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
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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
CHAPTER 1. INTRODUCTION 5
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
6 CHAPTER 1. INTRODUCTION
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.
CHAPTER 1. INTRODUCTION 7
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
8 CHAPTER 1. INTRODUCTION
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)
CHAPTER 1. INTRODUCTION 9
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
10 CHAPTER 1. INTRODUCTION
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
CHAPTER 1. INTRODUCTION 11
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
12 CHAPTER 1. INTRODUCTION
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.
CHAPTER 1. INTRODUCTION 13
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.
14 CHAPTER 1. INTRODUCTION
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
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
18 CHAPTER 2. AMS FRAMEWORK
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
CHAPTER 2. AMS FRAMEWORK 19
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
20 CHAPTER 2. AMS FRAMEWORK
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
CHAPTER 2. AMS FRAMEWORK 21
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.
22 CHAPTER 2. AMS FRAMEWORK
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
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
CHAPTER 2. AMS FRAMEWORK 23
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)
24 CHAPTER 2. AMS FRAMEWORK
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)
CHAPTER 2. AMS FRAMEWORK 25
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.
26 CHAPTER 2. AMS FRAMEWORK
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
CHAPTER 2. AMS FRAMEWORK 27
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.
28 CHAPTER 2. AMS FRAMEWORK
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.
CHAPTER 2. AMS FRAMEWORK 29
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.
30 CHAPTER 2. AMS FRAMEWORK
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.
CHAPTER 2. AMS FRAMEWORK 31
−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.
32 CHAPTER 2. AMS FRAMEWORK
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,
CHAPTER 2. AMS FRAMEWORK 33
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
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
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.
36 CHAPTER 2. AMS FRAMEWORK
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
CHAPTER 2. AMS FRAMEWORK 37
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.
38 CHAPTER 2. AMS FRAMEWORK
no CF is used. As Fig. 2.15 shows, the MSFE coarse-scale operator outperforms the
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
CHAPTER 2. AMS FRAMEWORK 39
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
Anisotropic factor α
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
Anisotropic factor α
MSFE−ILUMSFE−BILUMSFV−ILUMSFV−BILU
(c) Layered 15o
100
101
102
103
0
2
4
6
8
10
12
14
16
18
20
Anisotropic factor α
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.
40 CHAPTER 2. AMS FRAMEWORK
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.
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,
42 CHAPTER 2. AMS FRAMEWORK
(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.
CHAPTER 2. AMS FRAMEWORK 43
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.
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
44 CHAPTER 2. AMS FRAMEWORK
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].
CHAPTER 2. AMS FRAMEWORK 45
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
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
48 CHAPTER 2. AMS FRAMEWORK
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
CHAPTER 2. AMS FRAMEWORK 49
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
50 CHAPTER 2. AMS FRAMEWORK
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
54 CHAPTER 3. MONOTONE MSFV METHOD
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
CHAPTER 3. MONOTONE MSFV METHOD 55
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
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
CHAPTER 3. MONOTONE MSFV METHOD 57
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
58 CHAPTER 3. MONOTONE MSFV METHOD
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.
CHAPTER 3. MONOTONE MSFV METHOD 59
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.
60 CHAPTER 3. 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
CHAPTER 3. MONOTONE MSFV METHOD 61
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.
62 CHAPTER 3. MONOTONE MSFV METHOD
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.
CHAPTER 3. MONOTONE MSFV METHOD 63
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.
64 CHAPTER 3. MONOTONE MSFV METHOD
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
CHAPTER 3. MONOTONE MSFV METHOD 65
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
66 CHAPTER 3. MONOTONE MSFV METHOD
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.
CHAPTER 3. MONOTONE MSFV METHOD 67
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.
68 CHAPTER 3. MONOTONE MSFV METHOD
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
(a) Fine-scale reference
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.
CHAPTER 3. MONOTONE MSFV METHOD 69
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||
m−MSFV (TPFA) L2−norm
m−MSFV (LBC) L2−norm
m−MSFV (TPFA) L∞−norm
m−MSFV (LBC) L∞−norm
(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||
m−MSFV (TPFA) L2−norm
m−MSFV (LBC) L2−norm
m−MSFV (TPFA) L∞−norm
m−MSFV (LBC) L∞−norm
(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
70 CHAPTER 3. MONOTONE MSFV METHOD
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.
CHAPTER 3. MONOTONE MSFV METHOD 71
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.
72 CHAPTER 3. MONOTONE MSFV METHOD
0 500 1000 1500 2000
0
20
40
60
(a) Fine-scale reference (b) Original MSFV
(c) m-MSFV (TPFA) (d) m-MSFV (LBC)
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
CHAPTER 3. MONOTONE MSFV METHOD 73
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
16
18
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
0.5
1
1.5
2
2.5
3
3.5
4
ε
||e
v||
m−MSFV (TPFA) L2−norm
m−MSFV (LBC) L2−norm
m−MSFV (TPFA) L∞−norm
m−MSFV (LBC) L∞−norm
(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||
m−MSFV (TPFA) L2−norm
m−MSFV (LBC) L2−norm
m−MSFV (TPFA) L∞−norm
m−MSFV (LBC) L∞−norm
(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 %
critical coarse−scale nodes
critical coarse−scale interfaces
critical dual grid edges
(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.
74 CHAPTER 3. MONOTONE MSFV METHOD
0 500 1000 1500 2000
10
20
30
40
50
60 0
0.2
0.4
0.6
0.8
1
(a) Fine-scale reference
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.
(a) Fine-scale reference (b) Original MSFV
(c) m-MSFV (TPFA) (d) m-MSFV (LBC)
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.
CHAPTER 3. MONOTONE MSFV METHOD 75
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||
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 1 1.20
0.5
1
1.5
ε
||e
v||
m−MSFV (TPFA) L2−norm
m−MSFV (LBC) L2−norm
m−MSFV (TPFA) L∞−norm
m−MSFV (LBC) L∞−norm
(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||
m−MSFV (TPFA) L2−norm
m−MSFV (LBC) L2−norm
m−MSFV (TPFA) L∞−norm
m−MSFV (LBC) L∞−norm
(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 %
critical coarse−scale nodes
critical coarse−scale interfaces
critical dual grid edges
(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.
76 CHAPTER 3. MONOTONE MSFV METHOD
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.
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
CHAPTER 3. MONOTONE MSFV METHOD 77
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.
78 CHAPTER 3. MONOTONE MSFV METHOD
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)
82 CHAPTER 4. AMS WITH WELL MODELING
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
CHAPTER 4. AMS WITH WELL MODELING 83
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
84 CHAPTER 4. AMS WITH WELL MODELING
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)
CHAPTER 4. AMS WITH WELL MODELING 85
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.
86 CHAPTER 4. AMS WITH WELL MODELING
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
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)
88 CHAPTER 4. AMS WITH WELL MODELING
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
CHAPTER 4. AMS WITH WELL MODELING 89
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).
90 CHAPTER 4. AMS WITH WELL MODELING
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
CHAPTER 4. AMS WITH WELL MODELING 91
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
92 CHAPTER 4. AMS WITH WELL MODELING
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.
CHAPTER 4. AMS WITH WELL MODELING 93
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.
94 CHAPTER 4. AMS WITH WELL MODELING
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.
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.
CHAPTER 4. AMS WITH WELL MODELING 97
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
98 CHAPTER 4. AMS WITH WELL MODELING
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.
CHAPTER 4. AMS WITH WELL MODELING 99
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
(d) SPE 10 top layer
50 100 150 200
20
40
60−4
−2
0
(e) SPE 10 bottom layer
Figure 4.7: Fine-scale reference pressure solution for the scenario with c = 1.
100 CHAPTER 4. AMS WITH WELL MODELING
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
(d) SPE 10 top 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
(e) SPE 10 bottom layer
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.
CHAPTER 4. AMS WITH WELL MODELING 101
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
(d) SPE 10 top 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
(e) SPE 10 bottom layer
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.
102 CHAPTER 4. AMS WITH WELL MODELING
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
(d) SPE 10 top 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
(e) SPE 10 bottom layer
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.
CHAPTER 4. AMS WITH WELL MODELING 103
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
104 CHAPTER 4. AMS WITH WELL MODELING
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
CHAPTER 4. AMS WITH WELL MODELING 105
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
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.
CHAPTER 4. AMS WITH WELL MODELING 107
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.