Multi-point flux approximation L-method in 3D: Numerical ... · Multi-point flux approximation L-method in 3D: Numerical convergence and application to two-phase flow through porous

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Multi-point flux approximation L-method in3D: Numerical convergence and application to

two-phase flow through porous mediaMarkus Wolff, Yufei Cao, Bernd Flemisch, Rainer Helmig and

Barbara Wohlmuth

Abstract. We thoroughly investigate the multi-point flux approximation L-method in 3D andapply this method to several realistic two-phase flow problems. Detailed transmissibility ma-trices for the flux calculation are presented, a novel selecting criterion for proper L-stencils isproposed, and a simple boundary handling approach is suggested. We further study the nu-merical convergence of the L-method through two FVCA6 benchmark tests on highly skewedand non-planar grids. The results show that the new criterion is very robust for challengingcases and the boundary treatment is sufficiently accurate. The simulations for the advection-dominated Buckley-Leverett problem and the capillary-diffusion-dominated McWhorter prob-lem exhibit the superiority of the L-method over the two-point flux approximation on generalnon-K-orthogonal grids. For complex porous media flow in natural geological formations, theefficiency of the numerical method is crucial. Thus, local grid adaptivity using a refinementindicator is presented and applied to two infiltration scenarios which illustrate the performanceof the L-method for complex physical processes (capillary pressure, gravity, etc.) and hetero-geneity/anisotropy.

Keywords. multi-point flux, L-method, three dimensions, numerical convergence, local gridrefinement, refinement indicator, two-phase flow, anisotropy, heterogeneity.

AMS classification. 65M08, 76S05, 76M12.

1 Introduction

Cell-centered finite volume techniques are quite popular in reservoir simulation tosolve the elliptic flow equation. A physically intuitive control volume formulationis the so-called two-point flux approximation (TPFA) which is widely used due to itssimplicity and efficiency. However, it is consistent if and only if the simulation gridsare K-orthogonal [26]. For general non-K-orthogonal grids, the TPFA does not workproperly due to the O(1) error in its solution [2, 47]. Mesh refinement also cannot re-duce the error as long as it does not improve the K-orthogonality of the grid. This wasone of the prime motivations for the independent development of multi-point flux ap-proximations (MPFA) from the research group of Aavatsmark et al. [3] and full-tensor

Contract/grant sponsor: DFG; contract/grant number: GRK 1398/2; EXC 310/1.

2 M. Wolff, Y. Cao, B. Flemisch, R. Helmig and B. Wohlmuth

flux continuous schemes from the research group of Edwards et al. [18]. Both meth-ods are similar, but differ in the choices of geometrical points and control volumes.The extension from TPFA to MPFA gives a correct discretization of flow equations forgeneral grids and anisotropic permeability tensors.

There are many variants of the MPFA method, including the O(η)-method [17, 37],the L-method [8] and others [16, 41, 43]. The most popular is the O(0)-method, of-ten simply called O-method. A class of MPFA O-methods can be found in [1] forquadrilaterals, in [4, 31] for hexahedral grids and in [23] for general three-dimensionalunstructured meshes. The convergence of the family of O-methods is tested and dis-cussed in [6, 7, 20, 37] by presenting numerical convergence results and in [9, 29, 30,45] by giving theoretical proofs. The M-matrix monotonicity of the MPFA methodsis discussed in [19, 34, 35]. It is an important property for multi-phase flow to avoidunphysical oscillations in the discrete solution.

In this paper, the MPFA L-method is investigated. It has some advantages comparedto the O-variant. The flux stencils are smaller, the domain of convergence as well asthe domain of monotonicity is larger and the treatment of hanging nodes and thus thecapability for local adaptivity is naturally included. The first introduction of the L-method is given in Aavatsmark et al. [8] for quadrilateral grids in two dimensions.Aavatsmark et al. [5] further extends it to three dimensions. An alternative but closelyrelated scheme with anisotropy-favoring triangulation is presented in Pal and Edwards[36]. Mundal et al. [33] have tested the L-method for different heterogeneous, highlyanisotropic, elliptic problems from the FVCA5 benchmark on triangular and quadri-lateral grids. The boundary influence on the convergence rate and the geometricalinterpretation of the original transmissibility-based criterion in the case of homoge-neous media are studied in Cao et al. [14]. The first theoretical convergence proof ofthe L-method with homogeneous media on uniform grids is given in Cao et al. [15]through the equivalence to a modified conforming finite element method. Stephansen[42] further investigates the convergence on general grids by linking the L-method tomimetic finite differences (MFD).

In this work, the numerical convergence of the L-method in 3D is studied by com-parison with the TPFA and the MFD method [12]. Further, it is applied to two-phaseflow in porous media and tested and validated on standard benchmark problems aswell as on more realistic test scenarios. For solving the non-linear coupled two-phaseflow equations, the numerical method has to be efficient. Thus, we introduce meshadaptivity with hanging nodes in combination with the MPFAL L-method. In contrastto the O-method, the L-method naturally allows for hanging nodes and thus is quiteattractive in the case of adaptive mesh refinement.

The paper is organized as follows. In Section 2, the implementation of the MPFAL-method in 3D is illustrated in detail. The standard criterion for choosing a properL-stencil for flux calculation is discussed, and a new more robust criterion is proposed.Moreover, a simplified boundary handling approach is presented. Section 3 thoroughlyinvestigates the numerical convergence of the L-method through two benchmark ex-

3D MPFA L-method: Numerical convergence and application. 3

amples, where convergence rates of pressure and normal velocity are studied. Addi-tionally, the different selection criteria of the L-stencil are tested and the MPFA methodis compared to the TPFA and MFD methods. Details for the use of non-conforming,adaptive grids with the L-method are given in Section 4, where the focus is on differenttypes of interaction volumes around hanging nodes and boundary treatment. In Section5, the L-method is applied to four typical two-phase flow problems addressing differentphysical regimes like the advection-dominated case, the capillary-diffusion-dominatedcase or the capillary pressure/gravity-dominated case as well as homogeneous, hetero-geneous and anisotropic porous media. The 3D MPFA L-method is implemented inthe simulation toolbox DuMux [21], which is based on the DUNE framework [11].For the convergence study the sparse direct solver PARDISO [39, 40] is used as linearsolver.

2 The MPFA L-method in 3D

In this section, the MPFA L-scheme is introduced for the second-order elliptic equation

−∇ · (K∇p) = q, in Ω ⊂ R3, (2.1)

where Ω is the bounded Lipschitz domain of interest, K : Ω → R3×3 is a sym-metric and uniformly positive definite permeability tensor, p is the pressure and q isthe source term. Different from [5], the detailed geometrical information for fourL-stencils of each sub-interface and their transmissibility matrices are described andgiven here. Additionally to the L-shape selection criterion given in [5], we introducea new transmissibility-based criterion which is supposed to be robust also for verychallenging grids. A simple but sufficiently accurate treatment of fluxes through eachsub-interface connecting to or lying on the boundary is introduced. This allows aneasy implementation of more complex domain shapes and boundary configurations(especially in 3D) and thus provides the possibility to apply the method to realisticscenarios.

2.1 Details of the scheme

Let Th be a hexahedral partition of the domain Ω. The principle of the control volumeformulation is the local mass conservation of Equation (2.1) over each grid cellK ∈ Th(the so-called control volume)

−∫∂K

K∇p · nds =∫Kqdx, (2.2)

where ∂K is the boundary of the cell K which is the union of six cell faces ei, i =1, . . . , 6 (see Figure 1), n is the unit outer normal vector on ∂K. If the numerical


approximation of the flux −∫eiK∇p · nds through each cell face ei is denoted as f i,

then the local control volume formulation (2.2) can be written as

6∑i=1

f i =

∫Kqdx. (2.3)

z

z

z

z

x2

x3

x4

x5

x0x1f 1

f 11

f 12

f 13

f 14

Figure 1: Control volume K and its face fluxes.

Various developments of the numerical fluxes f i lead to different cell-centered finitevolume schemes. For example in Figure 1, the TPFA method approximates the fluxf1 as a whole using the information of the two neighboring cells of the face e1, whilethe L-method calculates f1 by the sum of the fluxes through four sub-interfaces, i.e.,f1 =

∑4j=1 f

1j using the geometry and soil matrix properties of multiple grid cells.

In case of 3D, there are 4 possible L-stencils instead of 2 choices in 2D to calculatethe numerical flux through each sub-interface inside the domain, and each L-stencilcontains 4 interacting cells instead of 3 for 2D cases. Figure 2 shows the 4 possibilitiescorresponding to the sub-interface x1x4x0x5, where (a) and (b) are centered L-stencilswhich have a center cell interacting with the other three cells, and (c) and (d) arenamed non-centered L-stencils [5]. In the following the flux calculation through thesub-interface x1x4x0x5 is described including implementational details.

For each sub-volume, e.g. x1x1x4x2x3x5x0x6 in the L-stencil (a), a linear pressureis applied which leads to 16 degrees of freedom for the whole L-stencil. Thus, 16equations for the determination of the unknown coefficients α0j , α1j , α2j , α3j of thefour affine pressure functions pj(x) = α0j + α1jx + α2jy + α3jz, j = 1, 2, 3, 5 inthe four tetrahedra x1x2x3x1, x1x4x5x2, x2x4x6x3, x3x5x6x5 are required, see Figure2. Here, x = (x, y, z)T represents the coordinate vector of an arbitrary point in 3DEulerian space. The tetrahedra used to construct the linear affine functions and theirconstant gradients are listed in Table 1 for the four kinds of L-stencils (a) (b) (c) (d).To close the numerical scheme, the physically meaningful conditions in Table 2 areforced to be satisfied for each L-stencil.

Denote the fluxes through the 3 sub-interfaces x1x4x0x5, x2x4x0x6, x3x5x0x6 in-side the L-stencil (a) as fa,1, fa,2, fa,3, then a small local algebraic system for the fluxvector fa = (fa,1, fa,2, fa,3)

T = Taua can be derived by using the aforementioned con-straints. Here, Ta is the so-called transmissibility matrix; ua = (p1, p2, p3, p5)

T is the


(a) (b)

2

5

3

1

x1 x2

x5

x1x2

x3

x4

x5

x6

fa,1

x0

21

4

6

x1x2

x4

x6

x1x2

x3

x4

x5

x6

fb,1

x0

(c) (d)

2

5

1

4x1x2

x4

x5

x1x2

x3

x4

x5

x6 x7

fc,1

x0

2

3

1

6

x1x2

x6

x1

x2

x3

x4

x5

x6x7

fd,1

x0

Figure 2: Four L-stencils for the flux calculation through sub-interface x1x4x0x5.

local pressure (solution) vector for the L-stencil (a), where pj is the function value ofpj(x) at the cell center xj , j = 1, 2, 3, 5. For completeness, the calculation of Ta forthe L-stencil (a) will be described below in detail (see also [5]). Due to the similarity,the derivation of the transmissibility matrices Tb,Tc,Td are omitted but the matrixforms are given in the appendix.

By applying the 3 flux continuity conditions, we get

fa,1 = −nT1 K1∇p1 = −nT1 K2∇p2,

fa,2 = −nT2 K1∇p1 = −nT2 K3∇p3,

fa,3 = −nT3 K1∇p1 = −nT3 K5∇p5,

(2.4)

where ni, i = 1, 2, 3, are the normal vectors of the 3 sub-interfaces having the lengthequal to the sub-interface area, and Kj , j = 1, 2, 3, 5, are the permeabilities of the 4interacting cells. Using the tetrahedra given in Table 1, the pressure gradient on theinteracting cell j can be expressed as

∇pj =1Tj

3∑k=1

νjk(pjk − pj), j = 1, 2, 3, 5, (2.5)

where pjk is the pressure value at the continuity point xjk, and νjk is the inwardnormal vector to the tetrahedron face opposite to xjk. The length of νjk is equal to


`````````````L-stencilsub-interface

x1x4x0x5 x2x4x0x6 x3x5x0x6

(a)x1x2x3x1

x1x4x5x2

x1x2x3x1

x2x4x6x3

x1x2x3x1

x3x5x6x5

(b)x1x4x5x1

x1x2x3x2

x1x2x3x2

x2x4x6x4

x1x2x3x2

x3x5x6x6`````````````L-stencil

sub-interfacex1x4x0x5 x2x4x0x7 x3x5x0x6

(c)x1x3x4x1

x1x2x5x2

x1x2x5x2

x2x4x7x4

x1x3x4x1

x3x5x6x5

(d)x1x2x5x1

x1x3x4x2

x1x2x5x1

x2x4x7x3

x1x3x4x2

x3x5x6x6

Table 1: The two tetrahedra used to calculate the flux through each of the three sub-interfaces for the four L-stencils (a) (b) (c) (d), see Figure 2.

conditions /degrees of freedom

number total equations

each sub-interface(×3)

flux continuityfull pressure continuity

13 16

each interacting cell (×4) pressure 1

Table 2: Required constraints for constructing the MPFA L-scheme in 3D.

twice the area of the face to which it is normal. This face area can be easily expressedas cross product of the two vectors that span the tetrahedron face. The coefficient Tjis given by the triple product of the three vectors that span the tetrahedron and is equalto 6 times the volume of the tetrahedron. For more details on νjk and Tj , see [1]. Inparticular for the L-stencil (a), we have

p11 = p1, p12 = p2, p13 = p3,

p21 = p4, p22 = p1, p23 = p5,

p31 = p2, p32 = p4, p33 = p6,

p51 = p6, p52 = p5, p53 = p3.

As done in [5], we further eliminate the additional unknowns p3+i, i = 1, 2, 3 atthe edge centers by using the linear pressure variation of the interacting cell 1. Letri = x3+i − x1, i = 1, 2, 3, and r1ki = νT1kri/T1, i, k = 1, 2, 3. Inserting these


expressions into Equation (2.5) we get

p3+i = ∇p1 · (x3+i − x1) + p1 =

3∑k=1

r1ki(p1k − p1) + p1, i = 1, 2, 3, (2.6)

where x1, x3+i are the coordinate vectors of points x1 and x3+i.From (2.4)-(2.6), it is easy to see that the flux through each sub-interface of the L-

stencil (a) can be spanned by the pressures at the points xi, i = 1, 2, 3, and the cellcenters xj , j = 1, 2, 3, 5. The first equality in (2.4) gives the matrix expression of theflux vector across the 3 sub-interfaces as

fa = Cva +Dua, C ∈ R3×3, D ∈ R3×4, (2.7)

where va = (p1, p2, p3)T is the pressure vector of pressure continuity points x1, x2 and

x3. After introducing the quantity ωijk = nTi Kjνjk/Tj , the matrices can be writtenas

C =

−ω111 −ω112 −ω113

−ω211 −ω212 −ω213

−ω311 −ω312 −ω313

, D =

ω111 + ω112 + ω113 0 0 0ω211 + ω212 + ω213 0 0 0ω311 + ω312 + ω313 0 0 0

.

Furthermore, an algebraic system of the form

Ava = Bua, A ∈ R3×3, B ∈ R3×4 (2.8)

can be derived from the second equality of the flux continuity equations (2.4), with

A =

ω111 − ω122−ω121r111 − ω123r112

ω112 − ω121r121−ω123r122

ω113 − ω121r131−ω123r132

ω211 − ω232r111−ω233r113

ω212 − ω231−ω232r121 − ω233r123

ω213 − ω232r131−ω233r133

ω311 − ω351r113−ω352r112

ω312 − ω351r123−ω352r122

ω313 − ω353−ω351r133 − ω352r132

,

B =

ω111 + ω112 + ω113+ω121(1− r111 − r121 − r131)+ω123(1− r112 − r122 − r132)

−(ω121+ω122 + ω123)

0 0

ω211 + ω212 + ω213+ω232(1− r111 − r121 − r131)+ω233(1− r113 − r123 − r133)

0 −(ω231+ω232 + ω233)

0

ω311 + ω312 + ω313+ω351(1− r113 − r123 − r133)+ω352(1− r112 − r122 − r132)

0 0 −(ω351+ω352 + ω353)

.

Finally, the transmissibility matrix Ta for the L-stencil (a) is obtained

Ta = CA−1B+D ∈ R3×4. (2.9)


2.2 Criterion for choosing the proper L-stencil

The choice of a proper L-stencil for each sub-interface is crucial for the robustnessof the L-method. As the selection of one of the four cases (see Figure 2) has to bemade a priori, an appropriate criterion is needed. In [5, 33] the sign property of thetransmissibility coefficients of a L-stencil is identified to be an important criterion.Considering a certain flux face on a regular grid the idea is that all transmissibilitycoefficients of cells located at one side of this flux face should have the same sign,leading to fluxes in the same direction, while all transmissibility coefficients on theother side should have the opposite sign. Therefore, [5] proposes to choose the L-stencil with the smallest sI value, where

sI = |TI,11 −TI,12|. (2.10)

The capital index I = a, b, c, d denotes the type of the L-stencil, TI,1i are the transmissi-bility coefficients of cell i for flux sub-interface 1 (x1x4x0x5). In [5] the authors showthat for parallelepiped grids the sI-criterion always yields a robust L-stencil. However,for irregular grids, which are not close to K-orthogonal grids, a good choice of theL-stencil is more difficult. For grids with large skewness, none of the transmissibilitiesof the four L-cases might fulfill the sign criterion. Especially in the 3D case, it is notobvious if the smallest sI value also leads to the most robust stencil (see Section 3).Thus, a new criterion is developed from the following ideas: (1) in the K-orthogonalcase the chosen transmissibility should satisfy the sign property; (2) it should includemore than the two neighboring cells at the respective flux sub-interface to better coverthe 3D behavior; (3) it should lead to a consistent flux stencil for the cases in which thesign property cannot give a unique choice. We propose to choose the L-stencil withthe largest tI value, where

ta = |Ta,11 +Ta,13 +Ta,15|,tb = |Tb,12 +Tb,14 +Tb,16|,tc = |Tc,11 +Tc,15|,td = |Td,11 +Td,13|.

(2.11)

On one hand the TI,1i in (2.11) tend to have the same sign if the absolute of their sumis larger. Thus, it is more likely that the chosen transmissibility shows the expectedsign property. On the other hand the transmissibility which has the largest impact onthe flux through sub-interface x1x4x0x5 is always preferred. This is in line with oneof the basic ideas of the L-method, namely to choose the flux stencil according to thepreferred direction of flow. If the contributing cells lead to higher fluxes through asub-interface, the orientation of the L-shape most likely is in better agreement with thedirection of the streamlines. This leads to a consistent choice of the L-stencils even inthe case of irregular grids with large skewness and thus increases the robustness of theMPFA L-method (see Section 3.1). However, it is not obvious which criterion yieldsmore accurate results. In our numerical examples, the new criterion is more robust.


2.3 Boundary handling

In Aavatsmark et al. [8], the boundary handling of the L-method in 2D is shortlymentioned where the concept of ghost cells is proposed. Especially, the flux throughthe second half edge near the boundary is difficult to derive in a consistent way andtherefore the O-method is suggested. Cao et al. [14] further studies the influence ofdifferent boundary treatments using the O-method on the convergence rate. Insteadof ghost cells or the O-method, we combine the TPFA discretization with the conceptof weighting factors to handle the boundary region. Although this method is rathersimple, it is still sufficiently accurate for many applications.

(a) (b) (c)

z

z

z

z

x2

x3

x4

x5

x0

x1 n

21

z

zx2

x3

x4

x5

x6

x7

x8 x0

f1f2

21

z

zx2

x3

x4

x5x7

x8 x0

f1

Figure 3: Boundary handling of three cases: the flux calculation through the cell facex2x3x4x5 (a) on the boundary, (b) intersecting with one boundary face, (c) intersectingwith two boundary faces.

As seen from Figure 3, there are three boundary situations we need to take intoaccount. In case (a) the cell face x2x3x4x5 lies on the boundary, so it is straightforwardfor the Neumann conditions to get the flux. For the Dirichlet boundary, a two-point

approximation− |KTn|

|x0 − x1|(p0−p1)|s| is used, where n is the unit outer normal vector

of the cell face, |KTn| is the norm, p0 and p1 are the pressures at the face center x0and the cell center x1, |x0 − x1| is the distance between the two centers, and |s| is thearea of the cell face. This discretization is exact for K-orthogonal grids, and its impacton the convergence of the whole domain is very small for non-K-orthogonal grids (seeSection 3).

Case (b) considers the cell face intersecting with one boundary face, where theflux f1 through the sub-interface x0x7x5x8 and the flux f2 through the sub-interfacex0x6x4x7 can be respectively obtained from two separate inner interaction volumesusing the L-method introduced in Sub-section 2.1. Then the flux f through the wholecell face x2x3x4x5 is computed by scaling the sum f1 + f2 with the area weighting

factor|s||sh|

, where |sh| is the area of the face x6x4x5x8. Similarly, when the cell face

contacts with two boundary faces as in case (c), the flux is easily approximated by

f = f1|s||sq|

, where |sq| is the area of the face x0x7x5x8.


3 Numerical convergence

The numerical examples in this section are designed by the benchmark session ofthe FVCA6 conference (June 2011, Praha [22]), where many interesting meshes areprovided and 16 contributed numerical schemes are compared. However, the MPFAL-method is not included. Therefore, two 3D benchmark problems are taken here tostudy the numerical convergence of the L-method in comparison with the TPFA andMFD methods. For both examples, the problem (2.1) on Ω = (0, 1) × (0, 1) × (0, 1)is considered, and two kinds of meshes are tested. One is the 3D version of Kershaw’s“z-mesh” [28] with planar faces. It is very challenging since some of its hexahedronshave faces that intersect at very acute angles which yield high skewness, see the lefttwo graphs in Figure 4. The other one is a random mesh with non-planar faces, seeFigure 4(right). Since a uniform refinement makes a grid asymptotically regular, mostnumerical schemes would finally converge. Therefore, the Kershaw and the randommesh are refined on each level by preserving the same local geometrical structure.

Figure 4: Kershaw meshes 8×8×8 and 16×16×16 (left), random meshes 4×4×4and 8× 8× 8 (right).

Throughout the rest of this section, the convergence order graphs of the pressure andthe normal velocity for various numerical simulations are shown and studied, wherendof is the number of degrees of freedom. The relative discrete, mesh dependent L2

norms of the error for the pressure ep and the normal velocity ev, are defined by

ep =

( ∑K∈Th |K|(pex,K − pK)2∑

K∈Th |K|p2ex,K

) 12

,

ev =

( ∑K∈Th

∑j∈∂K |K|(v⊥ex,j − v⊥j )2∑

K∈Th∑

j∈∂K |K|(v⊥ex,j)2

) 12

.

Here, pK is the discrete pressure solution of cell K ∈ Th. The analytical pressurevalue pex,K is evaluated at the cell center, |K| is the cell area. The analytical normalvelocity v⊥ex,j is evaluated by −K∇p · n at the face center, where n is the unit outernormal of the cell face j ∈ ∂K. The discrete normal velocity v⊥j is calculated at the


face center. For i ≥ 2, the convergence order cgl2(i) of the L2 norm of the error onthe solution between grid level i and i− 1 can be obtained as

cgl2(i) = −3log(em(i)/em(i− 1))

log(ndof(i)/ndof(i− 1)), m = p or v.

3.1 Benchmark test 1

Test 1 solves (2.1) with the permeability tensor

K =

1 0.5 00.5 1 0.50 0.5 1

(3.1)

which is mildly anisotropic but defines three principal directions not aligned with thecoordinate axes. Its exact solution is p = 1 + sin(πx) sin(π(y + 1

2)) sin(π(z + 13)).

Kershaw mesh

(a) (b)

102

104

106

10−3

10−2

10−1

100

101

ndof

ep

TPFA

MPFA L (sI)

MPFA L (tI)

MFD

102

104

106

10−3

10−2

10−1

100

101

ndof

ep

L (sI): te

L (sI): ie

L (tI ): te

L (tI ): ie

102

104

106

10−2

10−1

100

101

102

103

ndof

ev

TPFA

MPFA L (sI)

MPFA L (tI)

MFD

102

104

106

10−2

10−1

100

101

102

103

ndof

ev

L (sI): te

L (sI): ie

L (tI ): te

L (tI ): ie

Figure 5: Numerical comparison of the pressure (upper row) and the normal velocity(lower row) for test 1 on the Kershaw mesh: (a) convergence order of different meth-ods, (b) boundary effect on the convergence of the L-method using sI - and tI -criterion.


Test 1 is first run on the Kershaw mesh. Column (a) of Figure 5 shows the conver-gence rate of the pressure and the normal velocity for the TPFA, the MPFA L-methodusing sI- and tI-criterion and the MFD method. It can be seen that the TPFA doesnot converge at all, and its quantitative errors do not reduce with the refinement. The3D slice in the lower row of Figure 6(b) clearly presents the distortion of the TPFAresult which is affected by the anisotropy and the grid shape. In contrast, the MFDmethod converges very fast with order O(h2) for both the pressure and the velocity,and the surface graph and the slice in Figure 6(d) match well with the exact solution.However, the numerical solution of the L-method using the sI-criterion blows up forthe very fine mesh. Its convergence is mainly destroyed by extremely large errors insome local regions where the mesh has high skewness. One of those regions can beobserved in Figure 6(c).

(a) (b) (c) (d)p

Figure 6: 3D surface graphs (upper row) and the viewing mode of three intersect-ing orthogonal slices (lower row) of different methods: (a) exact solution, (b) TPFAmethod, (c) MPFA L-method (sI ), (d) MFD method for test 1 on the Kershaw mesh64× 64× 64.

It is obvious that the sI-criterion does not lead to proper L-stencils in some regionsof this difficult example, and this motivates the development of the tI-criterion (2.11).Figure 5(a) shows the improvement of the tI-criterion with convergence order O(h1)for both the pressure and the velocity. The highly colored regions disappear from theslice of the L-solution, see Figure 7. To analyze differences between the two criteriathe averaged size of the cell stencils is shown in Table 3 for different levels of theKershaw mesh. The average is based on the flux stencils of all cells not lying on theboundary. It can be observed that the tI-criterion leads to slightly larger flux stencils.Additionally, the average stencil using the sI-criterion decreases with increased refine-ment levels, which means that the flux stencils change for the finer grid. This change


coincides with the change in the convergence behavior (Figure 5) and the appearanceof the numerical instabilities. To investigate these effects in more details a simplifiedtest for the L-selection criteria is shown and discussed at the end of this subsection.Although the performance of the L-method in this test case is improved by using thetI-criterion, the MFD method is still superior. Concerning the simplified boundary

HHHHH

HHHHHCriterion

Grid(cells)

Kershaw 8(512)

Kershaw 16(4096)

Kershaw 32(32768)

Kershaw 64(262144 )

sI 15 15 14 14

tI 18 18 18 18

Table 3: Average size of the cell stencils resulting from the two L-selection criteria forbenchmark test 1 on the Kershaw mesh.

discretization of the L-method, the total error (te) on the whole domain Ω and the in-ner error (ie) on the inner domain (Ω without the boundary layer of cells) are fairlyclose for both criteria, see Figure 5(b). This confirms the good performance of thisboundary treatment. The first row of Figure 6 further tells us that this test case itself isnot challenging for the boundary approximation since the pressure distribution on theboundary of the TPFA method matches well with the exact solution. Thus, the TPFAin combination with the area weighting approach is quite often sufficient.

Figure 7: The viewing mode of three intersecting orthogonal slices of the MPFA L-method using the sI-criterion (left) and the tI-criterion (right) for test 1 on the Kershawmesh 64× 64× 64.

Comparison of the L-selection criteria with regard to the grid skewness To an-alyze the error resulting from the choice of the L-shape (Figures 5 and 7), a domaindiscretized by a 3 × 3 × 3 parallelepiped mesh is considered. Again Equation (2.1) issolved with the homogeneous anisotropic permeability tensor (3.1) which is visualizedas red ellipsoid in Figure 8.

Figure 9 shows the flux stencils of the center cell applying the two L-selection cri-teria for differently skewed grids. In case (a) with cubic cells the sI-criterion chooses


β

Figure 8: Visualization of the anisotropic permeability tensor (red ellipsoid) inside thetest domain.

a flux stencil of minimal size that is perfectly aligned with the permeability tensor.The tI-criterion chooses a larger stencil which is actually the maximal L-stencil. FromFigure 9(a) to 9(d) the angle β (Figure 8) is decreased. The stencil selected by thetI-criterion remains the same, whereas the stencil selected by the sI-criterion changes.The preferred direction of the flux stencil, which results from the sI-criterion, turnsfrom case (a) to case (d) compared to the direction of the permeability tensor. Thisbecomes even more obvious when looking down from the top (Figure 10). For thisvery skewed grid with β = 11.25, the preferred direction of the chosen flux stencil isopposite to the main direction of the permeability tensor. Therefore, the stencil mostlikely does not lead to a meaningful flux approximation. In contrast, the flux stencilusing the tI-criterion better covers the main direction of the permeability tensor, andthus, well supports the skewed grids.

To summarize, the sI-criterion tends to minimize the flux stencil while the tI-criterionresults in larger stencils. Comparing the shape of the stencils, the sI-criterion is moreselective with regard to certain directions. In the grid cells with large skewness it ap-pears that, the preferred direction of the flux stencil strongly differs from the directionof the streamlines. This also explains the large errors shown in Figure 6(c). In suchcases the cell stencil of the tI-criterion still leads to consistent fluxes. However, therobustness is increased at the cost of slightly larger stencils.

Random mesh

Next, we rerun test 1 on the random mesh to see the influence of the grid on theperformance of different numerical methods. From Figure 11(a) we can see that theL-method using the sI-criterion now can work properly withO(h1) convergence orderfor the pressure and the normal velocity, while the convergence rate of the velocityfor the MFD method is decreased compared to the Kershaw mesh. As expected, theTPFA cannot get the correct solution for any non-K-orthogonal grid whether with highor low skewness. For this test case, the two L-method selection criteria give similar


(a) (b) (c) (d)

Figure 9: Cell stencils for differently skewed parallelepiped grids using the sI-criterion(upper row) and the tI-criterion (lower row). The skewness of the grid is increasedfrom left to right by decreasing the angle β (Figure 8): (a) 90, (b) 45, (c) 22.5, (d)11.25.

(a) (b)

Figure 10: Top view on the cell stencils of the sI-criterion (a) and the tI-criterion (b)for β = 11.25 .

results. This can also be seen from the lower row of Figure 12 where the solutions ofthe L-method and the MFD method are quite similar and close to the exact result. Theinner pressure distribution of the TPFA does not catch the anisotropy correctly.

The most interesting aspect pointed out from the random mesh is that the non-planarsurfaces have a bigger impact on the accuracy of the simplified boundary approxima-tion of the L-method. For both selection criteria, the quantitative error of the normalvelocity is increased whereas the pressure is less sensitive to the boundary handling,see Figure 11(b). In comparison, the L-method using the tI -criterion yields a smallerboundary error for the normal velocity than the method using the sI -criterion.


(a) (b)

101

103

105

10−4

10−3

10−2

10−1

100

ndof

ep

TPFA

MPFA L (sI)

MPFA L (tI)

MFD

101

103

105

10−4

10−3

10−2

10−1

100

ndof

ep

L (sI): te

L (sI): ie

L (tI ): te

L (tI ): ie

101

103

105

10−3

10−2

10−1

100

ndof

ev

TPFA

MPFA L (sI)

MPFA L (tI)

MFD

101

103

105

10−3

10−2

10−1

100

ndof

ev

L (sI): te

L (sI): ie

L (tI ): te

L (tI ): ie

Figure 11: Numerical comparison of the pressure (upper row) and the normal velocity(lower row) for test 1 on the random mesh: (a) convergence order of different methods,(b) boundary effect on the convergence of the L-method using sI - and tI -criterion.

3.2 Benchmark test 3

Test 3 solves (2.1) with the exact solution p = sin(2πx) sin(2πy) sin(2πz) and thepermeability tensor

K =

1 0 00 1 00 0 103

. (3.2)

Obviously, the tensor has a relatively high anisotropy ratio, and its principal directionscoincide with the coordinate axes. So the Kershaw mesh is very non-K-orthogonalin the inner local regions, whereas the random mesh is quite close to a K-orthogonalgrid.

Kershaw mesh

Compared to test 1 on the Kershaw mesh, for test 3 the L-method quantitatively getsbetter solutions than the MFD method for both the pressure and the normal veloc-ity, see Figure 13(a). Since the Kershaw mesh is non-K-orthogonal in terms of the


(a) (b) (c) (d)p

Figure 12: 3D surface graphs (upper row) and the viewing mode of three intersect-ing orthogonal slices (lower row) of different methods: (a) exact solution, (b) TPFAmethod, (c) MPFA L-method (sI ), (d) MFD method for test 1 on the random mesh32× 32× 32.

tensor (3.2), the TPFA still fails to converge and its pressure solution exhibits un-dershoot/overshoot with the magnitude [−3.32, 3.32] as seen from the highly coloredregions in Figure 14(b). Compared to the full permeability tensor in test 1, the simula-tion of test 3 on the Kershaw mesh is not so challenging. Thus, the sI-criterion as wellas the tI-criterion lead to almost the same numerical results, see Figure 13(a). Again,the simplified L-boundary implementation is sufficiently accurate for both criteria.

Random mesh

Compared to the three other simulation cases, the random mesh with the diagonaltensor (3.2) is slightly better suited for the TPFA method. One obtaines O(h1) con-vergence for the pressure, but the relatively big error is amplified for the normal ve-locity resulting in a loss of convergence, see Figure 15(a). The L-method using thetI-criterion generates a larger error for the normal velocity than the sI-criterion, how-ever the difference is not significant. Both L-methods are still quantitatively better thanthe MFD method. Since the boundary faces of the random mesh are non-planar, theboundary errors in Figure 15(b) are bigger than those for test 3 on the Kershaw mesh.Although the TPFA solution in Figure 16 is more accurate than in the former cases,we still see some inner grid cells with small artificial oscillations.

As a conclusion, the robustness of the L-method is determined by the criterion forchoosing the proper L-stencil. The tI-criterion can handle all test cases, and it seemsmore robust than the sI-criterion. The boundary discretization proposed in Section 2.3


(a) (b)

102

104

106

10−3

10−2

10−1

100

101

ndof

ep

TPFA

MPFA L (sI)

MPFA L (tI)

MFD

102

104

106

10−3

10−2

10−1

100

101

ndof

ep

L (sI): te

L (sI): ie

L (tI ): te

L (tI ): ie

102

104

106

10−3

10−2

10−1

100

ndof

ev

TPFA

MPFA L (sI)

MPFA L (tI)

MFD

102

104

106

10−3

10−2

10−1

100

ndof

ev

L (sI): te

L (sI): ie

L (tI ): te

L (tI ): ie

Figure 13: Numerical comparison of the pressure (upper row) and the normal velocity(lower row) for test 3 on the Kershaw mesh: (a) convergence order of different meth-ods, (b) boundary effect on the convergence of the L-method using sI - and tI -criterion.

is demonstrated to be efficient and accurate enough. Most importantly, it is trivial toimplement, can be easily applied to more complex domain shapes, and can be usedin the case of adaptive mesh refinement (see Section 4). These examples show thesuperiority of the L-method over the TPFA, and moreover both the L-method andthe MFD approach provide qualitatively similar results. While the MFD method issuperior for the first test case with mild anisotropy and a full permeability tensor, theL-method yields better results for the second test case with strong anisotropy and adiagonal permeability tensor.

We point out that the L-method always gives rise to a purely pressure based sys-tem with one degree of freedom per element. This is not the case for the MFD andother mixed schemes. Usually, the assembly of flux and pressure degrees of freedomassociated with faces and elements results in a linear system that exhibits saddle pointstructure and, therefore, is approximately four times larger (for hexahedral meshes)and harder to solve. The classical remedy for this problem of the mixed approaches ishybridization by means of introducing face pressure degrees of freedom and allowingto locally eliminate the original unknowns. While this yields a symmetric positive def-inite system of linear equations, the number of degrees of freedom still is three timesas large as for a purely cell-centered method. In [44], a strategy is provided for the


(a) (b) (c) (d)

p

Figure 14: 3D surface graphs (upper row) and the viewing mode of three intersect-ing orthogonal slices (lower row) of different methods: (a) exact solution, (b) TPFAmethod, (c) MPFA L-method (sI ), (d) MFD method for test 3 on the Kershaw mesh64× 64× 64.

local elimination of the flux unknowns for the lowest-order mixed method yielding asystem with only element degrees of freedom. While this approach is attractive forsimplicial meshes, its application to hexahedral grids would involve larger patches ofelements for the solution of local problems and therefore larger stencils than for theL-method.


(a) (b)

101

103

105

10−2

10−1

100

ndof

ep

TPFA

MPFA L (sI)

MPFA L (tI)

MFD

101

103

105

10−2

10−1

100

ndof

ep

L (sI): te

L (sI): ie

L (tI ): te

L (tI ): ie

101

103

105

10−2

10−1

100

ndof

ev

TPFA

MPFA L (sI)

MPFA L (tI)

MFD

101

103

105

10−2

10−1

100

ndof

ev

L (sI): te

L (sI): ie

L (tI ): te

L (tI ): ie

Figure 15: Numerical comparison of the pressure (upper row) and the normal velocity(lower row) for test 3 on the random mesh: (a) convergence order of different methods,(b) boundary effect on the convergence of the L-method using sI - and tI -criterion.

(a) (b) (c) (d)

p

Figure 16: 3D surface graphs (upper row) and the viewing mode of three intersect-ing orthogonal slices (lower row) of different methods: (a) exact solution, (b) TPFAmethod, (c) MPFA L-method (sI ), (d) MFD method for test 3 on the random mesh32× 32× 32.


4 Grid adaptivityRegarding time-dependent coupled transport problems like multi-phase flow in porousmedia, the efficiency of the numerical method is crucial. Complex discretizationschemes like the MPFA L-method might not be applicable for large-scale problemson very fine meshes. One way to increase efficiency is to use locally refined grids.We adapt the grid by non-conforming grid refinement allowing one hanging node percell face. An advantage of the L-method is that the treatment of hanging nodes isstraightforward. Figure 17 shows the basic types of interaction volumes which can oc-cur around a hanging node. The flux faces corresponding to these interaction-volume

(Type 1)

1 2

3 4

5

•

(Type 2)

1 2

3 4

6

5

7

•

(Type 3)

1 2

3 4

5 6

•

(Type 4)

1

3

2

4 5

•

(Type 5)

1

2

34

6

5

7

••

•

(Type 6)

1

3

2

4

5

••

•

Figure 17: Interaction-volume-types around hanging nodes allowing only one hangingnode (HN) per cell face. (Type 1): one face HN; (Type 2): one edge HN, one coarsercell; (Type 3): one edge HN, two coarser cells; (Type 4): one edge HN, three coarsercells; (Type 5): three edge HNs, special case of (Type 2); (Type 6): three edge HNs,special case of (Type 4).

shapes are sketched in Figure 18. For each flux face of an interaction volume arounda hanging-node the flux can be calculated by applying one of the four L-stencils in-troduced in Figure 2. However, compared to a regular node only one or one of twoof these L-shapes can be suitable for certain flux faces. For instance, for the volume


Type 1, all four L-cases could be used for the flux calculation through faces 1-4, whileonly the stencil (a) or (b) could be used for the flux faces 9-12, see Figure 18. Let usconsider the L-stencil for face 9 of the volume Type 1, where cell number 5 is cell 1according to the numbering of Figure 2 and cell number 1 is cell 2. In this case onlycells 5, 1, 3, and 2 can be combined to the regular L-shape (b). The remaining threeL-shapes respectively combine cells 5 and 1 for the construction of the L-stencil (a),cells 5, 1, and 3 for the L-stencil (c), and cells 5, 1, and 2 for the L-stencil (d), whichlead to stencils consisting of less than 4 cells. Obviously, these degenerated stencilsare not feasible to derive a closed local system for the flux approximation through face9.

(Type 1)

•

12

34

9 10

1112

(Type 2)

•

12

34

5

78

9 10

1112

(Type 3)

•

12

34

5

7

8 9

1011

(Type 4)

•

1

34

9

12

Figure 18: Flux faces of the different interaction-volume-types shown in Figure 17.The numbering of the faces relates to the general case without hanging nodes (12faces per interaction-volume). (Type 1): 8 flux faces; (Type 2): 11 flux faces; (Type3): 10 flux faces; (Type 4): 5 flux faces; (Type 5): see (Type 2); (Type 6): see (Type4).

Boundary handling As explained in Section 2.3, the two-point flux approximationis used at Dirichlet boundaries and a weighting factor is used to obtain the fluxesthrough the faces intersecting with the boundary. If hanging nodes occur, the weight-ing becomes more complex. Considering two neighboring cells of the interaction vol-ume around a hanging node, the intersecting face of one (coarser) cell could touch aboundary while the intersecting face of the other (finer) cell could be completely insidethe domain. Figure 19 exemplarily shows such a situation. The blue shadowed areamarks a boundary face. The red area is the area through which the flux is calculatedby the MPFA interaction volumes inside the model domain (volumes that do not touchthe boundary). Each of the red arrows in Figure 19(a) represents the flux fi througha sub-interface i of the interface x5x6x3x4 calculated from one L-stencil. Like forthe non-adaptive case explained before we now want to weight the flux through theinterface x5x6x3x4 such that it approximates the flux through the interface x1x2x3x4.


(a)z

z

x1

x2

x3

x4

x5

x6

x7

x8

(b)z

zx1

x1x2

x2

x3x3

x4x4

x5

x5x6

x6

x7

x7

x8

x8

x9

x10

x11

x11

x12

Figure 19: Example for boundary treatment with hanging nodes. The blue shadowedarea marks the boundary faces. (a) MPFA flux area (red) and sub-fluxes (red arrows)- Fluxes are not calculated by the MPFA for the whole cell face! (b) MPFA flux area(red) only for the boundary cells.

According to Section 2.3 this means

f =|s|∑i|si|

∑i

fi (4.1)

where |si| is the area of the flux sub-interface i and |s| the area of face x1x2x3x4.However, instead of two cells sharing one face we now have five cells. Thus, we haveto weight separately for each of the five cells (Figure 19(b)). The two upper left cellsintersecting at sub-faces x8x11x12x4 and x11x7x3x12 are not located directly at the

boundary. For these cells, we have|s|∑i|si|

= 1.0. For the remaining cells, the flux is

approximated by Equation (4.1) with:• one right coarse cell: i = i ∈ x5x6x3x4 and |s| = area of face x1x2x3x4;• two lower left fine cells:

i = i ∈ x5x10x11x8 and |s| = area of face x1x9x11x8 , and

i = i ∈ x10x6x7x11 and |s| = area of face x9x2x7x11 .

Equation (4.1) is general and can be applied to any kind of boundary configuration,with fi being the sub-flux through one sub-interface calculated from non-boundaryMPFA interaction volumes, |s| the cell face area and |si| the area of sub-interface i. Ifa face does not intersect with a boundary, |s| =

∑i|si| and f =

∑ifi.

5 Two-phase flow applicationsIn this section the 3D MPFA L-method is applied to two-phase flow in porous media.Various discretization schemes, like the mimetic finite difference method used for the


comparison in Section 3, are able to handle elliptic equations with complex coeffi-cients on complex grids. However, in case of two-phase flow additionally the correctand conservative transport of phases, described by a parabolic transport equation, isimportant. Therefore, physical features like saturation discontinuities at fluid fronts orat heterogeneous material interfaces have to be reproduced by the numerical scheme.For regular K-orthogonal grids, the finite volume method with two-point flux approx-imation (TPFA) is a very powerful scheme, which is naturally very well suited toaccount for effects like saturation discontinuities at material interfaces (capillary pres-sure interface conditions [25]). For non-K-orthogonal meshes or at interfaces withhanging nodes, for example from local grid adaption, the correct flux approximation isdifficult or not possible. Alternative methods like MFD are very flexible with regard togrids and anisotropic coefficients. In the simplified case, in which we neglect capillary-pressure effects, the two-phase pressure equation (5.8) can be formulated in a similarform as the previously discussed one-phase equation (2.1). Thus, the application ofMFD and other mixed methods is straightforward. However, in the complex case withcapillary pressure, an additional term appears in the pressure equation. Correct treat-ment of capillary effects, especially in the case of heterogeneous capillary-pressurefunctions, is challenging. The flux approximation depends on the capillary-pressurevalues at the cell faces. To approximate these values at the heterogeneous materialinterfaces, additional equations, conditions and thus also degrees of freedoms could beadded to the system of equations or solved separately as an additional system of equa-tions. The latter is for example proposed in [27] for a MFE method. To the authors’knowledge, no MFD method has been published that solves two-phase problems onheterogeneous domains with capillary pressure and gravity. For the proposed MPFAmethod, no additional equations or degrees of freedom are needed. In the followingtest cases, we compare the MPFA L-method to the TPFA. The intention is to investi-gate the capabilities of the L-scheme by verifying the MPFA L-model and showing thesuperiority of the MPFA over the TPFA. The test cases are an advection-dominatedproblem (Buckley-Leverett-type problem [13]), a capillary-diffusion-dominated prob-lem (McWhorter-type problem [32]), and two DNAPL (Denser-Non-Aqueous-Phase-Liquid) infiltration scenarios. The tests include homogeneous as well as heteroge-neous, and isotropic as well as anisotropic porous media, without or with capillarypressure on structured and unstructured grids.

5.1 Two-phase model description

Assuming immiscible incompressible fluids under isothermal conditions, the two-phase flow system can be described by the mass conservation equations of each fluid


phase:

φ∂Sw

∂t+∇ · vw = qw, (5.1)

φ∂Sn

∂t+∇ · vn = qn. (5.2)

The two fluid phases are indicated by the subscripts w (wetting) and n (non-wetting),S is the saturation, φ is the porosity of the porous medium and q a source term. Withthe constitutive relation Sw + Sn = 1, Equations (5.1) and (5.2) can be combined toone pressure equation

∇ · vt = qt, (5.3)

where vt = vw + vn is the total velocity. To solve the system of the pressure equation(5.3) and the saturation transport equation (5.1), additional constitutive relations areneeded:

vα = −krα

µαK∇Φα, α = w, n, (5.4)

pc = pn − pw. (5.5)

Equation (5.4) is Darcy’s law for multi-phase flow (e.g., [24, 38]), which relates thefluid phase velocity vα to the phase potential Φα. Further parameters are the absolutepermeability of the porous medium K, the relative permeability of a phase krα, andthe dynamic fluid viscosity µα. Equation (5.5) relates the phase pressures pα to eachother by the capillary pressure pc. The capillary pressure as well as the relative phasepermeabilities are commonly assumed to be functions of the saturation (pc := pc (Sw),krα := krα (Sα)). The phase potentials are defined as

Φw = pw + %wg z,

Φn = pn + %ng z(5.6)

with g being the gravity constant and z the distance to the reference plane orthogonalto the direction of gravity. Inserting Equations (5.4)-(5.6) into Equation (5.3) yields

∇ · [−λtK (∇pw + fn∇pc)− (λw%w + λn%n) g∇z] = qt, (5.7)

where λα =krα

µαare phase mobilities, λt = λw+λn is the total mobility, and fα =

λαλt

are the phase fractional flow functions. Alternatively, the equation can be formulatedfor potentials [27, 46]

∇ · [−λtK∇Φw − fnλtK∇Φc] = qt, (5.8)

where the capillary potential is defined as

Φc = Φn −Φw = pc + (%n − %w)g z. (5.9)


The 3D MPFA L-model for two-phase flow used for the following examples is basedon the potential formulation (5.8). Advantages of the potential formulation in generaland combined with a MPFA method are discussed in Wolff et al. [46]. Pressure andsaturation equations are solved sequentially applying an IMPES (IMplicit PressureExplicit Saturation, e.g. [10]) algorithm.

5.2 Buckley-Leverett-type problem

qn

Sw = 0.8

pw

Sw,init = 0.2

q = 0

q = 0

Figure 20: Setup of the Buckley-Leverett problem. Blue arrow: wetting fluid inflow;red arrow: non-wetting fluid outflow.

The setup of this example is shown in Figure 20. The domain has a constant poros-ity, a constant permeability (φ = 0.2; K = kI, where k = 10−7 m2, I is the identitymatrix) and is initially saturated by a non-wetting fluid. From one side a wetting fluidis injected and displaces the non-wetting fluid. Both fluids have the same fluid proper-ties. Relative permeabilities are calculated using Brooks-Corey parameterizations withλ = 2.0 and residual saturations Swr = Snr = 0.2. Gravity as well as capillary effectsare neglected. The reference solution of this problem calculated on a very fine struc-tured grid is shown in Figure 21(b). To test the capabilities of the MPFA L-method, anunstructured grid (Figure 21(a)) is used. Figure 22 shows a comparison of the results

(a) (b)

Figure 21: Simulation grid of level 1 (a) and the reference solution (b) of the Buckley-Leverett problem.

of the TPFA method and the L-method. From top to bottom the grid shown in Figure21(a) is hierarchically refined from level 1 to level 4. It is obvious that the L-methodconverges to the reference solution with a limited number of grid refinements. In com-parison, the TPFA method is not able to calculate correct fluxes for the unstructuredgrid and exhibits a curved shock front (Figure 23).


(a) (b)

Sw

Figure 22: Water saturation of the Buckley-Leverett problem using (a) the TPFAmethod and (b) the MPFA L-method. From top to bottom the grid is refined fromlevel 1 to level 4.

(a)

(b) (c)

Figure 23: 2D slices in the x-z-plane, showing the water saturation of the Buckley-Leverett problem on the finest grid: (a) the reference solution, (b) the TPFA solution,(c) the MPFA L-solution.

5.3 McWhorter-type problem

Sw = 1.0

pn

Sw,init = 0.0

q = 0

q = 0 q = 0

Figure 24: Setup of the McWhorter problem. Blue arrow: wetting fluid inflow; redarrow: non-wetting fluid outflow.

The McWhorter problem is a capillary-diffusion-dominated flow problem. Themodel domain is initially fully saturated by a non-wetting fluid (Figure 24). All bound-


aries except one are no flow boundaries. The open boundary is a contact area betweenthe non-wetting fluid inside the domain and a wetting fluid outside the domain. Dueto the capillary-pressure gradient, counter-current flow develops. Porosity as well aspermeability are constant (φ = 0.3; K = kI, where k = 10−10 m2). Again bothfluids have the same fluid properties and relative permeabilities are parameterized byBrooks-Corey functions (λ = 2.0, Swr = Snr = 0). Gravity is neglected. The ref-erence solution is shown in Figure 25(b). Like for the Buckley-Leverett problem anunstructured grid is used (Figure 25(a)) and the L-method is compared to the TPFAmethod.

(a) (b)

Figure 25: Simulation grid of level 1 (a) and the reference solution (b) of theMcWhorter problem.

(a) (b)

Sw

Figure 26: Water saturation of the McWhorter problem using (a) the TPFA methodand (b) the MPFA L-method. From top to bottom the grid is refined from level 1 tolevel 4.

Also for the capillary-diffusion-dominated case the MPFA L-method converges tothe reference solution when the grid is refined (Figure 26(b)), while the front of theTPFA method is tilted following the direction of the skewed unstructured grid cells(Figure 27).


(a)

(b) (c)

Figure 27: 2D slices in the x-z-plane, showing the water saturation of the McWhorterproblem on the finest grid: (a) the reference solution, (b) the TPFA solution, (c) theMPFA L-solution.

5.4 DNAPL infiltration problem

Finally, we test the grid adaptive MPFA L-method on two DNAPL infiltration scenar-ios. In case 1 the porous medium is assumed to be heterogeneous but isotropic, in case2 it is assumed to be homogeneous but anisotropic.

The model domain of case 1 is shown in Figure 28. It is initially saturated withwater. DNAPL infiltrates from the top of the domain at the location of the red arrow.The bottom is impermeable and on the side boundaries we apply hydrostatic pressureconditions. Three low permeable lenses are marked by the anthracite boxes. The back-ground permeability and entry pressure are Kb = kbI with kb = 10−10 m2, pd,b = 500Pa. Permeabilities and entry pressures of the lenses are Kl = klI with kl = 10−14 m2,pd,l = 5000 Pa. Relative permeabilities and capillary pressure are parameterized byBrooks-Corey functions. Porosity as well as Brooks-Corey parameter λ are constantthrough the whole domain (λ = 2.0, φ = 0.4). The domain size is chosen to be 20 ×20 × 10 meters. We generate a structured coarse grid of 10 × 10 × 10 cells, which

qn

outflowpw = %wgz

outflowpw = %wgz

Sw,init = 1.0

q = 0

q = 0q = 0

Figure 28: Problem setup of the DNAPL infiltration example. The red arrow indicateswhere the DNAPL is released. The gray boxes are lenses of lower permeability andhigher entry pressure.


just resolves the lenses. For the TPFA model this grid is globally refined twice leadingto a grid of 40 × 40 × 40 cells. The adaptive MPFA L-model uses the coarse grid, butis allowed to locally refine the grid twice.

The results of DNAPL case 1 are plotted in Figure 29. Sub-figures 29(a) - 29(c)show the results of the TPFA method and Sub-figures 29(d) - 29(f) of the adaptiveMPFA L-method. Both methods show similar DNAPL saturation distributions. Theyaccount for the heterogeneities and also satisfy the interface condition due to thecapillary-pressure heterogeneity (see e.g., [25]). However, the TPFA is more diffu-sive than the L-method. Considering the pooling on top of the lenses, the DNAPLlayer from the L-model mainly consists of one cell layer while the DNAPL layer fromthe TPFA model consists of two or more cell layers. A possible reason is that dueto the larger flux stencil the ability of the L-method to deflect the flow at the lens ishigher. The same effect can be observed where the DNAPL flows round the corners ofthe lenses. The adaptive local grid refinement of the L-model is visualized in Figure

(a) t = 14 tend (b) t = 1

2 tend (c) t = tend

(d) t = 14 tend (e) t = 1

2 tend (f) t = tend

Sn

Figure 29: DNAPL infiltration case 1: (a) - (c) show the results of the TPFA method,(d) - (f) show the results of the adaptive MPFA L-method.

30. As can be observed from Figure 30(a), mostly the coarsest grid cells are used atthe early time of the simulation. Later, more and more cells are refined following theDNAPL distribution.

The setup of case 2 is similar to case 1 (Figure 28). DNAPL still infiltrates fromthe top at the location of the red arrow. Boundary conditions remain the same. How-ever, instead of the heterogeneities we now introduce anisotropy. The homogeneous


(a) t = 14 tend (b) t = tend

Figure 30: Locally adapted grids of the adaptive MPFA model for DNAPL case 1.

anisotropic permeability tensor is defined as

K =

10−10 0 −5× 10−11

0 10−10 5× 10−11

−5× 10−11 5× 10−11 5× 10−11

m2.

Further, a constant entry pressure pd = 5000 Pa is used in the entire domain. For thiscase where no lenses have to be resolved by the grid, a structured coarse grid of 2 ×2 × 2 cells is generated. This is globally refined to a 32 × 32 × 32 grid for the TPFAmodel. The adaptive MPFA L-model again uses the coarse grid but is allowed to refine4 times. Figure 31 shows the results of DNAPL case 2, where (a) and (b) are results ofthe TPFA model and (c) and (d) are of the adaptive L-model. The TPFA clearly fails toaccount for the anisotropy in the off-diagonal direction. The DNAPL just follows thedirection of gravity, which is the driving force, and pools at the impermeable bottom.The DNAPL distribution calculated by the L-model also flows towards the bottom ofthe domain but following the diagonal direction given by the off-diagonal entries of theanisotropic permeability tensor. Obviously, the TPFA can not be used to validate theadaptive L-method for this case. Thus, the results of the adaptive model are comparedto a reference solution using the L-method calculated on a structured grid of 32× 32×32 cells, which have the same size as the finest cells of the adapted grid. Additionally,a coarser structured grid of 16 × 16 × 16 cells is used, whose number of degrees offreedom has the order of the averaged number of degrees of freedom of the adaptedgrid. Figure 32 clearly shows that the adaptive scheme obtains the accuracy of the finestructured grid which can not be achieved by the coarser structured grid of a similarsize.

Two grids of the adaptive L-model for case 2 at t = 12 tend and t = tend are shown

in Figure 33. They clearly demonstrate the potential of grid adaptive methods withrespect to efficiency. Especially, when large parts of a model domain are not affectedby moving fluid fronts, the local grid refinement and coarsening can provide a fineresolution of the fronts while still having feasible grid sizes. This increase of efficiency



(c) t = 12 tend (d) t = tend

Sn

Figure 31: DNAPL infiltration case 2: (a) - (b) show the results of the TPFA methodand (c) - (d) of the adaptive MPFA L-method

also allows to use more complex methods like MPFA methods for realistic scenarioswhere they otherwise could be too expensive.

Refinement and coarsening indicator The quality and applicability of an adaptivemethod is highly affected by the criteria used for refinement and coarsening. In all ourexamples, the following local-gradient-based criterion is used:

refine =

true, if ∆SIi ≥ Erefine

false, else, (5.10)

coarsen =

true, if ∆SIi < Ecoarsen

false, else. (5.11)

Here, index i is the cell index and ∆SIi is defined as

∆SIi =max(∆Sij)

∆Smax, (5.12)

where ∆Sij is the saturation difference between cell i and a neighboring cell j and∆Smax is the global maximum of ∆Sij used for normalization. Further, the refinementthreshold Erefine is defined as

Erefine = εrefine∆Smax

Smax − Smin, (5.13)


(a) t = 12 tend

(b) t = tend

(c) t = 12 tend

(d) t = tend

(e) t = 12 tend

(f) t = tend

Sn

Figure 32: DNAPL infiltration case 2: Comparison of the results using a locally refinedgrid (c) - (d), a coarse structured grid (a) - (b), and a fine structured grid (e) - (f).


Figure 33: Locally adapted grids of the MPFA model for DNAPL case 2 at differentsimulation times.

where εrefine is a threshold value which has to be set (usually between 0 and 1) tocontrol the refinement, and Smax and Smin are the global maximum and minimum,respectively, of the saturation. Similarly, Ecoarsen is defined as

Ecoarsen = εcoarsen∆Smax

Smax − Smin, (5.14)

where εcoarsen <= εrefine is the threshold value to control the coarsening (usuallybetween 0 and 1) . In case of a pure shock front (e.g. if fα are linear functions)∆Smax = Smax − Smin and the adaption behavior is directly controlled by εrefine andεcoarsen. In all other cases εrefine and εcoarsen are additionally weighted to account for


the fluid distribution. Experience has shown that values of εrefine = 0.5, . . . , 0.9 andεcoarsen = 0.01, . . . , 0.5 usually lead to robust adaption behavior. For the DNAPL infil-tration examples we additionally force the grid to be refined at the area of the DNAPLinflow to correctly resolve the boundary condition.

6 Summary and conclusionsWe have generalized the well-known MPFA L-method in 3D for single and two-phaseflow systems to meshes with hanging nodes and applied these pressure-based locallyconservative schemes to realistic porous media flow scenarios. A simplified bound-ary treatment is proposed based on the combination of TPFA techniques and suitableweighting factors. This method is quite flexible and easily applicable to more com-plex boundary configurations and adaptively refined grids. We further show how theL-method has to be modified in the presence of hanging nodes which appear throughthe non-conforming grid refinement. As shown by our numerical results, the standardL-selection criterion fails in the case of meshes with highly skewed elements. This ob-servation motivates the introduction of a new selection criterion which is much morerobust and provides also in these cases quite accurate numerical results.

Firstly, the L-scheme has been extensively tested for the elliptic one-phase flowpressure equation. The numerical convergence has been investigated by means ofbenchmark test problems designed for the FVCA6 conference. Additionally to thegiven exact solution, the L-method has been compared to the TPFA and the MFDmethod. The L-method is able to solve both benchmark problems on all grids withsufficient accuracy. The robustness of the method is closely related to the choice of theL-stencil and thus to the L-selection criterion. In most cases both tested criteria leadto robust L-stencils. However, in case of strong non-K-orthogonality which appearsat regions of large skewness only the newly proposed criterion could obtain the cor-rect solution for benchmark test 1. As expected, the MPFA is superior to the TPFA incase of non-K-orthogonal grids. Regarding the convergence order, the MFD methodshows advantages compared to the L-method. For benchmark test 3 both methodsshow a O(h2) convergence of pressure and normal velocity, but for test 1 the MFDmethod converges with O(h2), while the L-method convergences only with O(h1).The impact of the simplified boundary treatment can be neglected in many cases, how-ever not in the situation of the random mesh with the non-planar surface which isused for test 3. Considering more realistic applications, for example in the field ofreservoir simulation, such complexity at the boundaries usually does not occur. Addi-tionally, important processes often take place somewhere in the domain and not at theboundaries. Therefore, in most application-relevant settings the simplified boundarytreatment can be justified.

Secondly, we have extended the 3D MPFA L-model to incompressible, isothermal,immiscible two-phase flow. In this case, a parabolic phase transport equation is weaklycoupled to the elliptic two-phase pressure equation, leading to additional requirements


for the numerical scheme. It has to be locally mass conservative and be able to accountfor discontinuous shock fronts as well as for diffusive fronts in capillary-pressure-dominated domains. An additional challenge is the treatment of material interfaces,e.g. between areas of different entry pressures. While there exists a variety of al-ternative methods like MFD that perform very well for the pressure equation withoutcapillary effects, only a few successful efforts have been undertaken up to now thatcan handle more complex settings. We have tested the MPFA L-method on two well-known benchmark problems for two-phase flow using unstructured, non-K-orthogonalgrids. In these tests, the TPFA fails as expected from the theory, while the MPFA L-method approximates the correct solution very well for both, the advection-dominatedas well as the capillary-diffusion-dominated case. Two more tests demonstrate the ap-plicability of the L-model to realistic problems like a DNAPL infiltration scenario. Inthe first case, the L-method is validated for the case of gravity-dominated flow includ-ing heterogeneities. Comparing the TPFA and the MPFA, both models show similarresults and account for the heterogeneity as well as they satisfy the capillary-pressureinterface condition. However, due to the larger flux stencil the L-method is superiorin modeling the sharp changes in direction of the streamlines due to blocking effectsat the heterogeneous interfaces. In the second case, the advantage of the L-method incase of an anisotropic porous medium leading to a non-K-orthogonal grid is obvious.

Realistic two-phase flow scenarios in porous media require a numerical model whichis flexible with regard to complex geometry of the domain and the domain boundaries,which is able to account for important two-phase flow specific features, and which isefficient enough to run simulations in reasonable time. We have demonstrated that theMPFA L-model can satisfy these requirements by combining it with the flexibility ofTPFA along the boundary and by using locally adaptive grids.

Acknowledgments. The authors would like to thank the German Research Foun-dation (DFG) for financial support of the projects within the Cluster of Excellencein Simulation Technology (EXC 310/1) at the University of Stuttgart and the Inter-national Research Training Group “Nonlinearities and Upscaling in Porous Media”(GRK 1398/2).

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Appendix

In this appendix, the detailed transmissibility matrices T = CA−1B +D for the L-stencils (b) (c) (d) in Figure 2 are also given which may be helpful to the researchersinterested in the L-method implementation. For the definition of the matrix elements,the reader is referred to Section 2.1.

L-stencil (b)

C =

−ω121 −ω122 −ω123

−ω221 −ω222 −ω223

−ω321 −ω322 −ω323

, D =

0 ω121 + ω122 + ω123 0 00 ω221 + ω222 + ω223 0 00 ω321 + ω322 + ω323 0 0

,

A =

ω121 − ω112−ω111r211 − ω113r212

ω122 − ω111r221−ω113r222

ω123 − ω111r231−ω113r232

ω221 − ω242r211−ω243r213

ω222 − ω241−ω242r221 − ω243r223

ω223 − ω242r231−ω243r233

ω321 − ω361r213−ω362r212

ω322 − ω361r223−ω362r222

ω323 − ω363−ω361r233 − ω362r232

,

B =

−(ω111+ω112 + ω113)

ω121 + ω122 + ω123+ω111(1 − r211 − r221 − r231)+ω113(1 − r212 − r222 − r232)

0 0

0ω221 + ω222 + ω223+

ω242(1 − r211 − r221 − r231)+ω243(1 − r213 − r223 − r233)

−(ω241+ω242 + ω243)

0

0ω321 + ω322 + ω323+

ω361(1 − r213 − r223 − r233)+ω362(1 − r212 − r222 − r232)

0 −(ω361+ω362 + ω363)

.

L-stencil (c)

C =

−ω111 − cω113r231r112−

cω113r211−cω113r221 −ω112 − cω113r231r122

−ω221 − cω223r112−cω223r132r211

−ω222 − cω223r132r221 −cω223r122

−ω311 − cω313r231r112−cω313r211

−cω313r221 −ω312 − cω313r231r122

,

D =

cω113r231(r112 + r122 + r132 − 1)+ω111 + ω112 + ω113

cω113(r211 + r221+r231 − 1) 0 0

cω223(r112 + r122 + r132 − 1)ω221 + ω222 + ω223+cω223r132(r211 + r221+

r231 − 1)0 0

cω313r231(r112 + r122 + r132 − 1)+ω311 + ω312 + ω313

cω313(r211 + r221+r231 − 1) 0 0

,


A =

ω111 − ω121+cω113(r231r112 + r211)−cω123(r112 + r132r211)

cω113r221 − ω122−cω123r132r221

ω112 + cω113r231r122−cω123r122

ω221 − ω243r214 + c(ω223−ω243r234)(r112 + r132r211)−cω242(r231r112 + r211)

ω222 − ω243r224 − ω241+c(ω223 − ω243r234)r132r221

−cω242r221

cr122(ω223 − ω243r234−ω242r231)

ω311 − ω353r113 + c(ω313−ω353r133)(r231r112 + r211)−cω352(r112 + r132r211)

cr221(ω313 − ω353r133−ω352r132)

ω312 − ω351 − ω353r123+c(ω313 − ω353r133)r231r122

−cω352r122

,

B =

c(ω113r231 − ω123)·(r112 + r122 + r132 − 1)+

ω111 + ω112 + ω113

c(ω113 − ω123r132)·(r211 + r221 + r231 − 1)−

(ω121 + ω122 + ω123)0 0

c(r112 + r122 + r132 − 1)·(ω223 − ω243r234−

ω242r231)

ω221 + ω222 + ω223−ω243(r214 + r224 + r234 − 1)+c(r211 + r221 + r231 − 1)·(ω223r132 − ω243r234r132

−ω242)

−(ω241+ω242 + ω243)

0

c(r112 + r122 + r132 − 1)·(ω313r231 − ω353r133r231−ω352) + ω311 + ω312+ω313 − ω353(r113+r123 + r133 − 1)

c(r211 + r221 + r231 − 1)·(ω313 − ω353r133−

ω352r132)0 −(ω351+

ω352 + ω353)

.

Here, the constant c =1

1 − r231r132.

L-stencil (d)

C =

−ω111 − cω113r212−

cω113r232r111−ω112 − cω113r232r121 −cω113r222

−ω211 − cω213r212−cω213r232r111

−ω212 − cω213r232r121 −cω213r222

−ω321 − cω323r111−cω323r131r212

−cω323r121 −ω322 − cω323r131r222

,

D =

cω113r232(r111 + r121 + r131 − 1)

+ω111 + ω112 + ω113cω113(r212 + r222 + r232 − 1) 0 0

cω213r232(r111 + r121 + r131 − 1)+ω211 + ω212 + ω213

cω213(r212 + r222 + r232 − 1) 0 0

cω323(r111 + r121 + r131 − 1) cω323r131(r212 + r222 + r232 − 1)+ω321 + ω322 + ω323

0 0

,

A =

ω111 − ω121+cω113(r212 + r232r111)−cω123(r111 + r131r212)

ω112 + cω113r232r121−cω123r121

cω113r222 − ω122−cω123r131r222

ω211 − ω233r114+c(ω213 − ω233r134)(r212 + r232r111)−

cω232(r111 + r131r212)

ω212 − ω231 − ω233r124+c(ω213 − ω233r134)r232r121

−cω232r121

cr222(ω213−ω233r134 − ω232r131)

ω321 − ω363r213+c(ω323 − ω363r233)(r111 + r131r212)−

cω362(r212 + r232r111)

cr121(ω323 − ω363r233−ω362r232)

ω322 − ω361 − ω363r223+c(ω323 − ω363r233)r131r222

−cω362r222

,


B =

c(ω113r232 − ω123)·(r111 + r121 + r131 − 1)+ω111 + ω112 + ω113

c(ω113 − ω123r131)·(r212 + r222 + r232 − 1)−(ω121 + ω122 + ω123)

0 0

ω211 + ω212 + ω213−ω233(r114 + r124 + r134 − 1)+c(r111 + r121 + r131 − 1)·

(ω213r232 − ω233r134r232 − ω232)

c(r212 + r222 + r232 − 1)·(ω213 − ω233r134−

ω232r131)

−(ω231+ω232 + ω233)

0

c(r111 + r121 + r131 − 1)·(ω323 − ω363r233 − ω362r232)

ω321 + ω322 + ω323−ω363(r213 + r223 + r233 − 1)+c(r212 + r222 + r232 − 1)·(ω323r131 − ω363r233r131

−ω362)

0 −(ω361+ω362 + ω363)

.

Here, the constant c =1

1 − r232r131.

Author informationMarkus Wolff, Universität Stuttgart, Department of Hydromechanics and Modeling ofHydrosystems, Pfaffenwaldring 61, 70569 Stuttgart, Germany.E-mail: [email protected]

Yufei Cao, Universität Stuttgart, Department of Hydromechanics and Modeling ofHydrosystems, Pfaffenwaldring 61, 70569 Stuttgart, Germany.E-mail: [email protected]

Bernd Flemisch, Universität Stuttgart, Department of Hydromechanics and Modeling ofHydrosystems, Pfaffenwaldring 61, 70569 Stuttgart, Germany.E-mail: [email protected]

Rainer Helmig, Universität Stuttgart, Department of Hydromechanics and Modeling ofHydrosystems, Pfaffenwaldring 61, 70569 Stuttgart, Germany.E-mail: [email protected]

Barbara Wohlmuth, Centre for Mathematical Sciences, Technical University of Munich,Boltzmannstr. 3, 85748 Garching, Germany.E-mail: [email protected]

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Multi-point flux approximation L-method in 3D: Numerical ... · Multi-point flux approximation L-method in 3D: Numerical convergence and application to two-phase flow through porous