Algebraic Multiscale Solver for Flow Problems in...

ALGEBRAIC MULTISCALE SOLVER FOR FLOW PROBLEMS IN

HETEROGENEOUS POROUS MEDIA

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

Yixuan Wang

December 2015

c© Copyright by Yixuan Wang 2016

All Rights Reserved

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I certify that I have read this thesis and that in my opinion it is fully

adequate, in scope and quality, as partial fulfillment of the degree of

Doctor of Philosophy.

(Hamdi Tchelepi) Principal Adviser




(Hadi Hajibeygi)




(Khalid Aziz)

Approved for the University Committee on Graduate Studies

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Abstract

Numerical simulations of multiphase flow in porous media lead to linear systems with

highly heterogeneous and anisotropic entries. The size of these linear systems is very

large and falls beyond the scope of classical iterative solvers. This has motivated ex-

tensive research on the development of solution techniques for such systems, including

multiscale methods to reduce the computational complexity.

In the first part of this study, an Algebraic Multiscale Solver (AMS) for the pres-

sure equations arising from incompressible flow in heterogeneous porous media is

developed. In addition to the fine-scale system of equations, AMS requires informa-

tion about the superimposed multiscale (dual and primal) coarse grids. The dual

coarse grid is used for the construction of the basis functions; the primal coarse grid

is used to build a conservative coarse-scale operator, and to help in the reconstruc-

tion of a conservative fine-scale velocity field as the last step of the solution process.

AMS employs a global solver only at the coarse scale and allows for several types

of local preconditioners at the fine scale to deal with the full spectrum of errors.

The convergence properties of AMS are studied for various combinations of global

and local stages for a wide range of challenging test cases. These include MultiScale

Finite-Element (MSFE) and MultiScale Finite-Volume (MSFV) methods as the global

stage, and Correction Functions (CF), Block Incomplete Lower-Upper (BILU) and

Incomplete Lower-Upper (ILU) factorizations as local stages. The performance of

the different preconditioning options is analyzed for a wide range of challenging test

cases. The best overall performance is obtained by combining MSFE and ILU as

the global and local preconditioners, respectively, followed by MSFV to ensure local

mass conservation. Comparison between AMS and a widely used Algebraic MultiGrid

(AMG) solver [1] indicates that AMS is quite efficient.

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A very important advantage of AMS is that a conservative fine-scale velocity

can be constructed using MSFV; however, the MSFV method is known to produce

non-monotone solutions. The second part of this dissertation tackles this challenge.

The causes of the non-monotone solutions are identified and connected to the lo-

cal flux across the boundaries of primal coarse cells induced by the basis functions.

We propose a monotone MSFV (m-MSFV) method based on a local stencil-fix that

guarantees monotonicity of the coarse-scale operator and the resulting approximate

fine-scale solution. Detection of non-physical transmissibility coefficients that lead

to non-monotone solutions is achieved using local information only and is performed

algebraically. For these critical primal coarse-grid interfaces, a monotone local flux

approximation, specifically, a Two-Point Flux Approximation (TPFA), is employed.

Alternatively, a local linear boundary condition can be used for the dual basis func-

tions to reduce the degree of non-monotonicity. The local nature of the two strate-

gies allows for ensuring monotonicity in local sub-regions, where the non-physical

transmissibility occurs. For practical applications, one can significantly reduce the

non-monotonicity of the conservative MSFV solutions by employing the m-MSFV

modifications for a small fraction of the domain.

In the third part of this study, AMS is extended to simulate flow in reservoirs

with complex well configurations. Two approaches are described to incorporate well

models into the AMS framework. The first approach is to use the well functions

introduced by Jenny and Lunati [2] to capture the well effects; the second approach

is to compute the basis functions based on an approximate Schur complement. The

coarse-scale operator can be constructed in either Finite Volume (FV) or Finite Ele-

ment (FE) formulation. The convergence properties and computational efficiency of

these approaches are evaluated by numerical simulations of various cases with simple

and complex wells.

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Acknowledgements

I would like to express my utmost gratitude to my advisor, Professor Hamdi Tchelepi,

for his constant support, patience and encouragement in my academic life. Without

his profound insight and masterful guidance, this work would not be possible. I have

learned a great deal from him not only in terms of knowledge, but also in the ways

to perform research. I am so honored that I could conduct my research under his

supervision.

Also, my deep gratitude is owed to Professor Hadi Hajibeygi, who was the post-

doc in our research group and is now a young professor in TU Delft, for providing

me with a comprehensive understanding of multiscale simulation techniques. His

encouragement and advice motivated me to resolve the challenges I encountered dur-

ing my entire PhD life. Professor Hajibeygi has taught me a lot of things, such as

presentation and paper writing skills. His great help and friendship are gratefully

acknowledged.

Sincere thanks to Professor Khalid Aziz, who read my dissertation carefully and

gave insightful feedback on improving its quality.

I wish to thank Dr. Hui Zhou for providing the initial code and answering my

questions via emails and calls. Many thanks to Dr. Abdulrahman Manea for many

constructive discussions about this work. In addition, I particularly appreciate Dr.

Xiaochen Wang and Dr. Yifan Zhou for their thoughtful discussions and kindly

encouragements.

I would like to express my special thanks to the Department of Energy Resources

Engineering at Stanford University, and all of its faculty, staff, and students. This

friendly environment made my five years here colorful.

I would like to thank the industrial affiliates of the Stanford University Reservoir

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Simulation (SUPRI-B) research programs for their financial support.

Finally, my endless gratitude also goes to my parents, for having me educated and

for supporting me from half a world away.

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Contents

Abstract v

Acknowledgements vii

1 Introduction 1

1.1 Multiscale Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Multiscale Finite Element Method . . . . . . . . . . . . . . . . 3

1.1.2 Multiscale Finite Volume Method . . . . . . . . . . . . . . . . 5

1.2 Linear Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Incomplete LU Factorization . . . . . . . . . . . . . . . . . . . 10

1.2.3 Algebraic Multigrid (AMG) . . . . . . . . . . . . . . . . . . . 11

1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Algebraic Multiscale Solver Framework 15

2.1 Algebraic Multiscale Method Formulation . . . . . . . . . . . . . . . 15

2.2 Fine-scale Velocity Reconstruction . . . . . . . . . . . . . . . . . . . . 23

2.3 Analysis of Correction Function . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Independent Local Stage . . . . . . . . . . . . . . . . . . . . . 26

2.3.2 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Sensitivity to Transmissibility Contrasts . . . . . . . . . . . . 32

2.3.4 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . 32

2.4 Local and Global Preconditioner . . . . . . . . . . . . . . . . . . . . . 34

2.4.1 AMS Global Stage: MSFV versus MSFE . . . . . . . . . . . . 36

2.4.2 AMS Global Stage: Local Boundary Conditions . . . . . . . . 40

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2.4.3 AMS Local Stage . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.4 AMS versus AMG . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Monotone Multiscale Finite Volume Method 51

3.1 Coarse-scale Transmissibility Coefficients . . . . . . . . . . . . . . . . 52

3.2 Monotone MSFV (m-MSFV) Method . . . . . . . . . . . . . . . . . . 60

3.2.1 Local TPFA Approach . . . . . . . . . . . . . . . . . . . . . . 60

3.2.2 Local Linear BC Approach . . . . . . . . . . . . . . . . . . . . 63

3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.1 Case 1: SPE 10 Bottom Layer . . . . . . . . . . . . . . . . . . 64

3.3.2 Case 2: SPE 10 Layers with Stretched Grid . . . . . . . . . . 70

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Algebraic Multiscale Solver with Well Modeling 79

4.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 Well Basis-Function Method . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.1 Prolongation Operator . . . . . . . . . . . . . . . . . . . . . . 84

4.2.2 Restriction Operator . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 Schur Complement Method . . . . . . . . . . . . . . . . . . . . . . . 90

4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.1 Convergence Rate . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.2 Computational Efficiency . . . . . . . . . . . . . . . . . . . . . 97

4.4.3 Scalability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 105

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 Conclusions and Future Work 113

Nomenclature 117

Bibliography 121

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List of Tables

2.1 Average total simulation time (sec) and number of iteration steps for

MSFE-BILU, MSFE-CF-BILU, and MSFE-MCF-BILU AMS solvers.

The three-stage (MSFE-CF-BILU) iterative procedure is not conver-

gent due to the CF sensitivity to the contrast in the transmissibility

field. However, if the modified CF (MCF) is used (Eq. (2.40)), i.e.,

MSFE-MCF-BILU, the procedure becomes convergent. Results are

shown on average for twenty statistically-the-same realizations. Also

shown in parentheses are the standard deviations. . . . . . . . . . . . 34

2.2 Five permeability sets (each with 20 equiprobable realizations) are used

for the numerical experiments of this section. Layered fields, i.e., sets

1 and 2, are generated for 4 different layering angles, each of which has

20 equiprobable realizations. . . . . . . . . . . . . . . . . . . . . . . . 35

2.3 The CPU time (sec) and iteration steps for MSFE-BILU and MSFE-

ILU preconditioned by GMRES. Iterations are stopped when the rela-

tive l2 norm of the residual is reduced by five orders of magnitude. . . 43

2.4 CPU time (sec) and iteration steps for SAMG on a patchy domain. . 46

2.5 CPU time (sec) and iteration steps for AMS on a patchy domain. . . 46

3.1 Relative errors of hybrid m-MSFV, m-MSFV(TPFA), m-MSFV(LBC)

and original MSFV for the SPE 10 top layer with stretched grids,

i.e.,∆x = 10∆y. In addition, the last two columns represent the

amount of TPFA coarse-scale interfaces and dual-grid boundaries using

LBC for all the methods. . . . . . . . . . . . . . . . . . . . . . . . . . 76

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4.1 Different options for the multiscale preconditioner M−1ms in the Schur

complement method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2 Well configurations, including well type, well control, and well index. 97

4.3 Well configurations, including well type, well control and well index. . 103

4.4 Iterations and solution phase CPU (sec) of different strategies for the

SPE 10 model with a coarsening ratio of 5× 5× 5. . . . . . . . . . . 107

4.5 Iterations and solution phase CPU (sec) of AMS and SAMG for the

SPE 10 test cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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List of Figures

1.1 Flow chart for a sequential solution scheme of coupled flow and trans-

port systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Primal (bold black) and dual (dashed blue) coarse cells. Fine-cells

belonging to a coarse cell (control volume) are shown in green. Fine-

cells that belong to a dual coarse cell are highlighted in light red. The

red circles denote the coarse nodes (vertices). . . . . . . . . . . . . . . 16

2.2 Ordering of the fine cells based on the imposed dual-coarse grid. Also

shown with bold solid lines is the primal coarse grid. . . . . . . . . . 18

2.3 Permeability field, fine-scale reference and MSFV pressure solution. . 25

2.4 Velocity divergence before and after reconstruction. . . . . . . . . . . 25

2.5 Natural logarithm of layered permeability field with ψ1 = 0.5 and

ψ2 = 0.02. Fine grid size is 100× 100 and coarse grid size is 10× 10. 27

2.6 Comparison between the solutions obtained from fine-scale reference,

MSFV, MSFV-CF and MSFV-BILU. Note that all solutions are con-

servative at the coarse-scale. Furthermore, error norms for MSFV (b),

MSFV-CF (c), and MSFV-BILU (d) are 5.13, 0.16 and 0.28, respectively. 28

2.7 Natural logarithm of permeability field for spectral analysis. . . . . . 30

2.8 Eigenvalues of MSFV, MSFV-CF, MSFV-BILU and MSFV-CF-BILU

iteration matrices for a simple homogeneous test case. . . . . . . . . . 30

2.9 Eigenvalues of MSFV, MSFV-CF, MSFV-BILU and MSFV-CF-BILU

iteration matrices for a heterogeneous test case. . . . . . . . . . . . . 31

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2.10 Natural logarithm of the permeability field for one of the 20 statistically-

the-same fields generated to analyze the computational efficiency of

CF. The domain consists of 128× 128× 64 fine and 16× 16× 8 coarse

grid cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.11 Natural logarithm of one realization of permeability set 1 with different

layering angles of 0, 15, 30 and 45 from left to right. For each

layering angle, 20 realizations are considered. . . . . . . . . . . . . . . 35

2.12 Natural logarithm of one (out of 20 statistically-the-same) realization

of the permeability set 3. . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.13 Natural logarithm of the permeability (a) and fine-scale pressure solu-

tion (b) for the SPE 10 bottom layer. . . . . . . . . . . . . . . . . . . 37

2.14 Iteration histories for AMS with different restriction schemes using the

SPE 10 bottom layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.15 Comparison of (a) total simulation time and (b) iteration steps for FE

and FV global solvers (i.e., restriction operator) on layered and patchy

permeability fields over 20 different realizations. Also shown in error

bars are the standard deviations. Clearly, the FE restriction operator

outperforms the FV one. . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.16 Comparison of MSFV and MSFE restriction operators for anisotropic

patchy and layered permeability fields (one realization for each) with

different local solvers (BILU and ILU). The convergence rate and total

simulation time (sec) are illustrated on the left and right columns,

respectively. For layered systems with orientation angles of 0o, 30o and

45o. The results are similar to the layered 15o case; therefore, they are

not shown here. Only the total simulation time of the MSFE operator

is illustrated since it has a high convergence rate compared with the

MSFV operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.17 Coarse grid size effect on average (a) total simulation time and (b)

iteration steps for permeability set 2 and 4. . . . . . . . . . . . . . . . 40

2.18 Solution phase time (i.e. excluding setup time) averaged over 20 equiprob-

able realizations for MSFV and MSFE restriction schemes with linear

and reduced boundary conditions. . . . . . . . . . . . . . . . . . . . . 41

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2.19 (a) Natural logarithm of the permeability and (b) pressure solution for

the full SPE 10 case. The grid contains 60×220×85 fine and 6×22×17

coarse cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.20 Iteration histories for MSFV and MSFE restriction operators with lin-

ear and reduced boundary conditions. Note that the MSFV with Re-

duced BC for local basis functions and BILU as the second stage solver

does not converge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.21 The average and error bar plots of (a) total simulation time and (b)

iteration steps for BILU and ILU comparison on layered and patchy

permeability fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.22 Iteration steps and total simulation time (sec) for GMRES precondi-

tioned by the MSFV (a) and MSFE (b) with CF, MCF, and ILU.

Results are averaged over 20 realizations of patchy permeability field

of set 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.23 Scalability analysis of AMS and SAMG. . . . . . . . . . . . . . . . . 46

2.24 Total, setup, and simulation times (sec) of AMS and SAMG as linear

solvers for permeability sets 1 and 3. Results are averaged over 20

statistically-the-same realizations for each case. . . . . . . . . . . . . 47

2.25 AMS algorithm chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Illustration of the basis function φid solved on dual-coarse cell ΩdD sub-

ject to reduced-dimensional boundary condition. Note that the basis

functions are always monotone and satisfy 0 ≤ φid ≤ 1, provided that

the mobility tensor λ is positive definite. . . . . . . . . . . . . . . . . 55

3.2 (left): Illustration of a 3 × 3 coarse- and 21 × 21 fine- grid domain.

The coarse cell i is highlighted in red, neighboring k and j on its South

and South-West sides. Also shown are the induced fluxes by the φj

(middle) and φk (right). Note that only the overlapping part of the

basis functions are plotted, and that for simplicity of the illustration a

homogeneous problem is used. . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Logarithm of permeability field for SPE 10 bottom layer. The domain

consists of 220× 60 fine- (not shown) and 20× 12 coarse- (shown) grid

cells. Two subdomains of the size 3× 3 coarse cells are highlighted. . 57

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3.4 (top-left): Logarithm of permeability field with coarse grid and coarse

nodes, extracted from Fig. 3.3. (top-right): part of the basis func-

tion φk overlapping with coarse cell i (coarse cell (10,3) in Fig. 3.3).

(bottom-left): basis function φi; (bottom-right): superimposed MSFV

pressure field, pms =∑φkp

ck, obtained for Ωh1. . . . . . . . . . . . . 58

3.5 (top-left): Logarithm of permeability field with coarse grid and coarse

nodes, extracted from Fig. 3.3, Ωh2. (top-right): part of the basis

function φk overlapping with coarse cell i (coarse cell (5,6) in Fig. 3.3).

(bottom-left): basis function φi; (bottom-right): superimposed MSFV

pressure field, pms =∑φkp

ck, obtained for Ωh2. Note that a non-

physical positive off-diagonal value of acik = 222.5 and small positive

value of acii = 0.65 are calculated for coarse-system entries, which also

clearly shows the i-th coarse-system row is not diagonally dominant. . 59

3.6 Fine-scale reference (left) and MSFV (right) solutions for the SPE 10

bottom layer heterogeneous test case. There exist 220 × 60 fine- and

20× 12 coarse- grid cells. Note that the MSFV superimposed solution

(right) entails several non-physical peaks. The permeability field is

also partly shown in the plots under the pressure solution. . . . . . . 60

3.7 Automatically detected critical interface (shown in bold red) where

acik 6≤ 0. The highlighted region with a pink rectangle shows the local

domain, where the transmissibility is calculated using the summation

of harmonically averaged values to replace with acik and acki. . . . . . . 62

3.8 Critical coarse node i and its neighboring faces Fij (indicated by red

solid lines) and edges Eij (indicated by yellow dash lines), j = 1, 2, 3, 4

for 2D domain. The black lines indicate the coarse volumes. . . . . . 62

3.9 Natural logarithm of the permeability (a) and fine-scale reference pres-

sure (b) for the SPE 10 bottom layer. . . . . . . . . . . . . . . . . . . 65

3.10 Original MSFV and m-MSFV pressure solutions for the SPE 10 bottom

layer, and the relative errors ep. The coarse-scale grids are indicated

by black lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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3.11 Streamline plots based on velocity fields reconstructed by fine-scale ref-

erence, original and m-MSFV pressure solutions for the SPE 10 bottom

layer. The coarse-scale grids are indicated by black lines. . . . . . . . 66

3.12 Histogram of ηij of the coarse-scale system Ac for original MSFV (a),

the reconstructed coarse-scale system for m-MSFV (TPFA) (b) and

m-MSFV (LBC) (c), respectively. . . . . . . . . . . . . . . . . . . . . 67

3.13 Pressure surface plots for fine-scale reference (a), original MSFV (b),

m-MSFV (TPFA) with ε = 0 (c) and ε = 0.7 (d) . . . . . . . . . . . . 68

3.14 Error measurements in pressure (a), velocity (b), residual (c) and the

computational complexity (d) with different threshold ε for the SPE

10 bottom layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.15 Permeability and fine-scale pressure solution for the SPE 10 top layer

with stretched grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.16 Original MSFV and m-MSFV pressure solutions for the SPE 10 top

layer with stretched grids, and the relative errors ep . . . . . . . . . . 71

3.17 Histogram of ηij of the coarse-scale system Ac for original MSFV (a)

and the reconstructed coarse-scale system for m-MSFV (LBC) (b),

respectively, for the SPE 10 top layer with stretched grids. Note that

m-MSFV (TPFA) eliminates all the positive indicators, therefore the

histogram is not shown. . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.18 Streamline plots based on velocity fields reconstructed by fine-scale

reference, original and monotone MSFV pressure solutions. . . . . . . 72

3.19 Error measurements in pressure (a), velocity (b), residual (c) and com-

putational complexity (d) with different threshold ε for the SPE 10 top

layer with stretched grids. . . . . . . . . . . . . . . . . . . . . . . . . 73

3.20 Original MSFV and m-MSFV pressure solutions for the SPE 10 bottom

layer with stretched grids, and the relative errors ep. . . . . . . . . . 74

3.21 Streamline plots based on velocity fields reconstructed by fine-scale ref-

erence, original and m-MSFV pressure solutions for the SPE 10 bottom

layer with stretched grids. The coarse-scale grids are indicated by black

lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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3.22 Error measurements in pressure (a), velocity (b), residual (c) and com-

putational complexity (d) with different threshold ε for the SPE 10

bottom layer with stretched grids. . . . . . . . . . . . . . . . . . . . . 75

3.23 Pressure distributions for fine-scale reference (a) and obtained by hy-

brid m-MSFV method (b) for the SPE 10 top layer with stretched

grids, i.e.,∆x = 10∆y. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.24 Velocity distributions for fine-scale reference (a) and obtained by hy-

brid m-MSFV method (b) for the SPE 10 top layer with stretched

grids, i.e.,∆x = 10∆y. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.1 Illustration of basis function and well basis function on a 2D dual coarse

grid with a homogeneous permeability field. The fine-scale cells with

a white circle represent a coarse-scale node, and the cells with a white

cross marker indicate the perforations of the well. . . . . . . . . . . . 84

4.2 Homogeneous permeability field with two wells located on the two sides

of the domain (the black dots indicate the well perforations): (a), the

fine-scale reference solution (b), the MSFV solution (c), and the MSFE

solution (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3 Well locations and natural logarithm permeability fields. The black

lines represent the well perforations and the white lines indicate the

dual coarse grid boundaries. . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Fine-scale reference pressure solution. . . . . . . . . . . . . . . . . . . 95

4.5 Iteration history of various linear solver options for different permeabil-

ity fields. The dashed lines indicate the use of an FE type of restriction

operator; the solid lines indicate that the restriction operator is based

on an FV formulation. The well index α =√kxky. . . . . . . . . . . 96

4.6 Well locations and permeability fields. The black lines represent the

perforations and the white lines indicate the dual coarse grid boundaries. 98

4.7 Fine-scale reference pressure solution for the scenario with c = 1. . . . 99




on an FV formulation. The well index α = 0.01√kxky. . . . . . . . . 100

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on an FV formulation. The well index α = 0.1√kxky. . . . . . . . . . 101




on an FV formulation. The well index α =√kxky. . . . . . . . . . . 102

4.11 Natural logarithm of one realization (out of 20 statistically-the-same)

of patchy permeability. A five-spot well pattern is considered and each

well penetrates all the layers in z direction. . . . . . . . . . . . . . . . 104

4.12 Natural logarithm of one realization of permeability set with different

layering angles of 0, 15, 30, and 45, from left to right. For each

layering angle, 20 realizations are generated for each case. A five-spot

well pattern is considered and each well penetrates all the layers in z

direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.13 Average iterations (a) and CPU time (b) of the 20 realizations for

patchy and layer domains. The linear solver strategies include the

well basis-function method with an finite volume restriction operator

(WF-FV), an finite element restriction operator (WF-FE), the Schur

complement method with an finite volume restriction operator (Schur-

FV) and an finite element restriction operator (Schur-FE), and SAMG. 106

4.14 Natural logarithm of the full SPE 10 model (a) and the simplified

version with the top 30 layers (b). A five-spot well pattern is considered

and each well penetrates all the layers in z direction. . . . . . . . . . 107

4.15 Natural logarithm of the patchy permeability field. . . . . . . . . . . 109

4.16 Well locations (shown in black dots) in the top layer of the z direction

for the scenarios with 4 and 16 wells. . . . . . . . . . . . . . . . . . . 109

4.17 The number of iterations (a) and the CPU time (b) in both setup and

solution phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.18 The scalability of well function constructions. . . . . . . . . . . . . . 111

xix

xx

Chapter 1

Introduction

In the oil and gas industry, reservoir simulation is an essential and powerful technique

for field development and operation. The computer software for reservoir simulation

is called a reservoir simulator. Given a geological model as input, the primary task of

reservoir simulation is to analyze flow through the porous formation, then to calculate

production profiles as a function of time for the wells in the reservoir. Reservoir

simulation allows the engineer to predict the reservoir performance under different

operating scenarios; therefore, reservoir simulation helps resolve a number of design,

operational, and troubleshooting problems during all stages in the development of

a field. The accuracy of simulating subsurface flow relies strongly on the level of

detail in the description of the reservoir formations. Formation properties such as

porosity and permeability typically display heterogeneity with high levels of variation

spanning multiple length scales. Small-scale heterogeneity may have an impact on

the flow and transport at larger scales and affect the flow dynamics in the entire

reservoir. Therefore, a high-resolution reservoir characterization is usually required to

achieve reliable simulation results. This leads to computational grid sizes on the order

of millions and more. Direct simulation on such a fine-scale heterogeneous reservoir

model is computationally demanding. Therefore, there is strong interest in multiscale

methods, which have been developed to reduce the computational complexity required

for large-scale reservoir simulation.

The objective of multiscale methods is to efficiently obtain an accurate approx-

imation of the fine-scale solution by reducing the number of degrees-of-freedom. In

1

2 CHAPTER 1. INTRODUCTION

multiscale methods, the global fine-scale problem is decomposed into large numbers

of local problems. The basis functions, which are numerical solutions of the local

problems, are used to construct accurate coarse-scale quantities. Once the coarse-

scale system is solved, the solution is mapped onto the fine scale using the basis

functions. Multiscale methods have many similarities with upscaling techniques [3–

10], but there are also significant differences. First, multiscale modeling is aimed at

obtaining the fine-scale solution, while upscaling is usually aimed at generating ap-

proximate coarse-scale solutions. Second, the coarse-scale system in multiscale meth-

ods is constructed numerically and dynamically using current fine-scale information

instead of pre-processed coarse-scale quantities as employed in upscaling. This makes

multiscale methods good candidates for general-purpose reservoir simulation.

In reservoir simulation application, multiscale methods, particularly the one pro-

posed by Jenny et al. [11], solve the nonlinear coupled flow (pressure and total ve-

locity) and transport (saturation) problems in a sequential manner, where the flow

and transport equations are treated separately and differently. As shown in Fig. 1.1,

each time step consists of an outer loop to solve the coupled system. In each time

step, the linearized pressure equation is solved by multiscale methods, then the pres-

sure solution is used to construct the velocity field, which is employed to solve the

transport problem. Most multiscale methods focus on building effective coarse-scale

operators for the flow problem (i.e., the pressure equation). The transport problem

is generally solved on the fine scale since the transport problem is intrinsically local.

The pressure equation is dominated by elliptic characteristics. It is quite difficult

to solve such a linear equation since the errors usually span a very wide range of

frequency spectrum. The quality of the pressure solution has a significant impact

on the transport solution and the stability of the nonlinear solution strategy. Hence,

the current challenge of developing a robust multiscale method is to efficiently obtain

physical and accurate solutions of the linearized pressure equation. To tackle this

challenge, we developed a general Algebraic Multiscale Solver (AMS) as an iterative

linear solver to deliver an efficient and accurate solution, and we propose a monotone

multiscale finite volume method to guarantee that the solution is physical (monotone)

for highly heterogeneous and anisotropic problems.

Next, we review a variety of multiscale methods for flow simulation and linear

CHAPTER 1. INTRODUCTION 3

Pressure loop

Saturation loop

Velocity calculation

Outer loop

Figure 1.1: Flow chart for a sequential solution scheme of coupled flow and transportsystems

solution strategies, which are the building blocks for this work.

1.1 Multiscale Methods

1.1.1 Multiscale Finite Element Method

The MultiScale Finite Element (MSFE) method, first introduced by Hou and Wu [12],

integrated the fine-scale information into a coarse-scale system by computing special

basis functions, which are numerical solutions of localized boundary-value problems

that capture the fine-scale effects. The MSFE method solves the following elliptic

problem

∇ · (λ · ∇u) = f on Ω. (1.1)

The main idea of MSFE is to construct basis functions that represent the fine-scale

information. In the context of finite-element methods, the weighting functions space

Vh is spanned by these basis functions as

Vh = spanφiK ; i = 1, ..., n;K ∈ Kh ⊂ H10 (Ω), (1.2)


where φiK denotes the basis function associated with node i in element K, n is the

number of nodes of element K, Kh is an element partition of domain Ω and H10 (Ω)

is the Hilbert functional space defined on Ω. These basis functions are solved locally

in each element with reduced boundary condition, such that∇ · (λ · ∇φiK) = 0 in ΩK

∇‖ · (λ · ∇φiK)‖ = 0 on ∂ΩK

φiK(xj) = δij ∀ j ∈ 1, ..., n,(1.3)

where the subscript ‖ indicates the vector (or operator) projected along the tangen-

tial direction of the element boundary ∂ΩK , and ΩK is the domain of element K,

superscript i denotes one node of that element, and xj represents the coordinate of

node j. After locally solving Eq. (1.3), the coarse-scale system can be constructed by

the basis functions, in a similar way as conventional finite-element methods. Once

the coarse-scale solution is obtained, the fine-scale approximation can be calculated

using the basis functions and the coarse-scale solution at node i (i.e., ui) as

u(x) =n∑i=1

φiK(x)ui if x ∈ ΩK . (1.4)

Hou and Wu [12] pointed out that large errors occur due to resonance when the

scale of oscillations in the fine-scale coefficient is close to the scale of the grid. In ad-

dition, the resonance error can be eliminated by improving the boundary conditions

of the basis functions. In order to tackle this issue, they proposed an oversampling

technique that imposes the reduced boundary condition on a sampled domain, which

is larger than the coarse element and then the basis functions are computed on that

sampled domain. They show that a good choice of the boundary condition deter-

mines that the local characteristics are well sampled into the basis functions and can

significantly improve the accuracy of multiscale methods. The investigations reported

in [13–16] demonstrate that the MSFE method has a good convergence rate for the

elliptic equation with highly oscillatory coefficients.

The major drawback of this method for reservoir simulation is that it cannot

deliver a mass conservative velocity field. Since existing multiscale methods use a

sequential strategy, having a conservative velocity field is crucial to achieve accurate


solutions of the transport problem. Later, Chen and Hou [17] presented a multiscale

formulation based on a mixed finite-element method and demonstrated the impor-

tance of a locally conservative algorithm for transport simulation. This family of

Mixed MultiScale Finite-Element (MMSFE) methods [18–22] can offer mass conser-

vative velocity fields for both fine- and coarse-scale grids.

1.1.2 Multiscale Finite Volume Method

In order to obtain approximate solutions that are strictly locally mass conservative

on the fine scale, the MultiScale Finite Volume (MSFV) [23] method was developed.

Compared with the MMSFE formulation, the MSFV method yields mass conservative

solutions using a smaller number of degrees of freedom.

Basis functions are employed to capture the fine-scale information, which are iden-

tical to those of the MSFE method [12]. The approximate pressure solution obtained

from MSFV guarantees local mass conservation, which can be used to reconstruct the

velocity field by solving local elliptic problems on primal coarse grids with Neumann

boundary conditions. Recent developments of the MSFV method include incorpo-

rating the effects of compressibility [24–27], gravity and capillary [28], complex wells

[29, 30], faults [31], fractures [32, 33], three-phase flow using the black-oil model

[34] and compositional displacements [35]. Furthermore, the efficiency of the method

has been enhanced by adaptive computation of the basis functions for multiphase,

time-dependent displacement problems [11, 36–38]. The first attempt to develop

an algebraic multiscale solver is the Operator-Based Multiscale Method (OBMM)

of Zhou and Tchelepi [25]. OBMM employs prolongation and restriction operators,

which are constructed in an algebraic manner, to capture the fine-scale information.

Applying these two operators to the original fine-scale mass balance equation leads to

a coarse-scale system, which can be solved for the coarse-scale pressure field. Then, a

fine-scale approximation can be obtained by prolongation of the coarse-scale solution.

This algebraic formulation reduces implementation complexity, especially for prob-

lems defined on unstructured grids, and allows for flexibility to incorporate complex

physics and easy integration of the method into existing reservoir simulators.

Although the MSFV and fine-scale reference results are in good agreement for a

wide range of test cases, it has been reported that the solution of the original MSFV


method deteriorates for channelized permeability fields [39], and large anisotropy [40].

As in all multiscale methods, in the MSFV framework, the coarse system is obtained

by using basis functions, which are numerically computed on local domains (with as-

sumed boundary conditions). The accuracy of the MSFV method is therefore strongly

dependent on the quality of the local boundary conditions. For some very challenging

problems, the method fails to provide accurate solutions. To resolve these limita-

tions, the iterative MSFV (i-MSFV) method was introduced by Hajibeygi et al. [41],

where the MSFV solution is iteratively improved by locally computed correction func-

tions [28] together with a fine-scale smoother. The i-MSFV method converges to the

fine-scale reference solution, and a conservative velocity field can be constructed after

any iteration level. The i-MSFV method [41] reduced the MSFV errors for many

challenging problems; however, for highly heterogeneous and anisotropic cases, it did

not perform satisfactorily. This issue was found to be due to the weak MSFV coarse

scale operator. To deal with this challenge, the Two-stage Algebraic Multiscale Linear

Solver (TAMS) was proposed [42]. Due to the importance of conservative solution,

once the iterations are stopped, the MSFV operator is employed as the last step in

the TAMS algorithm. TAMS consists of two stages, one local stage and one global

stage. In the global stage, low-frequency errors are resolved by the MultiScale (MS)

preconditioner. In the local stage, high-frequency errors are resolved by a local solver

or smoother like Block-ILU (BILU) [43]. With a combination of these two stages, all

the different frequency errors are resolved. In addition, TAMS is suitable for both

Finite Volume (FV) and Finite Element (FE) based approaches. Unlike other MSFV

methods, no Correction Function (CF) has been used in TAMS. Since CF has been

widely used to capture fine-scale source terms, it is necessary to develop a general

algebraic multiscale linear solver, which has the correction function to capture fine-

scale source terms. In addition, the best choice among a variety of possible local and

global stages had not been thoroughly investigated. These questions motivate the

development of the general Algebraic Multiscale Solver (AMS) described in Chapter

2.


1.2 Linear Solver

Numerical simulations of multiphase flow in porous media require solving a discretized

linear system of equations arising from the nonlinear governing equations for each

Newton iteration. There could be thousands of Newton iterations in a typical reser-

voir simulation run; hence, a large number of linear equations need to be solved.

These linear systems exhibit strong global coupling and can be quite difficult to

solve. Therefore, the linear solver often dominates the total computational effort in

practical reservoir simulations.

A system of linear equations can be written in matrix notation as

Ax = b. (1.5)

There are two basic classes of methods for solving such system. The first one is direct

methods. They essentially can be thought of as one variant of LU decomposition,

which is the process of finding a lower triangular matrix L and an upper triangular

matrix U , so that the coefficient matrixA can be represented by the product of L and

U . Sparse direct solvers can save memory and CPU time by exploiting the sparsity

pattern of the coefficient matrix A, such as the Thomas algorithm used in reser-

voir simulation applications for one dimensional problems [44]. In addition, different

ordering of unknowns of the sparse system of linear equations can affect the effi-

ciency of computational effort and memory storage for a direct solution method [45].

Theoretically, direct methods deliver an exact solution in a finite number of steps.

Unfortunately, this may not be true due to the rounding error, which is an error that

occurs in one step and accumulates during all following steps. Normally, direct solvers

are not practical for large-scale linear systems in terms of computation and memory

efficiency, which motivated the development of the second class of linear solvers, i.e.,

iterative linear solvers. Iterative methods compute a sequence of approximate solu-

tions, starting with an initial estimate, and continue the iterations until a stopping

criterion is satisfied (typically, the criterion is that error or residual is reduced to some

specified tolerance). In reservoir simulation, the unknowns are usually the changes

in variables from one Newton step to the next. Therefore, the zero vector could be

used as an appropriate initial guess for the final solution. The kernel of most modern


numerical techniques for iterative linear solvers is a combination of Krylov subspace

methods (i.e., acceleration procedures) and preconditioners. Freund et al. [46] provide

an overview of these methods. The advantage of iterative solvers is that only the ma-

trix and vector multiplications are needed, and explicit access to the matrix element

is not required. However, the efficiency of iterative solvers hinges on the quality of

the preconditioner. In this work, the Generalized Minimum Residual (GMRES) [47]

method is used as the Krylov subspace acceleration procedure. Next, we introduce

the basic concept of a preconditioner and some examples studied in this dissertation.

1.2.1 Preconditioner

The convergence rate of iterative linear solvers depends on the condition number and

the eigenvalue spectrum of the coefficient matrix. Preconditioners are used to reduce

the matrix condition number and accelerate the convergence rate [48]. Some stand-

alone algorithms, e.g., Incomplete Lower Upper (ILU) factorization, can be directly

applied as preconditioners. These preconditioners are single-stage preconditioners. In

some scenarios, several algorithms are combined to construct a multi-stage precon-

ditioner, such as Constrained Pressure Residual (CPR) [49, 50] used in the reservoir

simulation community. Although multi-stage preconditioners have some additional

burdens, they may lead to a much better numerical performance.

For a given matrixA, a preconditionerM−1 is a matrix that satisfies the following

two criteria: first, M−1A should have a smaller condition number than A. In other

words, M−1 should be close to A−1; second, M−1 should be cheap to compute [48].

Typically, we will solve the linear system My = b efficiently rather than calculate

the inverse of M explicitly, where b is a given right-hand-side (RHS) vector and y

is an intermediate vector. The preconditioner defined above is a left preconditioner

since M−1 is applied to the left side of matrix A. For a general sparse matrix system

of (1.5), it would be more advantageous to solve an equivalent linear system with a

much smaller condition number

M−1Ax = M−1b. (1.6)


Since Krylov subspace solvers (e.g., GMRES) are based on matrix-vector multiplica-

tions [48], there is no need to compute M−1A explicitly. Instead, an operation for

M−1A can be given as follows:

v←M−1Au, (1.7)

where u is a given vector and v is the result of the operation. The operation in

Eq (1.7) can be divided into two steps. First, a basic matrix-vector multiplication is

performed as:

u∗← Au, (1.8)

where u∗ is an intermediate vector. Then, Eq. (1.7) can be rewritten as:

v←M−1u∗. (1.9)

This second step is achieved by solving the linear system

Mv = u∗. (1.10)

A right preconditioner is performed in a similar manner, but the preconditioning

matrix M−1 is applied in the right side of matrix A. Therefore, the equivalent

equation to Eq. (1.5) becomes:

AM−1(Mx) = b, (1.11)

which can be also separated into two steps. In the first step, the iterative solver solves

the linear system

AM−1y = b (1.12)

with y = Mx. Since the condition number of AM−1 is smaller than the one of

the original matrix A, this linear system should be easy to solve. Each matrix-vector

operation involves one preconditioner call and one matrix-vector operation with A,

but in a reverse order compared with the left preconditioning operation. Once the

intermediate vector y is obtained, one more preconditioner call solves the following


equation for the original unknown vector x:

Mx = y. (1.13)

In the following sections, we will discuss several important single-stage preconditioners

used in this work.

1.2.2 Incomplete LU Factorization

For an n× n matrix, LU factorization requires approximately O(n2) operations with

natural ordering, which is not competitive with iterative solvers. Moreover, LU fac-

torization of a sparse matrix does not necessarily lead to sparse matrices; therefore,

LU factorization can take up a tremendous amount of storage. Hence, LU factor-

ization is not a practical option for large sparse matrices. However, Incomplete LU

(ILU) factorization is more attractive [48].

ILU factorization is one of the most popular preconditioning families. Compared

with LU decomposition, some non-zero elements in the L and U factors are ignored

to reduce the cost and the number of fill-ins (entries which change from an initial zero

to a non-zero value during the execution of an algorithm). ILU has many varieties

based on the level of fill-ins [48]. Among them, no fill-in ILU, i.e., ILU(0), is the

simplest one. In the ILU(0) factorization, the lower and upper triangular matrices

only keep non-zero elements whose positions have non-zero elements in the original

matrix. Therefore, if the lower and upper triangular matrices of ILU(0) are overlapped

together, we get a matrix with the same non-zero pattern as the original matrix.

The ILU(0) algorithm can be performed using the storage of the original matrix;

therefore, no significant additional memory is required. ILU(0) is a very simple and

fast factorization. However, the ILU(0) approximation to the original matrix can be

poor. In order to improve the approximation accuracy, ILU factorization algorithms

with fill-ins have been developed, e.g., ILU with threshold (ILUT) [51] and ILU with

fill-in level k (ILUK) [48].

The Block-ILU (BILU) preconditioner is the block extension of point-wise ILU

with level-of-fill. The algorithm of BILU with no fill-in, i.e., BILU(0), performs the

same operations as that of ILU(0), but all of the algebraic operations in ILU(0) are


mapped onto small (local) matrix operations for BILU(0). There are a number of

numerical techniques for approximately, or exactly, inverting the diagonal (pivot)

blocks. Manipulation on the block level introduces some overhead, but BILU may

benefit from the structure of the coefficient matrix and improve the efficiency and

robustness of the preconditioner.

1.2.3 Algebraic Multigrid (AMG)

For most reservoir simulation problems, the pressure system is usually formed and

solved in the first step. The pressure equation intrinsically has near-elliptic charac-

teristics, which displays long-range interactions of the pressure behavior. It is quite

challenging to solve such linear equations due to the fact that the errors usually span

a very wide range of the frequency spectrum. Therefore, solution strategies that can

efficiently remove both high-frequency (short-range and local) and especially low-

frequency (long-range and global) error components are necessary. For such pressure

system, the ILU preconditioned Krylov subspace solvers may stagnate after a few

iterations since the high-frequency errors can be successfully removed by the solver

during the first few iterations while the remaining low-frequency errors are difficult

to resolve [52]. On the contrary, multigrid methods, which are ideal linear solvers for

elliptic problems (the computational work and storage increase linearly with prob-

lem size) [53], resolve all the error components in an efficient manner by employing

simple (local) relaxation schemes (e.g., Gauss-Seidel, Jacobi, etc.) with a coarse-grid

correction.

Algebraic Multigrid (AMG) [54, 55] is one kind of multigrid methods. In the

setup phase, a hierarchy of coarse grids is generated based on the original coefficient

matrix in a purely algebraic manner. These coarse grids are designed for highly

discontinuous and anisotropic coefficients. Starting from the original linear system,

a series of coarse grid operators (i.e., discrete coarse-grid problems) are constructed

for each coarse level recursively. In this procedure, prolongation (interpolation) and

restriction operators are generated to transfer the information across this hierarchy

of algebraic problems. In the solution phase, the low-frequency errors on a finer grid,

which are difficult to resolve, are transformed into a coarser level, where they are

represented as high-frequency modes and are easier to tackle. These high-frequency


errors can be reduced with a simple (local) smoothing procedure performed at each

grid level, for example, by using a few iterations of the Gauss-Seidel method. Then,

a residual correction computed on a coarser grid is interpolated into a finer grid by

the prolongation operator to improve the solution on the fine scale. By traversing

these multiple grid levels in a particular sequence, the short- and long-range error

components are removed on a finer and coarser grid, respectively. The solution is

eventually achieved by cycling on the generated level hierarchy.

AMG does not depend on geometrical information and only requires the coeffi-

cient matrix. This allows AMG to be used as a black-box solver or preconditioner.

As a consequence, AMG can provide great flexibility for many reservoir simulation

applications, especially the ones involving generally unstructured grids, for two main

reasons. One reason is that normally, the gridding module in a reservoir simulator

is independent of the solver module, and the information exchange between differ-

ent modules is quite difficult unless the solver module only needs the information

from the coefficient matrix as AMG does. The other reason is that the discretization

of the pressure system on unstructured grids generally leads to strong discontinuity

and anisotropy in the coefficients, which makes it difficult to generate coarse-grid

operators for classic geometric-based multigrid methods; however, AMG can build

the operators without geometrical information [56]. Moreover, the local operations

in the algorithms of AMG can be performed concurrently across the computational

platform, which results in a high degree of parallelism.

Theoretically, AMG has some requirements for the characteristics of the pressure

matrix. The preferable matrix is an M -matrix, which has nonpositive off-diagonal

elements, and all eigenvalues with a nonnegative real part [57]. Only the matrices from

elliptic, or near-elliptic, systems can be solved effectively by AMG. Due to this fact,

AMG is preferably used as a preconditioner for iterative solvers rather than as a stand-

alone linear solver. For large-scale elliptic problems on anisotropic unstructured grids,

it is difficult to design an alternative algorithm that can outperform AMG. Therefore,

AMG is used as a benchmark preconditioner to evaluate our AMS strategy.


1.3 Dissertation Outline

This dissertation is organized as follows. In Chapter 2, we describe a general Algebraic

Multiscale Solver (AMS) to provide a linear solution strategy for the pressure equa-

tion, incorporating a correction function into our framework. We analyze the effects

of the correction function on the entire framework, and the benefits and drawbacks

are discussed. Then, systematic tests are performed to optimize AMS, considering

different restriction schemes, different local boundary conditions and different local

preconditioners. After that, we investigate the efficiency of AMS with the optimum

strategy by comparing it with the state-of-the-art linear solver, Algebraic Multigrid

Methods for Systems (SAMG) [1].

In Chapter 3, we identify the cause of the non-physical peaks (non-monotonicity)

associated with the MSFV solutions for highly heterogeneous problems. Then, we

propose two approaches to achieve monotone pressure solutions. For the first ap-

proach, a local TPFA method for the critical interfaces is used to calculate a positive

transmissibility and replace the original MPFA stencils on the coarse-scale system.

For the second approach, a Linear Boundary Condition (LBC) is employed as the

local boundary assumption to solve the basis functions only for the dual-coarse cells

associated with the critical coarse nodes. Using a variety of numerical examples, we

demonstrate the effectiveness of our monotone MSFV method.

In Chapter 4, the governing equation for the most common well models is de-

scribed. Two approaches are introduced to incorporate well models into the AMS

framework. The convergence and computational efficiency are analyzed by perform-

ing numerical simulations on various test cases. Finally, conclusions and possible

future directions are given in Chapter 5.


Chapter 2

Algebraic Multiscale Solver

Framework

2.1 Algebraic Multiscale Method Formulation

In this dissertation, we focus on heterogeneous and anisotropic problems for incom-

pressible single-phase flow, which serves as a prototype of the elliptic nature of the

pressure equation in reservoir simulation. The governing equation (mass conservation

equation) can be described by

∇ · (λ · ∇p) = ∇ · (ρgλ · ∇z) + q, (2.1)

where λ = k/µ is a positive-definite mobility tensor, k denotes the permeability

tensor, q represents source terms, g is the gravitational acceleration acting in the

∇z direction, and ρ is the density. The viscosity µ of the fluid is assumed to be

independent of pressure.

The MSFV grid consists of two sets of overlapping coarse grids, namely primal

and dual coarse grids, superimposed on the given fine grids (Fig. 2.1). There are nc

primal coarse cells (control volumes), ΩiC (i ∈ 1, · · · , nc), and nd dual-coarse cells

(local domains), ΩjD (j ∈ 1, · · · , nd).

15

16 CHAPTER 2. AMS FRAMEWORK

𝛀𝑪𝒊

𝛀𝑫𝒋

Figure 2.1: Primal (bold black) and dual (dashed blue) coarse cells. Fine-cells be-longing to a coarse cell (control volume) are shown in green. Fine-cells that belongto a dual coarse cell are highlighted in light red. The red circles denote the coarsenodes (vertices).

The basis functions in the MSFV and MSFE methods are obtained by solving∇ · (λ · ∇φij) = 0 in Ωj

D

∇‖ · (λ · ∇φij)‖ = 0 on ∂ΩjD

φij(xk) = δik ∀k ∈ 1, ..., nc,(2.2)

where φij denotes the basis function associated with coarse node i in dual coarse

block ΩjD [12, 23], xk represents the coordinates of coarse node k, and δik is the

Kronecker delta. The subscript ‖ indicates the vector (or operator) projected along

the tangential direction of the dual-coarse cell boundary, ∂ΩjD. The boundary con-

dition imposed in (2.2) for solving along the dual-coarse boundary is referred to as

the Reduced Boundary Condition (RBC). Alternatively, if one ignores the mobility

variation along the boundary, i.e., λ = I at ∂ΩjD, the formulation reduces to the Lin-

ear Boundary Condition (LBC). The Correction Functions (CF) [28], which account

for fine-scale RHS terms, are local particular solutions, and they are computed as

CHAPTER 2. AMS FRAMEWORK 17

follows: ∇ · (λ · ∇φ∗j) = ∇ · (ρgλ · ∇z) + q in Ωj

D

∇‖ · (λ · ∇φ∗j)‖ = ∇‖ · (ρgλ · ∇z)‖ + q on ∂ΩjD

φ∗j(xk) = 0 ∀k ∈ 1, ..., nc,(2.3)

where φ∗j is the CF for dual-coarse block ΩjD. Then, the approximate solution p′ is

obtained by using the superposition expression

p ≈ p′ =

nd∑j=1

[ nc∑i=1

φijpci + φ∗j

], (2.4)

where pci is the coarse-scale solution at coarse node i. The coarse-scale system is

constructed by first substituting Eq. (2.4) into Eq. (2.1), and then integrating over

the primal coarse control-volumes, ΩiC , which after using the divergence theorem can

be expressed as

ACpC = RC , (2.5)

with

AC(i, j) = −nd∑d=1

∫∂Ωi

C∩ΩdD

(λ · ∇φjd) · ~n dΓ (2.6)

and

RC(i) =

nd∑d=1

∫∂Ωi

C∩ΩdD

(λ · ∇φ∗d) · ~n dΓ−∫

ΩiC

r dv, i ∈ 1, ..., nc, (2.7)

entries. Here, ~n is the unit-normal vector pointing outward, and r represents the

RHS of Eq. (2.1). After solving Eq. (2.5) for the coarse-scale pressure, pC , Eq. (2.4)

is used again to obtain an approximate fine-scale solution, p′.

The problem (2.1) is well-posed for a d-dimensioanl computational domain Ω ⊂<d, subject to proper boundary conditions at ∂Ω ⊂ <d−1. The discrete form of

Eq. (2.1) at the given fine-scale, where the coefficients λ are computed using a finite-

volume Two-Point-Flux-Approximation (TPFA) scheme [44], can be written as

Ap = q. (2.8)

For 2D problems on structured grids, the dual-coarse grid divides the fine cells into

three categories: interior (white), edge (blue), and vertex (red) cells, as illustrated in


Fig. 2.2 [58, 59]. As the figure indicates, the vertices are the coarse-grid nodes, and

the edge cells are located on the boundaries of the dual-coarse cells. For 3D problems

on structured grids, an additional category is face cells. Finally, internal cells are

those that lie inside dual-coarse cells. For simplicity, the framework is described for

2D problems, although the implementation is in 3D.

Vertex Interior Edge

Figure 2.2: Ordering of the fine cells based on the imposed dual-coarse grid. Alsoshown with bold solid lines is the primal coarse grid.

A wirebasket reordered fine-scale system [59] can be expressed asAII AIE 0

AEI AEE AEV

0 AV E AV V

pIpEpV

=

qIqEqV

, (2.9)

where a local matrix Aij represents the contribution of cell j to the discrete mass

conservation equation of cell i. I,E and V denotes the interior, edge and vertex. The

gravitational source term, qG, is separated from the rest of the RHS term, q, because

qG requires special treatment according to Eq. (2.3). The reordered RHS vector is


therefore rewritten asqIqEqV

=

qGI

qGEqGV

+

qIqEqV

= B

qIqEqV

+ (I −B)

qIqEqV

, (2.10)

where B is a diagonal matrix with Bii = qGi /qi entries.

According to Eq. (2.3), only the tangential component of the gravitational source

term for edge cells is considered in the local problems for CF. Therefore, qGE is split

into tangential qGE‖ and normal qGE⊥ components. Thus, the RHS term used for CF

can be stated asqIq′EqV

=

qGI

qGE‖qGV

+

qIqEqV

= E

qGI

qGEqGV

+

qIqEqV

= (EB + I −B)

qIqEqV

, (2.11)

where E is a diagonal matrix with

E ii =

|~ne,i · ~ng| if i ∈ ℵedge1 otherwise

(2.12)

entries. Here, ℵedge is the set of edge cells, ~ne,i is the unit-vector tangent to the edge

cells at cell i, and ~ng is the unit vector parallel to gravitational acceleration. Finally,

the RHS term is expressed as: qIq′EqV

= E

qIqEqV

, (2.13)

where E = EB + I −B.

The matrix entries for interior cells are preserved in the approximate multiscale

operator. The stencil for edge cells, however, is modified to reflect the localization

assumption. In fact, the only source of error in the multiscale approximation is due to

the localization assumption. The reduced problem condition on edge cells is obtained

by setting AEI , and its corresponding part in AEE to zero. Finally, the multiscale


approximate system is expressed asAII AIE 0

0 AEE AEV

0 0 AC

p′I

p′Ep′V

=

qIq′ERC

, (2.14)

where ACp′V = RC is the coarse-scale system that is solved for p′V , where p′V

corresponds to pC in Eq. (2.5). It is clear that the multiscale system is upper-

triangular; hence, it is easy to invert. Note that TAMS by Zhou and Tchelepi [42]

did not account for qI and qE terms. In our AMS framework, we account for these

RHS terms, including gravitational effects.

Once the coarse system ACp′V = RC is solved, the pressure for the edges and

interior cells is obtained using backward substitution, i.e.,

p′E = −A−1EE(AEV p

′V − q′E)

p′I = −A−1II (AIEp

′E − qI) = A−1

II (AIEA−1EE(AEV p

′V − q′E) + qI).

(2.15)

Finally, the multiscale approximate solution, p′, is expressed asp′I

p′Ep′V

=

A−1II AIEA

−1EEAEV

−A−1EEAEV

IV V

p′V +

A−1II −A

−1II AIEA

−1EE 0

0 A−1EE 0

0 0 0

qIq′EqV

, (2.16)

where, IV V is an nc × nc identity matrix. The prolongation operator is defined as

P = G

A−1II AIEA

−1EEAEV

−A−1EEAEV

IV V

, (2.17)

where G is the permutation matrix that transforms the elements from wirebasket

ordering into natural ordering. Even though the multiscale formulation accounts for

the RHS terms, these terms do not appear in the prolongation operator. In fact, the


CF pressure in natural ordering pcorr is expressed as

pcorr = G

A−1II −A

−1II AIEA

−1EE 0

0 A−1EE 0

0 0 0

qIq′EqV

, (2.18)

which indicates that CF solves the same reduced problem for edge cells (in 3D for

face cells also) as the basis functions. The last column is zero because the vertices

are disconnected from both edge and interior cells. Since the permutation matrix is

orthogonal, i.e., GT = G−1, one can writeqIqEqV

= GTq. (2.19)

Finally, using Eqs. (2.13), (2.18), and (2.19), the CF pressure is related to the original

RHS vector as follows:

pcorr = G

A−1II −A

−1II AIEA

−1EE 0

0 A−1EE 0

0 0 0

EGTq. (2.20)

This equation can be simplified further by defining the correction operator, C, in

natural order, i.e.,

pcorr = Cq, (2.21)

where

C = G

A−1II −A

−1II AIEA

−1EE 0

0 A−1EE 0

0 0 0

EGT . (2.22)

Here, E is an extraction diagonal nf ×nf matrix (nf is the number of fine cells) that

includes the gravity term modification for edge cells. The modification of the gravity

term at the edge cells is employed only for the first iteration. For the rest of the

iteration steps, the RHS term is the residual in fulfillment of the governing equation.


Hence, no modification to the RHS is employed,

E =

(E − I)B + I ν = 0

I ν > 0, (2.23)

where I is the nf × nf identity matrix and ν is the iteration level.

The multiscale approximate solution expressed in Eq. (2.4) is stated algebraically

as

p′ = Pp′V + pcorr. (2.24)

Once the coarse-scale pressure p′V is obtained, Eq. (2.15) provides the edge and

interior pressures. To compute p′V , the following coarse-scale system is constructed

and solved

ACp′V = RC . (2.25)

Here,

AC =RAP , (2.26)

and

RC =Rq −RApcorr. (2.27)

The coarse-scale system of Eq. (2.25) is an algebraic description of Eq. (2.5). The

restriction operatorR is nc×nf , and can be based on finite-volume, or finite-element,

scheme. For the finite-volume operator, the fine-scale equations in a coarse cell are

simply summed up. Therefore, the entries of the MSFV restriction operator are

R(i, j) =

1 if Ωj

F ⊂ ΩiC

0 otherwise∀i ∈ 1, . . . , nc;∀j ∈ 1, . . . , nf . (2.28)

The condition ΩjF ⊂ Ωi

C is true if the fine cell j, ΩjF , belongs to the coarse control

volume i, ΩiC . The finite-element based restriction operator is the transpose of the

prolongation operator, i.e.,

R = PT . (2.29)

With the prolongation and restriction operators defined, one can solve the coarse-

scale system for p′V . Then, Eq. (2.24) is used to prolong the coarse-scale solution


back to the fine scale, i.e.,

p ≈ p′ =[P(RAP)−1R(I −AC) + C

]q. (2.30)

While the velocity u′ = −(λ · ∇p′) is conservative at the primal coarse scale by con-

struction, it is not conservative at the fine scale. Therefore, additional local Neumann

problems on primal coarse control volumes are solved in order to obtain a conservative

fine-scale velocity field, which is described in next section.

2.2 Fine-scale Velocity Reconstruction

Since the approximate multiscale solution, p′, is obtained using basis functions and

correction functions (Eq. (2.30)) that are constructed by solving local elliptic prob-

lems on dual coarse blocks, the fine-scale velocity calculated directly from p′ is not

continuous across dual coarse block boundaries. As a result, this velocity field cannot

be used to solve (nonlinear) transport problems (i.e., saturation equations). Since the

boundaries of primal-coarse blocks are in the interior of dual coarse blocks (Fig. 2.1),

the computed velocity field is continuous across these boundaries. Therefore, these

continuous fluxes can be used to impose Neumann boundary conditions for local prob-

lems, whose solution is locally conservative in each primal coarse block [23]. The only

prerequisite is that mass conservation is satisfied on the primal coarse scale, which

can be achieved by the application of the MSFV operator as the last step [42]. As

explained in [60], the reconstruction step is obtained algebraically by first reordering

the fine-scale system based on the primal coarse cells partitions, i.e.

Ap = q. (2.31)

For structured 2D problems, A is a block penta-diagonal matrix, which can be split

into block diagonal D, upper U and lower L parts, i.e.,

A = L+D+U . (2.32)


Each diagonal block ofD represents the transmissibilities between the fine cells within

coarse cell ΩiC . The corresponding off-diagonal blocks in L and U include the con-

nections with fine cells in the neighborhood of coarse cells ΩjC , j ∈ ℵi. Then, the

local problems with Neumann boundary conditions can be written as

D′p′′ = q− (A−D′)p′, (2.33)

whereD′ = D+E, and E is a diagonal matrix, defined as Eii =nf∑j=1

(Lij+Uij). Also,

(A−D′)p′ represents the flux across primal coarse cell boundaries, which corresponds

to Neumann boundary condition. Note that the local elliptic problems with Neumann

boundary conditions are singular. To solve this problem, pressure is fixed at one fine

cell in each coarse block ΩiC . Once p′′ is obtained, it is transformed into natural

ordering, p′′. Then, a fine-scale conservative velocity field can be reconstructed as

u =

−λ · ∇(p′′ − ρgz) on Ωi

C

−λ · ∇(p′ − ρgz) on ∂ΩiC

. (2.34)

In order to demonstrate the conservative reconstruction, the top layer of the SPE

10 [61] case is considered. The fine problem has 220 × 60 cells. We use a coarse

grid of 22 × 6. Pressure is fixed in the upper left cell (1,1) and the lower right cell

(220,60) with values of 1 and 0, respectively. Figure 2.3 shows the permeability field,

fine-scale reference pressure solution and MSFV solution p′. The absolute value of

the velocity divergence ‖∇·u‖ for each fine-scale cell is computed before and after the

reconstruction step. Figure 2.4 shows that the velocity divergence is non-zero on the

boundaries of the dual coarse blocks before reconstruction, and that it is zero for every

fine-scale cell, except for the source cell ((1,1) and (220,60)), after reconstruction.

2.3 Analysis of Correction Function

From Eq. (2.30), one can define the multiscale (MS) preconditioner with CF (which

is referred to as MSWC) as

M−1mswc = P(RAP)−1R(I −AC) + C. (2.35)


50 100 150 200

20

40

60

−4 −2 0 2 4 6 8

(a) Logarithm of permeability field

50 100 150 200

20

40

60

0 0.2 0.4 0.6 0.8 1

(b) Fine-scale reference

50 100 150 200

20

40

60

0 0.2 0.4 0.6 0.8 1

(c) MSFV

Figure 2.3: Permeability field, fine-scale reference and MSFV pressure solution.

50 100 150 200

20

40

60

0 0.05 0.1 0.15 0.2

(a) Before reconstruction

50 100 150 200

20

40

60

0 2 4 6 8 10

x 10−11

(b) After reconstruction

Figure 2.4: Velocity divergence before and after reconstruction.


This iterative procedure in combination with a fine-scale smoother, e.g., line re-

laxation [41], or GMRES preconditioner [60] was reported in the literature, where CF

played a major role in capturing the fine-scale RHS terms and the residual. However,

no detailed study of the computational efficiency of this procedure using different re-

striction operators with different local boundary conditions and smoothers has been

reported. Also, the exact role of CF is unclear. These issues are discussed in this

section.

2.3.1 Independent Local Stage

After some mathematical manipulation, Eq. (2.35) can be rewritten as

M−1mswc = P(RAP)−1R+ C −P(RAP)−1RAC

= M−1ms + C −M−1

msAC.(2.36)

In other words, the iterative procedure

pν+1 = pν +M−1mswc(q −Apν) (2.37)

is equivalent to the following two-stage iterative procedure

pν+1/2 = pν + C(q −Apν), (2.38)

pν+1 = pν+1/2 +M−1ms(q −Apν+1/2). (2.39)

These two steps are (1) updating the solution with the CF operator; (2) updating

with the multiscale preconditioner M−1ms = P(RAP)−1R, which does not involve

CF. Therefore, the operator C is a totally independent stage that does not affect the

MS preconditioner at all. This helps to quantify the impact of CF on the iterative

multiscale solution strategy. We show that CF is similar to other standard (local)

block preconditioners aimed at high-frequency errors.

A heterogeneous case with 100 × 100 fine and 10 × 10 coarse cells is considered.

The log-normally distributed permeability field with a spherical variogram and di-

mensionless correlation lengths of ψ1 = 0.5 and ψ2 = 0.02 is used. Also, the variance


and mean of ln(k) are 2 and 3, respectively. As depicted in Fig. 2.5, the angle be-

tween the long correlation length and the vertical domain boundaries is 45. The

pressure at (1,1) and (100,100) is fixed with the values of 1 and 0, respectively. In

this case, gravity acts in the y-direction with a constant value of ρg = 1. The BILU

block size is the same as the size of the dual coarse cells in order to provide the same

support as the CF. Figure 2.6 shows that both the CF and BILU improve the original

MSFV solution significantly. Also, the MSFV-CF solution is slightly better than the

MSFV-BILU solution. Denote p as the pressure solution and pfine as the fine-scale

reference, then the pressure solution error norms, defined as ‖p−pfine‖2/‖pfine‖2, for

MSFV, MSFV-CF, and MSFV-BILU are 5.13, 0.16, and 0.28, respectively. The main

difference between CF and BILU is the local boundary condition. In this case, the

reduced boundary condition captures the gravity effects quite well, if one employs

MSFV (with no iterations). Note that as long as MSFV is employed as the final

step, local mass conservation on the primal coarse grid is guaranteed regardless of

which local preconditioner is used. The choice of the local stage preconditioner is a

trade-off between accuracy and computational effort and is investigated in detail in

Section 2.4.

20 40 60 80 100

20

40

60

80

100

−1

0

1

2

3

4

5

6

7

8

Figure 2.5: Natural logarithm of layered permeability field with ψ1 = 0.5 and ψ2 =0.02. Fine grid size is 100× 100 and coarse grid size is 10× 10.


fine−scale solution

20 40 60 80 100

10

20

30

40

50

60

70

80

90

100−50

−40

−30

−20

−10

0

10

20

30

40

(a) Fine-scale reference

MSFV solution

20 40 60 80 100

10

20

30

40

50

60

70

80

90

100−50

−40

−30

−20

−10

0

10

20

30

40

(b) MSFVMSFV−CF solution

20 40 60 80 100

10

20

30

40

50

60

70

80

90

100−50

−40

−30

−20

−10

0

10

20

30

40

(c) MSFV-CF

20 40 60 80 100

10

20

30

40

50

60

70

80

90

100−50

−40

−30

−20

−10

0

10

20

30

40

(d) MSFV-BILU

Figure 2.6: Comparison between the solutions obtained from fine-scale reference,MSFV, MSFV-CF and MSFV-BILU. Note that all solutions are conservative at thecoarse-scale. Furthermore, error norms for MSFV (b), MSFV-CF (c), and MSFV-BILU (d) are 5.13, 0.16 and 0.28, respectively.


2.3.2 Spectral Analysis

The multiscale stage on its own (without CF or other smoothers) cannot converge

because rank(M−1ms)≤ nc, which implies that rank(M−1

msA)≤ nc; i.e., the coarse-scale

system cannot span the fine-scale space [62]. In fact, the multiscale stage resolves

only low-frequency errors. Therefore, local stages (smoothers) are required to remove

high-frequency errors. If only CF is considered as the local solver, the two-stage

iterative procedure (MSFV-CF) is not convergent. In fact, as shown in [41], MSFV

with CF (MSFV-CF) requires another local stage (smoother) to become convergent.

To illustrate the eigenvalue structure of MSFV alone and in combination with

local stages, such as the CF and BILU, homogeneous and heterogeneous 2D test

cases are considered. The fine and coarse grids are 40 × 40 and 4 × 4, respectively.

The BILU block size is the same as the size of the dual coarse cells (almost the same

support as CF). The pressure is fixed at (1,1) and (40,40). For the heterogeneous

case, a log-normally distributed permeability field with a spherical variogram (using

sequential Gaussian simulations [63]) is generated. The variance and mean are both

4 and the correlation lengths are 1/8 of domain size in each direction (Fig. 2.7).

For the homogeneous case, the permeability is unity. Figures 2.8 and 2.9 show that

using a multiscale strategy only does not guarantee convergence; however, when it

is combined with a local preconditioner such as BILU, all the eigenvalues are inside

the unit circle. If CF is used instead of BILU, i.e., MSFV-CF, some eigenvalues are

larger than unity. This is because CF shares the same reduced boundary condition

assumption as the basis functions in the MS system. Hence, local errors on dual-

coarse block boundaries cannot be removed by CF. To obtain a convergent iterative

scheme using CF, other local preconditioning stages (or GMRES) are required. If

BILU is used as an additional local stage, the three-stage AMS (i.e., MSFV-CF-

BILU) is convergent. Also, note that the maximum eigenvalue of this three-stage

AMS is smaller than that for the two-stage case of MSFV-BILU. Therefore, the

three-stage MSFV-CF-BILU converges faster, but each iteration is computationally

more expensive.


5 10 15 20 25 30 35 40

5

10

15

20

25

30

35

40−2

0

2

4

6

8

Figure 2.7: Natural logarithm of permeability field for spectral analysis.

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Re(λ)

Im(λ)

(a) MSFV

−8 −6 −4 −2 0−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re(λ)

Im(λ

)

(b) MSFV-CF

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re(λ)

Im(λ

)

(c) MSFV-BILU

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re(λ)

Im(λ

)

(d) MSFV-CF-BILU

Figure 2.8: Eigenvalues of MSFV, MSFV-CF, MSFV-BILU and MSFV-CF-BILUiteration matrices for a simple homogeneous test case.


−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Re(λ)

Im(λ)

(a) MSFV

−25 −20 −15 −10 −5 0−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re(λ)

Im(λ

)

(b) MSFV-CF

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re(λ)

Im(λ

)

(c) MSFV-BILU

−0.5 0 0.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re(λ)

Im(λ

)

(d) MSFV-CF-BILU

Figure 2.9: Eigenvalues of MSFV, MSFV-CF, MSFV-BILU and MSFV-CF-BILUiteration matrices for a heterogeneous test case.


2.3.3 Sensitivity to Transmissibility Contrasts

Another drawback of CF is that it is very sensitive to large contrasts in the transmis-

sibility field. This is because CF is the solution of lower dimensional problems on the

edges of dual-coarse cells with source terms. For example, if non-zero source terms

exist on the edge cells between two impermeable regions that cross the boundary, the

reduced problem is not solvable. This issue can slow down the convergence rate, and

even lead to divergence. To resolve these difficulties, we scale RHS terms of edge cells

(face in 3D) using a local factor, i.e.,qmI

qmEqmV

= E′

qIqEqV

, (2.40)

where qm is the modified RHS vector and E′ is a diagonal matrix with

E′ii =

Te,minTe,max

Eii if i ∈ ℵedge

Eii otherwise(2.41)

entries. Here, Te,min and Te,max are the local minimum and maximum values of the

transmissibility at the interfaces along edge e (in 3D, along a face). This approach

is purely local, and E′ can be calculated automatically based on the fine-scale trans-

missibility field and grid information. The correction function computed with the

modified RHS vector qm is referred to as a modified correction function (MCF), and

the effectiveness of MCF is shown in the following sections.

2.3.4 Computational Cost

As shown in the previous sections, in many settings, CF reduces the number of iter-

ations required to converge. It is important to note that this is beneficial if the gains

are worth the additional computational cost of CF. In order to examine the com-

putational cost of the (original and modified) CF stage, a log-normally distributed

permeability field generated by sequential Gaussian simulations [63] with 20 realiza-

tions is considered (see Fig. 2.10). The mean and variance of ln(k) are -1 and 4,


respectively. The correlation length is 1/8 of the domain size in each direction. The

fine-scale and coarse-scale grids consist of 128 × 128 × 64 and 16 × 16 × 8 cells, re-

spectively. Each BILU block contains 4 × 4 × 4 fine cells. The pressure is fixed at

the left and right faces with the values of 1 and 0, respectively. The iterations are

stopped once the reduction in the relative l2 norm of the residual is ten orders of

magnitude (i.e., ‖rk‖2/‖r0‖2≤ 10−10). A simple Richardson iterative scheme is used.

Table 2.1 indicates that the three-stage MSFE-CF-BILU with the original CF does

not converge. As discussed before, this is due to the high contrasts in the transmis-

sibility field at the local boundaries. On the other hand, the modification proposed

in Eq. (2.40) for CF leads to a convergent iterative scheme (i.e., MSFE-MCF-BILU).

Although the number of iterations is reduced by 19% when the MCF is used, the

computational cost of the solution phase (measured in CPU time) is increased by

46%. Further discussion on the efficiency of CF for a wide range of cases is shown

later.

Figure 2.10: Natural logarithm of the permeability field for one of the 20 statistically-the-same fields generated to analyze the computational efficiency of CF. The domainconsists of 128× 128× 64 fine and 16× 16× 8 coarse grid cells.

The study presented in this section shows that CF is an independent local stage

solver aimed at high-frequency errors. Hence, it can be replaced by other local solvers.

Also, CF helps the convergence rate, but it does not offset its additional computational

cost (based on total CPU). Moreover, CF cannot be used as a sole local-stage solver.

Additional local or global solvers, e.g., smoothers or GMRES, are required to make


MSFE-BILU MSFE-CF-BILU MSFE-MCF-BILUIteration steps 45.8 (±3.4) - 37.0 (±4.4)

Solve time (sec) 20.0 (±1.4) - 29.0 (±3.4)

Table 2.1: Average total simulation time (sec) and number of iteration steps forMSFE-BILU, MSFE-CF-BILU, and MSFE-MCF-BILU AMS solvers. The three-stage(MSFE-CF-BILU) iterative procedure is not convergent due to the CF sensitivity tothe contrast in the transmissibility field. However, if the modified CF (MCF) is used(Eq. (2.40)), i.e., MSFE-MCF-BILU, the procedure becomes convergent. Resultsare shown on average for twenty statistically-the-same realizations. Also shown inparentheses are the standard deviations.

the iterative procedure convergent. Furthermore, CF is sensitive to transmissibility

contrasts along edge cells.

Next, systematic numerical tests are provided to find the most effective combi-

nation of stages to solve the pressure equation. Also, the research code for AMS is

tested against a production-quality SAMG solver.

2.4 Local and Global Preconditioner

In this section, systematic tests are performed to find the best combination of local

and global stages. For the following experiments, five sets of log-normally distributed

permeability fields with spherical variograms are generated using sequential Gaussian

simulations [63]. For all the test cases, the variance and mean of ln(k) are 4 and

-1, respectively. The fine-scale grid size and dimensionless correlation lengths in

the x, y, z direction, i.e., ψx, ψy and ψz are shown in Table 2.2. Each set has 20

equiprobable realizations. For sets 1 and 2, 20 realizations with different orientation

angles (Fig. 2.11) of 0, 15, 30, and 45 degrees are considered. For sets 3, 4 and 5,

20 realizations of patchy domains are used (Fig. 2.12). In the following experiments,

GMRES preconditioned by AMS is employed as the iterative procedure. The pressure

is fixed on the left and right faces with dimensionless values of 1 and 0, respectively.

The iterative procedures are performed until the reduction in the relative l2 norm of

the residual is five orders of magnitude (i.e., ‖rk‖2/‖r0‖2≤ 10−5).

In the next three subsections, the coarse-scale restriction schemes, local boundary


Permeability set 1 2 3 4 5Fine-scale grid 1283 643 1283 643 323

ψx 0.5 0.5 0.125 0.125 0.125ψy 0.03 0.03 0.125 0.125 0.125ψz 0.06 0.01 0.125 0.125 0.125

Angle between ψx and y direction 0, 15, 30, 45 patchyVariance 4

Mean -1

Table 2.2: Five permeability sets (each with 20 equiprobable realizations) are usedfor the numerical experiments of this section. Layered fields, i.e., sets 1 and 2, aregenerated for 4 different layering angles, each of which has 20 equiprobable realiza-tions.

Figure 2.11: Natural logarithm of one realization of permeability set 1 with differentlayering angles of 0, 15, 30 and 45 from left to right. For each layering angle, 20realizations are considered.


Figure 2.12: Natural logarithm of one (out of 20 statistically-the-same) realization ofthe permeability set 3.

conditions, and second-stage local solvers are studied. Finally, in the last section, the

AMS efficiency as a linear solver is studied versus SAMG.

2.4.1 AMS Global Stage: MSFV versus MSFE

The performance of two different restriction schemes, which corresponds to MSFV

and MSFE global operators, is investigated. The permeability of the SPE 10 bottom

layer, which has channelized structures is considered (Fig. 2.13). This permeability is

selected because it is very challenging. The fine and coarse grids consist of 220× 60

and 22 × 6, respectively. Each BILU block contains 10 × 10 fine cells. Also, both

FV and FE restriction operators are employed with or without CF. The pressure is

fixed at the left and right sides with the values of 1 and 0, respectively. Figure 2.14

shows that if MSFV is used as the global stage solver, AMS does not converge. On

the other hand, if MSFE is used, AMS converges efficiently. No GMRES was used to

stabilize the iterations. Hence, the inefficient iterations associated with the previously

developed i-MSFV [41, 42] were mainly due to the weak coarse-scale MSFV operator.

CF is also considered, and it is shown that CF does not overcome the difficulties due

to the weak coarse-scale operator.

To study the performance of AMS with different global stage solvers, one should

consider the total CPU time, not just the iteration numbers. For this reason, perme-

ability sets 1 and 3 are considered. The fine and coarse grids contain 128× 128× 128

and 16× 16× 16 cells, respectively. ILU is employed as the local preconditioner, and


20 40 60 80 100 120 140 160 180 200 220

20

40

60

−6 −4 −2 0 2 4 6 8

(a)

20 40 60 80 100 120 140 160 180 200 220

20

40

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(b)

Figure 2.13: Natural logarithm of the permeability (a) and fine-scale pressure solution(b) for the SPE 10 bottom layer.

0 10 20 30 40 50 60−10

−8

−6

−4

−2

0

2

4

6

iterations

log10ε

MSFE−CF−BILU

MSFE−BILU

MSFV−CF−BILU

MSFV−BILU

Figure 2.14: Iteration histories for AMS with different restriction schemes using theSPE 10 bottom layer.


no CF is used. As Fig. 2.15 shows, the MSFE coarse-scale operator outperforms the

MSFV one for both permeability sets.

0

10

20

30

40

50

60

70

Tot

al s

imul

atio

n tim

e (s

ec)

Patchy Layered 0° Layered 15° Layered 30° Layered 45°

FEFV

(a)

0

10

20

30

40

50

60

70

Itera

tion

step

s


FEFV

(b)

Figure 2.15: Comparison of (a) total simulation time and (b) iteration steps for FEand FV global solvers (i.e., restriction operator) on layered and patchy permeabil-ity fields over 20 different realizations. Also shown in error bars are the standarddeviations. Clearly, the FE restriction operator outperforms the FV one.

Anisotropic cases are also considered by setting the factor α in ∆x = 2α∆y = α∆z

relation. One realization from each rotation angle of permeability set 2 and one re-

alization from permeability set 4 are chosen. Both ILU and BILU are employed as

local preconditioners, and the results are compared. The coarse grid is 8× 8× 8 and

each BILU block contains 4× 4× 4 fine cells. The convergence rate is defined as the

inverse of the number of iterations required for reducing the error by five orders of

magnitude. As shown in Fig. 2.16, the convergence rate of MSFV decreases monoton-

ically as α increases. However, with the MSFE coarse-scale operator, the convergence

rate does not decrease once α is greater than a certain value. The main message here

is that MSFE clearly outperforms MSFV for highly heterogeneous anisotropic prob-

lems. Note that although BILU has better convergence rates compared with ILU, it

has a more expensive setup phase and entails more operations per iteration.

For the above-mentioned anisotropic test cases, the coarsening factor is chosen

as 8× 8× 8 because this coarse-grid size leads to the most computationally efficient

performance for both FE and FV restriction schemes. For permeability sets 2 and

4, four different coarse-grid sizes are generated: 16 × 16 × 16, 8 × 8 × 8, 4 × 4 × 4


100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Con

verg

ence

rat

e

Anisotropic factor α

MSFE−ILUMSFE−BILUMSFV−ILUMSFV−BILU

(a) Patchy

100

101

102

103

0

2

4

6

8

10

12

14

16

18

20


Tot

al C

PU

tim

e (s

ec)

MSFE−ILUMSFE−BILU

(b) Patchy

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Con

verg

ence

rat

e


MSFE−ILUMSFE−BILUMSFV−ILUMSFV−BILU

(c) Layered 15o

100

101

102

103

0

2

4

6

8

10

12

14

16

18

20


Tot

al C

PU

tim

e (s

ec)

MSFE−ILUMSFE−BILU

(d) Layered 15o

Figure 2.16: Comparison of MSFV and MSFE restriction operators for anisotropicpatchy and layered permeability fields (one realization for each) with different localsolvers (BILU and ILU). The convergence rate and total simulation time (sec) areillustrated on the left and right columns, respectively. For layered systems withorientation angles of 0o, 30o and 45o. The results are similar to the layered 15o case;therefore, they are not shown here. Only the total simulation time of the MSFEoperator is illustrated since it has a high convergence rate compared with the MSFVoperator.


and 2 × 2 × 2. Also, ILU is employed as the local preconditioner. As shown in

Fig. 2.17, considering the total CPU time, the trade-off between the convergence rate

and computational cost is achieved at a coarsening factor of eight in each direction.

Generally, it is a good choice to chose the coarsening factor in each direction nearly

the square root of the number of cells in that direction.

16X16X16 8X8X8 4X4X4 2X2X20

5

10

15

20

25

30

Tot

al s

imul

atio

n tim

e

FE−0°FE−15°FE−30°FE−45°FE−patchyFV−0°FV−15°FV−30°FV−45°FV−patchy

(a)

16X16X16 8X8X8 4X4X4 2X2X20

50

100

150

200

250

300

Itera

tion

step

s

FE−0°FE−15°FE−30°FE−45°FE−patchyFV−0°FV−15°FV−30°FV−45°FV−patchy

(b)

Figure 2.17: Coarse grid size effect on average (a) total simulation time and (b)iteration steps for permeability set 2 and 4.

2.4.2 AMS Global Stage: Local Boundary Conditions

The effects of different local Boundary Conditions (BC), i.e. reduced BC and linear

BC, are studied here. Permeability sets 2 and 4 are considered. The coarse grid size

is 8× 8× 8, and ILU is employed as the local preconditioner. Figure 2.18 shows that

for patchy domains, the reduced and linear BC have similar performance. For layered

permeability fields, the linear BC improves the computational efficiency of MSFV.

However, the improved MSFV with linear BC is still not competitive with MSFE.

Clearly, the linear BC is advised for MSFV, while for MSFE the reduced problem is

as efficient as the linear BC.

Next, the full SPE 10 3D case (Fig. 2.19) is considered. The domain consists of

60× 220× 85 fine and 6× 22× 17 coarse grid cells. Also, each BILU block contains

4 × 4 × 5 fine cells. The pressure is fixed at the left and right faces with the values

of 1 and 0, respectively. In this case, AMS is used as a preconditioner for GMRES.


0

0.5

1

1.5

2

2.5

3

3.5

4

Solu

tion p

hase tim

e (

sec)


MSFE−ReducedBC

MSFE−LinearBC

MSFV−ReducedBC

MSFV−LinearBC

Figure 2.18: Solution phase time (i.e. excluding setup time) averaged over 20equiprobable realizations for MSFV and MSFE restriction schemes with linear andreduced boundary conditions.

The AMS performance with reduced and linear boundary conditions for both MSFV

and MSFE is shown in Fig. 2.20. From this figure, it is clear that MSFE with the

reduced boundary condition is more efficient than other options. For MSFV, the

linear boundary condition is better than the reduced boundary condition. In fact,

the MSFV-BILU-ReducedBC does not lead to a convergent iterative scheme for this

challenging test case. Also, note that when the linear BC is used, MSFE and MSFV

have comparable performance. However, MSFE with the reduced BC outperforms

the linear BC. To obtain an estimate about the efficiency of the AMS for this test

case, Table. 2.3 also shows the CPU time for MSFE-BILU and MSFE-ILU iterative

procedures with the reduced boundary conditions, from which it is also clear that

MSFE-ILU is more efficient.

2.4.3 AMS Local Stage

Overall, MSFE with the reduced boundary condition is found to be the most efficient

global-stage solver. Next, we investigate which local stage preconditioner is the best

overall choice. BILU is used as the second stage preconditioner in TAMS [62]. Here,


(a) (b)

Figure 2.19: (a) Natural logarithm of the permeability and (b) pressure solution forthe full SPE 10 case. The grid contains 60×220×85 fine and 6×22×17 coarse cells.

0 10 20 30 40 50−7

−6

−5

−4

−3

−2

−1

0

iterations

log

10ε

MSFE−BILU−ReducedBC

MSFV−BILU−ReducedBC

MSFE−BILU−LinearBC

MSFV−BILU−LinearBC

Figure 2.20: Iteration histories for MSFV and MSFE restriction operators with linearand reduced boundary conditions. Note that the MSFV with Reduced BC for localbasis functions and BILU as the second stage solver does not converge.


AMS strategy MSFE-BILU MSFE-ILUSetup phase time 26.91 13.24

Solution phase time 18.01 17.66Total simulation time 44.92 30.90

Iteration steps 26 82

Table 2.3: The CPU time (sec) and iteration steps for MSFE-BILU and MSFE-ILUpreconditioned by GMRES. Iterations are stopped when the relative l2 norm of theresidual is reduced by five orders of magnitude.

ILU is employed as the local preconditioner. Based on our experiments, the solution

time of BILU and ILU are comparable; however, ILU has a minimal setup time

compared with BILU. Hence, ILU outperforms BILU in terms of CPU time. A

comparison between ILU and BILU is performed for permeability sets 1 and 3. The

coarse grid and BILU blocks are 16× 16× 16 and 4× 4× 4, respectively. Figure 2.21

shows the although ILU employs many iterations to converge, its total CPU time is

much less than that of BILU for all the studied cases.

0

10

20

30

40

50

60

70

Tot

al s

imul

atio

n tim

e (s

ec)


ILU(0)BILU(0)

(a)

0

5

10

15

20

25

30

35

40

Itera

tion

step

s


ILU(0)BILU(0)

(b)

Figure 2.21: The average and error bar plots of (a) total simulation time and (b)iteration steps for BILU and ILU comparison on layered and patchy permeabilityfields.

Finally, in order to compare the efficiency of the iterative procedure including CF

and the proposed modified CF (MCF) with ILU, permeability set 4 is considered. The

fine and coarse grids contain 64× 64× 64 and 8× 8× 8 cells, respectively. GMRES


is also used in this case so that MSFV-CF is convergent. Figure 2.22 shows that

MSFV-ILU outperforms the other cases where CF is used as the second, or third,

stage solver. Due to the sensitivity of CF to high permeability contrasts, MSFV-CF

consumes a lot of iterations. The iterations converge faster when ILU is used as an

additional stage, and the efficiency will be further improved if the proposed modified

CF (MCF) is used instead of the original CF, i.e., MSFV-MCF-ILU. Nevertheless,

MSFV-ILU is still the most efficient combination. Similar results are obtained for

MSFE. The conclusion is that ILU is more efficient than CF and MCF.

MSFV−CF MSFV−CF−ILU MSFV−MCF−ILU MSFV−ILU0

10

20

30

40

50

60

iteration steps

total simulation time

(a)

MSFE−CF MSFE−CF−ILU MSFE−MCF−ILU MSFE−ILU0

10

20

30

40

50

60

iteration stepstotal simulation time

(b)

Figure 2.22: Iteration steps and total simulation time (sec) for GMRES precondi-tioned by the MSFV (a) and MSFE (b) with CF, MCF, and ILU. Results are averagedover 20 realizations of patchy permeability field of set 4.

On the basis of what we presented above, it is found that MSFE with ILU as

global and local stage solvers, respectively, lead to an efficient iterative multiscale

solver. For the local stage CF, MCF, ILU, and BILU were studied. Among these

choices, ILU was found to be the most efficient choice in terms of total CPU time.

Of course, several other choices for the local stage could be considered and studied.

However, our main message is that MSFE outperforms MSFV for both linear and

reduced local BC for the test cases we studied here. Also, we found that the CF does

not add significant improvements to the multiscale procedure.

Next, an optimum AMS procedure (on the basis of the presented study), i.e.,

GMRES preconditioned by the MSFE-ILU is tested against SAMG [1].


2.4.4 AMS versus AMG

To investigate the efficiency of AMS compared with SAMG, which is widely used in

the community, permeability set 5 (patchy field) is considered. As shown in Table

2.2, this problem set consists of 323 fine cells. To increase the size of the domain, but

keeping the same permeability statistics, a refinement procedure is employed, such

that each grid cell is divided into 8 cells (split into two in each direction) in each

refinement step. Employing this refinement procedure, four grid sets are generated

with 323, 643, 1283 and 2563 fine cells. For all the problem sizes, the coarsening

factor is kept constant with the value of 8× 8× 8. Dirichlet boundary conditions are

employed on the left and right faces with the values of 1 and 0, respectively. No-flow

boundary condition is applied on all other faces. Iterations are performed until the

relative l2 norm of the residual is reduced by five orders of magnitude. The MSFE

(with reduced boundary condition) and ILU are used as global and local solvers for

AMS. The SAMG library is obtained from Fraunhofer Institute SCAI, release version

25a1 of December 2010 [1]. It employs a single stand-alone V-cycle with a convergence

tolerance of 0.1 for the relative residual reduction and one Gauss-Seidel C-relaxation

sweep as pre- and post- smoothing steps on each level. Also, the coarsest system

is solved by a direct solver (Gaussian elimination). The CPU time and number of

iterations for SAMG and AMS are presented in Tables 2.4 and 2.5, respectively. It

is clear that the two methods perform similarly for this test case. Also, Table 2.5

indicates that the AMS convergence rate is independent of problem size, i.e., similar

to SAMG, AMS is a scalable solver. This fact is further illustrated in Fig. 2.23(a),

where the computational times for different problem sizes are normalized with that of

the 323 case and plotted for setup and solution phases. Note that Fig. 2.23(b) shows

SAMG is slightly above the ideal line for the setup phase, which reflects its complex

coarsening strategy.

The performance of both AMS and SAMG is also tested and compared for per-

meability sets 1 and 3. The coarse grid consists of 16 × 16 × 16 cells, and the same

strategy for AMS, i.e., MSFE with reduced BC as global and ILU as local stages, is

chosen. Note that the permeability set 1 is a layered field, for which the Cartesian

coarse grid is still used in the AMS coarse-scale solver. It is clear from Fig. 2.24

that SAMG outperforms AMS. The difference between the two is more severe for the


Problem size 32× 32× 32 64× 64× 64 128× 128× 128 256× 256× 256Setup phase 0.12 1.14 12.33 157.41

Solution phase 0.14 0.76 6.33 73.02Total 0.26 1.90 18.66 230.43

GMRES iterations 6 7 9 10

Table 2.4: CPU time (sec) and iteration steps for SAMG on a patchy domain.

Problem size 32× 32× 32 64× 64× 64 128× 128× 128 256× 256× 256Setup phase 0.29 2.12 17.00 147.96

Solution phase 0.10 0.87 8.01 55.91Total 0.39 2.99 25.01 203.87

GMRES iterations 22 21 21 18

Table 2.5: CPU time (sec) and iteration steps for AMS on a patchy domain.

1 8 64 5121

8

64

512

Nor

mal

ized

CP

U ti

me

Normalized problem size

Local linear systems Multiscale operators Setup phase Solution phase Linear reference

(a) AMS

1 8 64 5121

8

64

512

Nor

mal

ized

CP

U ti

me

Normalized problem size

Setup phase Solution phase Total simulation Linear reference

(b) SAMG

Figure 2.23: Scalability analysis of AMS and SAMG.


layered field, which demonstrates clearly the coarsening strategy of AMS needs to be

improved. In general, having an AMS solver that is competitive with SAMG is im-

portant for the following reasons. First, it is clear that both AMS and SAMG have a

considerable setup time. For time-dependent problems, AMS benefits from adaptive

updating of the basis functions. Second, AMS is a mass conservative iterative solver

when the MSFV operator is as employed the last step. This has been studied for

i-MSFV [38, 64].

0

5

10

15

20

25

30

35

40

45

CP

U ti

me

(sec

)


AMS setup phase timeAMS solution phase timeSAMG setup phase timeSAMG solution phase time

(a) Fine-scale grid is 128× 128× 128

0

50

100

150

200

250

300

350

400

CP

U ti

me

(sec

)


AMS setup phase timeAMS solution phase timeSAMG setup phase timeSAMG solution phase time

(b) Fine-scale size is 256× 256× 256

Figure 2.24: Total, setup, and simulation times (sec) of AMS and SAMG as linearsolvers for permeability sets 1 and 3. Results are averaged over 20 statistically-the-same realizations for each case.

2.5 Summary

In this chapter, a general Algebraic Multiscale Solver (AMS) for the pressure equation

was developed. We analyzed the role of the Correction Function (CF) in the context of

AMS, and we showed that CF can be seen as an independent local stage aimed at high-

frequency errors. As a local preconditioner, CF helps to capture the fine-scale RHS

(and residual) and accelerates the overall convergence rate. However, - on average -

the gain in convergence rate does not compensate for the additional computational

cost. Also, CF must be combined with other solvers (or smoothers) to guarantee

convergence. Furthermore, CF with the reduced boundary condition is sensitive to


transmissibility contrasts. A modification to CF is proposed and the improvement

of the modification was studied numerically. In general, other preconditioners, such

as ILU, are found to be more efficient than CF. For several highly heterogeneous

anisotropic problems, the MSFE restriction operator was found to be superior to the

MSFV one. The performance of AMS with many combinations of local and global

solvers was systematically tested for several problems. Overall, the best iteration

strategy of AMS is MSFE with reduced problem BC along with ILU. Once the residual

norm is reduced to a specified tolerance, the MSFV method is employed as the final

sweep to ensure the mass conservation. As a summary, the current AMS algorithm

is shown in Fig. 2.25.

Global stage:

Local stage:

𝑝𝜈+1/2 = 𝑝𝜈 +𝑀𝑀𝑆𝐹𝐸−1 𝑞 − 𝐴𝑝𝜈

𝑝𝜈+1 = 𝑝𝜈+1/2 +𝑀𝐼𝐿𝑈−1 𝑞 − 𝐴𝑝𝜈+1/2

𝑝 = 𝑝𝜈+1 +𝑀𝑀𝑆𝐹𝑉−1 𝑞 − 𝐴𝑝𝜈+1

Construct 𝑀𝑀𝑆−1 and 𝑀𝐼𝐿𝑈

−1

𝜈 = 0

Is 𝑞 − 𝐴𝑝𝜈+1 < 𝜖 ?

Yes

𝜈 = 𝜈 + 1

No

Figure 2.25: AMS algorithm chart

Our results show that the performance of AMS is comparable to the state-of-the-

art algebraic multigrid solver (SAMG) preconditioner for very large-scale problems.

Compared with SAMG, AMS can benefit from two aspects. First, AMS only requires

a few iterations to solve the pressure equations in practice. For the sequential strategy

used in the MSFV method, it is very important that the computed pressure solution

could guarantee local mass conservation. The violation of mass conservation often


results in non-physical and unbounded saturation fields when solving the hyperbolic

transport equations. Due to the fact that AMS can allow reconstruction of conserva-

tive velocity field after any iteration level if a MSFV global stage is applied as the last

step, only a small number of iterations are needed in the AMS framework to solve the

pressure equations for many practical applications; while SAMG requires a tight con-

vergence tolerance to ensure a mass conservative solution. Therefore, as an integral

part of the nonlinear reservoir simulator, AMS can enhance the overall efficiency of

the simulator using a relatively large (loose) convergence tolerance to solve the pres-

sure equation. Second, AMS can reduce the computational effort of the setup phase

for linear solutions by updating basis functions in an adaptive manner. Normally, a

nonlinear simulation run involves a large number of linear equations to be solved. It

is not necessary to recompute all the basis functions for each linear equations. The

basis functions are designed to capture local characteristics, as a result, only a small

fraction of basis functions need to be updated for the regions where fluid properties

significantly change over the simulation time [11, 36, 65]. These benefits are seen

by the extension of AMS to compressible flows in heterogeneous porous media, i.e.,

C-AMS, introduced by Tene et al. [27]. They showed that AMS is a competitive

solver for time-dependent (nonlinear) problems compared with SAMG, especially for

the cases which involve a large number of time steps. The overall efficiency is gained

by infrequently updating the selective basis functions to construct the prolongation

operators for each linear system, and requiring only a few iterations per linear solve

to achieve a good quality of approximate pressure solution for practical purposes.

While all studies presented in this work were done on single-processor machines,

AMS is amenable for massively parallel computations of the setup phase since basis

functions are computed independently. For the local-stage solver, an efficient and

robust solver for parallel computations is needed. ILU(0) is found to be efficient

for our single-processing computations. However, it may not be an efficient solver

for parallel computations. Detailed investigation of the proper components for local

and global stages for parallel computation can be found in the work by Manea et

al. [66]. They showed that AMS had a good scalability in the setup phase on multi-

core architectures, which indicates AMS is an efficient solver for large-scale problems

since the setup phase that takes significant portion of the total simulation time can


be completed efficiently on a parallel computation platform.

Moreover, AMS described in this chapter employs a structured Cartesian coarse

grid, which is not efficient specially for layered permeability fields. Improving the

coarsening strategy to account for the fine-scale transmissibility field is also a topic

of ongoing research.

Chapter 3

Monotone Multiscale Finite

Volume Method

In the development of Algebraic Multiscale Solver (AMS) described in Chapter 2, the

coarse-scale symmetric-positive-definite system of MSFE is used to reduce the error

norm to arbitrarily small values, while MSFV is employed only at the final stage to

obtain a conservative velocity field. Having a conservative velocity field is a critical

requirement for solving the nonlinear transport equations accurately and efficiently.

Moreover, local mass conservation allows for adaptive computations and the use of

relatively loose tolerances as a function of time [38, 41, 64]. However, the solutions

obtained from the MSFV method are non-monotone (non-physical) for the problems

with large contrasts in the local permeability and anisotropy in the transmissibility.

Thus, in the context of a multiscale linear solver, the final step of using MSFV to

ensure local conservation must be performed in a manner that minimizes the degree

of nonmonotonicity in the reconstructed fine-scale pressure solution. To improve the

quality of MSFV solutions for slightly heterogeneous and grid-aligned anisotropic

coefficients, a Compact-MSFV (C-MSFV) operator was proposed [67]. While the C-

MSFV was effective for many grid-aligned anisotropic problems, it does not overcome

the problem with nonmonotonicity for highly heterogeneous anisotropic fields. For

heterogeneous problems, some improvements were observed by changing Boundary

Conditions (BC) for all local problems [68].

51

52 CHAPTER 3. MONOTONE MSFV METHOD

In this chapter, the cause of the non-physical peaks associated with the MSFV op-

erator for highly heterogeneous problems is identified clearly and resolved. The peaks

are associated with the discretization stencil of coarse nodes that are surrounded by

low-permeability regions. It is shown that for these critical coarse nodes, integration

of the flux induced by the dual basis functions can result in negative transmissibilities

for the coarse-scale pressure system. A monotone MSFV (m-MSFV) method is de-

vised on the basis of local stencil-fix approach, which guarantees the monotonicity of

the MSFV solution. The critical interfaces with non-physical transmissibility values

for the coarse-scale system are detected algebraically. Then, a local Two-Point-Flux-

Approximation (TPFA) scheme is used to calculate the coarse-scale entries for the

critical coarse interfaces only. In addition, the Linear Boundary Condition (LBC)

can be employed for the basis function calculations of the critical regions. The LBC-

based m-MSFV reduces the norm of non-physical peaks (reducing nonmonotonicity).

In contrast to the TPFA-based approach, however, the LBC-based m-MSFV cannot

remove all the negative (non-physical) transmissibilities from the coarse-scale system.

The local nature of m-MSFV allows it to be employed adaptively in space and

time. A histogram of the critical interfaces is calculated based on a normalized value

of the non-physical transmissibility coefficients. Then, based on a threshold value,

only critical interfaces with large values are detected and fixed. This threshold-based

approach allows for minimizing the trade-off between the accuracy and monotonicity

of the solutions.

3.1 Coarse-scale Transmissibility Coefficients

In this section, we analyze the MSFV coarse-scale operator in detail to identify the

cause of nonmonotonicity in MSFV solutions. For simplicity, we ignore the gravity

and derive the coefficients in the coarse-scale operator from the following elliptic

pressure equation:

−∇ · (λ · ∇p) = q, (3.1)

where the highly heterogeneous mobility (assumed diagonal) tensor and the source

terms are denoted with λ and q, respectively. The discrete form of (3.1) at the given

fine-scale (denoted here on by superscript f), where the coefficients λ are computed

CHAPTER 3. MONOTONE MSFV METHOD 53

using a finite-volume Two-Point-Flux-Approximation (TPFA) scheme [44], can be

written as

Afpf = qf , (3.2)

where entries of the transmissibility matrix Af are afij|i 6=j = − λij ·~nij

δxij· ~nijδAij. Here,

λij, δAij and δxij are the harmonically averaged permeability, differential element

cross section area and the distance between the computational nodes i and j, respec-

tively. Also, the normal unit vector ~nij points out of volume i at its cross section

with cell j. Note that afij = afji and afii = −∑nf

j=1,j 6=i afij, where nf is the number of

fine-scale finite volumes, also hold. In our implementation, the positive definite mo-

bility tensor leads to non-positive off-diagonal (afij|i 6=j ≤ 0) and non-negative diagonal

(afii ≥ 0) entries for the transmissibility matrix.

As described in Chapter 2, the locally computed basis functions φk associated

with coarse node k are used to prolong the coarse-scale solution onto the fine-scale

resolution. Basis functions are first computed on dual-coarse cells, ΩdD, and then

assembled for all dual cells, nd, i.e.,

φk =

nd∑d=1

φkd. (3.3)

Note that the correction functions at the fine-scale is an independent stage and cannot

modify the coarse-scale system matrix; hence, we do not consider it in our analysis

for this chapter. The fine-scale pressure field is then constructed as follows:

pf ≈ pms =nc∑k=1

φkpck, (3.4)

The basis functions are local solutions of the governing equation (2.2) and the

reduced-dimensional problem condition can be stated as

∇‖ · (λ · ∇φkd)‖ = 0 on ∂ΩdD, (3.5)

which has been widely used in the multiscale literature. The subscript ‖ denotes

the normal projection (operator or vector) with respect to the boundary. The for-

mulation can be reduced to the Linear Boundary Condition (LBC) if the mobility


is assumed constant along the boundary, i.e., λ = I at ∂ΩdD. Note that the basis

functions computed with either of the two local boundary conditions are monotone

with numerical values between 0 and 1, i.e., 0 ≤ φk(x) ≤ 1 ∀x ∈ Ω, k = 1, 2, ..., nc,provided that the fine-scale mobility coefficients λ are positive definite. Therefore, in

the superposition pms =∑φkp

ck, p

ms would violate the monotonicity property if and

only if the pck violates this property. Hence, all the non-physical peaks are associated

with non-physical pck values. This important fact guides us to the cause of the non-

physical peaks in the MSFV solution, pms. That is, the properties of the coarse-scale

system control the monotone behavior.

This important fact guides us to the cause of the non-physical peaks in the MSFV

solution, pms, through the following two important Lemmas.

Lemma 1. If λ tensor is positive definite, the basis functions φk are monotone,

0 ≤ φk(x) ≤ 1 ∀x ∈ Ω holds, and the normal outgoing flux induced by φkd at external

face of ΩdD, i.e., ∂Ωd

D is nonnegative.

Proof. Since basis functions are conservative solutions of symmetric-positive-definite

elliptic systems, with no external neither boundary source terms, constructed based

on TPFA scheme at local dual-coarse cells, similar as in the fine-scale system, the

solutions of Eq. (2.2), i.e., basis functions φk, is always monotone and 0 ≤ φk(x) ≤1 ∀x ∈ Ω holds. Especially, φkd at each boundary cell of Ωd

D also has a numerical

value between 0 and 1, while for the cells not belonging to ΩdD, its numerical value

is zero, i.e., φkd(x 6∈ ΩdD) = 0. Hence, if the mobility is positive definite, the outgoing

fluxes at each external face of each boundary cell is nonnegative.

Lemma 2. If non-physical peaks are present in the MSFV solution, they are solely

due to non-physical coarse-scale solutions.

Proof. Therefore, in the superposition pms =∑φkp

ck, if pms violates the monotonicity

property if and only if the pck violates this property. Hence, non-physical peaks are

all associated with the non-physical pck values. Next, we analyze the properties of

the coarse-scale system, which result in non-physical coarse-scale pressure solution

pck.

The superposition expression is substituted into Eq. (3.1), and integrated over

coarse control volume boundaries. After applying the Gauss integral rule, one obtains


x3

x1

x2

1

Ω𝐷𝑑

𝜙𝑖𝑑

Figure 3.1: Illustration of the basis function φid solved on dual-coarse cell ΩdD subject

to reduced-dimensional boundary condition. Note that the basis functions are alwaysmonotone and satisfy 0 ≤ φid ≤ 1, provided that the mobility tensor λ is positivedefinite.

the coarse-scale system as

ACpC =

∫Ω

q dΩ, (3.6)

where the coarse-scale transmissibility matrix entries acij in AC are

acij = −∫∂Ωi

C

(λ · ∇φj) · ~ni dΓ. (3.7)

Here, ~ni is the unit normal vector pointing out of the control volume (coarse-cell) i.

Note that φj =∑nd

d=1 φjd. Mass conservation leads to

acii = −nc∑

j=1,i 6=j

acij = −∫∂Ωi

C

(λ · ∇φi) · ~ni dΓ, (3.8)

since∑nc

j=1 φjd = 1. Note that the coarse-scale system in MSFV is not guaranteed to

be symmetric, i.e.,

acij = −∫∂Ωi

C

(λ · ∇φj) · ~ni dΓ 6= acji = −∫∂Ωj

C

(λ · ∇φi) · ~nj dΓ, (3.9)

since the coefficients are integrals of different functions over different control volume

boundaries. This is in contrast to the symmetric-positive-definite MSFE coarse-scale

operator. A coarse-scale system that has positive-definite mobility tensors at the fine


j

i

k k

l

j

i

Fine cells Dual-coarse edge cells Coarse-node cells Coarse-grid boundaries Induced flux

Figure 3.2: (left): Illustration of a 3× 3 coarse- and 21× 21 fine- grid domain. Thecoarse cell i is highlighted in red, neighboring k and j on its South and South-Westsides. Also shown are the induced fluxes by the φj (middle) and φk (right). Note thatonly the overlapping part of the basis functions are plotted, and that for simplicityof the illustration a homogeneous problem is used.

scale is expected to yield negative off-diagonal, acij ≤ 0, and positive diagonal, acii ≥ 0

values. Next, we study the integrals (3.7) and (3.8) and investigate the situations

that may violate these conditions.

In order to study the coarse-scale transmissibility coefficients, a 3× 3 coarse-grid

problem in 2D is considered and shown in Fig. 3.2. We study the transmissibility

coefficients between cell i and two of its neighboring cells j and k.

For the South-West neighboring cell, j, the flux induced by the basis function φj,

acij, satisfies the physical property of acij ≤ 0 because the both boundary segments of

control volume i experience incoming fluxes. Note that the net induced flux (for any

heterogeneous field) from j to i is always nonnegative.

On the other hand, the fluxes induced by the basis function associated with cell k,

φk, must be computed along many (four in 2D) overlapping segments. For cell i, some

of these fluxes are incoming and some others are outgoing. For many heterogeneous

cases, the net incoming flux to the control volume i is positive, leading to a negative

off-diagonal entry, which is desirable. Figure 3.3 shows the SPE 10 bottom layer

permeability field which consists of 220× 60 fine cells. The MSFV coarse grid is also

shown in the figure for a coarsening ratio of 11× 5.

Figure 3.4 shows an extracted rectangular subdomain from Fig. 3.3, Ωh1, between

(88, 5) ≤ (x, y) ≤ (121, 20). The location of this extracted domain is highlighted in


2200

60

−2

0

2

4

Figure 3.3: Logarithm of permeability field for SPE 10 bottom layer. The domainconsists of 220 × 60 fine- (not shown) and 20 × 12 coarse- (shown) grid cells. Twosubdomains of the size 3× 3 coarse cells are highlighted.

Fig. 3.3. Figure 3.4 also shows that the central coarse cell (10, 3) of this subdomain

has a net incoming flux induced by the basis function of its southern neighboring

cell (10, 2), together with the interpolated pressure field only for the associated local

domain, i.e., pms in Ωh1. To obtain this interpolated solution, a test case is solved

subject to no-flow Neumann condition on all boundaries and Dirichlet condition of

p = 1 and p = 0 at fine cells (1, 60) and (220, 1), respectively. Note that due to

the positive diagonal and negative off-diagonal coarse-system entries corresponding

to this local sub-region, the interpolated solution is physical.

If for a heterogeneous field, the net incoming flux to the cell i is negative, then

off-diagonal entries acik become positive. This situation happens when the coarse node

xi lies in a low-permeability region, compared with the other boundary cells between

i and k. There are other scenarios that would cause the same situation, e.g., if a shale

layer (with very low permeability) crosses the boundary cells between i and k. Note

that in such cases, the reduced-problem local boundary condition, between the cells

i and k, would lead to a solution with a constant value of one (since the Dirichlet

condition at node k is not effective). This constant unity solution, which is then used

as a Dirichlet condition for the internal cells, leads to a non-physical outgoing induced

flux from the control volume. An example of such a case is illustrated in Fig. 3.5,

where the domain Ωh2 is extracted again from (and highlighted in) Fig. 3.3 for cells

belonging to (33, 20) ≤ (x, y) ≤ (66, 35) interval. The integral incoming flux induced

by φk over the faces of the control volume i is negative, which leads to a positive

off-diagonal value of acik = 222.5 for the coarse-scale system. The total outgoing

fluxes induced by the basis function of i, φi, over its own control volume is too small


i

k k

i

k

i

Figure 3.4: (top-left): Logarithm of permeability field with coarse grid and coarsenodes, extracted from Fig. 3.3. (top-right): part of the basis function φk overlappingwith coarse cell i (coarse cell (10,3) in Fig. 3.3). (bottom-left): basis function φi;(bottom-right): superimposed MSFV pressure field, pms =

∑φkp

ck, obtained for Ωh1.


k

k

i

k

ii

Figure 3.5: (top-left): Logarithm of permeability field with coarse grid and coarsenodes, extracted from Fig. 3.3, Ωh2. (top-right): part of the basis function φk over-lapping with coarse cell i (coarse cell (5,6) in Fig. 3.3). (bottom-left): basis functionφi; (bottom-right): superimposed MSFV pressure field, pms =

∑φkp

ck, obtained for

Ωh2. Note that a non-physical positive off-diagonal value of acik = 222.5 and smallpositive value of acii = 0.65 are calculated for coarse-system entries, which also clearlyshows the i-th coarse-system row is not diagonally dominant.

(acii = 0.65), which indicates that the corresponding row in the coarse-scale system is

not diagonally dominant. This is closely related to the fact that the coarse node lies

in a region with very low permeabilities (blue contour plot in Fig. 3.5). Note that

the other cells (especially the boundary cells) have higher permeability values. As a

result, the superimposed MSFV solution entails non-physical peaks (as shown in Fig.

3.5).

Figure 3.6, which is for the SPE 10 bottom layer, indicates that the original MSFV

strategy leads to non-physical solutions at several locations. From this figure, it is

clear that the peaks are located in regions with high contrasts in the permeability

between the neighboring cells. In the next section, we describe a monotone MSFV

method.


pf p0

Figure 3.6: Fine-scale reference (left) and MSFV (right) solutions for the SPE 10bottom layer heterogeneous test case. There exist 220× 60 fine- and 20× 12 coarse-grid cells. Note that the MSFV superimposed solution (right) entails several non-physical peaks. The permeability field is also partly shown in the plots under thepressure solution.

3.2 Monotone MSFV (m-MSFV) Method

In this section, to ensure the monotonicity of the MSFV solution, two approaches are

proposed. The first one is a local TPFA approach, which automatically detects the

interfaces with non-physical transmissibility coefficients for the coarse-scale system.

Only for these critical coarse-scale interfaces, a local stencil-fix is employed, where the

more stable TPFA stencil is used to calculate the connectivity of the adjacent coarse

cells. The second approach is based on employing a Linear Boundary Condition

(LBC) to solve the basis functions. Similarly to the local TPFA approach, after

detecting the critical coarse-scale interfaces, an LBC is locally applied for the dual-

coarse cell boundaries perpendicular to the critical coarse control volume interfaces,

while the reduced boundary condition is still used for the other interfaces.

3.2.1 Local TPFA Approach

This approach is based on local utilization of a physical flux calculation only for

critical faces to ensure monotonicity of the MSFV solution. First, the coarse cell


interfaces with negative transmissibility values, i.e., acik 6≤ 0, are detected. Then,

instead of using the basis functions to provide the acik values from Eq. (3.7), the

transmissibility field between the cells i and k are calculated with TPFA which guar-

antees that acik ≤ 0. Figure 3.7 shows the highlighted pink region used to obtain an

effective transmissibility coefficient at the interface between i and k. The procedure

to calculate TPFA-based acik is as follows. First, harmonically averaged transmissi-

bility factors among columns of the highlighted pink cells are calculated. Then, the

values are summed to compute acik. Therefore,

acik =Nx∑i=1

1Ny∑j=1

1

kij

∆x

∆y, (3.10)

where kij, ∆x and ∆y represent fine-scale permeability, gridblock size in x and y

directions, respectively. To ensure conservation, the symmetric entry acki is also up-

dated with the same value as for the acik. Here, the new coarse-scale transmissibilities

for the critical faces is computed based on averaging the fine-scale permeability field.

Other options such as flow-based upscaling are also possible and can be incorporated

into our monotone strategy, provided that they guarantee acik <= 0. In this work, we

focus on our permeability-based strategy. In fact, a slightly positive value acij does

not necessarily lead to non-monotone solutions, and only the acij with relatively large

positive values matter and have to be modified.

In order to quantify the critical acij, an indicator ηij for each positive off-diagonal

entry acij of the coarse-scale coefficients matrix Ac is used. We define ηij = acij/ωi,

where ωi represents the maximum absolute value of all the negative off-diagonal acij

in row i. The coarse node with an interface with ηij > ε is considered critical, where

ε is a user-specified threshold value. Then, all the neighboring interfaces associated

with the critical coarse node are replaced by TPFA stencils. Algorithm 1 summarizes

how the local TPFA approach is integrated in the MSFV procedure.


1 2 … 𝑖 … … 𝑁𝑥

2

……

𝑗

……

𝑁𝑦

𝐱

Δ𝐱𝒚

Δ𝒚

Figure 3.7: Automatically detected critical interface (shown in bold red) where acik 6≤0. The highlighted region with a pink rectangle shows the local domain, where thetransmissibility is calculated using the summation of harmonically averaged values toreplace with acik and acki.

𝒊

𝟏

𝟐 𝟑

𝟒

𝒊𝟏

𝒊𝟐 𝒊𝟑

𝒊𝟒

Figure 3.8: Critical coarse node i and its neighboring faces Fij (indicated by red solidlines) and edges Eij (indicated by yellow dash lines), j = 1, 2, 3, 4 for 2D domain.The black lines indicate the coarse volumes.


Algorithm 1 local TPFA approach integrated with MSFV classical procedure

1: Construct coarse and dual-coarse grids2: Compute basis functions φi3: Construct coarse-scale system, Eq. (3.6)4: Specify a threshold value ε5: for i = 1 to nc do6: if ηij > ε then7: Cancel the coarse-scale flux through all Fij, j = 1, 2, 3, 4 faces (see Fig. 3.8)8: Calculate T cij, i.e., TPFA transimissibilities for the faces Fij9: Modify the coarse system entries as following:

10: acij ← acij − T cij11: acii ← acii + T cij12: acji ← acji − T cij13: acjj ← acjj + T cij14: end if15: end for16: Solve this modified coarse-scale system17: Obtain prolongated solution using Eq. (3.4)18: Reconstruct conservative fine-scale velocity field consistently

3.2.2 Local Linear BC Approach

In addition to the local TPFA approach, the non-monotonicity of the MSFV pressure

solution can be mitigated by locally using an LBC instead of the reduced BC. For

the LBC approach, once the critical interface (i.e., the one with ηij > ε) is detected,

a linear BC is used for the corresponding dual coarse grid boundary crossing the

detected interface. For the remaining boundaries, the reduced BC is used. Then, the

basis functions affected by the linear BC are recomputed, and the coarse-scale system

is reconstructed. Afterward, the fine-scale solution is obtained by interpolating the

coarse-scale solution with the modified basis functions. Finally, the conservative fine-

scale velocity field can be constructed similarly as in the classical MSFV method.

The local TPFA approach guarantees monotonicity of the solution since the TPFA

flux is used over the coarse interfaces. The local LBC approach reduces the degree

of non-monotonicity; however, it cannot guarantee a monotone solution. In addition,

the choice of the threshold value, ε, is a trade-off between the computational effort,

quality of the solution, and the degree of monotonicity in the pressure field.


3.3 Numerical Results

In this section, several test cases are solved to illustrate the proposed m-MSFV

method. To quantify the accuracy of m-MSFV, relative errors of pressure, veloc-

ity and residuals, in terms of L2 and L∞ norms, are used. These norms are defined

as

‖ep‖= ‖pm − pf‖/‖po − pf‖, (3.11)

‖ev‖= ‖vm − vf‖/‖vo − vf‖, (3.12)

‖er‖= ‖rm − b‖/‖ro − b‖, (3.13)

where pm, vm and rm denote the pressure, velocity and residual from m-MSFV; po,

vo and ro denote pressure, velocity and residual from original MSFV; pf , vf and b

represent the fine-scale reference pressure, velocity, and RHS vector (source term).

All the pressure plots are scaled by the boundary pressure condition. The local TPFA

and LBC approaches are referred to as “m-MSFV(TPFA)” and “m-MSFV(LBC)”,

respectively.

3.3.1 Case 1: SPE 10 Bottom Layer

The first example is the SPE 10 bottom layer case with 220× 60 fine cells and 22× 6

coarse cells. The pressure is fixed at (220, 0) and (0, 60) with the non-dimensional

values of 1 and 0, respectively; and no-flow boundary conditions are specified on all

the boundaries. The threshold value ε = 0 indicates all the coarse-scale interfaces

with positive indicators ηij are considered as critical. The permeability and fine-scale

reference pressure solution are shown in Fig. 3.9. Since the problem is elliptic, the

pressure should be bounded by the pressure values at boundaries (i.e., 0 and 1).

However, as shown in Fig. 3.10(a), the original MSFV pressure exceeds these bounds

at several locations, which indicates that the obtained solution is nonmonotone. A

strictly monotone MSFV pressure can be obtained by using m-MSFV(TPFA), as

shown in Fig. 3.10(b). In this case, the m-MSFV(LBC) can also reduce the level of

nonmonotonicity significantly as shown in Fig. 3.10(c); however, this approach does


20 40 60 80 100 120 140 160 180 200 220

10

20

30

40

50

60−5

0

5

(a) Natural logarithm of the permeability

20 40 60 80 100 120 140 160 180 200 220

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(b) Fine-scale reference pressure

Figure 3.9: Natural logarithm of the permeability (a) and fine-scale reference pressure(b) for the SPE 10 bottom layer.

not guarantee that the solution is monotone. Figure 3.11 shows the streamlines as-

sociated with fine-scale pressure obtained using the original and monotone MSFV

methods. As shown in Fig. 3.11(b), the non-physical MSFV pressure leads to circu-

lations in the velocity field, which can decrease the stability of the entire nonlinear

simulation procedure. On the contrary, there exist no circulations in the velocity field

reconstructed by the monotone MSFV pressure. In addition, as seen from the pres-

sure errors, the m-MSFV method can deliver a monotone pressure solution without

sacrificing accuracy.

Figure 3.12 shows the histogram of ηij corresponding to the coarse-scale systems Ac

of the original MSFV, m-MSFV(TPFA) and m-MSFV(LBC) methods. Note that the

original coarse-scale system Ac (Fig. 3.12(a)) has many positive indicators which span

a wide range. These positive values lead to severely non-monotone pressure solution.

With the modifications of m-MSFV(TPFA), the positive indicators are reduced to a

limited range with small values, which are acceptable to obtain a monotone solution.

If zero indicators are desired, additional loops of detection and modification can be

performed as described in Algorithm 1. On the other hand, with the modification of

m-MSFV (LBC), even though this approach can eliminate some positive indicators,

many areas with long-range indicator values still remain. These values may result in

a non-monotone solution. Note that the remaining indicators cannot be eliminated

by additional modification loops. That is the reason why m-MSFV(LBC) can reduce

the level of non-monotonicity, but cannot guarantee to fully resolve the issue for all

the problems.

For practical purposes, strictly monotone pressure may not be required; there-

fore the threshold value ε provides a way to balance the degree of monotonicity


20 40 60 80 100 120 140 160 180 200 220

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(a) Original MSFV (‖ep‖2= 0.197; ‖ep‖∞= 3.815)

20 40 60 80 100 120 140 160 180 200 220

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(b) m-MSFV(TPFA) (‖ep‖2= 0.035; ‖ep‖∞=0.071)

20 40 60 80 100 120 140 160 180 200 220

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(c) m-MSFV(LBC) (‖ep‖2= 0.052; ‖ep‖∞=0.122)

Figure 3.10: Original MSFV and m-MSFV pressure solutions for the SPE 10 bottomlayer, and the relative errors ep. The coarse-scale grids are indicated by black lines.

0 40 80 120 160 200

0

20

40

60

(a) Fine-scale reference (b) Original MSFV

0 40 80 120 160 200

0

20

40

60

(c) m-MSFV (TPFA) (d) m-MSFV (LBC)

Figure 3.11: Streamline plots based on velocity fields reconstructed by fine-scalereference, original and m-MSFV pressure solutions for the SPE 10 bottom layer. Thecoarse-scale grids are indicated by black lines.


0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

ηij

Count

(a)

0 0.02 0.04 0.06 0.08 0.10

2

4

6

8

10

ηij

Co

un

t

(b)

0 0.2 0.4 0.6 0.80

5

10

15

20

ηij

Co

un

t

(c)

Figure 3.12: Histogram of ηij of the coarse-scale system Ac for original MSFV (a),the reconstructed coarse-scale system for m-MSFV (TPFA) (b) and m-MSFV (LBC)(c), respectively.


and the computational cost of the m-MSFV method. Figure 3.13 shows that m-

MSFV (TPFA) with ε = 0 guarantees that the pressure solution is strictly monotone.

When the threshold is loosened to ε = 0.7, the pressure solution still does not en-

counter severe non-monotone regions, while the computational effort is reduced by

50% compared with the ε = 0 case. Figure 3.14 shows the accuracy of the m-MSFV

050

100150

200

0

20

40

60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


050

100150

200

0

20

40

60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Original MSFV

050

100150

200

0

20

40

60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) m-MSFV (TPFA) with ε = 0

050

100150

200

0

20

40

60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(d) m-MSFV (TPFA) with ε = 0.7

Figure 3.13: Pressure surface plots for fine-scale reference (a), original MSFV (b),m-MSFV (TPFA) with ε = 0 (c) and ε = 0.7 (d)

method with respect to different strategies and indicates that both m-MSFV(TPFA)

and m-MSFV(LBC) have comparable error norms for pressure and velocity. The m-

MSFV(LBC) approach results in slightly better residual estimates, since it preserves

the MPFA stencil at coarse-scale, and just simplifies the heterogeneous field at the

dual coarse cell boundaries.


0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ε

||e

p||

m−MSFV (TPFA) L2−norm

m−MSFV (LBC) L2−norm

m−MSFV (TPFA) L∞−norm

m−MSFV (LBC) L∞−norm

(a)

0 0.2 0.4 0.6 0.8 10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

ε

||e

v||





(b)

0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

ε

||e

r||





(c)

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

ε

Perc

enta

ge o

f m

odific

ation %

critical coarse−scale nodes

critical coarse−scale interfaces

critical dual grid edges

(d)

Figure 3.14: Error measurements in pressure (a), velocity (b), residual (c) and thecomputational complexity (d) with different threshold ε for the SPE 10 bottom layer


3.3.2 Case 2: SPE 10 Layers with Stretched Grid

In this case, both SPE 10 top and bottom layers with stretched grid are examined.

The fine-scale and coarse-scale grids are 220 × 60 and 22 × 6, respectively. The

global boundary conditions are the same as in Case 1. The fine-scale grid has an

aspect ratio of 10, i.e., ∆x = 10∆y. First, for SPE 10 top layer, the permeability

field, fine-scale reference, original MSFV and m-MSFV pressure solutions are shown

in Figs. 3.15 and 3.16. Even though there are no significant peaks in the original

MSFV pressure solution, the resulting streamlines of the original MSFV still have

circulations. Also, in this case, the m-MSFV (TPFA) approach is using TPFA for

almost the entire domain. Therefore, the pressure solution is not accurate. However,

m-MSFV(TPFA) does guarantee monotonicity of the pressure distribution, which

is indicated by the circulation-free streamlines (Fig. 3.18(c)). Circulations can be

observed in the streamlines of m-MSFV (LBC) as shown in Fig. 3.18(d), which implies

that m-MSFV (LBC) cannot guarantee a monotone solution in this case. Moreover,

the non-monotone solution for original MSFV and m-MSFV (LBC) can be identified

by Fig. 3.17, which indicates that the long-range positive indicators of the coarse-

scale system may lead to unphysical multiscale solutions. The range indicates the

difference between the maximum and minimum value of ηij. Figure 3.17 shows that

the indicators’ values span a long-range, e.g., varying from 0 to 1.5, therefore, the

solution is non-monotone. If the indicators’ values are limited within a small range,

e.g., from 0 to 0.1, then the solution is expected to be monotone.

0 500 1000 1500 2000

10

20

30

40

50

60−5

0

5

(a) Natural logarithm of the permeability

0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(b) Fine-scale reference pressure

Figure 3.15: Permeability and fine-scale pressure solution for the SPE 10 top layerwith stretched grids.

Similarly, as shown in Fig. 3.20, the original MSFV is severely nonmonotone for

the SPE 10 bottom layer with stretched grids, and the m-MSFV (LBC) mitigates the

issue. However, it cannot fully resolve the non-monotonicity in the pressure solution.


0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(a) Original MSFV (‖ep‖2= 0.015; ‖ep‖∞= 0.148)

0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(b) m-MSFV(TPFA) (‖ep‖2= 0.252; ‖ep‖∞=0.407)

0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(c) m-MSFV(LBC) (‖ep‖2= 0.034; ‖ep‖∞=0.169)

Figure 3.16: Original MSFV and m-MSFV pressure solutions for the SPE 10 top layerwith stretched grids, and the relative errors ep

0 0.5 1 1.5 2 2.50

10

20

30

40

50

60

ηij

Co

un

t

(a) Original coarse-scale system Ac

0 0.5 1 1.50

5

10

15

20

25

30

35

ηij

Co

un

t

(b) Coarse-scale system Ac with linear BC

Figure 3.17: Histogram of ηij of the coarse-scale system Ac for original MSFV (a)and the reconstructed coarse-scale system for m-MSFV (LBC) (b), respectively, forthe SPE 10 top layer with stretched grids. Note that m-MSFV (TPFA) eliminatesall the positive indicators, therefore the histogram is not shown.


0 500 1000 1500 2000

0

20

40

60



Figure 3.18: Streamline plots based on velocity fields reconstructed by fine-scalereference, original and monotone MSFV pressure solutions.

The m-MSFV (TPFA) becomes a global TPFA scheme; therefore, it loses accuracy

as indicated in the streamline plots shown in Fig. 3.21. In addition, Figs. 3.19 and

3.22 show the accuracy of the m-MSFV method with respect to different strategies

and indicate that both m-MSFV(TPFA) and m-MSFV(LBC) have comparable error

norms for pressure and velocity.

Note that the streamlines given by m-MSFV(LBC) honor the fine-scale refer-

ence quite well for the region where no circulations occur. Therefore, it is bene-

ficial to apply m-MSFV(LBC) first, then employ m-MSFV(TPFA) for the places

where m-MSFV(LBC) fails to resolve non-physical peaks. Hence, combining both

m-MSFV(LBC) and m-MSFV(TPFA) can achieve circulation-free and conservative

fine-scale velocity fields without losing accuracy for anisotropic problems. For the

SPE 10 top layer with stretched grids, m-MSFV(LBC) is applied first resulting in the

pressure and velocity distributions as shown in Fig. 3.16(c) and Fig. 3.18(d). From

Fig. 3.18(d), m-MSFV(LBC) cannot fully resolve the circulations for some particular

regions but results in streamlines that are quite close to fine-scale reference in most

regions. In order to remove the circulations, the m-MSFV(TPFA) approach can be

employed for the regions where m-MSFV(LBC) is not adequate. With the combi-

nation of both approaches, we can obtain the fine-scale pressure and velocity fields

shown in Figs. 3.23 and 3.24. In additional, the pressure, velocity, and residual errors

with respect to the fine-scale reference are given in Table 3.1, where we can see that


0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

ε

||e

p||





(a)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

ε

||e

v||





(b)

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

ε

||e

r||





(c)

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

ε

Pe

rce

nta

ge

of

mo

dific

atio

n %




(d)

Figure 3.19: Error measurements in pressure (a), velocity (b), residual (c) and com-putational complexity (d) with different threshold ε for the SPE 10 top layer withstretched grids.


0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1


0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(b) Original MSFV (‖ep‖2= 0.417;‖ep‖∞=4.969)

0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(c) m-MSFV(TPFA) (‖ep‖2= 0.338;‖ep‖∞=0.392)

0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(d) m-MSFV(LBC) (‖ep‖2= 0.043;‖ep‖∞=0.331)

Figure 3.20: Original MSFV and m-MSFV pressure solutions for the SPE 10 bottomlayer with stretched grids, and the relative errors ep.



Figure 3.21: Streamline plots based on velocity fields reconstructed by fine-scalereference, original and m-MSFV pressure solutions for the SPE 10 bottom layer withstretched grids. The coarse-scale grids are indicated by black lines.


0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ε

||e

p||





(a)

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

ε

||e

v||





(b)

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ε

||e

r||





(c)

0 0.2 0.4 0.6 0.8 1 1.20

10

20

30

40

50

60

70

80

90

100

ε

Perc

enta

ge o

f m

odific

ation %




(d)

Figure 3.22: Error measurements in pressure (a), velocity (b), residual (c) and com-putational complexity (d) with different threshold ε for the SPE 10 bottom layer withstretched grids.


the hybrid m-MSFV delivers the most accurate velocity field.

0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(a)

0 500 1000 1500 2000

10

20

30

40

50

60 0

0.2

0.4

0.6

0.8

1

(b)

Figure 3.23: Pressure distributions for fine-scale reference (a) and obtained by hybridm-MSFV method (b) for the SPE 10 top layer with stretched grids, i.e.,∆x = 10∆y.

(a) (b)

Figure 3.24: Velocity distributions for fine-scale reference (a) and obtained by hybridm-MSFV method (b) for the SPE 10 top layer with stretched grids, i.e.,∆x = 10∆y.

Error ‖ep‖2 ‖ep‖∞ ‖ev‖2 ‖ev‖∞ ‖er‖2 ‖er‖∞ interfaces LBCHybrid m-MSFV 0.034 0.187 0.926 0.585 0.059 0.019 14% 33%m-MSFV(TPFA) 0.252 0.407 8.165 8.816 0.309 0.052 95% –m-MSFV(LBC) 0.034 0.169 3.349 8.119 0.051 0.017 – 52%original MSFV 0.015 0.148 8.115 7.507 0.122 0.022 – –

Table 3.1: Relative errors of hybrid m-MSFV, m-MSFV(TPFA), m-MSFV(LBC) andoriginal MSFV for the SPE 10 top layer with stretched grids, i.e.,∆x = 10∆y. Inaddition, the last two columns represent the amount of TPFA coarse-scale interfacesand dual-grid boundaries using LBC for all the methods.

3.4 Discussion

This proposed TPFA strategy leads to local modifications of the coarse-scale operator.

Therefore, the coarse-scale operator still has global MPFA stencils in high flow-rate


regions, but for some few critical interfaces (mainly in low-permeable regions), the

stencils are changed into the TPFA type. We expect that the local TPFA strategy

would not sacrifice too much accuracy. The efficiency and accuracy of the proposed

strategy rely on the fact that the fix is local. However, for extremely challenging

problems, where the coarse-scale operator has high contrast in the coefficients, the

proposed strategy may detect large numbers of interfaces that need to be modified,

which results in the fix becoming nearly global. In the limit, if the fix is applied

globally, it would be equivalent to the TPFA coarse stencil everywhere, which leads

to monotone, yet inaccurate, solutions. Alternatively, attempts to modify the basis

functions can be seen in the literature, e.g., the work by Møyner and Lie [69]. However,

these fixes are not applicable in a general way to the MSFV framework, and they fall

beyond the scope of this work.

The way to calculate an optimum TPFA transmissibility is an open question.

Other options, such as flow-based upscaling, are also possible and can be incorporated

into our strategy. We choose the averaging of the fine-scale permeability strategy in

this work because it guarantees positive-definite entries into the coarse-scale operator.

Not all flow-based upscaling approaches can guarantee this property, but a more

sophisticated strategy (e.g., flow-based upscaling) would be worth investigating.

Normally, monotone pressure fields are considered a safe prerequisite for perform-

ing nonlinear transport computations. However, if the nonlinear pressure dependen-

cies are not severe, then strict monotonicity of the pressure is not needed. Therefore,

we use a threshold ε to determine the level of monotonicity. The optimal choice

for ε is problem-specific. A rule of thumb, based on our experiments as shown in

Fig. 3.14, 3.19 and 3.22, is that the optimal choice for ε is around 0.5 for the SPE 10

test cases. For practical applications, a user defined monotone tolerance should be

set. We recommend starting with 0.5. However, one can quickly test the behavior

using a few values for ε to obtain a better estimate for the specific problem class

under study.


Chapter 4

Algebraic Multiscale Solver with

Well Modeling

Chapter 2 described the general Algebraic Multiscale Solver (AMS) framework with-

out considering well models. However, accurate and computationally efficient model-

ing of complex wells is a prerequisite for field applications. In this chapter, AMS is

extended to allow for flow simulation in reservoirs with complex well configurations.

The first section explains the governing equations with the standard well model.

Then, two methods for well modeling in AMS are proposed and investigated. In the

first method, the multiscale operators (i.e., prolongation, restriction, and coarse-scale

operators) are enriched by using the well basis functions [2] to capture the influence

of wells on the flow regimes across the reservoir. In the second method, the mul-

tiscale operators are constructed based on a diagonally approximate Schur comple-

ment [49, 50, 52, 56, 70], which is a reduced linear system obtained by eliminating the

well constraints from the coupled reservoir-well system. The pressure for the reservoir

gridblocks is solved by multiscale operators in the reduced system, then the wellbore

pressure is updated with the well constraint equations. In both methods, a local

preconditioner is employed after the multiscale stage to resolve the high-frequency

errors. Finally, the performance of these methods is examined for different test cases

and conclusions are drawn based on the numerical results.

79

80 CHAPTER 4. AMS WITH WELL MODELING

4.1 Governing Equation

The pressure equation for single-phase incompressible flow with well modeling can be

written as

∇ · (λ · ∇p) + α(p− pw) = 0, (4.1)

where no other physics is considered such as gravitational effect or capillary pressure.

All these source terms can be dealt with by a local preconditioner in the AMS frame-

work, as discussed in Chapter 2. Due to the essentially singular nature of a well, the

wellbore pressure pw can differ significantly from the wellblock pressure, p, i.e., the

pressure of the gridblock perforated by the well. The flow rate from gridblock i into

a well penetrating that block is given by the Peaceman well model [71], which can be

written as

qwi = αi(pi − pwi ), (4.2)

where α is the well index and is usually defined as

αi =

2πkh/µ

ln(rorw

) + s

i

, (4.3)

where k and h denote the permeability and thickness of the gridblock i, and rw is

the wellbore radius. ro is the radial position where the reservoir gridblock pressure,

computed by the simulator, is equal to the analytical pressure obtained by assuming

a single-phase and steady-state radial flow. The skin term s is used to account for

damage or stimulation of the well. This term can also include a flow rate dependent

skin to account for non-Darcy effects, especially for high rate gas wells. Generally,

the local wellbore pressure pwi is computed by solving a transport equation within the

well with appropriate boundary conditions. If the viscous pressure loss is neglected

and the fluid density ρ is assumed to be constant along the well, the pwi for gridblock

i at depth zi can be related to a reference pressure pw,ref at the specific reference

depth zref through the hydrostatic condition

pwi = pw,ref + (zi − zref )gρ. (4.4)

CHAPTER 4. AMS WITH WELL MODELING 81

The gravitational effect is ignored in this study; therefore, each local wellbore pressure

pwi along a well is equal to the reference pressure (denoted as pw for simplicity)

associated with that well.

The treatment of pressure-constraint wells is trivial since the wellbore pressure pw

is explicitly specified. However, obtaining the unknown wellbore pressure for rate-

constraint wells requires solving additional constraint equations, described as∫x∈Ωw

α(x)(p(x)− pw(x))dx = Qw, (4.5)

where Ωw is the domain penetrated by the well, and Qw is known for each rate-

constraint well.

After discretization of the governing equations (4.1) and (4.5), we obtain the fine-

scale linear system coupled with the reservoir and well equations as

Ap = b, (4.6)

where the unknown vector to solve consists of the pressure for the reservoir gridblocks

pR, and the wellbore pressure for each well pW as

p =

[pR

pW

]. (4.7)

The reservoir pressure and wellbore pressure vectors are expressed as

pR = [p1, p2, ..., pnf]T (4.8)

and

pW = [pw1 , pw2 , ..., p

wnw

]T , (4.9)

where nf and nw are the numbers of fine-scale gridblocks and wells, respectively.

Similarly, the right-hand-side (RHS) vector b is divided into a reservoir part bR and

a well part bW as

b =

[bR

bW

], (4.10)


where

bR = [b1, b2, ..., bnf]T , (4.11)

and

bW = [bw1 , bw2 , ..., b

wnw

]T . (4.12)

According to Eq. (4.1), all the entries of the reservoir part of the RHS vector, bi = 0

(i = 1, ..., nf ), are zeros as no other source terms are considered in the reservoir.

In the well part, the entries, bwi (i = 1, ..., nw), represent either the known total

injection/production rate, or the known wellbore pressure depending on the operation

type of the wells. The fine-scale coefficient matrix A can be divided into four parts:

the reservoir part ARR, the coupling between the reservoir and the wells ARW ,

AWR, and the well part AWW , where the first subscript refers to the equation and

the second subscript refers to the variable. Here, R and W denote reservoir and well

related quantities, respectively. Thus, the fine-scale system is written as[ARR ARW

AWR AWW

][pR

pW

]=

[bR

bW

]. (4.13)

For rate-constraint wells, AWR and AWW represent the well constraint equations

in Eq. (4.5); for pressure-constraint wells, the corresponding local matrices become

AWR = 0 and AWW = I.

4.2 Well Basis-Function Method

This method constructs the coarse-scale operator based on the full system A (e.g.,

the coupled reservoir-well system) as AC = RAP . The prolongation operator, P ,

and restriction operator, R, are expanded to capture well effects by using well basis

functions. These functions were introduced by Jenny and Lunati [2] to account for the

new degrees of freedom represented by the wellbore pressures. Well basis functions

have the same support domain as the original basis functions used in the MSFV

method. For each well, there exists a well basis function in every dual coarse cell

perforated by that well. The basis function associated with the coarse-scale vertex

i and defined on the perforated dual coarse cell ΩjD, φij, is computed by solving the


following local problems with the presence of the well, but the wellbore pressure pw

is set to be zero:∇· (λ · ∇φij) +

nw∑γ=1

αγφij = 0 in Ωj

D

∇‖· (λ · ∇φij) +nw∑γ=1

αγφij = 0 on ∂Ωj

D

φij(xk) = δik ∀k ∈ 1, ..., nc

. (4.14)

Well basis functions share the same supporting domains (dual coarse cells) as the

basis functions and the well basis function defined on the perforated dual cell ΩjD,

φw,βj , β = 1, 2, ..., nw is obtained by solving

∇· (λ · ∇φw,βj ) +

nw∑γ=1

αγ(φw,βj − δβγ) = 0 in Ωj

D

∇‖· (λ∇φw,βj ) +nw∑γ=1

αγ(φw,βj − δβγ) = 0 on ∂Ωj

D

φw,βj (xk) = 0 ∀k ∈ 1, ..., nc

. (4.15)

If the fine-scale cell is penetrated by well γ, then αγ is the well index defined as

Eq. (4.3) in this cell; otherwise αγ = 0. This formulation is capable of dealing with

the scenario where multiple wells intersect in the same gridblock. Figure 4.1 illustrates

an example of one basis function and one well function on a 2D dual-coarse grid. The

well penetrates the middle of the coarse grid. The basis function at one coarse node

is equal to unity by definition, while the maximum value of the well basis functions

is always less than one unless the well index is infinity. The basis function and well

basis function still preserve the partition of unity, i.e.,nc∑i=1

φij +nw∑β=1

φw,βj = 1 on dual

coarse grid ΩjD.

Once the basis functions and well basis functions are computed, the multiscale

solution pms can be written as

pms =nc∑i=1

pci

nd∑j=1

φij +nw∑β=1

pwβ

nd∑j=1

φw,βj , (4.16)

where the coarse-scale pressure pci and the wellbore pressure pwβ form the vector that


Basis function

0

0.2

0.4

0.6

0.8

1

(a)

Well function

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(b)

Figure 4.1: Illustration of basis function and well basis function on a 2D dual coarsegrid with a homogeneous permeability field. The fine-scale cells with a white circlerepresent a coarse-scale node, and the cells with a white cross marker indicate theperforations of the well.

needs to be solved on the coarse-scale system. Next, we describe the prolongation,

restriction, and coarse-scale operators which are used to solve the coupled reservoir-

well system (Eq. (4.13)).

4.2.1 Prolongation Operator

The prolongation operator P is employed to interpolate the coarse-scale solution into

the fine scale as

p = Ppc (4.17)

where the coarse-scale solution pc consists of both coarse-scale reservoir pressure and

the wellbore pressure for all the wells, i.e., pc = [pc1, pc2, ..., p

cnc, pw1 , p

w2 , ..., p

wnw

]T , and

nc is the number of coarse grids. With Eq. (4.16), the fine-scale solution vector can

be written as

p ≈

[pms

pw

]=

[Φ Φw

0 I

][pc

pw

], (4.18)


where Φ and Φw are nf × nc and nf × nw matrices, which are the collection of the

basis functions and well basis functions, respectively. The 0 is an nw×nc zero matrix

and I is an nw × nw identity matrix. pc and pw consist of coarse-scale pressure for

each coarse node and wellbore pressure for each well. Therefore, the prolongation

operator can be defined as

P =

[Φ Φw

0 I

](4.19)

Note that the prolongation operator in Chapter 2 is defined as P = Φ, and here it is

expanded to capture the well effects by the use of well functions.

4.2.2 Restriction Operator

Since each well has strong local effects on its surrounding reservoir regions, each well-

bore pressure is treated as the coarse-scale unknown variable, and the well constraint

equations are kept the same for both fine- and coarse-scale systems. Similar to the

prolongation operator, the restriction operatorR is expanded to a (nc+nw)×(nf+nw)

matrix, which is written as

R =

[RCF RCW

RWF RWW

], (4.20)

where F and C denote fine-scale and coarse-scale quantities for reservoir gridblocks.

RCW andRWF are nc×nw and nw×nf matrices, andRWW is an nw×nw matrix,

respectively.

The restriction operator is not unique, and can be constructed based on either

Finite Volume (FV) or Finite Element (FE) numerical discretization schemes for

coarse-scale mass balance equations. Next, we describe the FV- and FE-based re-

striction operators.


Finite-Volume Restriction Operator

The finite-volume method is based on honoring the mass conservation in each control

volume. The fine-scale formulation for the reservoir part is∫Ωj

F

(∇· (λ · ∇p) + α(p− pw))dΩ = 0 (j = 1, ..., nf ), (4.21)

where ΩjF represents a fine-scale control volume. On the coarse-scale control volume

ΩiC , the finite-volume method requires∫

ΩiC

(∇· (λ · ∇p) + α(p− pw))dΩ = 0 (i = 1, ..., nc). (4.22)

The well constraint equations are kept the same for both fine- and coarse-scale; there-

fore,RCW andRWF become nc×nw and nw×nf zero matrices, respectively;RWW

is a nw × nw identity matrix. The operator RCF accounts for numerical integration

from fine-scale cells into coarse-scale blocks; the entry RCF (i, j) in RCF is given by

RCF (i, j) =

1 if Ωj

F ⊂ ΩiC

0 otherwise(i = 1, . . . , nc; j = 1, . . . , nf ) . (4.23)

Therefore, the finite-volume type restriction operator can be written as

R =

[RCF 0

0 I

]. (4.24)

Finite-Element Restriction Operator

The algebraic formulation of well function with an FV restriction scheme is equiva-

lent to the method proposed by Jenny and Lunati [2]. Alternatively, the coarse-scale

operator can also be constructed in the FE framework. In order to derive the restric-

tion operator based on the finite-element framework, we first define the finite-element

space using only basis functions expressed from a global point of view:

Vh = spanφA : A = 1, .., nc, (4.25)


where φA is in fact the Ath column of Φ. From Eq. (4.1), we can have∫Ω

φA(∇· (λ · ∇p) + α(p− pw))dΩ = 0 (A = 1, ..., nc), (4.26)

which is equivalent to the Galerkin finite element formulation. Recall that the ap-

proximate p is given by Eq. (4.16); so the left-hand-side (LHS) term of Eq. (4.26) can

be written as

LHS =

∫Ω

φA

∇· (λ · ∇(

nc∑i=1

pci

nd∑j=1

φij +nw∑β=1

pwβ

nd∑j=1

φw,βj )

+ α((nc∑i=1

pci

nd∑j=1

φij +nw∑β=1

pwβ

nd∑j=1

φw,βj )− pw)

dxa

≈nf∑a=1

φA(xa)

nf∑b=1

ARR(a, b)(nc∑i=1

pci

nd∑j=1

φij(xb) +nw∑β=1

pwβ

nd∑j=1

φw,βj (xb))

+nw∑i=1

ARW (a, i)pwi

∆V

=

nf∑a=1

φA(xa)

[[ARR(a, :)

] [Φ Φw

] [pcpw

]+[ARW (a, :)

]pw

]∆V,

(4.27)

whereA(a, :) indicates the row vector of row a in matrixA. Note that these equations

correspond to the nc discretized coarse-scale equations of Eq (4.26) for the reservoir

part. After canceling the control volume ∆V , this set of linear equations can be

written algebraically as

ΦT[ARR

] [Φ Φw

] [pcpw

]+ ΦT

[ARW

]pw = 0. (4.28)

Similarly, the discretized form of the well constraint equations (i.e., Eq. (4.5)) can be

expressed as

AWR(Φpc + Φwpw) +AWWpw = bw. (4.29)


Therefore, the full system, which includes coarse-scale reservoir equations and addi-

tional well equations, becomes[ΦT 0

0 I

][ARR ARW

AWR AWW

]︸︷︷︸

A

[Φ Φw

0 I

]︸︷︷︸

P

[pc

pw

]︸︷︷︸pc

=

[ΦT 0

0 I

]b, (4.30)

which suggests taking the restriction operator as

R =

[ΦT 0

0 I

]. (4.31)

Once the prolongation and restriction operators are constructed, the coarse-scale

system can be formulated as

Acxc = bc, (4.32)

where

Ac =RAP (4.33)

bc =Rb. (4.34)

Therefore, the multiscale preconditioner with well models is obtained as

M−1MSwell = P(RAP)−1R, (4.35)

and the AMS two-step iterative procedure becomes

pν+1/2 = pν +M−1MSwell(q −Ap

ν), (4.36)

pν+1 = pν+1/2 +M−1ILU (q −Apν+1/2). (4.37)

Note that the coarse-scale operator based on either FV or FE formulation preserves

the same strength of the source terms on both the fine- and coarse-scales. This can be

verified by a homogeneous two-dimensional problem with two wells on the two sides

of the domain, as shown in Fig. 4.2. The well on the left side is pressure-controlled,

and the other one is rate-controlled. In this case, both MSFV and MSFE methods

should give an exact solution in one iteration. As shown in Fig. 4.2, the pressure


errors ‖ep‖2= ‖pms − pf‖2/‖pf‖2 for the MSFV and MSFE solutions are 2.4× 10−12

and 4.8× 10−11, respectively. This indicates that the coarse-scale operator with well

functions using either FV or FE formulation honors the boundary conditions on the

fine-scale.

10 20 30

5

10

25

0

0.5

20

115

1.5

2

30

(a) Well locations

10 20 30

5

10

15

20

25

30

1

1.2

1.4

1.6

1.8

2

(b) Fine-scale reference solution

10 20 30

5

10

15

20

25

30

1

1.2

1.4

1.6

1.8

2

(c) MSFV solution

10 20 30

5

10

15

20

25

30

1

1.2

1.4

1.6

1.8

2

(d) MSFE solution

Figure 4.2: Homogeneous permeability field with two wells located on the two sides ofthe domain (the black dots indicate the well perforations): (a), the fine-scale referencesolution (b), the MSFV solution (c), and the MSFE solution (d).


4.3 Schur Complement Method

In the previous section, well basis functions were employed to capture well models

and to build the coarse-scale system. Alternatively, another approach, which is widely

used in classic reservoir simulation, is to solve the coupled system in two steps. The

first step is to eliminate the unknown well pressures by performing block Gaussian

elimination. This step is essentially the process of finding the Schur complement of

the matrix block AWW . The resulting linear system can be solved efficiently using

the AMS framework. Then, the wellbore pressure is updated by the well constraint

equations. In the second step, the entire coupled system is solved by a smoothing

step to capture the information missed by the multiscale operator and the decoupling

of the reservoir and well parts. In this method, we decompose the fine-scale system

into a reservoir part and a well part and rewrite it as[ARR ARW

AWR AWW

][pR

pW

]=

[bR

bW

]. (4.38)

Therefore, the reduced but equivalent system is

A∗RRpR = b∗R, (4.39)

where

b∗R = bR−ARW ·A−1WWbW . (4.40)

The resulting Schur complement matrix of the matrix AWW has reservoir equations

only, but with additional fill-ins, and it can be written as

A∗RR = ARR−ARW ·A−1WW ·AWR. (4.41)

The second term on the right hand side has the same size as ARR; however, A∗RRhas new fill-in terms compared with ARR. The number of induced terms in a cell

penetrated by a well is equal to the number of perforations in the well. It is likely

that many of the induced terms will occupy new fill-in positions. As a consequence,

the computational cost of calculating the full Schur complement can be quite large.

In fact, the row-sum preconditioner can be used as an approximation of the Schur


complement, and can be obtained efficiently as follows:

A′RR = ARR− diag(ARW ·A−1WW ·AWR · ~e), (4.42)

where ~e is the vector with all elements equal to unity and diag denotes the operation

of construction of a diagonal matrix. We do not need to work out ARWA−1WWAWR;

instead, we just calculate Eq. (4.42) from right to left. The interim results are always

vectors; therefore the computational cost is small. We subtract the row-sum result

of ARWA−1WWAWR from the diagonal of ARR. In the other words, the well terms

are approximately compensated for in the diagonal of matrix ARR. Therefore, A′RRis closer to an elliptical operator and is more suitable for multiscale modeling, as

demonstrated in the results section.

In this method, no well basis function is employed. The basis functions are com-

puted based on either ARR or A′RR, in the same way as the original basis functions

used in the MSFE and MSFV methods. In order words, the prolongation operator

is constructed by Eq. (2.17), but the fine-scale operator here is either ARR or A′RR,

which includes well effects in the diagonal entries. Therefore, the well effects are

captured by basis functions. The restriction operator R can be calculated based on

the finite-volume or a finite-element approach as Eq. (2.28) and (2.29). Similarly,

the fine-sale coefficient matrix used to build the coarse-scale operator can be either

ARR or A′RR. These four different options are summarized in Table 4.1. Once the

reservoir pressure is updated, the unknown wellbore pressure can be easily obtained

by performing back substitution with Eq. (4.38). The final step is to use a local pre-

conditioner (i.e., ILU(0)) to the entire linear system in order to capture the missing

information by the well models and the decoupling process. The solution strategy of

the Schur complement method is described in Algorithm 2.

4.4 Numerical Results

4.4.1 Convergence Rate

The numerical simulations in this section are performed on a 2D domain which is

discretized into 220 × 60 fine-scale cells with ∆x = ∆y = 1 for each cell. There are


Algorithm 2 AMS with Schur complement method1: ν = 02: Construct the prolongation operator P based on either ARR or A′RR3: Construct the restriction operator R from Eq. (2.28) or (2.29)4: Construct the multiscale preconditioner M−1

ms

5: Initialize pν =

[pνRpνW

]= 0

6: while (ν < maximum iteration number && not converged) do

7: calculate the full residual rν =

[rνRrνW

]:

rν = b−Apν8: calculate the residual for the reservoir part:

rνR = rνR −ARW ·A−1WW · rνW

9: calculate reservoir pressure change with multiscale preconditioner:δpνR = M−1

msrνR

10: calculate wellbore pressure change:δpνW = A−1

WW (rνW −AWRδpνR)

11: update the entire solution vector:

pν+1/2 = pν +

[δpνRδpνW

]12: update the entire solution vector with local preconditioner:

pν+1 = pν+1/2 +M−1local(b−Apν+1/2)

13: end while

option P Ac M−1ms

Schur-1 ARR RARRP PA−1c R

Schur-2 ARR RA′RRP PA−1c R

Schur-3 A′RR RARRP PA−1c R

Schur-4 A′RR RA′RRP PA−1c R

Table 4.1: Different options for the multiscale preconditioner M−1ms in the Schur

complement method.


20× 6 coarse-scale cells, which corresponds to an upscaling factor of 11× 11. No-flow

boundary conditions are imposed on the four sides of the 2D domain. The well index

for each well is simplified as α = c√kxky, where kx and ky are the permeability of

the well-block in the x and y directions, respectively. The constant c has a typical

value ranging from 0.1 to 1. Five isotropic permeability fields are considered: (1) ho-

mogeneous permeability; (2) patchy permeability generated by sequential Gaussian

simulations [63] with spherical variograms, the dimensionless correlation lengths in x

and y directions as ψx = ψy = 0.1, and the variance ln(k) as 4; (3) layer permeability

generated by sequential Gaussian simulations [63] with spherical variograms, the cor-

relation lengths in the x and y directions as ψx = 0.3 and ψx = 0.1, and the variance

of ln(k) as 4; (4) the top layer of the SPE 10 model; and (5) the bottom layer of the

SPE 10 model. Six solution strategies are investigated here: the well basis-function

method, the Schur complement method with four options (denoted as Schur-1, Schur-

2, Schur-3, and Schur-4, respectively), and AMG preconditioner. ILU(0) is used as

the local preconditioner to update the solution of the entire linear system. GMRES

is employed in conjunction with different preconditioners for the iterative procedure.

Simple Geometry

There are five wells with simple geometry in the domain, as shown in Fig. 4.3. The

well configurations are described in Table 4.2. Figure 4.4 indicates the fine-scale

reference pressure for each test case. Figure 4.5 shows the performance of different

solver strategies for the five permeability cases. Among the different options of the

Schur complement method, option 4 outperforms the others for all the test cases.

Also, the well basis-function method leads to a better convergence rate than the

Schur complement method. Moreover, the FE type of restriction operator improves

the convergence for the well function method and the Schur complement method with

options 3 and 4, compared with the FV type of restriction operator, especially for the

SPE 10 bottom layer case. Except for AMG, the well function method with FE-based

restriction operator provides the best linear solver strategy.


20 40 60 80 100 120 140 160 180 200 220

20

40

60

Well 1 Well 2 Well 3 Well 4 Well 5

(a) homogeneous case

50 100 150 200

20

40

60−5

0

5

(b) patchy case

50 100 150 200

20

40

60 −5

0

5

(c) layer case

50 100 150 200

20

40

60 −5

0

5

(d) SPE 10 top layer case

50 100 150 200

20

40

60−5

0

5

(e) SPE 10 bottom layer case

Figure 4.3: Well locations and natural logarithm permeability fields. The black linesrepresent the well perforations and the white lines indicate the dual coarse grid bound-aries.


50 100 150 200

20

40

60−0.5

0

0.5

1

1.5


50 100 150 200

20

40

600.4

0.6

0.8

1

1.2

(b) patchy case

50 100 150 200

20

40

60 0.9

0.95

1

(c) layer case

50 100 150 200

20

40

60 0

100

200

300


50 100 150 200

20

40

60−0.5

0

0.5

1

1.5


Figure 4.4: Fine-scale reference pressure solution.


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(b) patchy case

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(c) layer case

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


Figure 4.5: Iteration history of various linear solver options for different permeabilityfields. The dashed lines indicate the use of an FE type of restriction operator; thesolid lines indicate that the restriction operator is based on an FV formulation. Thewell index α =

√kxky.


Well No. type control value of rate or pressure c in well index1 Injector rate 1 12 Producer pressure 1 13 Producer rate 1 14 Injector rate 1 15 Producer rate 1 1

Table 4.2: Well configurations, including well type, well control, and well index.

Complex Geometry

In this section, the flow is driven by eight geometrically complex wells, which can be

rate or pressure-constrained depending on the flow scenario, as described in Table 4.3.

The locations of the eight wells in the domain are shown in Fig. 4.6. Wells 3, 6, and

8 have two branches. Well 3 intersects Wells 2 and 4; and Well 6 intersects Well 7.

The setting of the wells is challenging due to the fact that some wells penetrate the

dual boundaries, which deteriorates the quality of boundary conditions for calculating

basis functions and well basis-functions. In addition, the effects of the well index are

also investigated. Figures 4.8, 4.9, and 4.10 show the iterations of different linear

solver strategies for the five permeability cases with different well indexes where the c

is set to 0.01, 0.1, and 1, respectively. For the scenario with c = 0.01, the magnitude

of the well term is smaller compared with the convection term in the flow equation.

Therefore, the well function method and the Schur complement method have a similar

performance for all the permeability cases. As the well term becomes more significant,

the Schur complement method with option 4 leads to a better performance compared

to other options, and the well function method outperforms the Schur complement

method. In addition, the FE restriction operator gives faster convergence than the FV

restriction operator, especially for the permeability field with channelized structures

such as the SPE 10 bottom layer.

4.4.2 Computational Efficiency

The convergence analysis in the previous section demonstrates that the Schur-4 is

the optimum among Schur complement methods. In this section, we investigate the


20 40 60 80 100 120 140 160 180 200 220

20

40

60

Well 1 Well 2 Well 3 Well 4 Well 5 Well 6 Well 7 Well 8

(a) patchy

50 100 150 200

20

40

60−5

0

5

(b) patchy

50 100 150 200

20

40

60 −5

0

5

(c) layer

50 100 150 200

20

40

60 −5

0

5

(d) SPE 10 top layer

50 100 150 200

20

40

60−5

0

5

(e) SPE 10 bottom layer

Figure 4.6: Well locations and permeability fields. The black lines represent theperforations and the white lines indicate the dual coarse grid boundaries.


50 100 150 200

20

40

60

−4

−2

0

(a) patchy

50 100 150 200

20

40

60−1.5

−1

−0.5

0

0.5

(b) patchy

50 100 150 200

20

40

600.7

0.8

0.9

(c) layer

50 100 150 200

20

40

600

100

200


50 100 150 200

20

40

60−4

−2

0


Figure 4.7: Fine-scale reference pressure solution for the scenario with c = 1.


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(a) homogeneous

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(b) patchy

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(c) layer

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


Figure 4.8: Iteration history of various linear solver options for different permeabilityfields. The dashed lines indicate the use of an FE type of restriction operator; thesolid lines indicate that the restriction operator is based on an FV formulation. Thewell index α = 0.01

√kxky.


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(a) homogeneous

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(b) patchy

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(c) layer

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


Figure 4.9: Iteration history of various linear solver options for different permeabilityfields. The dashed lines indicate the use of an FE type of restriction operator; thesolid lines indicate that the restriction operator is based on an FV formulation. Thewell index α = 0.1

√kxky.


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(a) homogeneous

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(b) patchy

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG

(c) layer

0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


0 20 40 60 80 10010

−10

10−8

10−6

10−4

10−2

100

Iteration number

Rela

tive r

esid

ual

Well function

Schur−1

Schur−2

Schur−3

Schur−4

AMG


Figure 4.10: Iteration history of various linear solver options for different permeabilityfields. The dashed lines indicate the use of an FE type of restriction operator; thesolid lines indicate that the restriction operator is based on an FV formulation. Thewell index α =

√kxky.


Well No. type control value of rate or pressure c in well index1 Injector rate 1 12 Producer rate 1 13 Producer rate 10 14 Producer rate 1 15 Injector pressure 1 16 Producer rate 1 17 Producer rate 1 18 Producer rate 1 1

Table 4.3: Well configurations, including well type, well control and well index.

computational efficiency of this Schur complement method against the well function

method. We consider five sets of log-normally distributed permeability fields with

spherical variograms generated by sequential Gaussian simulations [63]. The variance

and mean of ln(k) are 4 and -1, respectively. For all the test cases, the fine-scale grids

and coarse-scale grids are 64× 64× 64 and 8× 8× 8, respectively. The gridblock size

on the fine-scale is set as ∆x = ∆y = ∆z = 1. The dimensionless correlation lengths

in the x, y, z directions are set as ψx = ψy = ψz = 0.125 for the patchy domain

(shown in Fig. 4.11), and ψx = 0.5, ψy = 0.03, ψz = 0.01 for the layer domains. In

addition, four different orientation angles of 0, 15, 30, and 45 degrees are considered

for the layer domains (shown in Fig. 4.12). Each set has 20 equiprobable realizations.

In the following experiments, GMRES preconditioned by the AMS is employed as the

iterative procedure. The iterative procedures are performed until the reduction in the

relative l2 norm of the residual is five orders of magnitude (i.e., ‖rk‖2/‖r0‖2≤ 10−5).

The difference between the well basis-function method and the Schur complement

method in the setup phase is that the first requires additional computational cost

for calculating the well basis, and the second takes some overhead to compute the

approximate Schur complement. However, the amount of well functions under this

well configurations is only about 2% of the amount of basis functions, and the ap-

proximate Schur complement is cheap to compute. Therefore, we can assume that the

CPU time of the setup phase is comparable between the well function method and the

Schur complement method. Hence, the computational cost in the solution phase (i.e.,

the iterative procedure) is our focus. Figure 4.13 shows that the method with the FE


Inj

Inj

Inj

Inj

Prod

Figure 4.11: Natural logarithm of one realization (out of 20 statistically-the-same) ofpatchy permeability. A five-spot well pattern is considered and each well penetratesall the layers in z direction.

restriction operator converges faster than the one with the FV restriction operator in

terms of both iteration numbers and CPU time. The well function method with the

FE restriction operator is the most computationally efficient for all the test cases.

Moreover, the well basis-function method and the Schur complement method are

tested for the full SPE 10 model and the simplified version with only the top 30 layers.

The fine-scale cells for those two cases are 60× 220× 80 and 60× 220× 30, and the

coarse-scale cells are 12 × 44 × 16 and 12 × 44 × 6, respectively, which correspond

to a coarsening ratio of 5 × 5 × 5. The five-spot well pattern is also used for the

SPE 10 cases, as illustrated in Fig. 4.14. In addition to the well basis-function and

Schur complement methods discussed above, we also investigate strategies with more

ILU smoothing steps, i.e., FE with two ILU steps and three ILU steps. Table 4.4

shows that the FV restriction operator leads to divergence for the full SPE 10 model.

For the top 30 layers, the FV restriction operator converges more slowly than the FE

restriction operator. For the local preconditioner, two steps of ILU achieve the fastest

convergence compared with a single step or three steps of ILU. The best strategy is

the well basis-function method with the FE restriction operator and two steps of the


Inj

Inj

InjInj

Prod

Inj

Inj

Inj

Prod

Inj

Inj

Inj

Prod

Inj

Inj

Inj

ProdInj Inj Inj

Figure 4.12: Natural logarithm of one realization of permeability set with differentlayering angles of 0, 15, 30, and 45, from left to right. For each layering angle,20 realizations are generated for each case. A five-spot well pattern is considered andeach well penetrates all the layers in z direction.

ILU local preconditioner for the SPE 10 model.

Next, we compare the best AMS strategy, i.e., WF(FE-ILUx2), with SAMG for the

SPE 10 test cases with 30 and 80 layers. As shown in Table 4.5, SAMG outperforms

AMS in terms of total CPU time. However, AMS is comparable to SAMG in the

solution phase for the top 30 layers case, and even better than SAMG in the solution

phase for the top 80 layers case. The computational burden of AMS in the setup

phase could be mitigated by parallel constructions of basis functions, which is outside

of the scope of this work.

4.4.3 Scalability Analysis

In practice, there are typically hundreds, or even thousands, of wells operating in

a field. In this section, we investigate the performance of AMS with respect to

the number of wells. A log-normally distributed permeability field with spherical

variograms generated by sequential Gaussian simulations [63] is considered and shown

in Fig. 4.15. The variance and mean of ln(k) are 4 and -1, respectively. The fine-scale

grids and coarse-scale grids are 128× 128× 128 and 16× 16× 16, respectively. The

gridblock size on the fine-scale is set as ∆x = ∆y = ∆z = 1. The dimensionless

correlation lengths in the x, y, and z directions are set as ψx = ψy = ψz = 0.125.

Four scenarios are generated with well numbers of 4, 16, 64, and 256, respectively. All


0

50

100

150

200It

era

tio

n n

um

be

rs


WF−FV

WF−FE

Schur−FV

Schur−FE

SAMG

(a) Iteration numbers

0

2

4

6

8

10

CP

U t

ime

(se

c)


WF−FV

WF−FE

Schur−FV

Schur−FE

SAMG

(b) CPU time for solution phase

Figure 4.13: Average iterations (a) and CPU time (b) of the 20 realizations for patchyand layer domains. The linear solver strategies include the well basis-function methodwith an finite volume restriction operator (WF-FV), an finite element restriction op-erator (WF-FE), the Schur complement method with an finite volume restriction op-erator (Schur-FV) and an finite element restriction operator (Schur-FE), and SAMG.


8

6

4

2

0

−2

−4

−6

InjInj

ProdInjInj

(a) Full SPE 10

8

6

4

2

0

−2

−4

−6

InjInj

Inj InjProd

(b) Top 30 layers of SPE 10

Figure 4.14: Natural logarithm of the full SPE 10 model (a) and the simplified versionwith the top 30 layers (b). A five-spot well pattern is considered and each wellpenetrates all the layers in z direction.

SPE 10 Strategies Iterations Solution CPU

Top 30 layers

WF(FV-ILU) 46 3.00WF(FE-ILU) 34 2.41

WF(FE-ILUx2) 21 2.07WF(FE-ILUx3) 18 2.23Schur(FV-ILU) 54 3.64Schur(FE-ILU) 40 2.95

Schur(FE-ILUx2) 27 2.66Schur(FE-ILUx3) 22 2.70

Full layers

WF(FV-ILU) - -WF(FE-ILU) 60 11.90

WF(FE-ILUx2) 33 9.55WF(FE-ILUx3) 34 12.09Schur(FV-ILU) - -Schur(FE-ILU) 87 18.16

Schur(FE-ILUx2) 60 16.30Schur(FE-ILUx3) 53 17.56

Table 4.4: Iterations and solution phase CPU (sec) of different strategies for the SPE10 model with a coarsening ratio of 5× 5× 5.


SPE 10 Strategies Iterations Setup CPU Solution CPU Total CPU

Top 30 layersAMS 21 7.40 2.07 9.47

SAMG 15 1.89 1.97 3.86

Top 80 layersAMS 33 15.12 9.55 24.67

SAMG 36 5.56 12.76 18.32

Table 4.5: Iterations and solution phase CPU (sec) of AMS and SAMG for the SPE10 test cases.

of the well penetrates all the layers in the z direction and are uniformly distributed

in the x − y plane. Figure 4.16 illustrates the well locations for the scenarios with

4 and 16 wells. Each well has the same flow rate, and the global flow rate for each

scenario is the same. The iterative procedure stops when the relative residual norm

is reduced by five-orders of magnitude. Figure 4.17(a) indicates that the well basis-

function method and SAMG are not sensitive to the increase in the total number of

wells, while the Schur method takes significantly more iterations as the amount of

wells increases. As a consequence, the computational cost of the Schur Complement

method in the solution phase becomes more expensive when a larger number of wells

is considered. Note from Fig. 4.17 that the well basis-function method spends a

comparable amount of time in the solution phase with SAMG, which indicates the

well functions capture the well effects effectively and the FE formulation is quite

efficient. The CPU time in the setup phase of all three solvers did not vary too much

for the different number of wells. The additional cost of the setup phase of the well

basis-function method compared with the one of the AMS algorithm without well

models lies in the construction of the well basis-functions. The well basis-function is

perfectly scalable as demonstrated in Fig. 4.18, where the CPU time of well basis-

function constructions is normalized by the case with 4 wells, and the CPU time

increases linearly with the number of wells. Therefore, we can expect that the well

basis-functions can be efficiently constructed in a parallel setting for the cases with

large numbers of wells.


−6

−4

−2

0

2

4

6

8

10

12

Figure 4.15: Natural logarithm of the patchy permeability field.

(a) 4 wells (b) 16 wells

Figure 4.16: Well locations (shown in black dots) in the top layer of the z directionfor the scenarios with 4 and 16 wells.


4 16 64 2560

20

40

60

80

100

120

140

The amount of wells

Ite

ratio

ns

WF−FE

Schur−FE

SAMG

(a) Iterations

4 16 64 2560

20

40

60

80

100

120

The amount of wells

CP

U t

ime

(se

c)

WF setup phase time

WF solution phase time

Schur setup phase timeSchur solution phase time

SAMG setup phase time

SAMG solution phase time

(b) CPU time (sec)

Figure 4.17: The number of iterations (a) and the CPU time (b) in both setup andsolution phases.


1 4 16 641

4

16

64

Normalized amount of wells

Norm

alized C

PU

for

well functions

CPU for well functions

Linear reference

Figure 4.18: The scalability of well function constructions.

4.5 Summary

In this chapter, we investigated two methods to incorporate well models into the

AMS framework. The first method is to use local well basis functions to capture the

well effects. Well basis functions have the same support as the dual basis functions,

which makes them easy to integrate into the existing data structure of the AMS

framework. This method introduces an additional overhead to calculate well basis-

functions, which increases the computational cost for the setup phase if the number of

wells is large. Since the computational effort of calculating well functions is scalable,

the setup phase is efficient in parallel computation for large-scale problems. The

second method is to first reduce the system into a reservoir part of equations, which

is approximate to the Schur complement of AWW , and then compute basis functions

and construct the coarse-scale operator based on the diagonally approximate Schur

complement. From all the cases we have examined, we found that the well basis-

function method outperforms the Schur complement method, and the FE restriction

operator is more robust and efficient than the FV restriction operator. This makes

the well basis-function method with the FE restriction operator as the best approach

to treat well models in the AMS framework. For some problems, one can achieve


additional efficiency by using multiple local preconditioner steps. However, there is

still a gap in the convergence rate between AMS and SAMG, which means AMS needs

further improvement.

Chapter 5

Conclusions and Future Work

Conclusions

In this dissertation, a general Algebraic Multiscale Solver (AMS) for the pressure

equation is developed. We incorporate the correction function into the AMS frame-

work and show that it can be represented as an independent local stage, which can

be entirely separated from the multiscale stage. As a local preconditioner, the correc-

tion function does help capture the fine scale right-hand-side term and accelerate the

convergence rate. However, the computational cost of the correction function usually

offset the gain in convergence. Therefore, we conclude that the correction function

is not necessary for the AMS framework. We proposed a flexible iterative multistage

multiscale method. In addition, we describe algebraically how AMS can allow recon-

struction of a conservative velocity field after any iteration level, which is crucial for

the hyperbolic transport problem. Then, we systematically tested the performance

of AMS and found that the finite-element restriction scheme is more robust and effi-

cient than the finite-volume scheme. To improve the computational efficiency of the

iterative procedure, the MSFE operator is recommended as a robust preconditioner,

the MSFV operator is employed at the end of the iterative procedure to obtain a con-

servative velocity field. For the finite-element restriction, the reduced local boundary

condition outperforms the linear boundary condition. Therefore, the optimum strat-

egy for AMS is to employ the finite-element restriction scheme, the ILU local pre-

conditioner, and the reduced local boundary condition. The performance of AMS is

113

114 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

quite comparable to the state-of-the-art algebraic multigrid (SAMG) preconditioner

for large-scale problems. Moreover, AMS is amenable to parallel computation, which

results in a powerful solver for large-scale systems.

We also developed a monotone MultiScale Finite Volume (m-MSFV) Method.

The m-MSFV is based on automatic detection of the local interfaces with negative

coarse-scale transmissibilities obtained from the integration of fluxes induced by the

basis functions. Two approaches are developed to fix the non-physical coarse-scale

transmissibility, namely, local TPFA and local linear BC approaches. For the first

approach, a local TPFA method for only critical interfaces is used to calculate a

positive transmissibility and replace the original MPFA stencils in the coarse-scale

system. For the second approach, a linear local boundary condition is used to solve the

basis function only for the dual-coarse cells associated with the critical coarse nodes.

Then, the coarse-scale system is reconstructed and solved. The local TPFA approach

guarantees the monotonicity of the reconstructed fine-scale solution. The local linear

BC can mitigate the level of non-monotonicity. Therefore, a hybrid strategy that

combines both approaches may be effective, whereby the local linear BC approach is

used to reduce the degree of non-monotonicity, and the local TPFA approach is used

to achieve the monotonicity for the regions where the linear BC cannot help. Since

this m-MSFV method only employs a local fix for critical coarse-cell interfaces that

lie in low-permeability regions, the transmissibility values have a small impact on

the flow activity. This helps the m-MSFV solution to be quite accurate with respect

to the fine-scale reference. Moreover, the m-MSFV method is able to optimize the

efficiency-monotonicity tradeoff adaptively. Using the m-MSFV method is expected

to improve the overall efficiency of sequential fully implicit simulations.

Finally, two approaches are investigated to deal with well models in the AMS

framework. Both approaches are easily integrated into a multi-stage iterative scheme,

i.e., a multiscale preconditioner as the global stage and ILU(0) as the local stage. The

difference between these approaches is only seen by the multiscale preconditioner

stage. The first approach is to enrich the multiscale operators by employing well

functions to capture well effects on local domains. The second approach is to con-

struct the multiscale operators based on a diagonally approximate Schur complement

associated with the well part of the equations, which is obtained by eliminating well

CHAPTER 5. CONCLUSIONS AND FUTURE WORK 115

constraint equations from the full system. The basis functions are computed based on

the approximate Schur complement with the same localization assumption as in the

original MSFV method. The coarse-scale system can be formulated in either finite

volume (FV) or finite element (FE) formulations. Numerical simulations of various

test cases for convergence analysis and computational efficiency demonstrate that the

well function method with the FE restriction is the most robust and efficient strat-

egy in the AMS framework. Additional local preconditioner steps can be employed

to achieve more computational efficiency for some challenging problems such as the

SPE 10 test case.

Future Work

Existing multiscale methods are all based on the sequential implicit method, which

has weaker coupling between flow and transport compared with the fully implicit

method. In practice, the fully implicit formulation is widely used for the coupled

systems associated with general-purpose reservoir simulation. Thus, the extension of

AMS for fully implicit systems will be of great interest.

The following two investigations are worthwhile. First, AMS can be used as

a preconditioner for the pressure equations in the two-stage Constrained Pressure

Residual (CPR) approach [49, 50]. Second, AMS could improve the linear solution

strategy for fully implicit models by employing the hierarchical grids for both flow

and transport equations. It would be beneficial if the saturation equations can be

integrated into the coarse-scale system and solved simultaneously with the pressure

equations on the coarse level. To achieve this goal, the prolongation and restriction

operators should be enlarged, since each gridblock contains multiple unknowns in a

fully implicit setting. The algorithm to construct these two operators for the flow

part could be the same as AMS presented in this dissertation. For the transport part,

one could employ the smoothed aggregation operators [72] or the adaptive operators

for multiscale modeling of transport problems developed by Zhou et al. [37]. One

interesting question is whether we can benefit from the mass conservation property

of MSFV methods to accelerate the solution of the fully implicit system. The key

challenge would be how to construct an accurate and efficient operator to prolong the

116 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

coarse-scale saturation solution to the fine-scale.

Another research direction is adapting AMS to generally unstructured grids. Since

AMS only requires fine-scale transmissibility and wirebasket reordering information,

it is suitable for unstructured grids which can be represented as a general graph.

One challenge of extending multiscale methods to unstructured grids is to design

an automated coarse-scale grids generator (or graph partitioner) to divide the entire

domain into a specified number of local domains that effectively capture the fine-scale

heterogeneities.

Nomenclature

AMS Algebraic Multiscale Solver

AMG Algebraic Multigrid

SAMG Algebraic Multigrid Methods for Systems

FE Finite Element

FV Finite Volume

MSFE Multiscale Finite Element

MSFV Multiscale Finite Volume

m-MSFV Monotone Multiscale Finite Volume

BC boundary condition

RBC reduced boundary condition

LBC linear boundary condition

CF Correction Function

ILU Incomplete Lower-Upper factorization

BILU Block Incomplete Lower-Upper factorization

GMRES generalized minimal residual

TPFA two-point flux approximation

117

MPFA multi-point flux approximation

RHS right-hand side

LHS left-hand side

A coefficient matrix

M−1 preconditioner

P prolongation operator

R restriction operator

AC coarse-scale coefficient matrix

RC coarse-scale RHS

δ Kronecker delta

k permeability tensor

λ mobility tensor

µ viscosity

∇ divergence operator

ℵedge the set of edge cells

p pressure

pcorr correction function vector

pci coarse-scale solution at node i

p′ approximate fine-scale solution

pms multiscale solution without correction function

ρ density

118

g gravitational acceleration

q source term

<d d dimensional space

acij coarse-scale coefficients matrix entry

nc the number of primal-coarse grids

nd the number of dual-coarse grids

nf the number of fine-scale grids

φji basis function associated with coarse node i in dual grid j

φ∗j correction function in dual grid j

ΩjD dual-coarse grid j

ΩiC primal-coarse grid i

Ω domain

∂Ω domain boundary

I,E,V interior, edge and vertex

119

120

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