = 0 ; i c < f l o w c o n n s u b s e t−>num blocks ( ) ; ++ i c )

c o n s t s i z e t i a = sd . conn [ i c ] . i a , i b = sd . conn [ i c ] . i b ; / / two c o n n e c t e d c e l l s

c o n s t ADsca la r p d i f = f l o w c e l l s u b s e t −>p r e s ( i b ) − f l o w c e l l s u b s e t −>p r e s ( i a ) ;

c o n s t s i z e t u p i = ( p d i f >= 0) ? i b : i a ; / / t a k e u p s t r e a m c e l l f o r bo th p h a s e s

FlowVars u p v a r s ( f l o w c e l l s u b s e t , u p i ) ;

R e s i d u a l I t e r a t o r r e s a = r e s i d u a l ( f l o w c e l l s u b s e t , i a ) ;

R e s i d u a l I t e r a t o r r e s b = r e s i d u a l ( f l o w c e l l s u b s e t , i b ) ;

/ / c a l c u l a t e phase f l u x e s

f l o w c o n n s u b s e t−>p h a s e f l u x ( i c , PHASE OIL ) = sd . conn [ i c ] . t r a n s * p d i f

* u p v a r s . m o b i l i t y b ( PHASE OIL ) ;

f l o w c o n n s u b s e t−>p h a s e f l u x ( i c , PHASE WATER) = sd . conn [ i c ] . t r a n s * p d i f

* u p v a r s . m o b i l i t y b (PHASE WATER) ;

/ / we s t i l l w r i t e i n t o f l o w c e l l s u b s e t r e s i d u a l

r e s a . v a l [ 0 ] += f l o w c o n n s u b s e t−>p h a s e f l u x ( i c , PHASE OIL ) ;

r e s a . v a l [ 1 ] += f l o w c o n n s u b s e t−>p h a s e f l u x ( i c , PHASE WATER) ;

r e s b . v a l [ 0 ] −= f l o w c o n n s u b s e t−>p h a s e f l u x ( i c , PHASE OIL ) ;

r e s b . v a l [ 1 ] −= f l o w c o n n s u b s e t−>p h a s e f l u x ( i c , PHASE WATER) ;

vo id FlowProblem : : c o m p u t e A l l P r o p e r t i e s ( )


FlowVars v a r s ( f l o w c e l l s u b s e t , i b ) ;


v a r s . s a t ( PHASE OIL ) = 1 . 0 − v a r s . s a t (PHASE WATER) ;

/ / c a l c u l a t e f o r m u l a t i o n volume f a c t o r

v a r s . b f v f ( PHASE OIL ) = exp ( sd . cO * ( v a r s . p r e s ( ) − sd . pSC ) ) ;

v a r s . b f v f (PHASE WATER) = exp ( sd .cW * ( v a r s . p r e s ( ) − sd . pSC ) ) ;

/ / v i s c o s i t i e s

v a r s . v i s c o s i t y ( PHASE OIL ) = sd .Kmu * exp ( sd . Cmuo * ( v a r s . p r e s ( ) − sd . pSC ) ) ;

v a r s . v i s c o s i t y (PHASE WATER) = sd .muw;

/ / c a l c u l a t e m o b i l i t i e s

c o n s t ADsca la r k ro = sd . kroo * pow ( v a r s . s a t ( PHASE OIL ) , sd . no ) ;

c o n s t ADsca la r krw = sd . krwo * pow ( v a r s . s a t (PHASE WATER) , sd . nw ) ;

v a r s . m o b i l i t y ( PHASE OIL ) = kro / v a r s . v i s c o s i t y ( PHASE OIL ) ;

v a r s . m o b i l i t y (PHASE WATER) = krw / v a r s . v i s c o s i t y (PHASE WATER) ;

v a r s . m o b i l i t y b ( PHASE OIL ) = v a r s . m o b i l i t y ( PHASE OIL ) * v a r s . b f v f ( PHASE OIL ) ;

v a r s . m o b i l i t y b (PHASE WATER) = v a r s . m o b i l i t y (PHASE WATER) * v a r s . b f v f (PHASE WATER) ;

WellProblem uses flow cell subset for properties from reservoir. The listing of the

WellProblem functions is below.

WellProblem : : Wel lProblem ( R e s i d u a l& r e s i d u a l , AD Independen tSe t& adX , W e l l I n f o * wi )

: SubProblem ( r e s i d u a l , adX ) , winfo ( wi )

m name = wi−>name , m type = WELL PROBLEM;

w e l l c e l l s u b s e t = new W e l l C e l l S u b s e t ( 2 , WELL CELL) ;

a d d S u b s e t ( w e l l c e l l s u b s e t , ”WELL CELL” ) ;

f l o w c e l l s u b s e t = n u l l p t r ;

vo id Wel lProblem : : b i n d S u b s e t s ( c o n s t s t d : : v e c t o r<SubProblem*> &vSubProblems )

f o r ( s i z e t i = 0 ; i < vSubProblems . s i z e ( ) ; ++ i )

i f ( f l o w c e l l s u b s e t == n u l l p t r && vSubProblems [ i ]−> t y p e ( ) == FLOW PROBLEM)

f l o w c e l l s u b s e t = s t a t i c c a s t <FlowProblem*>( vSubProblems [ i ] )−>f l o w C e l l S u b s e t ( ) ;

vo id Wel lProblem : : i n i t i a l i z e S u b P r o b l e m ( )

c o m p u t e A l l P r o p e r t i e s ( ) ;

vo id Wel lProblem : : addToRes idua l ( c o n s t d ou b l e d t )


WellVars w e l l v a r s ( w e l l c e l l s u b s e t , 0 ) ; / / s i n g l e p e r f o r a t i o n

R e s i d u a l I t e r a t o r r e s = r e s i d u a l ( f l o w c e l l s u b s e t , winfo−>r b l o c k ) ;

r e s . v a l [ 0 ] += w e l l v a r s . r a t e ( PHASE OIL ) ;

r e s . v a l [ 1 ] += w e l l v a r s . r a t e (PHASE WATER) ;

vo id Wel lProblem : : c o m p u t e A l l P r o p e r t i e s ( )

/ / compute BHP and r a t e f o r w e l l

Wel lVars w e l l v a r s ( w e l l c e l l s u b s e t , 0 ) ; / / s i n g l e p e r f o r a t i o n

FlowVars r e s v a r s ( f l o w c e l l s u b s e t , winfo−>r b l o c k ) ;

i f ( winfo−>wtype == PRODUCER)

c o n s t ADsca la r m o b i l i t y r a t i o = r e s v a r s . m o b i l i t y b (PHASE WATER) / r e s v a r s . m o b i l i t y b

( PHASE OIL ) ;

i f ( winfo−>wcon t ro lType == OIL RATE CONTROL )

w e l l v a r s . r a t e ( PHASE OIL ) = −winfo−>r a t e ;

w e l l v a r s . r a t e (PHASE WATER) = −w e l l v a r s . r a t e ( PHASE OIL ) * m o b i l i t y r a t i o ;

w e l l v a r s . p r e s ( ) = r e s v a r s . p r e s ( ) + w e l l v a r s . r a t e ( PHASE OIL ) / ( winfo−>WI *r e s v a r s . m o b i l i t y b ( PHASE OIL ) ) ;

e l s e / / BHP CONTROL ( o t h e r s n o t s u p p o r t e d i n p r o t o t y p e )

w e l l v a r s . p r e s ( ) = winfo−>BHP;

w e l l v a r s . r a t e ( PHASE OIL ) = winfo−>WI * r e s v a r s . m o b i l i t y b ( PHASE OIL ) * ( r e s v a r s .

p r e s ( ) − w e l l v a r s . p r e s ( ) ) ;

w e l l v a r s . r a t e (PHASE WATER) = w e l l v a r s . r a t e ( PHASE OIL ) * m o b i l i t y r a t i o ;

e l s e / / INJECTOR

w e l l v a r s . r a t e ( PHASE OIL ) = 0 . ;

c o n s t ADsca la r m o b i l i t y t o t a l = r e s v a r s . m o b i l i t y b (PHASE WATER) + r e s v a r s . m o b i l i t y b

( PHASE OIL ) ;

i f ( winfo−>wcon t ro lType == WATER RATE CONTROL)

w e l l v a r s . r a t e (PHASE WATER) = winfo−>r a t e ;

w e l l v a r s . p r e s ( ) = r e s v a r s . p r e s ( ) + w e l l v a r s . r a t e (PHASE WATER) / ( winfo−>WI *m o b i l i t y t o t a l ) ;

e l s e / / BHP CONTROL ( o t h e r s n o t s u p p o r t e d i n p r o t o t y p e )

w e l l v a r s . p r e s ( ) = winfo−>BHP;


w e l l v a r s . r a t e (PHASE WATER) = winfo−>WI * m o b i l i t y t o t a l * ( r e s v a r s . p r e s ( ) −w e l l v a r s . p r e s ( ) ) ;

Finally, the full system of sub-problems is assembled, initialized and solved in Simulator

(sd is a storage object):

vo id S i m u l a t o r : : i n i t i a l i z e ( c o n s t v e c t o r<SubProblemType> &s u b P r o b l e m s L i s t )

/ / c r e a t e a l l subp rob l ems

vSubProblems . r e s e r v e ( s u b P r o b l e m s L i s t . s i z e ( ) ) ;

s i z e t iw = 0 ;

/ / c r e a t e new subprob l ems ( w e l l and f low )

f o r ( s i z e t i s p = 0 ; i s p < s u b P r o b l e m s L i s t . s i z e ( ) ; ++ i s p )

i f ( s u b P r o b l e m s L i s t [ i s p ] == WELL PROBLEM)

vSubProblems . p u s h b a c k ( new WellProblem ( r e s i d u a l , adX , sd . w e l l s [ iw ++] ) ) ;

i f ( s u b P r o b l e m s L i s t [ i s p ] == FLOW PROBLEM)

vSubProblems . p u s h b a c k ( new FlowProblem ( r e s i d u a l , adX ) ) ;

/ / s e t−up max r e s i d u a l s i z e

s i z e t neqn = 0 ; / / number o f e q u a t i o n s f o r a l l s u b s e t s

f o r ( s i z e t i s = 0 ; i s < adX . n u m b e r o f s u b s e t s ( ) ; ++ i s )

neqn += adX . g e t s u b s e t ( i s ) . t o t a l e q n s ( ) ;

r e s i d u a l . r e s i z e ( neqn , 0 ) ;

/ / b ind and i n i t i a l i z e s u b s e t s

f o r ( s i z e t i = 0 ; i < vSubProblems . s i z e ( ) ; ++ i )

vSubProblems [ i ]−>b i n d S u b s e t s ( vSubProblems ) ;

vSubProblems [ i ]−> i n i t i a l i z e S u b P r o b l e m ( ) ;

/ / once we add s u b s e t s and s t a t u s e s we e n a b l e d e r i v a t i v e s

adX . i n i t i a l i z e ( ) ;

adX . i n i t i a l i z e o f f s e t s ( ) ;

adX . s a v e t o b a c k u p ( a d e t l : : X N ) ; / / s ave c u r r e n t s o l u t i o n

vo id S i m u l a t o r : : s o l v e ( )

/ / run s i m u l a t i o n from t ime = 0 ; run newton i t e r a t i o n s t e p by s t e p

/ / and u p d a t e p r o p e r t i e s f o r each t ime s t e p


do ub l e t c u r = 0 ;

w h i l e ( sd . TIME < sd . TOTAL TIME | | abs ( sd . TOTAL TIME − sd . TIME) < sd . EPS DIFF )

s u g g e s t T i m e s t e p ( ) ; / / we don t need s u g g e s t t i m e s t e p i f we r e t r a c t t i m e s t e p

/ / s o l v e newton e q u a t i o n f o r a t ime s t e p

w h i l e ( ! s o l v e T i m e S t e p ( ) && sd . DT > sd . EPS DIFF )

;

c o m p u t e A l l P r o p e r t i e s ( ) ; / / u p d a t e p r o p e r t i e s

adX . s a v e t o b a c k u p ( a d e t l : : X N ) ; / / s ave t h i s s t e p p r o p e r t i e s a s l a s t t ime s t e p

c o u t << ” F i n i s h e d s i m u l a t i o n ” << e n d l ;

implicit coupling framework for multi-physics reservoir

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