Finite-Difference Approximation for Fluid-Flow …...Finite-Difference Approximation for Fluid-Flow Simulation 777 rock. With the increasing computational resources and advances in

Transp Porous Med (2012) 94:775–793DOI 10.1007/s11242-012-0024-y

Finite-Difference Approximation for Fluid-FlowSimulation and Calculation of Permeabilityin Porous Media

Vahid Shabro · Carlos Torres-Verdín ·Farzam Javadpour · Kamy Sepehrnoori

Received: 24 January 2011 / Accepted: 16 May 2012 / Published online: 15 June 2012© Springer Science+Business Media B.V. 2012

Abstract We introduce a finite-difference method to simulate pore scale steady-state creep-ing fluid flow in porous media. First, a geometrical approximation is invoked to describe theinterstitial space of grid-based images of porous media. Subsequently, a generalized Laplaceequation is derived and solved to calculate fluid pressure and velocity distributions in theinterstitial space domain. We use a previously validated lattice-Boltzmann method (LBM)as ground truth for modeling comparison purposes. Our method requires on average 17 % ofthe CPU time used by LBM to calculate permeability in the same pore-scale distributions.After grid refinement, calculations of permeability performed from velocity distributionsconverge with both methods, and our modeling results differ within 6 % from those yieldedby LBM. However, without grid refinement, permeability calculations differ within 20 %from those yielded by LBM for the case of high-porosity rocks and by as much as 100 % inlow-porosity and highly tortuous porous media. We confirm that grid refinement is essentialto secure reliable results when modeling fluid flow in porous media. Without grid refinement,permeability results obtained with our modeling method are closer to converged results thanthose yielded by LBM in low-porosity and highly tortuous media. However, the accuracy ofthe presented model decreases in pores with elongated cross sections.

Keywords Pore scale · Permeability · Finite differences · Geometrical pore approximation ·Generalized Laplace equation

V. Shabro (B) · C. Torres-Verdín · K. SepehrnooriDepartment of Petroleum and Geosystems Engineering, The University of Texas at Austin,1 University Station C0300, Austin, TX 78712-0228, USAe-mail: [email protected]

F. JavadpourBureau of Economic Geology, Jackson School of Geosciences, The University of Texas at Austin,University Station, Box X, Austin, TX 78713-8924, USA

123

776 V. Shabro et al.

Abbreviations3D Three-dimensionalFDGPA Finite-difference geometrical pore approximationLBM Lattice-Boltzmann methodNS Navier–Stokes

List of Symbols

Variables��A A septa-diagonal matrix, representing the relevant �w for all grids, s�B Boundary condition representing the inlet and outlet pressures, Pa sd Digital equivalent of r , dimensionlessdmax Digital equivalent of rmax, dimensionlessfc(dmax) Calibration function, dimensionlessJ, Jz Mass flux, kg m−2 s−1

Jtot Total mass flux, kg m−2 s−1

K Permeability, m2[1 D = 1 Darcy = 9.869 × 10−13 m2]L Porous media (tube) length, mP, P1, P2, �P Pressure, PaPavg Average pressure, PaR Tube radius, mr Radial distance from the inner wall, mrmax The largest inscribed radius, mS(dmax) Area of the smallest possible cross-section for a given dmax, dimensionlessV Volume flux, m s−1

�w Weighting factor in the generalized Laplace equation, s

Greek Symbolsμ Viscosity, Pa sν Local fluid velocity, m s−1

ρ Fluid density, kg m−3

ρavg Average fluid density, kg m−3

τ Relaxation parameter, dimensionlessτr z Z -momentum across a surface perpendicular to the radial direction, kg m s−1

1 Introduction

Modeling fluid flow in porous media is essential in biology, earth sciences, and engineering.In the earth sciences, fluid flow in porous media is conventionally described by Darcy’s law,where permeability quantifies the ability of rocks to flow fluids in their interstitial space.There are several ways to relate permeability to other measurable properties of porous mediasuch as porosity, pore size distribution, grain size distribution, and tortuosity (Bear 1988).However, knowledge of the exact topology of the pore network in porous media is essentialto calculate permeability (Banavar and Johnson 1987; Rocha and Cruz 2010). There aretwo technological frontiers in the calculation of permeability at the pore-scale: accessiblecomputational resources and retrieving an adequately detailed spatial representation of the

123

Finite-Difference Approximation for Fluid-Flow Simulation 777

rock. With the increasing computational resources and advances in porous media-imagingcapabilities in the last two decades, several pore-scale characterization methods and numeri-cal models have been proposed to calculate macroscopic permeability (Banavar and Johnson1987; Bear 1988; Bryant et al. 1993; Pan et al. 2001; Guo and Zhao 2002; Rocha and Cruz2010).

Recent technological progress in X-ray tomography enabled researchers to capture imagesat the microscopic detail of porous media (Flannery et al. 1987) and even at sub-micronresolution (Larson et al. 2002; Javadpour et al. 2012). In parallel, several pore-scale fluidflow models have been developed to quantify porous media, including cellular automaton(Rothman 1988), lattice-Boltzmann method (LBM) (Guo and Zhao 2002; Arns et al. 2004;Jin et al. 2007), network models (Constantinides and Payatakes 1989; Bryant et al. 1993;Javadpour and Pooladi-Darvish 2004; Raoof and Hassanizadeh 2010), and random walk(Scheidegger 1958; Toumelin et al. 2007). In a grid-based image, cellular automaton, whichrepresent fluid (Rothman 1988), evolve in time with the implementation of local interactionrules. LBM (Guo and Zhao 2002; Arns et al. 2004; Jin et al. 2007) implements kinetic the-ory to solve the discrete Boltzmann equations with a scheme similar to cellular automaton.Pore-network models (Constantinides and Payatakes 1989; Bryant et al. 1993; Javadpourand Pooladi-Darvish 2004; Raoof and Hassanizadeh 2010) simplify pore geometry detailsof porous media into pore bodies and pore throats, usually represented by spheres and cylin-ders (Bryant et al. 1993), or converging–diverging geometries (Constantinides and Payatakes1989; Javadpour and Pooladi-Darvish 2004). Pore geometry simplifications make networkmodels a fast simulation option in comparison to other aforementioned methods, but the over-simplification of exact pore geometries sacrifices accurate fluid flow calculations. Anotherapproach is to sample a small section of the porous medium (i.e., unit cell), and calculatethe velocity field in the interstitial space domain by solving Navier–Stokes (NS) equation.Thereafter, upscaled (macroscopic) results are intended to represent those of the originalporous media (Javadpour 2009a; Javadpour et al. 2009; Shabro et al. 2010; Javadpour andJeje 2012). This approach has the potential to reduce the computational burden of modeling.In addition, porous media can be modeled as a pack of spherical grains (Jin et al. 2003) andpermeability of the resulting pack can be calculated using the aforementioned methods (Panet al. 2001; Jin et al. 2004).

We propose a new and fast method to model fluid flow in porous media which we referto as finite-difference geometrical pore approximation (FDGPA). In this model, geometricalparameters are defined in the interstitial space domain to characterize pore-scale images. Themethod calculates a local fluid flow resistance based on the smallest distance to the confiningwall and the largest inscribed radius. The local fluid flow resistance values are used in ageneralized Laplace equation to calculate the distributions of pressure and fluid velocity inthe interstitial space domain. The presented method relies on the geometrical pore approx-imation algorithm to take into account the viscous forces exerted from the confining wallon the fluid instead of directly solving for the NS equation at the pore space. The methodachieves fast calculation of the distributions of pressure and fluid velocity in the interstitialspace domain through approximation of pore geometry; however, the approximation theo-retically introduces an error in calculation of permeability in porous media. The geometricalpore approximation is applied in two-dimensional cross-sections in three Cartesian direc-tions. The three-dimensional (3D) structures of digital images are taken into account via theeffect of local fluid flow resistances at neighbors. The simplification of 3D structures intro-duces additional source of error in theory. The accuracy of the presented model is discussedin Sect. 6 and Appendix I. The approximation method is accurate in circular cross sections,but the error reaches to 50 % in cross sections with elongated conduit of fluid flow.

123


Given the fluid velocity distribution output by FDGPA, one can trace pore-scale stream-lines using the same streamline tracking approach described by Datta-Gupta and King (2007).The finite-difference method allows one to define complex inner boundary conditions on thesurface of mineral grains. For instance, the model allows us to enforce slip boundary condi-tions on grain surfaces at various fluid flow regimes (Shabro et al. 2009) with applications tonatural (Javadpour et al. 2007; Javadpour 2009b) and artificial (Roy et al. 2003) nanoporousmedia.

In what follows, we describe grid-based images representing porous media and introduce anew method to approximate pore geometries and to model fluid flow using a finite-differenceapproximation. We verified our model with the analytic solution of regular pore geometries.Subsequently, modeling results are validated by comparing them to results obtained withLBM for the same images. Next, we discuss the effects of grid refinement for FDGPA andLBM modeling results in the cases of low-porosity and high-porosity porous media. Finally,we appraise the two methods in terms of accuracy and the required computational resourcesfor calculation of permeability.

2 Digitized Porous Media

We use 3D grid-blocks in Cartesian coordinates to describe spatial pore-scale details of per-meable media. Each grid-block is either a pore or a grain. The 3D grid-blocks are obtainedfrom numerical grain pack simulations (Silin et al. 2003), or from the X-ray micro-tomog-raphy images (Jin et al. 2007; Shabro et al. 2010). To construct a numerical grain pack, agrain-radius distribution between 5 and 30 µm is employed as described in Table 1. After100 compaction steps, 1,573 spherical grains are generated in a 300×300×1, 000 µm3 boxand subsequently compacted into a 300×300×352 µm3 box. Table 1 describes the resultingquantitative and volumetric distributions. Figure 1 shows a schematic of the resulting pack.In the next step, the first 50- µm portion on each side of the pack is disregarded to avoidthe pack boundaries and the numerical grain pack between 50 and 250 µm in the X , Y , andZ directions is digitized using a 2- µm grid size which results in 106 grid blocks. Grid sizeshould be smaller than one-fifth of the smallest grain diameter to accurately describe poreimage details. The resulting pack (Case No. 1) has a porosity of 37.7 %. We implementnumerical uniform cementation (Silin et al. 2003) on the grains to reduce the porosity to14.5 % (Case No. 2). These two samples are adequate to illustrate and appraise the capa-bilities of the simulation model. Further, we use X-ray microtomographic images of naturalrocks, namely, dolomite, quartzose sandstone, Fontainebleau sandstone and Berea sandstone(Jin et al. 2007; Shabro et al. 2010) to compare our FDGPA model and LBM results.

Table 1 Summary of grain-sizedistributions for a synthetic pack

Grain radius (µm) Quantitative (%) Volumetric (%)

0.5 56 3

1 27 12

2 14 46

3 3 40

123


Fig. 1 Grain pack constructed within a cube with the distribution of grain sizes described in Table 1

3 Conservation of Mass and Momentum Equations

We digitize a combination of the mass conservation and momentum transfer equations usinga finite-difference stencil, and assuming creeping and incompressible fluid flow with no-slip inner boundary conditions in the interstitial space domain. These conditions allow usto neglect the effects of inertia and fluid compressibility. The solution invokes Neumann’sboundary conditions at the grain-pore surface and finite pressures at both inlet and outlet.

Conservation of mass under the assumption of no sink or source and incompressible fluidgives rise to

∇ · J = 0, (1)

where J is the mass flux. It indicates that the algebraic summation of the influx and outflowof each grid is equal to zero.

In theory, one can use the NS equation along with mass conservation to predict the spatialdistribution of fluid velocity in porous media using appropriate boundary conditions (Landauand Lifchitz 1887). However, complex pore geometries in porous media make the solutioncomputationally intractable. We first divide the irregular pore geometries to the form of manyinterconnecting cylindrical pores. Then, we use the analytical solution of the NS equation forcylindrical pore geometry for each pre-defined cylindrical section. Accordingly, in a cylindri-cal tube with no-slip conditions on the inner wall, the momentum flux distribution is given by

τr z =(

P2 − P1

2L

)· (R − r) , (2)

where τr z is the z-momentum across a surface perpendicular to the radial direction,R is thetube radius, r is the radial distance from the inner wall, P1 and P2 are the pressures at the

123


inlet and outlet of the tube, respectively, and L is the length of the tube. Subsequently, fluidflux in a tube is modeled by the so-called Hagen–Poiseuille equation:

Jz = −R2 ρ

8μ· P2 − P1

L, (3)

where Jz is the mass flux over the axial tube cross-section (z-direction), ρ is the average fluiddensity at Pavg[=(P1 + P2)/2], and μ is the fluid viscosity.

We hypothesize that Eq. 2 is applicable in cross sections of pore-scale images with com-plex geometries provided that R and r are defined by our geometrical pore approximationmethod (Sect. 4). Through the use of this hypothesis, the proposed model approximates fluidflow in porous media and provides an alternative approach to LBM to calculate permeabilityin digital images. We should note that the hypothesis simplifies the physical principles offluid flow to solve for fluid flow in irregular flow paths of digital images in a practical timeframe. Based on molecular kinetic theory, viscous forces in a fluid are caused by molecu-lar interactions between various layers of the fluid with different velocities and confiningwalls with zero velocity and no-slip boundary conditions (Bird et al. 2002). The collectiveeffect causes the fluid to resist movement against the applied pressure. This phenomenonis simplified in Darcy’s equation by assigning viscosity as the effective fluid property andpermeability as the geometrical consequence of confining walls in porous media.

The technical rationality of our approach is based on assigning relevant momentum to eachpart of the fluid according to its position in the cross section. We assume that momentum ateach grid is a function of the grid distance to the nearest confining wall. Momentum decreasesand fluid velocity increases as a grid-block recedes away from the wall, where there are morelayers of fluid hindering the wall effect. In so doing, the resulting momentum is translated toa relevant coefficient in Eq. 3 for each grid-block, which we refer to as the grid weightingfactor, �w. The grid weighting factor is linked to local fluid flow resistance in each grid. Theapproximation method assumes that the nearest confining wall provides the dominant viscousforce. The smallest distance to the confining wall and the largest inscribed radius are used tocalculate local fluid flow resistance. The approximation is exact for a circular tube; however,it introduces error for a non-circular tube as the geometry of the entire confining wall isimportant in calculating local fluid flow resistance in each grid. Appendix I demonstratesthat the maximum error occurs when modeling fluid flow in a confined image with elongatedpore cross sections using the geometrical pore approximation algorithm. In addition, localfluid flow resistance is calculated separately in three directions. The 3D structures of digitalimages are taken into account through the interaction of neighboring pressure with the localfluid flow resistance values calculated in three cross sections.

4 Geometrical Pore Approximation

The grid weighting factor uses two parameters: the largest inscribed radius, rmax,; and thedistance to the pore wall, r . Accordingly, the weighting factor ( �w), rmax, and r are defined forall grid-blocks representing pores; dmax and d are digital equivalents of rmax and r , respec-tively. These parameters are calculated at the cross sections perpendicular to all X , Y , andZ directions for all grid-blocks representing pores. A calibration function, fc(dmax), is alsodefined which is used in the fluid flow model as discussed in Sect. 5. The center of each gridis taken as the local coordination point, and the listed parameters are quantified in grid-sizeunits.

123


Fig. 2 Geometrical pore approximation on several cross-sections. a, b, and c show the smallest possiblecross-sections which result in dmax equal to 0.5, 1.5 and 3, respectively. In each item, the numbers in thecross-section represent d; other parameters are also shown in each block. In d and e, arbitrary cross-sectionsare shown with the engraved relevant d. There are three and four regimes with different parameters in d ande, respectively, which are indicated with different colors

To find d in each grid, its value is initialized to 1 which is the value for a pore surroundedby grains in all face sides. This condition is illustrated in Fig. 2a. Next, we check all theoutpost grids within a distance d of the original grid. If all the outpost grids are pores, thend is incremented an amount of 0.5; if not, the current value of d is taken as final. At the end,if d is 1, we assign 0.5 when all the outpost grids are grains, and assign 1 if there is at leastone grain in the surrounding grids. This corrective criterion discriminates an isolated porewithin mineral grains from a pore with at least one open face. Figure 3a shows a flowchartof the algorithm. The values of d in all grids representing grains are zero.

We use the calculated d values to find dmax for each grid. The algorithm compares the dvalue for the grid of interest to the d values in all neighboring grids. If the grid of interest hasthe highest d value, dmax is equal to d; if not, dmax is equal to the dmax of the neighboringgrid with the highest d value. This recursion is implemented as described by Fig. 3b. Again,the values of dmax in all the grain grids are zero.

The smallest possible cross sections which result in different dmax values are shown inFig. 2a–c for dmax = 0.5, 1.5 and 3, respectively. The area of such cross sections, S(dmax),defines the relevant rmax as rmax (dmax) = √

S (dmax) /π . This value is the equivalent tube

123


Fig. 3 Flowcharts of the algorithms used in this paper to calculate. a The distance of a point within theporous medium to the closest grain boundary and b the maximum inscribed radius in a pore. c Flowchart ofthe developed finite-difference method, FDGPA

radius with the same cross-sectional area. Likewise, in the same smallest possible crosssections the calibration function ( fc) is defined as

fc (dmax) =S(dmax)∑

1

(2dmaxd − d2). (4)

Figure 2 shows the values of d, dmax, rmax and fc for five different pore geometries and sizes.

5 Fluid Flow Model at the Pore Scale

Following Eqs. 2 and 3, we define the weighting factor (w) for each direction in each grid as

w = r2max

ρ

8μ·(2dmaxd − d2

)fc (dmax)

. (5)

123


The weighting factor represents local fluid flow resistance in each grid for each direction.Note that Eq. 5 is analogous to the local function of fluid flow velocity in a tube:

ν (r) = R2

8μ·(2Rr − r2

)R2 · P2 − P1

L, (6)

where ν is the local fluid velocity in a tube at distance r from the tube wall. Further, iftwo pore grids share a face, the weighting factor in the applicable direction is calculated asthe arithmetic average of the two respective weighting factors in each pore grid. Thus, theweighting factor, �w, has six elements for each grid, in the X−, X+, Y −, Y +, Z−, and Z+directions.

We define mass flux at a grid face as

J = w · (P2 − P1) /L = w · ∇ P (7)

and substitute it in Eq. 1 for each grid to construct a generalized Laplace equation of the form

∇ ( �w · ∇ P) = 0. (8)

Neumann’s boundary condition at grain boundaries is implemented by imposing null flowacross grain boundaries or equivalently setting �w to zero in the direction facing the grain.Initial pressures are set at the inlet and outlet faces via additional external grid-blocks facingthe grid-blocks of porous media. Using central difference derivations, the finite-differencemethod results in the linear system of equations

��A · �P = �B, (9)

where ��A is a septa-diagonal matrix, representing the relevant �w for all grids, and �B represents

the inlet and outlet pressures. ��A has a matrix dimension equal to the square of the total numberof grids and �B is a vector array with the number of grid elements. If we remove isolated pores

which are not connected to the flow path (hence their pressure is not identifiable), ��A becomessymmetric positive-definite. Consequently, Eq. 8 is solved via the conjugate gradient method(Shewchuk 1994) to calculate the pressure distribution in the interstitial space. We furtherdecrease the CPU time associated with the conjugate gradient solver by taking into account

the septa-diagonal structure of ��A. Mass flux and velocity field for all the grids are thencalculated using Eqs. 5 and 7. Finally, Darcy’s equation is used to calculate permeability, K ,from the calculated total flux, Jtot, in our model, i.e.,

Jtot = Kρ

μ· P2 − P1

L⇒ K = μL Jtot

ρ (P2 − P1). (10)

Figure 3 shows flowcharts for the model algorithms. When the spatial distribution ofpressure and, subsequently, the spatial distribution of fluid velocity are calculated, pore-scalestreamlines are drawn using the streamline tracking approach described by Datta-Gupta andKing (2007).

6 Results and Discussion

6.1 Model Comparison and Validation

We now discuss results obtained with FDGPA and compare them to LBM results to verifythe validity and accuracy of our model. LBM simulates fluid flow in porous media using

123


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

40 80 120 160 200

LBM-Z direction

LBM-Y direction

LBM-X direction

FDGPA-Z direction

FDGPA-Y direction

FDGPA-X direction

Per

mea

bilit

y (D

)

Cubic sample size (µm)

Fig. 4 Permeability calculations performed with LBM and FDGPA for Case No. 1 with porosity = 37.7 %.Relative differences between results obtained with the two methods are lower than 25 % in all cases

similar assumptions to those of FDGPA: steady-state condition, incompressible creepingflow, no-slip and no-flow boundary conditions at the pore-grain surfaces, and constant inletand outlet pressures. To this end, we use the D3Q19 LBM model implemented by UT Austin’sPore-Level Petrophysics Toolbox 1.00 (Jin et al. 2003, 2007) using the relaxation parameter,τ , equal to 0.65. The latter LBM software has been tested with experiments (Jin et al. 2007)and was compared to other pore-scale fluid flow modeling software (Shabro et al. 2010).Additionally, the accuracy of LBM has been previously verified elsewhere (Arns et al. 2004;Jin et al. 2007) and we use it as ground-truth for our modeling purposes.

Figure 4 shows permeability results obtained for Case No. 1 using both LBM and FDGPA.Relative differences between permeability results obtained with the two methods are lowerthan 25 %. Permeability variations vs. cubic sample size and directionality confirm the het-erogeneous and anisotropic nature of porous media. Moreover, we compared 81 CT-scanimages of various rocks with sizes between 603 and 1503 grids (Fig. 5). The majority ofthe comparisons shown in Fig. 5 indicate differences below 20 % between the two meth-ods; samples with low-porosity and high tortuousity entailed differences higher than 20 %.Appendix I entails the model comparison with the available analytical expressions in regularshapes. Next, we consider cases of low porosity and highly tortuous samples.

6.2 Grid Refinement

Sensitivity analysis was performed with both FDGPA and LBM for the case of grid refine-ment. Several levels of grid refinement were considered for FDGPA and LBM on a 403

sample size using Cases No. 1 and 2. Grid refinements from two through five levels indi-cate that each grid in the original image is refined to 8, 27, 64, and 125 grids in the newimages, respectively. Figure 6 shows that grid refinement is necessary to ensure convergenceof results. Four levels of grid refinement was deemed sufficient to ensure convergence inthe case of LBM (Thürey et al. 2006). In all cases described in Fig. 6, permeability resultsobtained with FDGPA and LBM converge with differences between 0 and 6 % after fourand five levels of grid refinement. It is notable that, without grid refinement, differences varybetween 9 and 14 % for Case No. 1, and between 73 and 100 % for Case No. 2.

123


0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

10

Relative difference between

Num

ber

of o

ccur

ence

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

Permeability calculated by LBM (D)

Per

mea

bilit

y ca

lcul

ated

by

FD

GP

A (

D)

FDGPA and LBM permeability results (%)

(a)

(b)

Fig. 5 a Comparisons between permeability calculations performed with FDGPA and LBM for 81 pore-scaleimages with sizes between 603 and 1503 grids. The dashed line represents a perfect match, with red linesrepresenting 20 % difference margins. Calculations in the majority of experiments exhibit differences lowerthan 20 % between the two methods. However, in low permeability cases which occur in low-porosity andhighly tortuous images, FDGPA yields higher permeability values than LBM, with differences higher than20 %. b Histogram of permeability differences between the two models for the 81 samples shown in a

In Case No. 1 (Fig. 6a), all grid refinement levels cause a decrease of permeability exceptfor one of the LBM results. In Case No. 2 (Fig. 6b), which comprises low porosity with tight,tortuous paths, LBM does not find connecting paths between the inlet and outlet and showszero permeability in two cases without grid refinement. Permeability results increase whengrid refinement is implemented with LBM in Case No. 2. By contrast, FDGPA calculates per-meability more accurately without grid refinement in these low-porosity and highly tortuousexample cases. FDGPA permeability results consistently decrease when grid refinement isperformed for both high- and low-porosity samples. Grid refinement leads to more accurate

123


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5

Grid refinement levels

Per

mea

bilit

y (D

)

0

2

4

6

8

10

12

14

16

18

1 2 3 4 5

Grid refinement levels

Per

mea

bilit

y (m

D)

(a)

(b)

FDGPA-Z direction

LBM-Z direction

FDGPA-Y direction

LBM-Y direction

FDGPA-X direction

LBM-X direction

FDGPA-Z directionLBM-Z directionFDGPA-Y directionLBM-Y directionFDGPA-X directionLBM-X direction

Fig. 6 Permeability results obtained after five grid refinement levels with FDGPA and LBM in the X , Y , andZ directions. Each calculation is refined to 8, 27, 64, and 125 grids for grid refinement levels 2, 3, 4, and 5,respectively, where one level indicates calculation without grid refinement. a and b correspond to samples inCases No. 1 and 2, respectively. In all cases, we observe that LBM and FDGPA calculations converge afterfour grid refinement levels

enforcement of the parabolic flow distribution in the fluid models, which in turn causes adecrease of permeability results. However, the increase in LBM permeability results is dueto the implicit requirement to have several pore grids in a throat to establish connection. Ifa path is connected with only one grid at some point in the image, LBM does not calculatethe path. Figure 7 shows the morphology of the sample in Case No. 2 and the resulting pore-scale streamlines in all directions. It is evident that there is a connecting path between theinlet and outlet in all three directions. In the X and Y directions, the flow inevitably shouldbe established through a tight path. This behavior results in zero permeability from LBMwithout grid refinement. However, the flow path in the Z direction is wide enough for LBMto enforce a connecting path without grid refinement.

When investigating pore-scale images of tight formations such as shales, it is importantto consider LBM grid refinement effects to establish tight connecting paths. Even though

123


0 10 20 30 40 50 60 70 80

0 10 20 30 40 50 60 70 80

010

2030

4050

6070

80

010

2030

4050

6070

800

1020304050607080

01020304050607080

x, µmy, µm

z, µ

m

x, µm

0 10 20 30 40 50 60 70 80

x, µm

y, µm

010

2030

4050

6070

80

y, µm

z, µ

m

01020304050607080

z, µ

m

10

20

30

40

50

60

70

80

z, µ

m

1020

3040

5060

7080

y, µm

1020

3040

5060

70 80

x, µm

(a)(b)

(c) (d)

Fig. 7 a 3D schematic of Case No. 2. Pore space is shown with blue blocks. Sample size is 403 grids withgrid size equal to 2 µm in all directions. Pore-scale streamlines of the sample in the X , Y , and Z directions areshown in b, c, and d, respectively. Streamlines identified with the red color spectrum show high-flow pathswhere the time-of-flight is low. Conversely, streamlines toward the blue color spectrum identify low-flowpaths where the time-of-flight is high. Each inlet grid generates one streamline toward the outlet

in high-porosity samples the paths with large throat sizes dominate flow, the tight paths arealso important; e.g., in Fig. 6a LBM permeability calculations in the X direction increasein the first step of refinement due to enforcement of tight paths. Conversely, grid refinementeffects on permeability calculations performed with FDGPA are consistent for all cases. Fig-ure 6 shows that permeability calculations decrease between 16 and 23 % for Case No. 1and between 37 and 49 % for Case No. 2 when grid refinement is performed in FDGPA;however in LBM, permeability calculations decrease between 6 and 36 % for Case No. 1,and increase to 133 % for Case No. 2 in the Z direction when performing grid refinement.Modeling Case No. 2 yields zero permeability in the X and Y directions of the original imagewhen calculations are performed with LBM.

The Cases No. 1 and 2 represent the extreme cases of high-porosity and highly tortuousrocks, respectively. We estimate the permeability in three rock images using FDGPA andLBM to compare the two models in moderate conditions of porosity and tortuosity. Figure 8shows the 3D schematic of three rocks namely, Limestone, Dolomite, and Fontainebleausandstone and Table 2 summarizes the permeability results for these three rock images in X ,

123


Fig. 8 3D schematics of X-ray images of: a Limestone with 603 grids, grid size equal to 4.64 µm and porosityof 23.5 %, b Dolomite with 1253 grids, grid size equal to 3.5 µm and porosity of 20.2 %, and c Fontainebleausandstone with 1253 grids, grid size equal to 3.5 µm and porosity of 19.8 %. Pore space is shown with blueblocks. Table 2 shows the permeability results for these three images using both FDGPA and LBM, and withgrid refinement

Y , and Z directions, with and without grid refinement. The differences between the perme-ability values estimated by the models are below 20 % for all the three rock images. Afterthree levels of grid refinement in the Limestone sample, the permeability values calculatedby the two models are within 5 % disparity. The permeability is estimated in the originalimage for Dolomite and Fontainebleau sandstone and the permeability results differ between0 and 17 % for these three rock images.

6.3 Computational Complexity

As emphasized above, ensuring reliable and accurate permeability calculations of porousmedia requires attention to the following properties:

123


Table 2 Summary of permeability results using both FDGPA and LBM for the three rock images shown inFig. 8

Limestone Dolomite Fontainebleau sandstone

Permeability using FDGPA in the X , Y and Z directions (D)

The original image 2.51, 1.22, 2.11 0.539, 1.74, 0.932 2.27, 2.71, 1.23

Grid refinement level 2 2.01, 1.10, 1.69



Permeability using LBM in the X , Y and Z directions (D)

The original image 2.73, 1.23, 2.30 0.550, 1.86, 0.773 2.36, 2.24, 1.23




0.001

0.01

0.1

1

10

100

1000

1000 10000 100000 1000000 10000000

Sim

ula

tio

n t

ime

(min

ute

s)

Number of grids

LBM- case No. 2FDGPA - case No. 2LBM - case No. 1FDGPA - case No. 1LBM - refinement - case No. 2FDGPA - refinement - case No. 2LBM - refinement - case No. 1FDGPA - refinement - case No. 1

Fig. 9 Simulation times for LBM and FDGPA calculations for Cases No. 1 and 2 and their associated grid-refinement steps. Computational time for FDGPA is on average six times faster than for LBM for both samples.At grid refinement level 4, FDGPA is computationally similar to LBM

1. Quality of images: the larger the image size and the higher the resolution, the more spatialdetails of a porous medium are captured in the calculations.

2. Computational capabilities: increased computational efficiency and memory enables themodeling of larger samples within practical CPU times.

3. Grid refinement: grid refinement is necessary to ensure the accuracy of modeling results.

High computational efficiency and memory are required to model high resolution and largeimages of porous media. The need of grid refinement also implies increased computa-tional requirements. Therefore, computational efficiency is the main limiting factor to secureaccurate and reliable pore-scale calculations.

LBM and FDGPA have similar memory requirements; however, as shown in Fig. 9, thesimulation time in FDGPA requires between 7 and 27 % of the CPU times used by LBMwithout grid refinement. When the ratio of the CPU times used by the two models are aver-aged among all cases, FDGPA is six times faster than LBM to calculate permeability forthe same pore-scale distributions. The computational time required for Case No. 1 is largerthan for Case No. 2 because more iterations are needed for low-porosity and highly tortuous

123


samples to ensure convergence of results. Grid refinement causes LBM to converge in feweriterations, especially for Case No. 2 (low porosity). Grid refinement with FDGPA increasesthe simulation time almost linearly in a log–log plot.

7 Summary

We introduced and successfully tested a fast pore-scale finite-difference fluid flow method(FDGPA) to quantify pore-scale geometry in an image of porous media. The method definesa weighting factor and solves a generalized Laplace equation to calculate the spatial distri-butions of pressure and fluid flow in porous media deterministically to subsequently tracepore-scale streamlines. The weighting factor represents the local fluid flow resistance in theporous media. FDGPA calculates permeability in high-porosity porous media within 20 %difference when compared to LBM results. In low-porosity, highly tortuous porous media,FDGPA finds all possible connecting paths where LBM requires additional grid refinementsto achieve the same level of connectivity. Grid refinement is necessary in both methods toensure convergence; permeability calculations are within 6 % difference between FDGPAand LBM in low- and high-porosity cases after four grid refinement levels. Grid refine-ments with FDGPA cause monotonically improved calculations of permeability toward finalconvergence. FDGPA is based on a hypothesis to find the local fluid flow resistances. Thehypothesis decreases in accuracy in elongated shapes with high-aspect ratios. Last but notleast, the simulation time associated with FDGPA is on average six times faster than that ofLBM.

Acknowledgments The authors thank Alexander Zazovsky from Schlumberger and Steven Bryant, MasaProdanovic, Pawel Matuszyk, and Tatyana Torskaya from the University of Texas at Austin for their instruc-tive comments and discussions. This study was supported by the University of Texas at Austin’s ResearchConsortium on Formation Evaluation and by the Advanced Energy Consortium (AEC).

Appendix I

The fluid velocity in the Z direction at location x from the center of infinite parallel slabsshown in Fig. A1 is (Bird et al. 2002)

νz =(R2 − x2

)2μL

�P. (A1)

Fig. A1 Schematic of a cross section of a circular tube and infinite parallel slabs

123


Table A1 Calculation of permeability in regular shapes based on analytical expressions, FDGPA and LBM

Dimensions Analytical

Permeability (D)FDGPA error LBM error

a = 100 µm 249 -1.7% -0.5%

a = 400 µm 5701 -17% -2.1%

b>>a

a = 20 µm b = 400 µm

1.69 -51% -17%

b>>a

a = 40 µm b = 180 µm

57.3 -44% -14%

a = 100 µm 47.1 -12% +18%

70.2 -9% -15%

10.9 -19% -38%

Grid size is equal to 10 µm and there is 402 grids in the total cross-section

In comparison, the fluid velocity in the Z direction at location x from the center of a circulartube is (Bird et al. 2002)

νz =(R2 − x2

)4μL

�P. (A2)

It is evident that 50 % error is induced if Eq. A2 is applied for the case of fluid flow in infiniteparallel slabs. Because FDGPA is based on the hypothesis that the nearest confining wallprovides the dominant viscous force and the smallest distance to the confining wall and thelargest inscribed radius are used along with Eq. 5 to calculate local fluid flow resistance,the model is accurate for circular cross sections and results up to 50 % error for the caseof elongated parallel slabs. Table A1 indicates that the estimated permeability based on thegeometrical pore approximation is accurate for a circular cross section. The permeabilityvalues are compared with the permeability values calculated from analytical expressions

123


(Sisavath et al. 2001a,b). The error in permeability increases as the cross sections changefrom circular to elongated shapes.

References

Arns, C.H., Knackstedt, M.A., Pinczewski, W.V., Martys, N.S.: Virtual permeametry on microtomographicimages. J. Petrol. Sci. Eng. 45, 41–46 (2004)

Banavar, J.R., Johnson, D.L.: Characteristic pore sizes and transport in porous media. Phys. Rev. B 35, 7283–7286 (1987)

Bear, J.: Dynamics of Fluids in Porous Media. Dover, Mineola (1988)Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. Wiley, New York (2002)Bryant, S.L., King, P.R., Mellor, D.W.: Network model evaluation of permeability and spatial correlation in a

real random sphere packing. Transp. Porous Med. 11, 53–70 (1993)Constantinides, G.N., Payatakes, A.C.: A three dimensional network model for consolidated porous media.

Basic studies. Chem. Eng. Commun. 81, 55–81 (1989)Datta-Gupta, A., King, M.J.: Streamline Simulation: Theory and Practice. Society of Petroleum Engi-

neers, Richardson (2007)Flannery, B.P., Deckman, H.W., Roberge, W.G., D’amico, K.L.: Three-dimensional X-ray microtomogra-

phy. Science 237, 1439 (1987)Guo, Z., Zhao, T.S.: Lattice Boltzmann model for incompressible flows through porous media. Phys. Rev.

E 66, 36304 (2002)Javadpour, F.: CO2 injection in geological formations: determining macroscale coefficients from pore scale

processes. Transp. Porous Med. 79, 87–105 (2009)Javadpour, F.: Nanopores and apparent permeability of gas flow in mudrocks (Shales and Siltstone). J. Can.

Petrol. Technol. 48, 16–21 (2009)Javadpour, F., Jeje, A.: Modeling filtration of platelet-rich plasma in fibrous filters. Transp. Porous

Med. 91, 677–696 (2012)Javadpour, F., Pooladi-Darvish, M.: Network modelling of apparent-relative permeability of gas in heavy

oils. J. Can. Petrol. Technol. 43, 23–30 (2004)Javadpour, F., Fisher, D., Unsworth, M.: Nanoscale gas flow in shale gas sediments. J. Can. Petrol. Tech-

nol. 46, 55–61 (2007)Javadpour, F., Shabro, V., Jeje, A., Torres-Verdín, C.: Modeling of coupled surface and drag forces for the

transport and capture. In: Multiphysics Conference 2009, Lille (2009)Javadpour, F., Moravvej, M., Amrein, M.: Atomic force microscopy (AFM): a new tool for gas-shale charac-

terization. J. Can. Petrol. Technol. (2012). doi:10.2118/161015-PAJin, G., Patzek, T.W., Silin, D.B.: Physics-based reconstruction of sedimentary rocks. Soc. Petrol. Eng. 83587-

MS (2003)Jin, G., Patzek, T.W., Dmitry, B., Silin, D.B.: Direct prediction of the absolute permeability of unconsolidated

and consolidated reservoir rock. Soc. Petrol. Eng. 90084-MS (2004)Jin, G., Torres-Verdín, C., Radaelli, F., Rossi, E.: Experimental validation of pore-level calculations of static

and dynamic petrophysical properties of clastic rocks. Soc. Petrol. Eng. 109547-MS (2007)Landau, L.D., Lifchitz, E.M.: Fluid Mechanics. Pergamon Press, Oxford (1987)Larson, B.C., Yang, W., Ice, G.E., Budai, J.D., Tischler, J.Z.: Three-dimensional X-ray structural microscopy

with submicrometre resolution. Nature 415, 887–890 (2002)Pan, C., Hilpert, M., Miller, C.T.: Pore-scale modeling of saturated permeabilities in random sphere pac-

kings. Phys. Rev. E 64, 66702 (2001)Raoof, A., Hassanizadeh, S.M.: A new method for generating pore-network models of porous media. Transp.

Porous Med. 81, 391–407 (2010)Rocha, R.P., Cruz, M.E.: Calculation of the permeability and apparent permeability of three-dimensional

porous media. Transp. Porous Med. 83, 349–373 (2010)Rothman, D.H.: Cellular-automaton fluids: a model for flow in porous media. Geophysics 53, 509 (1988)Roy, S., Raju, R., Chuang, H.F., Cruden, B.A., Meyyappan, M.: Modeling gas flow through microchannels

and nanopores. J. Appl. Phys. 93, 4870 (2003)Scheidegger, A.E.: The physics of flow through porous media. Soil Sci. 86, 355 (1958)Shabro, V., Javadpour, F., Torres-Verdín, C.: A generalized finite-difference diffusive-advective (FDDA) model

for gas flow in micro- and nano-porous media. World J. Eng. 6, 7–15 (2009)

123

http://dx.doi.org/10.2118/161015-PA


Shabro, V., Prodanovic, M., Arns, C.H., Bryant, S.L., Torres-Verdín, C., Knackstedt, M.A.: Pore-scale model-ing of two-phase flow. In: XVIII International Conference on Computational Methods in Water Resources,Barcelona (2010)

Shewchuk, J.R.: An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. CarnegieMellon University, Pittsburg (1994)

Silin, D.B., Jin, G., Patzek, T.: Robust determination of the pore space morphology in sedimentary rocks. Soc.Petrol. Eng. 84296-MS (2003)

Sisavath, S., Jing, X.D., Zimmerman, R.W.: Laminar flow through irregularly-shaped pores in sedimentaryrocks. Transp. Porous Med. 45, 41–62 (2001a)

Sisavath, S., Jing, X.D., Zimmerman, R.W.: Creeping flow through a pipe of varying radius. Phys. Flu-ids 13, 2762–2772 (2001b)

Thürey, N., Pohl, T., Rüde, U., Oechsner, M., Körner, C.: Optimization and stabilization of LBM free surfaceflow simulations using adaptive parameterization. Comput. Fluids 35, 934–939 (2006)

Toumelin, E., Torres-Verdín, C., Sun, B., Dunn, K.J.: Random-walk technique for simulating NMR mea-surements and 2D NMR maps of porous media with relaxing and permeable boundaries. J. Magn.Reson. 188, 83–96 (2007)

123

Documents

Finite-Difference Approximation for Fluid-Flow …...Finite-Difference Approximation for Fluid-Flow Simulation 777 rock. With the increasing computational resources and advances in