Lattice-Boltzmann simulations in reconstructedparametrized porous media

BENJAMIN AHRENHOLZ*, JONAS TOLKE† and MANFRED KRAFCZYK‡

Institute of Computer Applications in Civil Engineering, Technical University of Braunschweig, Pockelsstr.3, 38106 Braunschweig, Germany

(Received 30 September 2005; in final form 22 May 2006)

Computations of flows in explicitly resolved porous media reported in the literature so far are based onbinarized porous media data mapped to uniform Cartesian grids. The voxel set is directly being used asthe computational grid and thus the geometrical representation is usually only first-order accurate dueto stair-case patterns. In this work, we pursue a more elaborate approach: starting from a highlyresolved tomographic grey value data set we utilize a Marching Cube algorithm to reconstruct thesurface of the porous medium as a set of planar triangles. The numerical resolution of the Cartesian gridfor the simulation can then be chosen independently from the voxel set. As we take into account thesubgrid distances between the nodes of the Cartesian grid and the planar triangle surfaces, one canutilize a second-order accurate lattice Boltzmann flow solver to efficiently compute, e.g. permeabilities.As these interpolation-based no-slip boundary conditions are not mass preserving, we also present alocal modification of the no-slip boundary condition restoring mass conservation. Our numerical resultsdemonstrate that for saturated flow simulations this coupled approach allows a substantial accelerationof saturated flow computations in porous media.

Keywords: Lattice-Boltzmann; Porous media; Reconstructed geometry; Marching cubes

1. Introduction

Numerical models based on the lattice Boltzmann equation

(LBE) have matured as a well established tool for solving

complex problems in fluid dynamics. A popular LBE

model for simulating flows in porous media is the well

known lattice Bhatnagar–Gross-Krook (LBGK) model

(Qian et al. 1992), in combination with a simple bounce

back scheme for the solid–fluid boundaries (Ginzburg and

d’Humieres 2003). This approach has two disadvantages.

First, its numerical stability is relatively limited. Second,

the simplified nature of the bounce back scheme implies

that the true position of the wall is depending on the value

of the relaxation time used for the simulation, i.e. the

permeability of a porous medium becomes viscosity

dependent (Ginzbourg and Adler 1994). These problems

can be drastically reduced by using a multiple-relaxation-

time model (d’Humieres 1992, Lallemand and Luo 2000,

d’Humieres et al. 2002), which not only improves

numerical stability but also eliminates the viscosity

dependence of the permeability (Ginzburg and d’Humieres

2003). Furthermore, the simple bounce back scheme has to

be replaced by a boundary condition whose convergence

behaviour is consistent with the methods bulk behaviour,

e.g. Bouzidi et al. (2001). Simulations investigating flow

in porous media using multi relaxation time (MRT) and

higher-order boundary conditions have been carried out

for simple analytic geometries such as body centred arrays

of spheres (Ginzburg and d’Humieres 2003), as well for

polydisperse sphere packings (Pan et al. 2006). In this

paper, we combine a MRT-LB model with second-order

boundary conditions for the simulation of single-phase

flow in reconstructed porous media. The paper is organized

as follows: in Section 2, we describe the reconstruction of

the geometry of porous media by using a Marching Cube

algorithm to decouple the numerical grid from its voxel-

based data. Section 3 describes the numerical model and its

implementation. In Section 4.1 numerical experiments

for a body centred cubic array of spheres to validate the

approach have been carried out. In Section 4.2 flow

International Journal of Computational Fluid Dynamics

ISSN 1061-8562 print/ISSN 1029-0257 online q 2006 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/10618560601024694

*Corresponding author. Email: ahrenholz@cab.bau.tu-bs.de.†Email: toelke@cab.bau.tu-bs.de‡Email: kraft@cab.bau.tu-bs.de

International Journal of Computational Fluid Dynamics, Vol. 20, No. 6, July 2006, 369–377

simulations in reconstructed parameterized porous media

using second-order boundary conditions have been

performed and demonstrate the efficiency of our approach.

2. Decoupling the numerical grid from the

tomography-based voxel set

2.1 Motivation

Most fluid simulations in resolved porous media that are

based on natural or real-world geometries are derived from

tomography scans or similar extraction methods. These

tomography scans are usually converted to binary voxel

sets, which are directly utilized as the computational grid

and thus have the same resolution as the voxel matrix. The

immediate consequence of this method is an implicit

coupling of the resolution of the geometry with the

computational (i.e. numerical) grid. This has two distinct

disadvantages: (a) an intrinsical stair-case representation

of the geometry; and (b) convergence studies are hard to

perform, because from a numerical point of view, it would

be necessary to have scans of different resolutions. To

evade these limitations we are introducing an approach,

which is based on voxel geometries (Lehmann et al. 2006)

represented by grey values and that will reconstruct a plane

triangle mesh to obtain a second-order accurate geometric

representation of the pore space. A sample of a porous

medium (Kaestner et al. 2005) with a dimension of 8003

voxels and a triangulated section of the same sample are

shown in figures 1 and 2.

2.2 Isocontour algorithm

For triangulation one can utilize different types of

isocontour/isosurface algorithms. In this work, we

adopted a variant of the Marching Cube family, which

for our application is favourable due to its ability to

conserve topological quantities to a large extent (Lewiner

et al. 2003) and which is designed to avoid cracks in the

surface. Marching Cubes (Lorensen and Cline 1987) is a

computer graphics algorithm for extracting a polygonal

mesh of an isosurface from a 3D discrete scalar field. The

algorithm successively scans the scalar field, groups 8

voxels to an imaginary cube and determines the polygons

whose cutting plane represents the isosurface of the

predefined scalar value in this cube. This computation can

be accelerated by creating an index array of the 28 ¼ 256

possible polygon configurations which serve as a look-up

table in all subsequent cube computations. The look-up

table can be further condensed by symmetry arguments to

as few as 15 unique cases. These basic configurations are

shown in figure 3.

Finally, each vertex of the generated polygons is placed

on the appropriate position along the cube’s edge by

linearly interpolating the two scalar values of its

corresponding two nodes. The result is a mesh of triangles

representing the geometric structure of a voxel set.

Especially, in the case of a natural porous medium with

its arbitrary geometrical complexity it is crucial to have a

consistent mesh of triangles, because cracks will lead to

leaks which again will lead to wrong surface orientations

Figure 1. X-ray scan of a porous media measured at HASYLAB (courtesy of Kastner and Lehmann).

B. Ahrenholz et al.370

(a pore may be interpreted as solid and vice versa). It

turned out that even our implemented type of Marching

Cubes can lead to wrong triangle configurations under

certain conditions. In that case, we are forced to correct

the mesh by checking the connectivity of the triangle

mesh. If a triangle has less than three neighbouring faces,

it is part of rare triangulation configuration, which can

appear in very narrow channels. Then it is safe to delete

the affected triangle to keep the topological quantity. After

the geometric reconstruction the triangulation is used to

generate the numerical grid, whose resolution can now be

freely chosen. While the grid is adapted to the mesh, we

calculate the distances from the Cartesian grid nodes to the

triangles describing the outer walls. These distances will

be used later as so-called q-values for the boundary

conditions (see Section 3.2). A recursive fill algorithm

then separates fluid, boundary and solid nodes for the

upcoming simulation.

2.3 Calculation of subgrid distances

To apply higher-order boundary conditions based on linear

or quadratic interpolations (Bouzidi et al. 2001) or the

even more complex multi-reflection boundary scheme

Figure 3. Representation of the 15 base configurations of the Marching cube algorithm.

Figure 2. Section of a triangulated surface of a porous medium. Figure 4. Boundary nodes with subgrid distances qi.

LB simulation in porous media 371

(Ginzburg and d’Humieres 2003), the projected distances

between the boundary grid node and the reconstructed

surface along the links of the stencil are required. A

representation of boundary nodes (quadratic dots) in pore

space with their corresponding links qi, consistent with the

D3Q19 model, is shown in figure 4.

The links and their length have only been calculated once

as a part of the pre-processing, as the geometry of the porous

medium is assumed not to depend on time. For calculations

with voxel scans up to a size of 8003 the triangulation

algorithm in the pre-processing step required between 100

and 200 million triangles for a reconstructed surface for the

medium under consideration and generally depends on the

mediums topology as well its porosity. The time consump-

tion is moderate: triangulation and computation of the

subgrid distances requires about 4 h on a 2 GHz Opteron and

32 GB memory. The algorithm has an optimum complexity

of NðlogNÞ where N is the number of triangles.

3. The LB method for single-phase flow with second-

order accuracy

3.1 The multi relaxation time model (MRT)

In the following section x represents a 3D vector in space and

f a b-dimensional vector, where b is the number of

microscopic velocities. We use the so-called d3q19 model

(Qian et al. 1992) with the following microscopic velocities,

where c is a constant velocity related to the speed of sound by

c2s ¼ c 2=3. The generalized Lattice Boltzmann (GLB)

equation using the multi-relaxation time model introduced

by d’Humieres (1992) and Lallemand and Luo (2000) is

used in this paper in a slightly modified version (Tolke et al.

2006). The LBE is given by

f iðt þ Dt; xþ eiDtÞ ¼ f iðt; xÞ þVi;

i ¼ 0; . . . ; b2 1;ð1Þ

where Dt is the time step, the grid spacing is Dx ¼ cDt and

the collision operator is given by

V ¼ M21SððMfÞ2meqÞ: ð2Þ

The matrix M given in Appendix A transforms the

distributions into moment space. The resulting moments

m ¼ Mf are labelled as

m ¼ ðdr; e; 1; jx; qx; jy; qy; jz; qz; 3pxx; 3pxx; pww;

pww; pxy; pyz; pxz;mx;my;mzÞ;

where dr is a density variation related to the pressure

variation dp by

dp ¼ c2sdr; ð3Þ

and ðjx; jy; jzÞ ¼ r0ðux; uy; uzÞ the momentum, where r0

is a constant reference density. The moments

e; pxx; pww; pxy; pyz; pxz are related to the stress tensor by

sxx ¼ 2 1 2sn

2

� � 1

3eþ pxx 2 r0u

2x

� �ð4aÞ

syy ¼ 2 1 2sn

2

� � 1

3e2

1

2pxx þ

1

2pww 2 r0u

2y

� �ð4bÞ

szz ¼ 2 1 2sn

2

� � 1

3e2

1

2pxx 2

1

2pww 2 r0u

2z

� �ð4cÞ

sxy ¼ 2 1 2sn

2

� �ð pxy 2 r0uxuyÞ ð4dÞ

syz ¼ 2 1 2sn

2

� �ð pyz 2 r0uyuzÞ ð4eÞ

sxz ¼ 2 1 2sn

2

� �ð pxz 2 r0uxuzÞ: ð4fÞ

Here, sn is a collision rate explained later. The other

moments of higher-order have no physical meaning for the

incompressible Navier–Stokes equations. The vector meq

is composed of the equilibrium moments given by

meq0 ¼ Dr ð5aÞ

meq1 ¼ e eq ¼ r0 u2

x þ u2y þ u2

z

� �ð5bÞ

meq3 ¼ r0ux ð5cÞ

meq5 ¼ r0uy ð5dÞ

meq7 ¼ r0uz ð5eÞ

meq9 ¼ 3peq

xx ¼ r0 2u2x 2 u2

y 2 u2z

� �ð5fÞ

meq11 ¼ peq

zz ¼ r0 u2y 2 u2

z

� �ð5gÞ

{ei; i ¼ 0; . . . ; 18} ¼

0 c 2c 0 0 0 0 c 2c c 2c c 2c c 2c 0 0 0 0

0 0 0 c 2c 0 0 c 2c 2c c 0 0 0 0 c 2c c 2c

0 0 0 0 0 c 2c 0 0 0 0 c 2c 2c c c 2c 2c c

8>><>>:

9>>=>>;;

B. Ahrenholz et al.372

meq13 ¼ peq

xy ¼ r0uxuy ð5hÞ

meq14 ¼ peq

yz ¼ r0uyuz ð5iÞ

meq15 ¼ peq

xz ¼ r0uxuz ð5jÞ

meq2 ¼m

eq4 ¼m

eq6 ¼m

eq8 ¼m

eq16 ¼m

eq17 ¼m

eq18 ¼ 0: ð5kÞ

The matrix S is a diagonal collision matrix composed of

relaxation rates {si;i; . . . ;b2 1}, also called the eigen-

values of the collision matrix M21SM. The rates different

from zero are

s1;1 ¼2se s2;2 ¼2se

s4;4 ¼ s6;6 ¼ s8;8 ¼2sq

s10;10 ¼ s12;12 ¼2sp

s9;9 ¼ s11;11 ¼ s13;13 ¼ s14;14 ¼ s15;15 ¼2sn

s16;16 ¼ s17;17 ¼ s18;18 ¼2sm:

The relaxation rate sn is related to the kinematic viscosity

n by

sn ¼1

3 nc 2Dt

þ 12

: ð6Þ

The other relaxation rates se; se ; sq; sp and sm can be freely

chosen in the range ½0;2� and may be tuned to improve

accuracy and/or stability (Lallemand and Luo 2000). The

optimal values depend on the specific system under

consideration (geometry, initial and boundary conditions)

and cannot be computed in advance for general cases.

These values will be discussed in more detail in the

sections dealing with the numerical examples. Using

either a Chapman-Enskog expansion (Frisch et al. 1987)

or an asymptotic expansion using the diffusive scaling

(Junk et al. 2005), it can be shown that the LB method is a

scheme of first-order in time and second-order in space for

the incompressible Navier–Stokes equations.

3.2 Boundary conditions

The macroscopic flow quantities can only be set implicitly

via incoming particle distribution functions on the

boundary nodes. A well-known and simple way to

introduce no-slip walls is the so-called bounce back

scheme which allows spatial second-order accuracy if the

boundary is aligned with one of the lattice vectors ei and

first-order otherwise (Ginzburg and d’Humieres 2003). As

we have arbitrarily shaped objects, we use the modified

bounce back scheme developed in Bouzidi et al. (2001)

and Lallemand and Luo (2003) for velocity boundary

conditions, which is second-order accurate for arbitrarily

shaped boundaries (figure 5).

Here, we identify two cases:

(i) the wall has a distance less then 0.5 eiDt from the

node; and

(ii) the wall has a distance between 0.5 and 1.0 eiDt from

the node.

f tþ1IA ¼ ð1 2 2qÞ·f tiF þ 2q·f tiA 2 6

eiuw

c2; 0:0 , q , 0:5

ð7Þ

f tþ1IA ¼

2q2 1

2q·f tIA þ

1

2q·f tiA 2 3

eiuw

qc2; 0:5 # q # 1:0:

ð8Þ

Distributions at time t/t þ 1 are post/pre-collision values,

qeiDt is the distance to the wall and uw is the velocity at

the wall. Therefore, we obtain second-order accurate

results in space even for curved geometries (Geller et al.

2006). For a detailed discussion of LBE boundary

conditions we refer to Ginzburg and d’Humieres (2003).

In contrast to the simple bounce back scheme the use of

this interpolation based no-slip boundary conditions result

in a notable mass loss across the no-slip lines. To

circumvent this problem we transfer the mass difference to

1 q

F A W

q<1/2

2q

D

fi,Ft fi,A

t

1 q

F A W

q>1/2

2q-1

fI,At fi,A

t

F A WD

fI,At+1

F A W

fI,At+1

D

D

t+1

t

Figure 5. Interpolations for second-order bounce back scheme.

LB simulation in porous media 373

the rest particle distribution. This results in a bounce back

scheme which is conservative in mass (and thus pressure)

while introducing a higher-order disturbance of the stress

tensor which does not change the results significantly.

The results obtained with the first-order bounce back were

inferior to the second-order scheme which highlights the

importance of a proper geometric resolution of the flow

domain.

Pressure boundary conditions are implemented by

setting the incoming distributions as (Thurey 2003)

f I ¼ 2f i þ feqI ðP0; uÞ þ f

eqi ðP0; uÞ ð9Þ

where P0 is the prescribed pressure, eI ¼ 2ei and u is

obtained by extrapolation.

3.3 Implementation

A simple Lattice Boltzmann algorithm can be

implemented easily, however, more advanced approaches

in terms of accuracy, speed and memory consumption

require careful programming. The simulation kernel used

for the numerical experiments has been implemented

using the Fortran 90 standard. The efficiency of matrix-

based data structures and its convenient syntax for

numerical problems were the determining factor. The

parallelization follows a distributed memory approach

using a Message Passing Interface (MPI-Forum 2006).

The kernel has been optimized with respect to speed in

the first place, by e.g. using two arrays for storing the

distribution functions. Collision and propagation have

been combined into a single nested loop. The array

containing the subgrid distances (q-values) is indexed by a

list as they are only needed for the fluid–solid boundary

nodes. The simple bounce back scheme does not require

the exchange of all neighbouring distributions due to its

local nature. Therefore, it would be sufficient to exchange

only the five distributions pointing into communication

direction. However, while using the interpolated bounce

back schemes it is necessary to exchange also the

distributions necessary for the interpolation, which will

slightly, increase the parallelization overhead.

4. Numerical experiments

4.1 Body centred periodic array of spheres

Before running simulations in natural porous media with

the introduced framework we validated the approach with

an analytic geometry for which a semi-analytical solution

is available: a body centred periodic array of spheres as

shown in figure 6. For this configuration one can derive

a solution following the work of Hasimoto (1959) and

Sangani and Acrivos (1982).

The drag force of one sphere can be computed as:

F ¼ Cd·6pmuda ð10Þ

where Cd is the dimensionless drag, m the dynamic

viscosity and ud the Darcy velocity. The drag can be

determined as a function of the solid volume fraction c by

a series expansion:

Cd ¼X30

n¼0

anxn; x ¼

c

cmax

� �1=3

;

c ¼8pa3

3L3; cmax ¼

ffiffiffi3

pp

8:

ð11Þ

The drag only depends on the characteristics of the

geometry, which are defined by L and a (see also figure 7).

Figure 6. A BCC array of spheres in periodic space.

Figure 7. 2D projection of the sphere array.

B. Ahrenholz et al.374

The permeability of the medium can then be derived by

considering the region depicted in figure 7. Taking into

account that the region contains two spheres (Adler 1992)

in sum we can compute the average pressure gradient

in z-direction as

2›z p ¼2F

L3: ð12Þ

The Darcy velocity using equation (17) is then

u ¼k

m

2F

L3: ð13Þ

Substituting equation (10) in equation (13) we obtain for

the intrinsic permeability k

k ¼L3

12paCd

: ð14Þ

The dimensionless permeability k* is

k* ¼k

a2¼

1

12pCd

·L

a

� �3

: ð15Þ

The geometry used in the simulation has been modelled

using a common geometric modelling tool, such as

AutoCAD (2006) or Microstation (2006). The triangle

mesh was generated using the grid generation tool as

described in Section 2. Therefore, it was easy to create

grids in different resolutions using always the same input-

model. The smallest grid used 103 nodes and the largest up

to 1203.

The setup for the simulation is a LB-simulation core

using the MRT-model and the “magic”-relaxation

parameters given below according to Ginzburg (2006).

These parameters eliminate the permeability dependence

on viscosity for simulations using Stokes flow (i.e.

neglecting the second-order terms of the equilibrium

distributions) for simple bounce back and reduce them

substantially in the case of interpolation-based bounce

back by relaxing the even and odd moments differently:

se ¼ se ¼ sp ¼ sn; sq ¼ sm ¼ 8ð2 2 snÞ

ð8 2 snÞð16Þ

Permeability is determined by measuring the volume

averaged fluid velocity (Darcy velocity) after reaching a

stationary state. From Darcy’s law one can calculate the

fluid permeability:

ud ¼ 2k

rnð›z pþ rgÞ ð17Þ

which is then compared to the theoretical solution given

by Hasimoto (1959) and Sangani and Acrivos (1982).

Figures 8 and 9 show the comparison for different values

of x. Figure 8 reveals that the accuracy of the simulation

using the linear interpolated bounce back schemes for a

resolution of about 143 grid nodes is comparable to the one

from the truncated series expansion of the theoretical

solution (relative error below 1%).

The variations of the numerical results even for low

resolutions can be explained by the type of setup. The

numerical grid is intentionally not aligned to the geometry

but is created using a small offset, which is always as large

as half the grid spacing. This displacement of the grid

results in a non-symmetrical setup and is considered to fit

better to a natural porous medium and more importantly is

supposed to avoid the cancellation and therefore the

reduction of errors due to a symmetrical setup. At higher

resolutions this out of alignment will have less impact to

the obtained results, which explains the relatively high

error at low resolutions. The newly introduced simple

mechanism for a mass conservative interpolated bounce

back scheme gives the same order of convergence, but it is

not as accurate as the classical boundary condition.

However, it may be useful for setups where a mass

conservative scheme is mandatory and quadratic conver-

gence is required. The simple bounce back scheme cannot

compete here; for an accurate solution one would require

resolutions beyond 1003 which would require almost

three-orders of additional CPU-power and memory.

1,0E–02

1,0E–01

1,0E+00

1,0E+01

1,0E+02

10 100

resolution [voxel]

rel.

erro

r [%

]

N–2

N–1

Accuracy of Ref.-Sol.

K [SBB]

K [LIBB]

K [LIBB FIX]

Figure 8. Plot comparing the accuracy of simple and linear interpolatedBB with x ¼ 0:76.

1,0E–01

1,0E+00

1,0E+01

1,0E+02

10 100 1000

resolution [voxel]

rel.

erro

r [%

]

K [SBB]

K [LIBB Fix]

K [LIBB]

N–1

Accuracy of Ref.-Sol.

N–2

Figure 9. Plot comparing the accuracy of simple and linear interpolatedBB with x ¼ 0:96.

LB simulation in porous media 375

4.2 Convergence study for a natural porous medium

Present neutron tomography or X-rays from synchrotron

scanners are capable of scanning grey value images reaching

voxel resolutions of a few microns. Thus, one can perform

fluid simulations in natural porous media with voxel

resolutions up to 109 voxels. However, the demands of a

simulation with a computational grid of the same size are very

high. To demonstrate thegain of decoupling the numerical grid

from the geometrical voxel description we performed a

convergence study based on tomography scans of sand cubes.

The data sources were subsets of the sand sample introduced in

Section 2 with a size of 2503 which corresponds to a size of

1.25 mm which is approximately 5–7 times the diameter of a

typical pore diameter. Multiple simulations at different

resolutions have been carried out and the corresponding results

are shown in table 1 and figure 10. The resolution shown in the

first column indicates the factor in grid size from the original

voxel-based geometry resolution of 250. The reference

permeability of the medium has been calculated using a

Richardson extrapolation.

4.3 Computational issues

At the highest resolution used in the simulations, which was

4363, the memory consumption is quite large. Even this is a

single flow simulation and basically only one set of 19

double precision values are necessary, the array containing

the q-values for each boundary node consumes also 19 float

values. Additional ghost nodes, which are mandatory for

non-local interpolation schemes in combination with

parallelization cause additional hardware resources. How-

ever, the time used for the collision and propagation steps of

the LB scheme is comparably large. Thus, a very good

parallel efficiency is still observed; especially in the case of

asynchronous communication.

Based on the results of the convergence study, we find that

using a lower grid resolution than the original voxel set, one

can obtain results which are of a comparable accuracy as the

reference solution, using full resolution and simple bounce

back. Thus, from the corresponding runtimes we can see that

the computational efficiency rises by at least one order of

magnitudewhen using the parameterized reconstruction. For

the porous medium used in our simulations we achieve a

relative error below 10% in permeability using a grid with a

resolution factor of 0.5 and interpolated bounce back. With a

simple bounce back scheme one has to use at least a

resolution factor of 1.5 to obtain a result of comparable

accuracy. That leads to a gain in efficiency-saving CPU time,

memory consumption, etc.—by a factor of 16.

5. Conclusions

In this work, we demonstrate that second-order accuracy and

a significant speed-up in convergence can be achieved not

only for flow simulations based on CAD-type geometries,

but also from scanned data such as porous media flow in

sand. Contrary to LBGK models with single-relaxation-time

Combination with the simple bounce back scheme, the MRT

model in combination with higher-order boundary con-

ditions are more accurate and due to its faster convergence to

a steady state far more efficient. The additional costs in terms

of pre-processing and decreased locality of the boundary

stencil are more than balanced by the increase in numerical

efficiency. Finally, we would like to point out that for the

simulation of flows in deformable porous media (fluid–

structure-interaction) the parameterization of the geometry

is mandatory to consistently compute the structural

deformation including contact problems.

Acknowledgements

Financial support by the Deutsche Forschungsgemeinschaft

in the framework of the GermanLattice-BoltzmannResearch

Group is gratefully acknowledged. Also many thanks to

Peter Lehmann and Anders Kastner from ETH Zurich for

providing the high-quality X-ray scans of sand samples.

Table 1. Relative permeability at different resolutions.

Resolution Voxel rel. K (SBB) rel. K (LIBB) rel. error SBB [%] rel. error LIBB [%]

1.75 436 3.8137092E 2 05 4.1587915E 2 05 8.915E þ 00 1.222E 2 011.5 374 3.7605696E 2 05 4.1800425E 2 05 1.045E þ 01 6.300E 2 011.0 250 3.5834207E 2 05 4.2637626E 2 05 1.591E þ 01 2.581E þ 000.75 187 3.4072388E 2 05 4.3666093E 2 05 2.191E þ 01 4.876E þ 000.5 125 2.9753718E 2 05 4.6620740E 2 05 3.960E þ 01 1.090E þ 010.3 76 1.7191929E 2 05 5.4151301E 2 05 1.416E þ 02 2.329E þ 01

1,0E–01

1,0E+00

1,0E+01

1,0E+02

1,0E+03

10 100 1000

resolution [grid nodes]

rel.

erro

r [%

]

N–2

N–1

rel. error LIBB

rel. error SBB

Figure 10. Plot comparing absolute permeability depending on theresolution.

B. Ahrenholz et al.376

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Appendix A: Transformation matrix M

1· ð1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Þ

c2· ð21 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1Þ

c4· ð1 22 22 22 22 22 22 1 1 1 1 1 1 1 1 1 1 1 1Þ

c· ð0 1 21 0 0 0 0 1 21 1 21 1 21 1 21 0 0 0 0Þ

c3· ð0 22 2 0 0 0 0 1 21 1 21 1 21 1 21 0 0 0 0Þ

c· ð0 0 0 1 21 0 0 1 21 21 1 0 0 0 0 1 21 1 21Þ

c3· ð0 0 0 22 2 0 0 1 21 21 1 0 0 0 0 1 21 1 21Þ

c· ð0 0 0 0 0 1 21 0 0 0 0 1 21 21 1 1 21 21 1Þ

c3· ð0 0 0 0 0 22 2 0 0 0 0 1 21 21 1 1 21 21 1Þ

c2· ð0 2 2 21 21 21 21 1 1 1 1 1 1 1 1 22 22 22 22Þ

c4· ð0 22 22 1 1 1 1 1 1 1 1 1 1 1 1 22 22 22 22Þ

c2· ð0 0 0 1 1 21 21 1 1 1 1 21 21 21 21 0 0 0 0Þ

c4· ð0 0 0 21 21 1 1 1 1 1 1 21 21 21 21 0 0 0 0Þ

c2· ð0 0 0 0 0 0 0 1 1 21 21 0 0 0 0 0 0 0 0Þ

c2· ð0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 21 21Þ

c2· ð0 0 0 0 0 0 0 0 0 0 0 1 1 21 21 0 0 0 0Þ

c3· ð0 0 0 0 0 0 0 1 21 1 21 21 1 21 1 0 0 0 0Þ

c3· ð0 0 0 0 0 0 0 21 1 1 21 0 0 0 0 1 21 1 21Þ

c3· ð0 0 0 0 0 0 0 0 0 0 0 1 21 21 1 21 1 1 21Þ

26666666666666666666666666666666666666666666666664

37777777777777777777777777777777777777777777777775

LB simulation in porous media 377