9th International Forum on Reservoir Simulation, Abu Dhabi, 9–13 December 2007
Multipoint flux approximation methods for
quadrilateral grids
Ivar Aavatsmark
Centre for Integrated Petroleum Research, University of Bergen, Norway
Abstract
In this paper, we discuss two multipoint flux approximation (MPFA) methods: The O-method and the L-method. The first of these methods is the
classical MPFA method and the most intuitive method. The second method
is fairly new, is less intuitive, but seems to be more robust. Convergence and
monotonicity properties of both methods are discussed. Finally, error estimates when applying the two-point flux approximation (TPFA) method to
skew parallelogram grids are presented. These estimates are based on the flux
expressions of the O-method.
Contents
1 Introduction
2
2 Multiphase flow
3
3 Interaction volume
4
4 The
4.1
4.2
4.3
O-method
6
Derivation of the equations . . . . . . . . . . . . . . . . . . . . . . . . 7
Uniform grid and homogeneous medium . . . . . . . . . . . . . . . . 12
Extension to three dimensions . . . . . . . . . . . . . . . . . . . . . . 14
5 The
5.1
5.2
5.3
5.4
5.5
5.6
L-method
Derivation of equations . . . . . . . . . .
Uniform grid and homogeneous medium
Boundary conditions . . . . . . . . . . .
Extension to three dimensions . . . . . .
Uniform parallelepiped grid . . . . . . .
Local grid refinements . . . . . . . . . .
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16
18
21
24
24
26
29
6 Properties
30
6.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1
2
Multipoint flux approximations
7 Error using two-point flux
32
7.1 Interpretation of two-point flux . . . . . . . . . . . . . . . . . . . . . 33
7.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8 Future works
38
9 Bibliography
38
1
Introduction
Reservoir simulation is for most field cases performed on corner-point grids. However, these grids are usually not orthogonal. Traditional reservoir simulators allow
nonorthogonal grids to be used in the model [60, 56], but the discretization is only
correct if the grid directions are aligned with the principal directions of the permeability tensor K. The principal directions are orthogonal for a symmetric tensor.
In this paper we discuss multipoint flux approximation (MPFA) methods. They
are designed to give a correct discretization of the flow equations for general nonorthogonal grids as well as for general orientation of the principal directions of the
permeability tensor. We limit the discussion to quadrilateral grids. There is a whole
class of MPFA methods for such grids. Here, we only consider the methods known as
the O-method and the L-method for quadrilateral grids in two and three dimensions.
These were introduced and discussed in [2, 4, 5, 6, 8, 11, 14, 15, 18, 29, 30, 44, 53, 66].
Related MPFA methods are discussed in [18, 29, 30, 52], see also [33]. Generalizations to grids with faults and local refinements are discussed in [16, 17, 20], see also
[11, 14].
Important properties for any numerical method are convergence and monotonicity. Convergence proofs of the O-method are given in [13, 42, 43, 68, 69], see
also [34, 49, 59, 67]. For other MPFA methods, convergence proofs do not exist.
However, convergence rates have been tested numerically for the O-method as well
as for other MPFA methods, see [10, 11, 12, 14, 30, 32, 54, 55].
Numerical solutions of elliptic equations may experience spurious oscillations,
even if the solution converges to the correct solution. Also, at no-flow boundaries, spurious extrema may occur. Such oscillations and such no-flow boundary
extrema violate the maximum principle. To avoid such a behavior in the solution,
one therefore needs to establish conditions which ensure that the method satisfies
the maximum principle. A method satisfying a discrete maximum principle is called
monotone. Monotonicity of MPFA methods is discussed in [14, 29, 30, 48, 50, 51, 52].
Techniques to speed up the solution are presented in [1, 21, 22, 23, 24, 25, 40,
45, 46], and hybrid solution techniques are discussed in [9, 35]. The O-method has
been applied to well modeling [19] and to upscaling [31].
The MPFA methods are by no means restricted to quadrilateral grids. The
methods were introduced and discussed for two-dimensional unstructured grids in
[5, 7, 26, 27, 28, 61, 62, 63] and for three-dimensional unstructured grids in [39].
Unstructured grids are not discussed in this paper.
In [41, 58, 64, 65] the MPFA O-method is compared to other local conservative
discretization methods, see also [47].
Introduction
3
This document is organized as follows. First we discuss how MPFA transmissibility coefficients are applied for multiphase flow, and discuss the concept of interaction
volumes which is the basis for all MPFA methods. Then we derive and discuss the
O-method [2, 8, 30] and the L-method [11, 14, 15] for quadrilateral grids in two and
three dimensions. We also discuss convergence and monotonicity properties of the
methods [12, 14, 51]. Finally, we present error estimates when applying the twopoint flux approximation (TPFA) method to skew parallelogram grids [3]. These
estimates are based on the flux expressions of the O-method.
2
Multiphase flow
A control-volume formulation in reservoir simulation involves computing the flux of
a phase α
Z
α
fi = −
(λα K grad uα ) · n dS
(1)
Si
through some surface Si of a control volume. Here, λα is the relative mobility of
phase α, uα is the phase potential and n is the unit normal to the surface. The
permeability tensor K is assumed to be symmetric and positive definite.
Let w = Kn. Since K is positive definite, w points to the same side of the
surface as n does. Conventionally, the relative mobility is evaluated upstream, based
on the sign of −(w · grad uα ). This convention is also adopted here. The evaluation
of the flux is then reduced to the problem of how to calculate the integral
Z
fi = −
(K grad u) · n dS,
(2)
Si
where now the phase dependency has been removed.
For one-dimensional problems the flux (2) may be approximated by a two-point
flux stencil
fi ≈ Ti (u1 − u2 ),
(3)
where Ti is the transmissibility of interface i and u1 and u2 are potential values at
the cell centers of the adjacent cells 1 and 2, see Fig. 1. Introducing the relative
mobility for multiphase flow, the flux (1) is approximated by
fiα ≈ λα (uαi )Ti (u1 − u2 ),
(4)
where λα (uαi ) is evaluated at node 1 if Ti (u1 −u2 ) is positive and at node 2 otherwise.
In the method outlined in this paper, for multidimensional problems the flux (2)
will be approximated by a multipoint flux expression
X
fi ≈
ti,j uj ,
(5)
j∈J
P
where j∈J ti,j = 0. The coefficients ti,j are called transmissibility coefficients. The
quantity uj is the potential value at the center of cell j, see Fig. 2. The set J
depends on the grid. For a two-dimensional quadrilateral grid, J consists of some
of the numbers of the six cells shown in Fig. 2. The presented method is called a
4
Multipoint flux approximations
1
2
4
3
1
2
5
6
Figure 2: Multipoint flux approximation.
Figure 1: Two-point flux approximation.
multipoint flux approximation method or MPFA method for short. There is a whole
class of MPFA methods.
Using the approximation (5), the flux (1) will be approximated by
X
fiα ≈ λα (uαi )
ti,j uαj ,
(6)
j∈J
where λα (uαi ) is evaluated at node 1 if j∈J ti,j uαj is positive and in node 2 otherwise.
While the purpose of this paper is to construct flux expressions of the form (5)
which then may be used in multiphase flow equations, it is convenient to discuss
these flux expressions for the model equation
P
div q = Q,
q = −K grad u,
(7)
where q is the flow density and Q is a source density. We will discuss convergence
and monotonicity for (7) after the derivation of the methods.
3
Interaction volume
In one-dimensional problems, the traditional transmissibility is calculated as a harmonic average between the two neighboring cells 1 and 2. The underlying principle
is the continuity of flux and potential.
Consider Fig. 1. Let the permeability in cell i be ki , and let ∆xi be the length
of the cell. Assume for the moment that the potential u is linear in each cell. Let
ū1 be the value of u at the interface between cell 1 and cell 2. Equating the flux on
each side of the interface gives the equation
−k1
ū1 − u1
ū1 − u2
= k2 1
.
1
∆x1
∆x2
2
2
(8)
Define Ti = ki /∆xi , i = 1, 2. We can solve for ū1 in equation (8) to get
ū1 =
T1 u1 + T2 u2
.
T1 + T2
(9)
Inserting this expression back into, say, the left-hand side of equation (8) gives for
the flux expression
f = −2T1
T1 u1 + T2 u2 − (T1 + T2 )u1
2
=
(u1 − u2 ),
T1 + T2
1/T1 + 1/T2
(10)
Interaction volume
5
P
Q
R
Figure 3: Control volumes (solid) and
interaction volumes (dashed).
Figure 4: Two interaction volumes.
which shows that the transmissibility between the two cells should be approximated
by the harmonic average of the cell transmissibilities. Note that the assumption of
linear potentials in the cells was only used while calculating the transmissibilities.
Once these have been found, only the cell center values ui are used.
To calculate the tranmissibility coefficients in the expression (5), we apply the
same principles as for one-dimensional problems.
Consider the grid shown by the solid lines in Fig. 3. The permeability is assumed
to be constant in each cell (control volume). The potential is evaluated at the center
of each cell. We introduce a dual grid by drawing lines from each cell center to the
midpoints of the cell surface. The dual grid may be extended outside the primary
grid. The cells of the dual grid are termed interaction volumes. The interaction
volumes devide the cell interfaces in two parts for two-dimensional problems and in
four parts for three-dimensional problems. Each part will be termed a subinterface.
MPFA methods are constructed such that the local interaction between the cells
of the interaction volume will determine the transmissibility coefficients for all subinterfaces inside an interaction volume. Having found the transmissibility coefficients
for all subinterfaces, the transmissibility coefficients for the cell interfaces are found
by assembling the contributions from those subinterfaces which constitute a cell interface. This is shown in Fig. 4, where the subinterfaces P Q and QR constitute the
cell interface P R. In three dimensions, four subinterfaces constitute a cell interface.
Thus, the transmissibility coefficients of a cell interface have contributions from
two neighboring interaction volumes in two dimensions and from four neighboring
interaction volumes in three dimensions.
Inside an interaction volume, we apply the same continuity principles as for
one-dimensional problems: Continuity of potential and flux across the subinterfaces
in the interaction volume. We assume that the potential is described by a linear
function in each subcell in the interaction volume. However, it is not possible to
require continuity of flux and potential everywhere at all interfaces of the interaction
volume. For example, in two dimensions, a linear approximation of the potential
in each of the four subcells involved leads to 4 · 3 = 12 degrees of freedom. Flux
continuity at the four subinterfaces gives four conditions. Full potential continuity at
the interfaces gives eight conditions. In addition the linear functions have to honor
the cell center values, and this gives four conditions. In total, there are 16 conditions
and 12 degrees of freedom. This problem may be solved in different ways. Here,
we discuss two methods, the O-method and the L-method, where relaxed continuity
6
Multipoint flux approximations
x3
x̄2
x4
x̄4
x̄1
x2
x̄3
x1
Figure 5: Interaction volume (bounded by the dashed lines).
conditions are applied.
The linear assumption for the potential is only used while calculating the transmissibility coefficients. Afterwards, only the cell center values are used in the discrete
equations.
Note that the transmissibility coefficients only depend on the grid geometry and
the permeability. Hence, the calculation of the transmissibility coefficients may be
done in advance as in a preprocessor.
4
The O-method
In this section the equations for the MPFA O-method in two dimensions are derived.
Consider the four quadrilateral cells with a common vertex in Fig. 5. The cells have
cell centers xk , and the edges have midpoints x̄i . The points are enumerated locally
as shown in Fig. 5. Lines are drawn between the cell centers and the midpoints of
the edges (shown as dashed lines in Fig. 5). These lines bound an area around each
vertex which is called an interaction volume. Hence, the interaction volume in Fig.
5 is the polygon with corners x1 x̄1 x2 x̄4 x4 x̄2 x3 x̄3 .
Within the interaction volume, there are four half cell edges. Below, it will
be shown how to determine the flux through these half edges from the interaction
between the four cells. When the fluxes through the four half edges in an interaction
volume around a vertex are determined, the procedure may be repeated for the
interaction volumes of the other vertices. In this way, the flux through all the half
edges in a grid will be determined. When the fluxes through the two half edges of
an entire edge are known, they may be added to form an expression for the flux
through the entire edge.
This procedure also holds for the half edges at the boundary of a domain if the
boundary conditions are given as homogeneous Neumann conditions. Outside the
active cells a strip of artificial cells is put with vanishing permeability. The same
procedure as described above for the interaction volumes around the vertices at the
boundary, then gives the flux through the half edges at the boundary. More general
boundary conditions are discussed in [32].
We now show how the fluxes through the four half edges in an interaction volume
may be determined. In each of the four cells of the interaction volume, the potential
u is expressed as a linear function. The value of the potential in each cell center
determines one of the coefficients in each cell for these linear functions. The linear
function determines the flux through the half edges of the cell and the potential at the
half edges. We require that the fluxes through the half edges in an interaction volume
The O-method
7
n2 ν (k)
2
ν1
x2
ν3
x̄2
x3
(k)
ν1
x1
x̄1
n1
xk
ν2
Figure 7: Normal vectors in cell k.
Figure 6: Triangle with edge normals
ν i.
are continuous, and that the potentials at the midpoints of the edges are continuous.
This yields eight equations for the determination of the unknown coefficients of the
linear functions in the cells.
Since the curve connecting the cell centers with the continuity points (this curve
is identical to the boundary of the interaction volume) constitutes a stylized “O”,
this method is called the O-method.
Every linear function is described by three coefficients in two dimensions, but one
of them is already determined by the potential value at the cell center. Altogether
there are therefore eight unknown coefficients for the linear functions. They are
determined by the eight continuity equations. Note that the continuity principles
used here, are exactly the same as the principles used to derive the classical two-point
flux formula.
Every cell is shared among four different interaction volumes. The representations of linear functions for the potential in a cell, may vary from one interaction
volume to another. This does not cause any difficulties, since the linear functions
are only used to determine an expression for the flux. In the resulting difference
equations, only the potential value of the cell centers appears.
4.1
Derivation of the equations
For each interaction volume, the linear functions in each cell may be determined in
the following way. On a triangle with corners xi , i = 1, 2, 3, any linear function may
be described by
3
X
u(x) =
ui φi (x).
(11)
i=1
Here, ui is the value of u(x) at vertex i, and φi (x) is the linear basis function defined
by φi (xj ) = δi,j . The gradient is easily calculated to be
grad φi = −
1
ν i,
2F
(12)
where F is the area of the triangle, and ν i is the outer normal vector of the edge
located opposite of vertex i, see Fig. 6. The length of ν i equals the length of the
8
Multipoint flux approximations
edge to which it is normal. For these normal vectors the following relation holds
3
X
ν i = 0.
(13)
i=1
Thus, the gradient expression of the potential in the triangle may be written in the
form
3
1 X
1 grad u = −
ui ν i = −
(u2 − u1 )ν 2 + (u3 − u1 )ν 3 .
(14)
2F i=1
2F
Now consider the grid cell in Fig. 7. The grid cell has index k, and its center is
xk . Using local indices, the midpoints on the edges are denoted x̄1 and x̄2 , and the
associated normals on the connection lines between the cell center and the midpoints
(k)
(k)
of the edges are denoted ν 2 and ν 1 , see Fig. 7. Later, it will be suitable to let
(k)
the vectors ν i point in the direction of increasing global cell indices. In this cell
we therefore reverse the direction of these vectors. Using the formula (14) on the
triangle xk x̄1 x̄2 , yields
grad u =
1 (k)
(k)
ν 1 (ū1 − uk ) + ν 2 (ū2 − uk ) ,
2Fk
(15)
where ūi = u(x̄i ), i = 1, 2, and uk = u(xk ). Each of the edges can be associated
with a global direction, defined through the unit normal ni . It is convenient to also
let ni point in the direction of increasing global cell indices. The flux through half
(k)
edge i as seen form cell k is denoted fi , and may now be determined from the
gradient of the potential in the cell. For the fluxes associated with the cell in Fig.
7, the following expression appears
"
#
(k)
Γ1 nT
f1
1
K k grad u
(k) = −
Γ2 nT
f2
2
h
i ū − u 1 Γ1 nT
1
k
(k)
(k)
1
=−
,
Kk ν1 ν2
ū2 − uk
2Fk Γ2 nT
2
(16)
where Γi is the length of half edge i. By defining the matrix
"
#
(k)
(k)
h
i
T
T
1 Γ1 nT
1
Γ1 n1 K k ν 1 Γ1 n1 K k ν 2
(k)
(k)
1
Gk =
ν2 =
T Kk ν1
(k)
(k) ,
Γ
n
2Fk 2 2
2Fk Γ2 nT
Γ2 nT
2 Kkν2
2 Kkν1
(17)
equation (16) may be written in the form
"
#
(k)
f1
ū1 − uk
.
(k) = −Gk
ū2 − uk
f2
(18)
Now consider the interaction volume in Fig. 8. Here, the normal vectors of the edges
are denoted n1 , n2 , n3 , and n4 . With these normal vectors, the matrix Gk is defined
The O-method
9
(4)
ν2
(3)
ν2
(2)
ν2
(1)
ν2
x3
x̄3
x1
n4
x̄4
n1
(1)
ν1
x4
x̄2
n2
n3
(3)
ν1
(4)
ν1
(2)
ν1
x̄1
x2
Figure 8: Normal vectors with local numbering in an interaction volume.
for all the four cells. Thus,
#
"
(1)
ū1 − u1
f1
,
(1) = −G1
ū3 − u1
f3
#
"
(3)
ū2 − u3
f2
,
(3) = −G3
u3 − ū3
f3
#
(2)
u2 − ū1
f1
,
(2) = −G2
ū4 − u2
f4
#
"
(4)
u4 − ū2
f2
.
(4) = −G4
u4 − ū4
f4
"
(19)
(20)
Here, as before, uk = u(xk ) and ūi = u(x̄i ), see Fig. 8. Compared to cell 1, the
(4)
(4)
(3)
(2)
directions of ν 1 , ν 2 , ν 1 , and ν 2 have been reversed (see Fig. 8). The differences
ū1 − u2 , ū3 − u3 , ū2 − u4 , and ū4 − u4 therefore appear in the expressions (19) and
(20) with opposite signs.
The continuity conditions for the fluxes now yield
(1)
(2)
(4)
(3)
(3)
(1)
(2)
(4)
f1 = f1 = f1 ,
f2 = f2 = f2 ,
(21)
f3 = f3 = f3 ,
f4 = f4 = f4 .
Using the expressions (19) and (20), these equations become
(1)
(1)
(2)
(2)
(3)
(3)
f1 = −g1,1 (ū1 − u1 ) − g1,2 (ū3 − u1 ) = g1,1 (ū1 − u2 ) − g1,2 (ū4 − u2 ),
(4)
(4)
f2 = g1,1 (ū2 − u4 ) + g1,2 (ū4 − u4 ) = −g1,1 (ū2 − u3 ) + g1,2 (ū3 − u3 ),
(3)
(3)
(1)
(1)
f3 = −g2,1 (ū2 − u3 ) + g2,2 (ū3 − u3 ) = −g2,1 (ū1 − u1 ) − g2,2 (ū3 − u1 ),
(2)
(2)
(4)
(22)
(4)
f4 = g2,1 (ū1 − u2 ) − g2,2 (ū4 − u2 ) = g2,1 (ū2 − u4 ) + g2,2 (ū4 − u4 ).
The equations (22) contain the edge values ū1 , ū2 , ū3 , and ū4 . Tacitly we have here
used the same expression for the edge value of the cells at each side of an edge, and
thereby implicitly demanded continuity for the potential at the points x̄1 , x̄2 , x̄3
and x̄4 .
10
Multipoint flux approximations
If the matrix Gk is diagonal for all cell indices k, the grid is called K-orthogonal.
The system of equations (22) is then no longer coupled, and the flux through the
edges can be determined by eliminating the edge values ūi . This gives a two-point
flux expression. If the grid is not K-orthogonal, the edge values ūi may still be
eliminated in each interaction volume. The procedure is described in the following
way:
The fluxes of the system of equations (22) can be collected in the vector f defined
by f = [f1 , f2 , f3 , f4 ]T . The system of equations further contains the potential
values of the cell centers u = [u1 , u2 , u3 , u4 ]T and the potential values at the
midpoints of the cell edges v = [ū1 , ū2 , ū3 , ū4 ]T . The expressions on each side of
the left equality sign of (22) can therefore be written in the form
f = Cv + F u.
(23)
The expressions on each side of the right equality sign in the system of equations
(22) may after a reorganization be written in the form
Av = Bu.
(24)
Hence, v may be eliminated by solving equation (24) with respect to v and substituting v = A−1 Bu into (23). This gives the flux expression
f = T u,
(25)
T = CA−1 B + F .
(26)
where
The entries of the matrix T are called transmissibility coefficients. Equation (25)
gives the flux through the half edges expressed by the potential values at the cell
centers of an interaction volume.
Having determined the flux expression for all half edges, the two flux expressions
of the two half edges which constitute an edge, can be added. This is shown in Fig.
9, where the cells 1, 2, 3, and 4 constitute one interaction volume, and the cells 1,
2, 5, and 6 constitute the other. The flux stencil of the edge between cell 1 and 2
will therefore consist of the six cells of the figure.
If two neighboring cells have vanishing permeability, the corresponding row in
the matrix A vanishes, and hence, the matrix A is singular. Because there is no
need to determine the flux across the interfaces of cells with vanishing permeability,
the system may be reduced, and this will remove the singularity. However, it is more
favorable to retain the system of unknowns and redefine the matrix A such that it
becomes nonsingular. This is easily done by setting the diagonal elements of the
vanishing rows in the matrix A equal to 1. The new system of equations has the
same transmissibility coefficients as the reduced system for the interfaces between
cells with nonvanishing permeability .
If cell k in Fig. 7 is a parallelogram, the expression for the matrix Gk , equation
(17), is simplified. For a parallelogram-shaped cell with index k, we denote the
(k)
normal vectors of the edges with ai , i = 1, 2. These have lengths equal to the length
(k)
of the edges. The normal vectors are shown in Fig. 10. Obviously, Γi ni = ai /2
The O-method
11
(k)
a2
(k)
ν2
x6
x5
x2
x1
x3
x4
Figure 9: Flux stencil.
(k)
(k)
ν1
(k)
a1
xk
Figure 10: Normal vectors in parallelogram cells.
(k)
and ν i = ai /2. Further, Fk = Vk /8, where Vk is the area of cell k. It follows that
for a parallelogram-shaped cell,
i
iT
h
1 h (k)
(k)
(k)
(k)
Gk =
(27)
K k a1 a2 .
a2
a
Vk 1
(k) (k) Letting J k = a1 , a2 , it follows that Vk = |det J k |, and equation (27) becomes
Gk =
1
JT
KkJ k.
|det J k | k
(28)
Hence, for a parallelogram cell, the tensor Gk is symmetric. Equation (28) is a
congruence transformation. Thus, the tensor Gk , as given by (27), is symmetric
and positive definite if and only if K k has these properties. If the tensor Gk is
diagonal for all cell indices k, i.e., if
T
(k)
(k)
i 6= j,
(29)
ai
K k aj = 0,
then the grid is K-orthogonal.
In the matrix Gk , it is sometimes useful to perform a splitting, such that anisotropy and grid skewness appear in one matrix and the mesh distances in another. If
(k)
(k)
∆ηk is the length of a1 and ∆ξk is the length of a2 , then for a parallelogram grid,
Gk =
1
Dk H k Dk ,
∆ξk ∆ηk
(30)
where
T
1
n1 n2 K k n1 n2
det[n1 , n2 ]
"
#
T
nT
K
n
n
K
n
1
k
1
k
2
1
1
,
=
T
det[n1 , n2 ] nT
K
n
n
k 1
2
2 K k n2
Hk =
(31)
12
Multipoint flux approximations
and
D k = diag(∆ηk , ∆ξk ).
(32)
(k)
Here, ni is the unit normal vector which is parallel with ai , see Fig. 10. If H k is
diagonal, the grid is K-orthogonal.
4.2
Uniform grid and homogeneous medium
In this section, we assume that the medium is homogeneous and that the grid is a
uniform parallelogram grid. The tensor Gk is then identical for all cells. It is further
symmetric und positive definite. Let the matrix elements of Gk be
"
# T
T
a
Ka
a
Ka
1
1
2
a c
1
1
Gk =
=
,
(33)
T
c b
V aT
1 Ka2 a2 Ka2
where V = det[a1 , a2 ]. The system of equations (22) then reads
f1
f2
f3
f4
= −a(ū1 − u1 ) − c(ū3 − u1 ) = a(ū1 − u2 ) − c(ū4 − u2 ),
= a(ū2 − u4 ) + c(ū4 − u4 ) = −a(ū2 − u3 ) + c(ū3 − u3 ),
= −c(ū2 − u3 ) + b(ū3 − u3 ) = −c(ū1 − u1 ) − b(ū3 − u1 ),
= c(ū1 − u2 ) − b(ū4 − u2 ) = c(ū2 − u4 ) + b(ū4 − u4 ).
These equations can be

2a 0
 0 2a
A=
 c −c
−c c
written in the form (24), where



c −c
a+c a−c
0
0
 0
−c c 
0
a − c a + c

,
.
B
=
b + c
2b 0 
0
b−c
0 
0 2b
0
b−c
0
b+c
(34)
(35)
Here, the matrix A has a block structure with 2 × 2 submatrices. It is therefore
easy to determine an analytical expression for the inverse of A. A simple calculation
yields


2b − c2 /a
−c2 /a
−c
c
 −c2 /a
1
2b − c2 /a
c
−c 


A−1 =
(36)
2

2
−c
c
2a − c /b
−c2 /b 
4(ab − c )
c
−c
−c2 /b
2a − c2 /b
and


2 + c/a 2 − c/a −c/a
c/a
1  c/a
−c/a 2 − c/a 2 + c/a
.
A−1 B = 
c/b 
4  2 + c/b −c/b 2 − c/b
c/b
2 − c/b −c/b 2 + c/b
The fluxes through the
the form (23), where

−a 0 −c
 0
a
0
C=
 0 −c b
c
0
0
(37)
four half edges in the interaction volume can be written in

0
c
,
0
−b


a+c
0
0
0
 0
0
0
−(a + c)
.
F =

 0
0
−(b − c)
0
0
b−c
0
0
(38)
The O-method
13
j
4
1
3
5
6
j
2
i
Figure 11: Flux stencil in i direction.
2
3
3 1
4
3
2
6
4
1
5
i
Figure 12: Flux stencil in j direction.
4
2
5
4
3
6
1
8
2
7
9
1
Figure 13: Local numbering of cells
and edges in an interaction volume.
Figure 14: Local numbering in the cell
stencil.
For the fluxes, the expression f = T u follows, where
T = CA−1 B + F


2a + c − c2 /b −2a + c + c2 /b
−c + c2 /b
−c − c2 /b
(39)
1  c + c2 /b
c − c2 /b
2a − c − c2 /b −2a − c + c2 /b
.
= 
2
2
2
2


−c + c /a
−2b + c + c /a
−c − c /a
4 2b + c − c /a
2
2
2
c + c /a
2b − c − c /a
c − c /a
−2b − c + c2 /a
Notice that the row sums of the matrix (39) are zero. Since in this example the same
matrix T = {ti,j } appears for all interaction volumes in the grid, the flux across each
entire edge can be determined through a combination of the elements of the matrix
(39). The local numbering of the edges and the cells of the interaction volume is
shown in Fig. 13, while the local cell numbering in the flux stencils is shown in Figs.
11 and 12.
The expression for the flux through an entire edge in i direction (see Fig. 11)
is constructed from the flux across half edge 1 in the upper interaction volume and
from the flux across half edge 2 in the lower interaction volume. The flux in the i
direction reads
fi = (t1,1 + t2,3 )u1 + (t1,2 + t2,4 )u2 + t1,4 u3 + t1,3 u4 + t2,1 u5 + t2,2 u6
c2
c
c
c
c
= a−
(u1 − u2 ) +
1+
(u5 − u3 ) −
1−
(u4 − u6 ).
2b
4
b
4
b
(40)
The expression for the flux through an entire edge in j direction (see Fig. 12) is
constructed from the flux across half edge 3 in the right interaction volume and from
the flux across half edge 4 in the left interaction volume. The flux in j direction
14
Multipoint flux approximations
f2
f1
f3
f4
Figure 15: Flux through the cell edge
of a cell.
Figure 16: Three-dimensional interaction volume (bold lines) with 8 subcells and 12 interfaces (thick lines).
reads
fj = (t3,1 + t4,2 )u1 + (t3,3 + t4,4 )u2 + t4,3 u3 + t4,1 u4 + t3,2 u5 + t3,4 u6
c
c
c
c
c2
(u1 − u2 ) −
1−
(u5 − u3 ) +
1+
(u4 − u6 ).
= b−
2a
4
a
4
a
(41)
Finally, we may construct the difference equation of a cell. The difference equation is
given for the equation (7), where the fluxes are shown in Fig. 15. If the neighboring
cells are numbered as shown in Fig. 14, the expression for the outflux out of the cell
becomes
9
X
f1 + f2 − f3 − f4 =
mi (ui − u1 ),
(42)
i=2
where
m2 = −a +
c2
,
d
m3 = −
c
c
1+
,
2
d
m4 = −b +
c2
,
d
m5 =
c
c
1−
2
d
(43)
and
mi+4 = mi ,
i = 2, 3, 4, 5.
(44)
Here, d = 2ab/(a + b). The equation (42) gives one row in the matrix of coefficients
of the system of difference equations for the equation (7).
4.3
Extension to three dimensions
The principles of the MPFA O-method carry over to corner-point grids in three
dimensions in a straightforward manner. In three dimensions, an interaction volume
contains eight subcells and twelve interfaces, see Fig. 16. The linear functions in
the eight cells are described by 32 coefficients. Eight of these are determined by
the potential values at the cell centers. The rest of them are determined by the
two continuity conditions at each of the twelve interfaces: the flux is required to be
continuous at the interfaces, and the potential is required to be continuous at the
interface midpoints.
The generalization of the equations of section 4.1 to three dimensions is straightforward. Equation (15) generalizes to
3
1 X (k)
grad u =
ν (ūi − uk ).
Vk i=1 i
(45)
The O-method
15
x4
x3
x̄3
x̄2
xk
β
x̄1
x1
Figure 17: Hexahedral cell with tetrahedron spanned by the cell center xk
and the midpoints x̄i , i = 1, 2, 3, of
three surfaces.
α
x2
Figure 18: Bilinear coordinates of a
surface.
Here, Vk is the triple product
Vk = (x̄1 − xk )(x̄3 − xk )(x̄3 − xk ) ,
(46)
which is equal to six times the volume of the tetrahedron xk x̄1 x̄2 x̄3 of Fig. 17
(assuming the points x̄i , i = 1, 2, 3, form a right-handed system as seen from the
tetrahedron vertex xk ), and
(k)
νi
= (x̄j − xk ) × (x̄k − xk ),
i, j, k cyclic.
(47)
(k)
The vector ν i is the inner normal vector to the tetrahedron surface spanned by the
points xk , x̄j , x̄k , having length equal to twice the area of this surface (assuming
the points x̄i , i = 1, 2, 3, describe a right handed system).
When expressing the fluxes as in (16), we have taken into consideration that a
three-dimensional cell described by its eight corners, generally does not have planar
surfaces. The unit normal vector ni of an interface is therefore not a constant across
the interface. This can be accounted for by integrating the normal vector over the
interface of the subcell in question.
To determine this integral, consider the interface in Fig. 18. Let the corners of
the interface be xk , k = 1, . . . , 4, and let the surface which is spanned by these
points, be described by the bilinear function
x(α, β) = β αx4 + (1 − α)x3 + (1 − β) αx2 + (1 − α)x1 ,
(48)
where (α, β) ∈ [0, 1] × [0, 1]. The corners xi , i = 1,R2, 3, 4, may not lie in the same
plane. We now calculate the surface integral n̂ = S n dσ, where S is the surface
given by α ≤ 21 and β ≤ 21 . This surface is the lower left quarter in Fig. 18. A
straightforward integration gives
Z
Z 1/2 Z 1/2 ∂x ∂x
n̂ =
n dσ =
×
dα dβ
∂α ∂β
0
0
S
Z 1/2 Z 1/2
=
β(x4 − x3 ) + (1 − β)(x2 − x1 )
0
0
(49)
× α(x4 − x2 ) + (1 − α)(x3 − x1 ) dα dβ
1h
=
9(x2 − x1 ) × (x3 − x1 ) + 3(x2 − x1 ) × (x4 − x2 )
64
i
+ 3(x4 − x3 ) × (x3 − x1 ) + (x4 − x3 ) × (x4 − x2 ) .
16
Multipoint flux approximations
Figure 19: Flux stencil of an interface
(shaded) for the O-method in 3D.
Figure 20: Cell stencil of a cell
(shaded) for the O-method in 3D.
Figure 21:
Interaction
(dashed) of the O-method.
Figure 22: L-shaped coupling between
three cell centers and triangle-shaped
domain containing two half edges.
volume
The quantity n̂ is the integrated normal vector over the interface of the subcell at
the corner x1 . The vector n̂ has length equal to the area of the subcell interface.
In the three-dimensional O-method, the flux stencil contains 18 cells (see Fig.
19), and the cell stencil contains 27 cells (see Fig. 20).
5
The L-method
The L-method uses the same concepts as the O-method. To compute the transmissibility coefficients, the same dual grid defining the interaction volumes is constructed,
see Fig. 21. However, for the L-method, we do not define continuity points at the
interfaces, but require full potential continuity at the interfaces inside the interaction volumes. Linear potential functions are still applied in each subcell. For the 2D
case, there are then three conditions per half edge (two for potential continuity and
one for flux continuity). Not counting the cell center value, two degrees of freedom
per subcell are available. Hence, the half-edge conditions can be met if three subcells
(with six degrees of freedom) and two half edges (with six conditions) are applied to
compute the transmissibility coefficients. A triangle-shaped area containing three
subcells and two half edges is shown in Fig. 22. Since the curve connecting the three
cell centers constitutes a stylized “L”, this method is called the L-method. As the
O-method, the L-method is exact for linear potential fields on arbitrary grids.
Two triangle-shaped areas constitute an interaction volume. Obviously, the triangles can be arranged in two ways, as shown in Fig. 23. When computing the
transmissibility coefficients in each triangle by the L-method, we obtain transmissibility coefficients of each half edge in the interaction volume. As in all MPFA
methods, the transmissibility coefficients of the entire edges may then be obtained
The L-method
17
(a)
(b)
Figure 23: Two triangles covering the interaction volume. (a) Short diagonal. (b)
Long diagonal.
Figure 24: Cell stencil containing seven points.
by assembling the transmissibility coefficients of the half edges.
Since the triangles can be arranged in different ways, we need a criterion for how
to place the triangles in which the transmissibility coefficients are computed. For a
uniform parallelogram grid in a homogeneous, isotropic medium, the obvious way to
place the triangles is such that the diagonal edge of the triangles is the shorter of the
two possible alternatives, as in Fig. 23(a). With this choice, the flux stencil of each
entire edge contains only the four closest neighboring cells of the edge. Likewise,
the cell stencil contains only the center cell and its six closest neighboring cells, see
Fig. 24.
Figure 25 shows the equivalent case for a uniform rectangular grid in a homogeneous, anisotropic medium. The cell stencil has diagonal contributions only in the
direction of the highest permeability.
However, in the general case, it is not so obvious how to place the triangles.
This is solved in the following way. For each half edge in the interaction volume,
we consider the two triangles which may cover this half edge. In Fig. 26, the two
triangles which may be used for the computation of the transmissibility coefficients
of the lower half edge (shown in bold) are drawn. For this half edge, we number
the cells as shown in Fig. 26, and the transmissibility coefficients of the lower half
edge are denoted tj , where j is the cell index. The transmissibility coefficients are
computed for each triangle I, I = 1, 2. To decide which triangle to use, we consider
the quantities
sI = |t1 − t2 |
(50)
for each of the two triangles. We choose the set of transmissibility coefficients which
yields the smallest sI value.
The reason for applying this selection criterion is the following. When s1 < s2 ,
we can for moderate skewness expect that for triangleP
1 it holds that sgn t1 = sgn t3
while for triangle 2 it holds that sgn t1 = sgn t4 . Since j tj = 0, this means that for
triangle 1, the two transmissibility coefficients to the left of the lower half edge will
balance the transmissibility coefficient to the right of the half edge. But for triangle
2, the transmissibility coefficients along the diagonal (cells 1 and 4) will balance
18
Multipoint flux approximations
(a)
(b)
Figure 25: Permeability and cell stencil in rectangular grid. (a) Ellipse xT K −1 x = 1
of the permeability tensor. (b) Seven-point cell stencil.
3
4
3
4
1
2
1
2
(a)
(b)
Figure 26: Possible triangles for lower half edge (shown in bold). (a) Triangle 1. (b)
Triangle 2.
the transmissibility coefficient of the mid cell (cell 2). The sign property of the
transmissibility coefficients of triangle 1 is therefore likely to give the most stable
discretization. In Section 5.2 we show the above sign property for homogeneous
media and uniform parallelogram grids.
If, in Fig. 26, s1 < s2 , the triangle of Fig. 26(a) is chosen for the computation
of the transmissibilities of the lower half edge. Although the left half edge is also
contained in this triangle, this does not mean that this triangle is chosen for the
computation of the transmissibilities of the left half edge. For each half edge, the
same procedure as above is followed.
However, in case of a uniform parallelogram grid on a homogeneous medium,
the above procedure always chooses only two triangles per interaction volume. If
the medium of Fig. 23 is homogeneous and isotropic, the triangles of Fig. 23(a) are
chosen by the procedure. This is shown in Section 5.2.
5.1
Derivation of equations
In this section, we derive the equations for calculating the transmissibility coefficients
of the L-method.
Let half edge i be part of the boundary of cell j. The flux through this half edge
is approximated by
fi = −n̂T
(51)
i K j ∇Uj .
Here, n̂i is the normal vector of the half edge, having length equal to the length
of the half edge. The tensor K j is the permeability of cell j, and Uj is a linear
approximation of the potential in cell j.
To obtain a linear approximation of the potential U in a cell, consider Fig. 27.
Here, we omit the cell index j. Let x0 be the location of the cell center, and let x̄k ,
The L-method
19
x2
ν3
x̄2
x̄3
x̄1
x3
x0
Figure 27: Triangle spanned by a cell
center x0 and continuity points x̄k ,
k = 1, 2.
ν6
ν5
x̄2
ν4
ν 7 x̄1
ν2
ν1
x1
Figure 28: “Triangle” spanned by the
cell centers xj , j = 1, 2, 3. Also shown
are the edge midpoints x̄i , i = 1, 2,
the common corner of the three cells
x̄3 and the vectors ν k , k = 1, . . . , 7.
k = 1, 2, be points on the edges. Define u0 = U (x0 ) and ūk = U (x̄k ), k = 1, 2. Then
ū1 − u0
X∇U =
,
(52)
ū2 − u0
where
(x̄1 − x0 )T
X=
.
(x̄2 − x0 )T
To get expressions for the inverse of the matrix X, we introduce the matrix
0 1
R=
.
−1 0
(53)
(54)
The matrix R satisfies R−1 = RT = −R. When a vector is multiplied by the matrix
R, the vector is rotated an angle −π/2. Hence, the vectors ν 1 and ν 2 , defined by
ν 1 = R(x̄2 − x0 ),
ν 2 = −R(x̄1 − x0 ),
(55)
are normal to the distance vectors x̄2 − x0 and x̄1 − x0 , pointing into the triangle
spanned by x0 , x̄1 and x̄2 (assuming that the points form a right-handed system).
Now, X[ν 1 , ν 2 ] = T I, where
T = det X = (x̄1 − x0 )T R(x̄2 − x0 ) = ν T
1 Rν 2 .
It follows that
X −1 =
and
1
ν1 ν2
T
2
1X
∇U =
ν k (ūk − u0 ).
T k=1
(56)
(57)
(58)
Equation (58) is identical to (15), which we used in the derivation of the O-method.
We may now apply the formulae (51) and (58) for each cell in the triangles of an
20
Multipoint flux approximations
interaction volume. Consider the “triangle” shown in Fig. 28. Here, xj , j = 1, 2, 3,
are cell centers, and x̄i , i = 1, 2, are midpoints of the edges. In the following, we
construct expressions for the fluxes through the half edges x̄1 x̄3 and x̄2 x̄3 . For this
purpose we utilize the normal vectors ν k , k = 1, . . . , 7, whose lengths are equal to
the lengths of the lines they are normal to. These vectors are shown in Fig. 28, but
with only correct orientation due to visualization purposes. Further, let n̂i , i = 1, 2,
be normal vectors of the half edges with midpoints x̄i , i = 1, 2, having length equal
to the length of the half edges they are normal to. The direction of the vectors n̂i ,
i = 1, 2, also define the sign of the flux through these edges.
Utilizing the formulae (51) and (58), the fluxes through the half edges of cell 1
are given by
1 T
n̂ K 1 ν 1 (ū1 − u1 ) −
T1 1
1
f2 = − n̂T
K 1 ν 1 (ū1 − u1 ) −
T1 2
f1 = −
1 T
n̂ K 1 ν 2 (ū2 − u1 ),
T1 1
1 T
n̂ K 1 ν 2 (ū2 − u1 ),
T1 2
(59)
(60)
where T1 = ν T
1 Rν 2 . From (58) it also follows that the value ū3 = U1 (x̄3 ) is given
by
ū3 = u1 + (x̄3 − x1 )T ∇U1 = u1 − (Rν 7 )T ∇U1
1 T
1
ν Rν 2 (ū2 − u1 ).
= u1 + ν T
7 Rν 1 (ū1 − u1 ) +
T1
T1 7
(61)
We may also express the flux through half-edge 1 as seen from cell 2. Utilizing (61),
this flux expression reads
1
1 T
n̂1 K 2 ν 3 (ū3 − u2 ) − n̂T
K 2 ν 4 (ū1 − u2 )
T2
T2 1
1 T
1 T
1 T
= − n̂1 K 2 ν 3 u1 − u2 + ν 7 Rν 1 (ū1 − u1 ) + ν 7 Rν 2 (ū2 − u1 )
T2
T1
T1
1
K 2 ν 4 (ū1 − u2 ),
− n̂T
T2 1
f1 = −
(62)
where T2 = ν T
3 Rν 4 . Finally, we may express the flux through half edge 2 as seen
from cell 3. Utilizing (61), this flux expression reads
1 T
1
n̂2 K 3 ν 5 (ū2 − u3 ) − n̂T
K 3 ν 6 (ū3 − u3 )
T3
T3 2
1
= − n̂T
K 3 ν 5 (ū2 − u3 )
(63)
T3 2
1
1 T
1
− n̂T
ν 7 Rν 1 (ū1 − u1 ) + ν T
Rν 2 (ū2 − u1 ) ,
2 K 3 ν 6 u1 − u3 +
T3
T1
T1 7
f2 = −
where T3 = ν T
5 Rν 6 . To simplify the expressions in (59) (60) (62) and (63), we
introduce the quantities
ωijk =
1 T
n̂ K j ν k ,
Tj i
χijk =
1 T
ν Rν k .
Tj i
(64)
The L-method
21
Further, we introduce the vector quantities
 
u1
f1

,
u = u2  ,
f=
f2
u3
ū
v= 1 .
ū2
(65)
Then the equations (59) and (60) can be written
f = Cv + Du,
where
−ω111 −ω112
C=
,
−ω211 −ω212
ω111 + ω112 0 0
D=
.
ω211 + ω212 0 0
(66)
(67)
We now make use of the fact that the half-edge fluxes should be continuous, i.e., we
require that the expressions for f1 ((59) and (62)) are equal and that the expressions
for f2 ((60) and (63)) are equal. This requirement yields the equation
where
Av = Bu,
(68)
ω111 − ω124 − ω123 χ711
ω112 − ω123 χ712
A=
ω211 − ω236 χ711
ω212 − ω235 − ω236 χ712
(69)
and
ω111 + ω112 + ω123 (1 − χ711 − χ712 ) −ω123 − ω124
0
B=
.
ω211 + ω212 + ω236 (1 − χ711 − χ712 )
0
−ω235 − ω236
(70)
Since the equations (66) and (68) contain the vector v, whose components are the
edge values of the potential, we have tacitly required that the potential is continuous
across the half edges x̄1 x̄3 and x̄2 x̄3 . If K 1 = O, the fluxes vanish, and the halfedge transmissibilities of the triangle are zero. For K 1 6= O, the matrix (69) may be
assumed to be nonsingular, and equation (68) then gives v = A−1 Bu. Substituted
into equation (66) this yields
f = T u,
(71)
where
T = CA−1 B + D.
(72)
Equation (72) contains the transmissibility coefficients of the half edges in the triangle of Fig. 28.
In the special case when the grid is K-orthogonal, the bilinear forms n̂T
1 K 1ν 2,
T
T
T
n̂1 K 2 ν 3 , n̂2 K 1 ν 1 and n̂2 K 3 ν 6 vanish, and the method degenerates to the conventional two-point flux-approximation method.
5.2
Uniform grid and homogeneous medium
To illustrate the L-method with the selection parameter (50) in the case of a uniform
parallelogram grid in a homogeneous medium, we introduce the same quantities a,
b and c, as defined in (33):
T 1 a c
a1 a2 K a1 a2 ,
=
(73)
c b
V
22
Multipoint flux approximations
2
3
3
4
1
2
4
1
Figure 29: Local numbering and positive edge direction in an interaction volume.
where V is the area of each cell and ai , i = 1, 2, are the normal vectors of the
parallelogram sides, having length equal to the side on which
√ it is normal, Since K
is positive definite, it follows that a > 0, b > 0 and |c| < ab. Cells and interfaces
are numbered locally as in Fig. 29. The transmissibility coefficients of the half edges
may then be computed as in Section 5.1. Choosing the triangles as in Fig. 23(a),
the transmissibility matrix


a −(a − c)
0
−c
1 c
0
a−c
−a
R

T R = {τi,j
}= 
(74)
0
−(b − c) −c 
2 b
c
b−c
0
−b
results, while choosing the triangles as in Fig. 23(b), the transmissibility matrix


a + c −a −c
0
1 0
c
a −(a + c)
L

T L = {τi,j
}= 


0
2 b + c −c −b
0
b
c −(b + c)
(75)
results.
The selection parameter (50) is given by |2a ± c| for the first two rows and |2b ± c|
for the last two rows of the matrices T R and T L . If c < 0, then |2a + c| < 2a − c
and |2b + c| < 2b − c and hence, the criterion chooses the triangles yielding T L ,
and the half-edge transmissibilities will be given by the elements of T L . Likewise, if
c > 0, then |2a − c| < 2a + c and |2b − c| < 2b + c and hence, the criterion chooses
the triangles yielding T R , and the half-edge transmissibilities will be given by the
elements of T R .
However, the sign of the half-edge transmissibilities will not always be as expected. Consider e.g. the transmissibilities of the lower half edge, i.e., the elements
of the first row in (74) and (75). Without loss of generality, let c < 0. For moderate
skewness, i.e., for |c| < a, the signs of the transmissibilities are as shown in Fig.
30. In Fig. 30(a) both transmissibilities on one side of the lower edge have the same
sign, and therefore this candidate is the natural choice. For large skewness, i.e., for
|c| > a, the signs of the transmissibilities are as shown in Fig. 31. Now there is
no choice for which both transmissibilities on one side of the lower edge have the
same sign, and one may argue that there is no natural choice. However, also for
this case, the selection criterion chooses the candidate coming from T L , i.e., the
transmissibilities of Fig. 31(a).
The L-method
23
+
+−
+
+−
(a)
(b)
Figure 30: Sign of the elements of the first row of T L (75) and T R (74). −a < c < 0.
(a) Elements of T L . (b) Elements of T R .
+
−−
+
+−
(a)
(b)
Figure 31: Sign of the elements of the first row of T L (75) and T R (74). c < −a.
(a) Elements of T L . (b) Elements of T R .
In [51], it was shown that a necessary condition for the M-matrix property of ninepoint MPFA schemes is that c ≤ min{a, b}. It was also noted that this inequality is a
practical bound for general monotonicity. Therefore, the unfortunate sign property
mentioned above should not be seen as a defect of the L-method. Rather, it should
be seen as inevitable for all nine-point MPFA schemes.
Numbering the cells as in Fig. 32, the flux from cell 1 to cell 2, denoted f1 , and
the flux from cell 1 to cell 4, denoted f2 , are
f1 = (τ1,1 + τ2,3 )u1 + (τ1,2 + τ2,4 )u2 + τ1,4 u3 + τ1,3 u4 + τ2,1 u8 + τ2,2 u9
= (a − 21 |c|)(u1 − u2 ) + 41 (|c| + c)(u8 − u3 ) + 14 (|c| − c)(u4 − u9 ),
f2 = (τ3,1 + τ4,3 )u1 + (τ3,3 + τ4,4 )u4 + τ4,3 u5 + τ4,1 u6 + τ3,2 u2 + τ3,4 u3
= (b − 12 |c|)(u1 − u4 ) + 41 (|c| + c)(u6 − u3 ) + 14 (|c| − c)(u2 − u5 ).
(76)
These expressions are valid for both signs of c, i.e., they are valid whether the
half-edge transmissibilities are given by T R or T L . Finally, we may construct the
difference equation of a cell. The difference equation is given for the equation (7).
The cell stencil of cell 1 reads, confer Fig. 15,
9
X
mk uk = f1 + f2 − f3 − f4 .
(77)
k=1
Substituting equation (76) into (77) yields the following expressions for the elements
of the cell stencil,
m2
m3
m4
m5
= m6
= m7
= m8
= m9
P
m1 = − 9k=2 mk
= −t1,1 + t2,2 − t2,3
=
t1,3 + t2,3
= −t1,3 + t1,4 − t2,1
=
−t1,4 − t2,2
=
= −a + |c| ,
= − 12 (|c| + c),
= −b + |c| ,
= − 12 (|c| − c),
(78)
2(t1,1 + t2,1 ) = 2(a + b − |c|).
Note that each of the flux stencils in (76) are four-point stencils and that the cell
stencil in (78) is a seven-point stencil. Also note that the matrix of coefficients given
24
Multipoint flux approximations
5
4 f2
1
6
7
3
f1 2
8
9
Figure 32: The fluxes f1 and f2 with the local cell numbering.
by the elements (78) is an M-matrix for
|c| ≤ min{a, b}.
5.3
(79)
Boundary conditions
No-flow boundary conditions are easily handled by adding a strip of ghost cells
around the domain, using zero permeability in the ghost cells. If other Neumann
boundary conditions apply, a special set of equations has to be derived for the
interaction volumes covering the boundary. This is straightforward.
Ghost cells may also be applied for Dirichlet boundary conditions, using the
potential values at the cell centers outside the boundary. However, if conditions
on the boundary are to be applied, there seems to be no outright solution for this.
For every second boundary half edge it seems difficult to determine the flux in a
consistent way. This is a weakness of the method, but it can easily be solved by
applying the O-method in the interaction volumes covering the boundary [32].
In reservoir simulation, no-flow boundary conditions are prevailing.
5.4
Extension to three dimensions
The principles of the L-method carry over to corner-point grids in three dimensions
in a straightforward manner. As for the O-method, an interaction volume contains
eight subcells and twelve interfaces, see Fig. 16. For the L-method, full continuity is
required at the subinterfaces, and the potential is approximated by linear functions
in each subcell. Hence, we now have four conditions per subinterface (three for
potential continuity and one for flux continuity). Each subcell has three degrees
of freedom, not counting the cell center value. Hence, applying four subcells (with
twelve degrees of freedom) and three subinterfaces (with twelve conditions), the
associated transmissibility coefficients can be computed.
In 2D the transmissibility coefficients are computed by considering the two “triangles” covering a subinterface, see Fig. 26. In 3D the transmissibility coefficients
are computed by considering four-cell groups covering a subinterface. We term such
a group of four cells an L-stencil, and Fig. 33 shows the four L-stencils of the lower
back subinterface. They are here shown for a uniform parallelepiped grid, but the
grid may be a conventional corner point-grid.
The L-method
25
Figure 33: The four L-stencils of the lower back subinterface in an interaction volume. Top left A. Top right B. Bottom left C. Bottom right D.
Figure 34: Flux stencil of an interface
(shaded) for the L-method in 3D.
Figure 35: Cell stencil of a cell
(shaded) for the L-method in 3D.
The four cells of an L-stencil consists of the two cells sharing the subinterface
and two more cells. These additional two cells are selected such that they have
one corner in common with the subinterface and such that they do not share any
interface. Thereby, each L-stencil has four cells with three interfaces, see Fig. 33.
For each of the twelve subinterfaces in an interaction volume (see Fig. 16), there are
four L-stencils.
The generalization of the equations of section 5.1 is straightforward. We apply
the selection parameter sI (equation (50)) for each of the four L-stencils, and choose
the set of transmissibility coefficients associated with the smallest sI value. For uniform parallelepipeds on homogeneous media (discussed below), this criterion always
yields transmissibilities that are of equal sign on the opposite sides of the interface.
Having found the transmissibility coefficients for each subinterface in an interaction volume, the expressions are added up, to yield flux expressions for entire
interfaces. The flux stencil may then contain between six and ten cells, depending
on the L-stencils which are chosen for each of the subinterfaces. Likewise the cell
stencil may contain between 13 and 19 cells. The stencils containing a maximum
number of cells are shown in Figs. 34 and 35.
26
Multipoint flux approximations
7
a3
5
8
6
a2
1
a1
Figure 36: The normal vectors a1 , a2
and a3 .
5.5
3 2
4
Figure 37: Local cell numbering in the
interaction volume.
Uniform parallelepiped grid
In this section, we assume a uniform parallelepiped grid on a homogeneous medium.
For this case, we can derive analytical expressions for the stencils of the L-method.
To derive these stencils, it is convenient to introduce the quantities a, b, c, d, e, f ,
defined by


a d e
d b f  = 1 a1 a2 a3 T K a1 a2 a3
(80)
V
e f c
Here, V is the volume of a parallelepiped cell, and a1 , a2 and a3 are the cell normals
in i, j and k direction, respectively, the length of each normal being equal to the area
of the parallelepiped face to which it is normal. These normal vectors are shown in
Fig. 36. Note that the quantities c and d do not have the same meaning here as in
the sections 4.2 and 5.2.
Since the right-hand side of equation (80) is a congruence transformation, it
follows that a > 0, b > 0 and c > 0.
Figure 37 shows an interaction volume in the grid. The volume contains eight
interacting cells. They are numbered locally as shown in Fig. 37. These eight cells
determine the fluxes through the twelve subinterfaces of the interaction volume.
For each subinterface, there are four possible L-stencils for computation of the flux
through the subinterface. For the lower back interface of the interaction volume, the
four possible L-stencils are shown in Fig. 33.
We index the four L-stencils of a subinterface with A, B, C and D. The flux
expressions of each of the L-stencils in Fig. 33 are given by
fA
fB
fC
fD
=
=
=
=
1
[du1
4
1
[du2
4
1
[du2
4
1
[du1
4
+ (a − d + e)u3 − au4 − eu7 )],
+ au3 − (a + d − e)u4 − eu8 )],
+ (a + e)u3 − (a + d)u4 − eu7 )],
+ (a − d)u3 − (a − e)u4 − eu8 )].
(81)
We now compute the selection indicator (50) for the fluxes (81). Since the flux
expressions use the local numbering of Fig. 37, the indicator here reads sI = |t3 − t4 |,
The L-method
i direction
27
j direction
k direction
Figure 38: Cells of the flux stencils for the case d < 0, e < 0 and f < 0. Left f1 .
Center f2 . Right f3 .
I = A, B, C, D. For each of the above flux expressions, the indicator sI has the value
sA
sB
sC
sD
=
=
=
=
1
(2a
4
1
(2a
4
1
(2a
4
1
(2a
4
− d + e),
+ d − e),
+ d + e),
− d − e).
(82)
Here, we would like the coefficients of cells 1 and 7 to be positive and the coefficients
of cells 2 and 8 to be negative. This is accomplished if we choose the L-stencil with
the smallest indicator s. In the same way, we may compute the flux expressions
for the four L-stencils of each subinterface in the interaction volume. Using this
criterion, the correct L-stencil is chosen. Having found the correct L-stencil for
each of the twelve subinterfaces, we may compute the flux expressions of the entire
interfaces. Let f1 , f2 and f3 be the flux of an entire interface in i, j and k direction,
respectively. The derivation gives for the flux in i direction
f1 = a − 21 (|d| + |e|) (ui,j,k − ui+1,j,k )
+ 41 (|d| + d) (ui,j−1,k − ui+1,j+1,k ) + 14 (|d| − d) (ui,j+1,k − ui+1,j−1,k )
(83)
1
1
+ 4 (|e| + e) (ui,j,k−1 − ui+1,j,k+1 ) + 4 (|e| − e) (ui,j,k+1 − ui+1,j,k−1 ) ,
for the flux in j direction
f2 = b − 21 (|d| + |f |) (ui,j,k − ui,j+1,k )
+ 41 (|d| + d) (ui−1,j,k − ui+1,j+1,k ) + 14 (|d| − d) (ui+1,j,k − ui−1,j+1,k )
+ 41 (|f | + f ) (ui,j,k−1 − ui,j+1,k+1 ) + 14 (|f | − f ) (ui,j,k+1 − ui,j+1,k−1 ) ,
and for the flux in k direction
f3 = c − 21 (|e| + |f |) (ui,j,k − ui,j,k+1 )
+ 14 (|e| + e) (ui−1,j,k − ui+1,j,k+1 ) + 14 (|e| − e) (ui+1,j,k − ui−1,j,k+1 )
+ 14 (|f | + f ) (ui,j−1,k − ui,j+1,k+1 ) + 14 (|f | − f ) (ui,j+1,k − ui,j−1,k+1 ) .
(84)
(85)
Each of the flux stencils is a ten-point stencil, containing the cells of Fig. 34. However, since one of the expressions |d| + d or |d| − d vanishes, and similar with the
28
Multipoint flux approximations
Figure 39: Cell stencil for the case d < 0, e < 0 and f < 0.
expressions for e and f , each of the flux stencils reduces to a six-point stencil. The
cells of the flux stencils for the case d < 0, e < 0, f < 0 are shown in Fig. 38.
The flux stencils determine the cell stencil. For the equation (7), the cell stencil
reads
Z
−
div(K grad u) dτ ≈ 2(a + b + c − |d| − |e| − |f |)ui,j,k
Ωi,j,k
− (a − |d| − |e|) (ui−1,j,k + ui+1,j,k )
− (b − |d| − |f |) (ui,j−1,k + ui,j+1,k )
− (c − |e| − |f |) (ui,j,k−1 + ui,j,k+1 )
− 12 (|d| + d) (ui−1,j−1,k + ui+1,j+1,k )
− 21 (|d| − d) (ui−1,j+1,k + ui+1,j−1,k )
− 21 (|e| + e) (ui−1,j,k−1 + ui+1,j,k+1 )
− 21 (|e| − e) (ui+1,j,k−1 + ui−1,j,k+1 )
− 21 (|f | + f ) (ui,j−1,k−1 + ui,j+1,k+1 )
− 12 (|f | − f ) (ui,j−1,k+1 + ui,j+1,k−1 ) .
(86)
As equation (86) shows, the cell stencil is the 19-point stencil of Fig. 35. As for the
flux stencils, however, due to vanishing coefficients, it reduces to a 13-point stencil.
The cells of the cell stencil for the case d < 0, e < 0, f < 0 are shown in Fig. 39.
Equation (86) shows that the matrix of coefficients is an M-matrix if
|d| + |e| ≤ a,
|d| + |f | ≤ b,
|e| + |f | ≤ c.
(87)
From equation (80) it follows that d = e = f = 0 for K-orthogonal grids. Hence,
the inequalities (87) will be satisfied for grids which are close to K-orthogonal.
Monotonicity of MPFA methods in 3D is a topic with ongoing research. However,
the 2D results derived in [51] indicate that the inequalities (87) are optimal, in
the sense that no other 27-point control-volume method has a larger monotonicity
region.
The L-method
1
29
2
1
3
1
2
Figure 40: Interaction volume used by
the O-method for the case of LGR.
5.6
3
1
2
Figure 41: Support of the flux stencil for subinterface 1 used by the Lmethod.
Local grid refinements
In reservoir simulation, local grid refinements (LGR) are often added to structured
grids to improve accuracy near wells and other features. Interfaces between cells at
different resolutions introduce unstructured connectivity that provides a challenging
test for MPFA methods.
We consider a two-dimensional Cartesian grid with local grid refinement using a
refinement ratio of two as shown in Figs. 40 and 41. This grid contains interaction
volumes involving either three or four cells. Those with four cells can be handled in
a straightforward manner using the L-method or the O-method.
The treatment of interaction volumes involving only three cells is more challenging because the normal vectors for subinterfaces 1 and 2 are parallel as shown
in Fig. 40. For the O-method, the flux expressions are computed simultaneously
using a linear approximation for potential within each cell. Hence, the flux across
subinterfaces 1 and 2 will be identical, even if the permeability varies between cells
2 and 3. If cell 2 or cell 3 has low permeability, then flow will be restricted on all
subinterfaces in the interaction volume.
The flexibility in the choice of flux stencil for the L-method circumvents this
limitation of the O-method. Figure 41 shows the “triangle” which has to be selected
for the computation of the flux across subinterface 1. This choice does not restrict
the flow through subinterface 1 by a low permeability in cell 3. However, if the
permeability is homogeneous within the interaction volume, the L-method and the
O-method yield identical results.
The situation is even more critical in three-dimensional Cartesian grids. Consider
a case with local grid refinement as shown in Fig. 42. An interaction volume with
center at the hanging node between the coarse cell and the fine cells contains subcells
of these five cells. In the O-method, we have three degrees of freedom per cell (not
counting the cell center value) and two continuity conditions for each of the eight
interfaces. This yields 3·5 = 15 degrees of freedom, but 2·8 = 16 conditions. Hence,
the O-method cannot be utilized for this grid.
There is no such restriction for the L-method. An L-stencil consisting of cells 1,
2, 3 and 4 will yield flux expressions for the subinterfaces I, II and III in Fig. 42.
For this L-stencil, there are no continuity conditions on the interface between cell 1
and cell 3. This is the only L-stencil applicable for subinterface I.
A discussion of convergence for local grid refinement of the L-method can be
found in [11, 14].
30
Multipoint flux approximations
4
1
3
III
I II
2
Figure 42: Three-dimensional Cartesian grid
with local grid refinement.
6
Figure 43: Rough grid.
Properties
In this section we discuss convergence and monotonicity properties of the O- and
L-method.
6.1
Convergence
The convergence property for the equation (7) has been discussed in several papers
[10, 11, 12, 13, 14, 42, 43, 54, 55, 68, 69]. In reservoir simulation, the medium is
often strongly heterogeneous on all simulation scales. Therefore, when grids are
refined, one cannot expect the grids to be more and more smooth. To account for
this difficulty, the concept of rough grids has been introduced. Rough grids are grids
which are not smooth on any refinement level. To generate rough grids, one usually
generates a fairly smooth grid with some typical mesh distance h. This grid is then
disturbed, such that all corner points are moved some random distance of O(h), see
Fig. 43.
On rough grids, the L2 convergence rates for potential and flow density in (7)
using the O- and the L-method are O(h2 ) and O(h), respectively, as long as the
solution is smooth. However, due to the heterogeneities in the medium, one cannot
expect the solution to be smooth. If the potential belongs
to the Sobolev
space
H 1+α , α > 0, the convergence rates become O hmin{2,2α} and O hmin{1,α} for the
potential u and the flow density q, respectively.
The above convergence rates are only valid for moderate anisotropy ratios or grid
geometries. If the grid aspect ratio is larger than, say, 1:10 for an isotropic medium,
the convergence rate is reduced. For increasing aspect ratios, the convergence rates
drop, and eventually convergence is lost. The reduction in convergence rate is larger
for the O-method than for the L-method. At an aspect ratio of 1:100, the L-method
still converges fairly well. The O-method, however, may lose convergence for difficult
grids at this aspect ratio.
Properties
31
This means that the L-method converges fairly well for typical reservoir simulation grids, whereas the O-method requires nicer grids to guarantee convergence.
In regions where the solution is not smooth, it is favorable to apply local grid
refinement. For logically Cartesian grids in three dimensions, this can only be accomplished by the L-method. Hence, also for this purpose, the L-method seems to
be the best choice.
6.2
Monotonicity
Numerical solutions of (7) may experience spurious oscillations, even if the solution
converges to the correct solution. Also, at no-flow boundaries, spurious extrema may
occur. Such oscillations and such no-flow boundary extrema violate the maximum
principle [36, 37, 38, 57]. To avoid such a behavior in the solution, one therefore
needs to establish conditions which ensure that the method satisfies the maximum
principle. A method satisfying a discrete maximum principle is called monotone. A
thorough discussion of monotonicity for general MPFA methods is given in [51].
If the matrix of coefficients for the numerical solution of (7) is an M-matrix,
the method is monotone. The inequalities (79) and (87) are the conditions for the
M-matrix property of the L-method on uniform grids and homogeneous media in
two and three dimensions, respectively. Although the M-matrix property is only
sufficient for monotonicity, numerical tests indicate that these inequalities are also
necessary for monotonicity of the L-method [51].
It is much harder to establish monotonicity conditions for the O-method. On
uniform grids and homogeneous media in two dimensions (see equations (43) and
(44)), the O-method only yields an M-matrix if c = 0 or if a = b = |c|. In the first
case, the grid is K-orthogonal, which means that the method degenerates to a twopoint flux method. In the second case, the problem degenerates to a one-dimensional
problem in the direction of one of the grid diagonals. These cases are therefore
not very interesting. It is, however, possible to establish sufficient monotonicity
criteria, without requiring that the M-matrix property is fulfilled. Such criteria
are given in [51], and they show that for uniform grids in homogeneous media, the
two-dimensional O-method is monotone if
c2
2ab
|c| 2 −
<
.
(88)
ab
a+b
In the three-dimensional case, the conditions for monotonicity of the O-method has
not yet been established.
The monotonicity regions of the L- and the O-method for two-dimensional uniform grids in homogeneous media are compared in Fig. 44, for the case a ≤ b. The
diagrams show that both methods are monotone if the grid is K-orthogonal (c = 0).
However, if |c| is large (i.e., the grid is very skew, or if the principal axes of the permeability tensor strongly deviates from the grid directions), then monotonicity is
lost for the O-method whereas the L-method may remain monotone. Both methods
are nonmonotone if
√
min{a, b} < |c| < ab.
(89)
This is not a defect of the methods. No nine-point control-volume method can be
expected to be monotone in the parameter region (89), see [51].
32
Multipoint flux approximations
a/b
a/b
1
O
1
L
0.8
0.8
0.6
0.6
0.4
|c| =
√
0.4
ab
0.2
0
0
O
L
|c| =
√
ab
0.2
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
|c| /b
(a)
0.6
0.8
1
|c| /b
(b)
Figure 44: Monotonicity regions of the O- and L-method in two-dimensional uniform
grids on homogenous media.
√ The monotone regions are above the curves in question.
The elliptic bound is |c| = ab. (a) Curves calculated with the inequalities (79) and
(88). (b) Curves for which a discrete maximum principle holds, see [51].
In the nonmonotone parameter region, spurious oscillations may occur and spurious extrema may occur on no-flow boundaries. Typically, oscillations are much
larger for the O-method than for the L-method. On the other hand, no-flow boundary extrema are more striking for the L-method than for the O-method [14].
Neither the O-method nor the L-method should be used outside their monotonicity regions. The above inequalities only give monotonicity regions for uniform
grids in homogeneous media. However, in [51] local conditions for monotonicity
in two-dimensional quadrilateral grids and heterogeneous media are given. They
may be used to check if a method is monotone on a given grid. However, in the
three-dimensional case, no such conditions are available (except for the M-matrix
conditions (87)).
7
Error using two-point flux
Two-point flux stencils [60] are widely used in control-volume discretizations of the
equation (7). Two-point flux approximations (TPFA) are physically intuitive, and
the discretization of the equation (7) results in a positive definite M-matrix. Hence,
discretization with two-point flux stencils always yields convergence to a nice-looking
solution. However, for non-K-orthogonal grids, TPFA gives an error in the solution
which does not vanish as the grids are refined.
In the MPFA O-method, the flux stencil reduces to a two-point flux stencil if
and only if the grid is K-orthogonal. Therefore, any two-point flux stencil applied
to a skew grid may be interpreted as a multipoint flux stencil using a permeability
which leaves the grid K-orthogonal. It follows that the solution which the TPFA
method converges to, is the solution of the equation (7) with a different permeability tensor K, and that this permeability tensor may be found by interpreting
the TPFA method as a K-orthogonal MPFA O-method. The permeability of the
Error using two-point flux
33
n2 ν 2
θ
θ
η
ξ
ν1
n1
∆η
∆ξ
Figure 45: Parallelogram with normal vectors n1 and n2 and tangential vectors ν 1
and ν 2 .
MPFA method is then the permeability in the problem of the converged solution,
whereas the permeability of the TPFA method is the apparent permeability in the
problem which the user thinks he is solving. Below, we show how to find the true
permeability for skew parallelogram grid cells, given the apparent permeability used
in the TPFA approximation. A more thorough discussion may be found in [3], see
also [70].
7.1
Interpretation of two-point flux
The flux through the edges of a parallelogram cell is determined by the geometric
quantities shown in Fig. 45. Here, ν 1 is a unit vector pointing in the positive ξdirection, and ν 2 is a unit vector pointing in the positive η-direction. Further, n1
is a unit vector normal to the edge on which ξ is constant, and n2 is a unit vector
normal to the edge on which η is constant. The positive direction of the vectors n1
and n2 is determined such that ni · ν i = cos θ > 0 for i = 1, 2, where θ is the angle
between ni and ν i , i = 1, 2. If ν 1 is rotated a positive angle from n1 , the angle θ
is positive. The vectors ν 1 and n2 are orthogonal, and so are the vectors n1 and
ν 2 . The quantities ∆ξ and ∆η are the edge lengths in ξ-direction and η-direction,
respectively.
The Eclipse reservoir simulator [60] uses with the Newtran keyword a two-point
flux formula. With the above quantities, the flux through the right and the top
edges of a cell is given by
2 det[n1 , n2 ] λ1 (∆η)2 ∆pξ
f=
.
∆ξ∆η
λ2 (∆ξ)2 ∆pη
(90)
Here, λ1 and λ2 are the user-specified permeabilities in the ξ- and the η-direction,
respectively. Further, ∆pξ and ∆pη are the pressure drops in ξ- and η-direction, measured from the cell center to the midpoint of the right and the top edge, respectively.
The determinant of the matrix [n1 , n2 ] satisfies
det[n1 , n2 ] = n1 · ν 1 = n2 · ν 2 = cos θ.
(91)
34
Multipoint flux approximations
e2
n2 ν 2
θ
φ
e1
φ
θ
ν1
n1
Figure 46: Vectors ni , ν i and ei , i = 1, 2. The vector ν 1 is rotated an angle θ from
n1 , and the vector e1 is rotated an angle φ from ν 1 .
In the MPFA O-method, the flux through the right and the top edges is given by
2
∆pξ
f=
DHD
,
(92)
∆pη
∆ξ∆η
where the matrices H and D are given by (31) and (32):
"
#
T
nT
Kn
n
Kn
1
1
2
1
1
,
H=
T
T
det[n1 , n2 ] n2 Kn1 n2 Kn2
D = diag(∆η, ∆ξ).
(93)
(94)
By requiring that the flux expressions (90) and (92) are equal, the permeability
tensor K may be calculated from the user-specified permeabilities λ1 and λ2 .
Equating the expressions (90) and (92) yields the equation
H = det[n1 , n2 ] diag(λ1 , λ2 ),
(95)
i.e.,
2
nT
1 Kn1 = (det[n1 , n2 ]) λ1 ,
2
nT
2 Kn2 = (det[n1 , n2 ]) λ2 ,
(96)
nT
1 Kn2 = 0.
The last equation in the system (96) is the condition for K-orthogonality of the
grid. Only those permeability tensors which satisfy this equation, allow a consistent
two-point flux approximation. The equations (96) show that when K is a solution
of these equations, then λi = kKni k2 / cos θ, i = 1, 2.
The system of equations (95) or (96) yields an interpretation of the quantities
λ1 and λ2 as the cell’s apparent permeabilities. By using the flux expression (90),
the user is actually simulating on a medium with the cell permeability tensor K,
determined by (95) or (96).
We now show how the permeability tensor K is determined by the apparent
permeabilities λ1 and λ2 . Let the eigenvalues of K be k1 and k2 with associated
unit eigenvectors e1 and e2 , respectively. The eigenvector e1 is rotated an angle φ
from the vector ν 1 , see Fig. 46. Since K is symmetric, the vectors e1 and e2 are
Error using two-point flux
35
orthogonal. In the coordinate system of the eigenvectors, the permeability tensor is
given by K e1 e2 = diag(k1 , k2 ), and the edge unit normal vectors are
cos(θ + φ)
=
,
− sin(θ + φ)
(n1 )e1 e2
(n2 )e1 e2
sin φ
=
.
cos φ
(97)
Hence, in the coordinate system of the eigenvectors, the system of equations (96)
reads
k1 cos2 (θ + φ) + k2 sin2 (θ + φ) = λ1 cos2 θ,
k1 sin2 φ + k2 cos2 φ = λ2 cos2 θ,
k1 cos(θ + φ) sin φ − k2 sin(θ + φ) cos φ = 0.
(98)
To solve the system of equations (98), it is suitable to introduce the following new
quantities: The anisotropy ratio
k1
κ= ,
(99)
k2
the apparent anisotropy ratio
µ=
λ1
,
λ2
(100)
and the ratio between the mean permeability and the mean apparent permeability
ρ=
k1 k2
λ1 λ2
1/2
.
(101)
With these quantities, the system of equations (98) may be written
ρ κ1/2 cos2 (θ + φ) + κ−1/2 sin2 (θ + φ) = µ1/2 cos2 θ,
ρ κ1/2 sin2 φ + κ−1/2 cos2 φ = µ−1/2 cos2 θ,
κ1/2 cos(θ + φ) sin φ − κ−1/2 sin(θ + φ) cos φ = 0.
(102)
(103)
(104)
The system of equations (102), (103) and (104) is solved as follows. From equation
(104) the anisotropy ratio κ may be expressed,
κ=
sin(θ + φ) cos φ
.
cos(θ + φ) sin φ
(105)
Dividing equation (102) by equation (103) yields
κ cos2 (θ + φ) + sin2 (θ + φ)
= µ.
κ sin2 φ + cos2 φ
(106)
36
Multipoint flux approximations
Substituting expression (105) into equation (106) gives
sin(θ + φ) cos φ
cos2 (θ + φ) + sin2 (θ + φ)
cos(θ + φ) sin φ
µ=
sin(θ + φ) cos φ 2
sin φ + cos2 φ
cos(θ + φ) sin φ
=
sin(θ + φ) cos2 (θ + φ) cos φ + sin2 (θ + φ) cos(θ + φ) sin φ
sin(θ + φ) sin2 φ cos φ + cos(θ + φ) sin φ cos2 φ
=
sin(θ + φ) cos(θ + φ)
sin φ cos φ
=
sin 2(θ + φ)
.
sin 2φ
(107)
Hence,
sin 2θ cos 2φ + cos 2θ sin 2φ = µ sin 2φ,
(108)
i.e.,
tan 2φ =
sin 2θ
.
µ − cos 2θ
(109)
Finally, the ratio ρ is determined from equation (103),
cos2 θ
ρ=
κ sin2 φ + cos2 φ
1/2
κ
.
µ
(110)
Using the expression (105) for κ and the expression (107) for µ, one finds
1/2
sin(θ + φ) cos φ
 cos(θ + φ) sin φ 
cos2 θ


ρ=
 sin(θ + φ) cos(θ + φ) 
sin(θ + φ) cos φ 2
2
sin φ + cos φ
cos(θ + φ) sin φ
sin φ cos φ
1/2
cos(θ + φ) cos2 θ
cos2 φ
=
sin(θ + φ) sin φ cos φ + cos(θ + φ) cos2 φ cos2 (θ + φ)

=
(111)
cos2 θ
sin(θ + φ) sin φ + cos(θ + φ) cos φ
= cos θ.
The expressions (105), (109) and (111) contain the solution of the system of equations (102), (103) and (104). The solution reads
φ=
1
sin 2θ
arctan
,
2
µ − cos 2θ
κ=
tan(θ + φ)
,
tan φ
ρ = cos θ.
(112)
Error using two-point flux
37
K
φ
θ
Figure 47: Illustration of the example (equation (117)). Principal directions of the
permeability tensor K are denoted by arrows. The anisotropy ratio κ is indicated
by the length ratio of the arrows.
Here, |θ| < π/2 and µ > 0. The solution satisfies |φ| < π/4 and κ > 0. Further, the
solution obeys the symmetry relations φ(−θ, µ) = −φ(θ, µ) and κ(−θ, µ) = κ(θ, µ).
Having found the solution (112), we may also express the permeability tensor K
in coordinates aligned with the grid. In the coordinate system of the eigenvectors
e1 and e2 , the tensor reads
K e1 e2 = ρ(λ1 λ2 )1/2 diag κ1/2 , κ−1/2 .
(113)
In the coordinate system of the orthogonal vectors ν 1 and n2 , the tensor is given by
K ν 1 n2 = R(−φ)K e1 e2 R(φ),
where
cos φ
R(φ) =
− sin φ
sin φ
.
cos φ
(114)
(115)
Hence,
K ν 1 n2 =
"
ρ(λ1 λ2 )1/2
κ1/2 cos2 φ + κ−1/2 sin2 φ
κ1/2 − κ−1/2 cos φ sin φ
#
κ1/2 − κ−1/2 cos φ sin φ
κ1/2 sin2 φ + κ−1/2 cos2 φ
. (116)
Similarly, in the coordinate system of the orthogonal vectors n1 and ν 2 , the tensor
reads K n1 ν 2 = R(−(θ + φ))K e1 e2 R(θ + φ), i.e., K n1 ν 2 is obtained from equation
(116), by replacing φ with θ + φ.
7.2
Example
Assume a grid skewness angle θ = 10◦ and an apparent anisotropy ratio µ = 2. The
solution (112) gives the following results, written with two digits,
φ = 8.9◦ ,
κ = 2.2,
ρ = 0.98.
(117)
Hence, the data given by the user, indicate that he is simulating a system where the
principal axes are aligned with the grid directions, and where the anisotropy ratio
38
Multipoint flux approximations
is 2. In reality, however, he is simulating a system where the first principal axis is
rotated an angle φ = 8.9◦ from the first grid direction, and where the anisotropy
ratio is κ = 2.2, see Fig. 47. Also, the mean permeability is 2% smaller than his
data indicate.
In the coordinate system of the vector pairs (ν 1 , n2 ) and (n1 , ν 2 ), the permeability tensor (compare (116)) reads
1/2
K ν 1 n2 = (λ1 λ2 )
1.44 0.12
,
0.12 0.69
1/2
K n1 ν 2 = (λ1 λ2 )
1.37 0.24
.
0.24 0.75
(118)
If, for example, λ1 = 2 and λ2 = 1 in some unit system, then K e1 e2 = diag(k1 , k2 ) =
diag(2.06, 0.94) and
K ν 1 n2
8
2.03 0.17
=
,
0.17 0.97
K n1 ν 2
1.94 0.34
=
.
0.34 1.06
(119)
Future works
The O-method and the L-method are promising methods in reservoir simulation.
However, there are still some results for these methods which need to be accomplished.
Firstly, the selection condition of the L-method is not based on physical principles. While this condition works well for “nice” media and grids, it may not be
optimal in the general case. Here, more work is needed to establish the “optimal”
MPFA method.
Secondly, we need to establish monotonicity conditions for MPFA methods for
three-dimensional corner-point grids. The methodology which has been used for
two-dimensional problems, does not easily carry over to three-dimensional cases.
Thirdly, we need to work out constructive gridding techniques which honor the
monotonicity conditions. This will guarantee that the constructed grid is feasible
for the method chosen, such that we can trust the solution. Monotonicity seems
especially important for simulation of multiphase flows.
9
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