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SPE 90084

Direct Prediction of the Absolute Permeability of Unconsolidated andConsolidated Reservoir Rock

Guodong Jin, SPE, UC Berkeley; Tad W. Patzek, SPE, UC Berkeley / Lawrence Berkeley National Laboratory; and Dmitry B.

Silin, SPE, Lawrence Berkeley National Laboratory

Copyright 2004, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the SPE Annual Technical Conferenceand Exhibition held in Houston, Texas, U.S.A., 2629 September 2004.

This paper was selected for presentation by an SPE Program Committee follow-ing review of information contained in an abstract submitted by the authors(s).Contents of the paper, as presented, have not been reviewed by the Society ofPetroleum Engineers and are subject to correction by the author(s). The material,as presented, does not necessarily reflect any position of the Society of PetroleumEngineers, its officers, or members. Papers presented at SPE meetings are subjectto publication review by Editorial Committees of the Society of Petroleum Engi-neers. Permission to copy is restricted to an abstract of not more than 300 words.Illustrations may not be copied. The abstract should contain conspicuous acknowl-edgement of where and by whom the paper was presented. Write Librarian, SPE,P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-214-952-9435.

AbstractA procedure of estimating the absolute rock permeabil-

ity directly from a microscopic 3D rock image has beendeveloped. Both computer-tomography and computer-generated images of reconstructed reservoir rock samplescan be used as input. A physics-based depositional modelserves to reconstruct natural sedimentary rock, and gen-erate 3D images of the pore space at an arbitrary resolu-tion. This model provides a detailed microstructure of therock, and makes it possible to calculate the steady statevelocity field in the single-phase fluid flow. In particu-lar, using our model, one can analyze unconsolidated rockswhose micro-tomographic images cannot be obtained. Thelattice-Boltzmann method is used to simulate viscous fluidflow in the pore space of natural and computer-generatedsandstone samples. Therefore, the permeability is calcu-lated directly from the sample images without convertingthem into a pore network or solving Stokes’ equation ofcreeping flow. We have studied the effect of compactionand various styles of cementation on the microstructureand permeability of reservoir rock. The calculated perme-ability is compared with the Kozeny-Carman formula andexperimental data.

IntroductionThe quantitative prediction of the continuum flow de-

scriptors of porous media, such as the absolute perme-ability, the relative permeabilities, the capillary pressures,the formation resistivity, etc., is essential in earth sciencesand – in particular – in petroleum engineering. Usually,the theoretical prediction of the absolute rock permeabil-ity is performed in two steps: (1) A model of the rockmicrostructure is formulated, and (2) a discretized fieldequation, such as Poisson’s or Stokes’ equation, is numer-ically solved on this model.1

Rock flow properties cannot be predicted without an ac-curate 3D representation of the rock microstructure. Sev-eral approaches have been proposed to reconstruct the3D microstructures of natural rock: (1) Experimental;2–6(2) Statistical;7–12 and (3) Process- or physics-based.13–18The experimental approach is necessary, but it is time-consuming, expensive, and not applicable to damaged corematerial or drill cuttings. As an alternative, computer-based rock models have become increasingly attractive,because of their low cost and high speed, as well as theability to overcome the present resolution constraints ofexperiment. Quantitative comparisons between computer-generated and microtomographic rock images have shownthat the process-based models reproduce the 3D geometryof natural sedimentary rock much better than the stochas-tic models. The process-based models are also superior intheir predictions of the pore space connectivity1,19–21 and,thus, the rock permeability.22,23

In this paper, we obtain the virtual samples of reservoirrock by applying the physics-based reconstruction proce-dure introduced in Refs.17,18 The essence of our approachis to build virtual samples of real sedimentary rock by (a)simulating the dynamic geological processes of grain sedi-mentation and compaction, (b) modeling the result of di-agenesis, and (c) reproducing the mechanical behavior of

2 G. JIN, T. W. PATZEK AND D. B. SILIN SPE 90084

the simulated rock. In particular, our method accounts forthe translations and rotations of rock grains during theirsettling and compaction.

Once the detailed microstructure of a reservoir rock sam-ple is obtained, it is possible to derive the macroscopicflow properties of this sample by, for example, solving nu-merically the continuum flow equations governing the fluidtransport,24,25 or using the pore-network models.15,26–28The numerical solution of the continuum flow equationscan be extraordinarily challenging when all the details ofthe stunningly complex 3D rock microstructure are ac-counted for. The steady-state solution of Poiseuille’s flowon a disordered network of cylindrical ducts is easy to ob-tain, but the proper extraction of a pore network froma microscopic, 3D rock image is exceedingly difficult andnon-unique, even if we neglect errors introduced by imageprocessing and interpretation, and concentrate purely onrock geometry.29,30

In recent years, the lattice-Boltzmann method(LBM)31–35 has progressed into an established nu-merical scheme for simulating fluid flow and modelingfluid physics. This method relies on solving discrete kineticequations for the flow of fluid particles, and it recoversthe macroscopic continuity and momentum equations foran incompressible fluid in the double asymptotic limit ofsmall Knudsen and Mach numbers.36 One of the mostadvertised advantages of LBM is its flexibility in handlingcomplex flow geometries.37 The intricacies associatedwith the complicated boundary conditions can easily behandled in terms of particle reflections and bounces atsolid sites. In principle, LBM is suitable for simulations ofhydrodynamic flow in any domain. However, its “simple”treatment of complex geometries makes it particularlyuseful in the simulation of single-phase flow in porousmedia, see, e.g., Refs.24,37–45

In this paper, we apply LBM to model the viscous flowof a single fluid in the pore space of imaged and recon-structed samples of sedimentary rock. The absolute rockpermeability is derived directly from the simulated velocityfield. First, a simple rock, consisting of identical sphericalgrains, is used to validate our LB model for single-phasefluid flow. By combining the rock compaction model withthe fluid flow simulation, the permeability is evaluated atdifferent stresses. The results of these simulations are com-pared with the Kozeny-Carman formula and available ex-perimental data. Second, beginning with an unconsoli-dated non-uniform grain packing, we study the influenceof diagenesis on the rock microstructure and, thus, on theabsolute permeability. The relation between the perme-ability, porosity, and diagenetic alteration is studied bydepositing increasing amounts of cement on grain surfaces.

Models of porous mediaIn general, sedimentary rock formation process is clas-

sified into three stages.46 First, detrital rock fragments(grains) are deposited from flowing air or water to forman unconsolidated grain packing. Second, the packing isburied and compacted by the overburden rock. Third, flow

Fig. 1— An unconsolidated random packing of 9111 identicalgrains after gravity sedimentation. The grain diamater is d =3 mm. The arrow shows the direction of compaction.

Fig. 2— An unconsolidated random packing of 14000 grainsafter gravity sedimentation. The grain diameters are uniformlydistributed between dmin = 0.14 mm and dmax = 0.26 mm.

of warm water causes dissolution, transport, and nucle-ation of different rock minerals. These coupled, complexphysico-chemical processes, which take place in the buriedsediments, are called diagenesis.47 Diagenetic alterationscement a grain packing into solid rock, simultaneouslymodifying the pore space geometry and connectivity.

An efficient method of modeling the fundamental geolog-ical processes of rock formation is described in Refs.17,18Here, we only describe the reconstructed rock samples andthe discretized images of their pore space, on which weperform the LB simulations of viscous fluid flow.

Unconsolidated grain packings. Two computer-generated unconsolidated random grain packings aftergravity-driven sedimentation are shown in Fig. 1 andFig. 2. These two packings have already reached gravity

SPE 90084 DIRECT PREDICTION OF ABSOLUTE PERMEABILITY OF UNCONSOLIDATED AND CONSOLIDATED. . . 3

Fig. 3— The pore space in a 2 cm× 2 cm× 2 cm sub-volume(x ∈ [2.0, 4.0] cm, y ∈ [2.0, 4.0] cm, and z ∈ [1.0, 3.0] cm) of thegrain packing in Fig. 1. The voxel resolution is 200 micronsand the porosity is about 41.9%.

equilibrium. Their dimensions are 6 cm×6 cm×6.6 cm inFig. 1 and 5 mm×5 mm×4.6 mm in Fig. 2. Different grainsize distributions are used in the packings: The uniformgrain size d = 3 mm in Fig. 1 is used to match numericallythe experiments by Wyllie and Gregory,48 and a uniformgrain size distribution bounded by dmin = 0.14 mm anddmax = 0.26 mm in Fig. 2 is used to model Fontainebleausandstone.1

For the purpose of fluid flow simulation, each sampleis discretized into a three-dimensional array of identicalsmall cubes, called voxels. We use the convention that avoxel is an element of the pore space if its center is inthe pore space. Otherwise, it is an element of the solidskeleton. In order to eliminate or reduce the image gener-ation boundary effects, only a fixed volume in the middlepart of both grain packings is selected for discretization:x ∈ [2.0, 4.0] cm, y ∈ [2.0, 4.0] cm, and z ∈ [1.0, 3.0] cmin Fig. 1 and x ∈ [2.0, 3.0] mm, y ∈ [2.0, 3.0] mm, andz ∈ [2.0, 3.0] mm in Fig. 2. The orientation of the Carte-sian coordinate system used in this analysis is shown inFig. 1.

The discretized images of the pore space in the cen-tral parts of both grain packings are shown in Fig. 3 andFig. 4. The voxel resolutions of 200 microns and 10 mi-crons, respectively, will be used in the analysis of thesepackings. The initial voxel-based porosity (the ratio of thenumber of voxels in the void space to the total number ofvoxels) of is 41.9% in Fig. 3 and 40.4% in Fig. 4.

Consolidated grain packings. Compaction and cemen-tation processes modify the microstructure of the initialgrain packing. Fig. 5 displays the pore space of the central1 mm×1 mm×1 mm sub-volume of the compacted grainpacking, obtained after the packing in Fig. 2 was com-pacted uniaxially in the z-direction to reduce the bulk vol-ume by 18.8%. So, the voxel-based porosity of this volumeis now 28.5%. Comparing this image with the image be-fore compaction, see Fig. 4, one observes that compactioncauses some grain interpenetration, and large pores be-

Fig. 4— The pore space in a 1 mm × 1 mm × 1 mm sub-volume (x, y, z ∈ [2.0, 3.0] mm)of the grain packing in Fig. 2.The voxel resolution is 10 microns and the porosity is about40.4%.

Fig. 5— The pore space in a 1 mm×1 mm×1 mm sub-volume(x, y, z ∈ [2.0, 3.0] mm) of the compacted grain packing, ob-tained after the packing in Fig. 2 is uniaxially compacted in thez-direction to reduce the bulk volume by 18.8%. The porosityis now 28.5%.

come interconnected through smaller ones.Cementation is the process of mineral nucleation and

precipitation, which binds the rock grains. Depending ontheir chemical and crystallographic properties, cements areprecipitated on the grain surfaces of the compacted rock,and increase the specific surface area and tortuosity of therock. However, it is not an easy task to predict the locationand chemical composition of the various cements inside thepore space of consolidated rock. In the modeling, we as-sume that cements, independently of their composition,are homogeneously deposited on the grain surfaces withinthe sample according to the same cement overgrowth algo-rithm.17,18,49 In this algorithm, the rate of cement growthis related to the direction and grain size:

∆(r) =(

R

R0

)ξ

min(κl(r)ζ , l(r)

)(1)

In Eq. (1), ∆(r) = L(r) − R0, is the increment of cementthickness along the direction r measured from the grain


Grain

Cement

Pore Space

)(rL

)(rl

Fig. 6— Notation in the cement overgrowth algorithm. Thedark grey zone is the original grain, the light gray zone is thecement, and the white area is the void space.

Fig. 7— The pore space in the central 1 mm× 1 mm× 1 mmsub-volume (x, y, z ∈ [2.0, 3.0] mm) after cement has beendeposited on the grain surfaces in the compacted grain packingin Fig. 5. The porosity is about 18.0%.

center, where L(r) is the distance from the grain centerto the pore-cement interface, Fig. 6. The original grainradius and the mean grain radius in the whole sample aredenoted by R and R0. The distance between the originalgrain surface and the plane of the polyhedron (polygon in2D) into which the grain is inscribed is denoted by l(r).The parameter ξ controls the effect of grain size on the rateof cement overgrowth, whereas the exponent ζ controls thegrowth direction. Finally, the coefficient κ determines theporosity reduction. More details about the effects of themodel parameters can be found in Refs.17,18,49

For example, Fig. 7 depicts the pore space of the 1 mm×1 mm×1 mm sub-volume, after cement has been depositedon the grain surfaces in the compacted grain packing shownin Fig. 5. The porosity is about 18.0%. The parametersused in Eq. (1) are ξ = 1.0, ζ = −1.0 and κ = 1.30 ×10−4. This type of cementation preserves a relatively highporosity even when the pore space becomes disconnected.

The lattice Boltzmann methodIn the lattice Boltzmann method (LBM), the fluid par-

ticles are modeled by a time-dependent distribution mov-ing on a regular lattice. Starting from an initial state, ateach time step the particle distributions propagate fromone lattice node to the neighboring ones, and redistributethemselves locally subject to the conservation of mass andmomentum. To simulate fluid flow, the lattice is identicalwith the discretized rock image, or the voxel image, suchas the one shown in Fig. 7. Each lattice node is put inthe center of a voxel in the pore space image. A generaldescription of LBM can be found in Refs.31–35 Here wejust describe our implementation of the method, and focuson the practical aspects of the simulations.

Basic equations. The particle distribution Fi(x, t) givesthe probability of finding a particle with the lattice velocityci (defined below), in the location x, and at time t. Itsevolution is computed by the discrete lattice Boltzmannequation (LBE) with the Bhatnagar-Gross-Krook (BGK)approximation:31,50,51

Fi(x + ci∆t, t + ∆t) = (1 − ω)Fi(x, t) + ωF ei (x, t) (2)

where the BGK relaxation parameter ω controls the rateof approach to the dynamic equilibrium. In this simula-tion, the value of ω is typically selected from the inter-val: 0.50 < ω < 2, where the upper limit is dictated bythe consideration of numerical stability of LBM.39,52 Thelocal equilibrium distribution, towards which the particledistribution is relaxed, is denoted by F e

i (x, t), and a singletime step during which the particle distributions travel ex-actly one node spacing is denoted by ∆t. Since, by design,we track the motion of every particle as it hops from onelattice node to another in one time step, the LBM imple-mentation is Lagrangian, and the evolution equation (2)has a particularly simple form.

The hydrodynamic fields, such as the local macroscopicmass density ρ and the fluid velocity u, are defined for agiven node in terms of moments of the particle distribu-tions Fi(x, t) by

ρ(x, t) =n∑

i=0

Fi(x, t) (3)

u(x, t) =∑n

i=0 Fi(x, t)ci

ρ(x, t)(4)

where the number of admissible lattice velocities per latticenode is denoted by n. That is, the particles at each nodecan move along n different directions or stay still.

The number of admissible velocities at each node is re-lated to the connectivity assumption adopted in the simu-lations. The standard connectivity numbers are 6, wheretwo voxels are connected if and only if they have a com-mon face, 18, where two voxels are connected if they havea common face or a common edge, or 26, where two voxelsare connected if they have a common face, edge, or vertex.Intermediate connectivity hypotheses can be adopted as


1

3

4 2

5

0

12

7

16

17

14

18

11

15

13

9

8

10

6

Fig. 8— Lattice structure of the lattice Boltzmann model with19 velocity directions per lattice node. Each arrow representsa direction along which the particle distributions can move.The x direction is defined from 0 to 11, y from 0 to 13 and zfrom 0 to 1.

well. Here, we use the 18-connectivity. Fig. 8 shows thecorresponding lattice model, which has 19 velocity direc-tions per lattice node, including a zero velocity.

In the model, each lattice node is connected to its sixnearest and twelve diagonal neighbors at a distance of

√2a,

where a is the voxel resolution. Particles can only stay atthe node or move along the fixed links connecting the lat-tice nodes, and interact at the nodes. All particles associ-ated with a given direction have the same lattice velocity.All admissible lattice velocities are:

c0, c1,6, c11,12 = (0, 0, 0), (0, 0,±a), (±a, 0, 0)c13,14, c2,3,7,8 = (0,±a, 0), (±a, 0,±a) (5)

c4,5,9,10, c15,16,17,18 = (0,±a,±a), (±a,±a, 0)

The local equilibrium particle distribution F ei (x, t) de-

pends on the values of ρ and u, and is chosen specificallyto recover the macroscopic Navier-Stokes equation. One ofthe forms of F e

i (x, t) is given by51

F ei = wiρ

(1 +

1c2s

(ci · u) +1

2c4s

(ci · u)2 − 12c2

s

u2

)(6)

where the lattice speed of sound cs is assigned the dimen-sionless value of

√1/3, and the weight factor wi are

wi =

1/3 i = 01/18 i = 1, 6, 11 − 141/36 i = 2 − 5, 7 − 10, 15 − 18

(7)

The kinematic viscosity of the simulated fluid ν and itspressure expressed in lattice units, respectively, are ν =(2−ω)/(6ω) and P (x, t) = c2

sρ(x, t). Dimensionless latticeunits are used in the simulation for all the variables andresults. However, the physical units are related to latticeunits by a scaling similar to the one proposed in Ref.,53ν′ = νa2/∆t′ and P ′ = P∆m′/(a∆t′2), where a is the

voxel resolution corresponding to the lattice spacing, ∆t′ isthe physical time associated with one lattice time step, and∆m′ is the physical mass associated with a unit of latticemass. Similarly, u′ = ua/∆t′ and ρ′ = ρ∆m′/a3, whereu is the dimensionless velocity obtained in the simulation.The primes denote that the variables are expressed in theirphysical units.

The initial and boundary conditions. The initializa-tion of the fluid flow in LBM is crucial for numerical test-ing, and also for transient flows, when it is desired to calcu-late how an initial velocity field evolves immediately afterthe start-up. However, initialization is not very importantfor studying long-term behavior and steady-state flows.54Our purpose is to calculate the absolute permeability ofthe porous medium in steady state flow. Thus, the initialconditions in the simulation are set u = 0 for the flow ve-locity, and ρ = 1 for the fluid density in the whole domain.

Two kinds of boundary conditions are used in the LBMsimulations here: a no-slip boundary condition at the solid-fluid interfaces, and a prescribed flow at the inlet and out-let faces. The complete bounce-back scheme55,56 is used tosimulate the no-slip boundary condition at the solid-fluidinterfaces. This bounce-back scheme requires that when aparticle distribution streams to a solid boundary node, itscatters back to the node it came from with no relaxation.This treatment is independent of the direction of the par-ticle distributions, and it is very easy to apply in the com-plicated geometries. Other no-slip boundary treatmentshave been proposed to improve the numerical accuracy ofLBM at the solid boundaries.54,57–61 However, they aredifficult to implement for the complicated geometries ofporous media.61

In the LBM simulation, a flow can be driven by thebody forces23,43 or the pressure45,53,58,61 prescribed onthe flow boundaries. A good agreement between the solu-tions corresponding to these two flow boundary conditionshas been found in the creeping-flow regime, i.e., for thelow Reynolds number flows.62,63 Here, we use the uniformbody force to drive the flow. The uniform body force is de-signed to produce the same momentum input into the flowas the true pressure gradient ∆P/L1. To implement thisscheme, we add a constant value ∆Fi(x, t) to the particledistributions moving along the pressure gradient, and placea corresponding penalty on the counter-particle distribu-tions. For example, assuming the pressure drop is alongthe x−direction, the constant value will be added in eachof i = 2, 8, 11, 15 and 18 directions, whereas the penaltywill be added in each of i = 7, 3, 12, 17 and 16 directions.Assuming that the magnitude of ∆Fi(x, t) be equal for allthese directions, we have

|∆Fi(x, t)| =110

∆P

L1(8)

In addition, a periodic boundary condition is applied atthe inlet and outlet to allow the departing particle distri-butions to reenter the flow domain through the inlet.


x y

z

0

Sample

Fig. 9— Schematic diagram of fluid flow through a rock sam-ple. The fluid, driven by a uniform body force, enters fromthe left (inlet) and proceeds rightwards to the outlet in thex−direction. Additional layers of void space are added at theinlet and outlet faces.

Scaling of the absolute permeability. The single-phase, low Reynolds number flow through a porous rocksample is described by Darcy’s law, which can be writtenin the form:

q′ = − k′

ρ′ν′∆P ′

L′1

(9)

where k′ is the permeability of the porous medium, ρ′ is thefluid density, ν′ is the fluid kinematic viscosity, ∆P ′/L′

1 isthe total pressure drop along the sample length L′

1, andq′ is the volumetric fluid flux through the porous medium,q′ = φu′. Here φ is the sample porosity, and u′ is thevelocity averaged over all lattice nodes in the pore space.Using the scaling forms mentioned above, and Eqs. (3),(4) and (8), the permeability of the sample can be deriveddirectly from the LB simulation as follows,

k′ = −a2φν

∑Nj=1

∑ni=0 Fi(x, t)ci∑N

j=1

∑ni=0 ∆Fi(x, t)ci

(10)

Here the number of the pore space nodes within the sampleis denoted by N . Since only permeability is scaled backto physical units in the following discussion, the prime willbe dropped for brevity.

The numerical simulation procedure. Driven by theuniform body force, a single fluid enters a rock sample fromthe inlet face and flows steadily towards the outlet face. Innumerical simulations, an additional 5 layers of void spaceare added at both the inlet and outlet faces, so that theperiodic boundary condition can be applied. The otherfour faces, parallel to the overall flow direction, see Fig. 9,are set to solid walls with the no-slip boundary conditionimposed on them.

In a LB simulation, fluid flow through the sample occursonly in the connected (percolating) part of the pore space.In other words, the disconnected pore space does not con-tribute to the sample permeability. Identification of thepercolating pore space is achieved by clustering the porespace using the Hoshen-Kopelman algorithm.64,65 To beconsistent with the nineteen-velocity lattice model, shown

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.29

10

11

12

13

14

15

Relaxation parameter ω

Perm

eability,×

10−

5cm

2

Fig. 10— Effect of the relaxation parameter ω on the averagecalculated permeability of the initial unconsolidated sample,shown in Fig. 3. The average permeability is defined as themean value of the permeabilities in the x, y and z directions.

in Fig. 8, the 18-neighbor rule is used here: face-to-faceand edge-to-edge.

The iterations are run to steady state. It is assumed thatthe flow has stablized when the change of the inferred per-meability meets the condition ||kt/kt+1|− 1| ≤ 10−6. Herekt is the inferred permeability at time t. To assess possi-ble anisotropy of the rock, the simulations are performedindividually for each of the x−, y− and z−directions.

Results and analysisAs described above, the diagenetic geological processes,

such as compaction and cementation, usually alter the mi-crostructure of sedimentary rock and, thus, affect the rockmacroscopic transport properties. In the following sec-tions, we apply our approach to evaluate the permeabilityof natural and reconstructed porous media, and investigatethe effects of compaction and various modes of cementationon the absolute permeability. The results are comparedwith experimental data and the Kozeny-Carman formula.However, first we analyze the sensitivity of the simulationresults with respect to variations of key computational pa-rameters.

The influence of the LB model parameter ω andthe voxel resolution a. In the lattice Boltzmann sim-ulation, the calculated permeability depends on the BGKrelaxation parameter ω, whose value is typically in therange 0.50 < ω < 2 due to stability requirements. In prac-tice, ω is usually chosen to be smaller than 2. In order todetermine a suitable value of ω, we have conducted a seriesof simulations on the initial unconsolidated rock sample,shown in Fig. 3.

The dependence of the calculated average permeabilityon the BGK relaxation parameter ω is shown in Fig. 10.The average permeability is defined as the mean value of


0 5 10 15 20 2535

35.5

36

36.5

37

37.5

38

38.5

39

Resolution a, microns

Perm

eability,D

arc

y

Fig. 11— Effect of the voxel resolution a on the calculatedpermeability of the 0.6 mm × 0.6 mm × 0.6 mm subsample(x, y, z ∈ [2.2, 2.8] mm) in the model sample SI, shown inFig. 2. The relaxation parameter ω is 1.05 in the simulation.

the permeabilities in the x−, y− and z−directions. Oneobserves that the calculated permeability decreases as ωincreases. Compared with the experimental permeability,ke = 9.54×10−5 cm2, we found that the simulation result,ks = 9.62 × 10−5 cm2, gives a good estimation of the per-meability when ω = 1.05. The relative error, defined bye = |ks − ke|/ke × 100%, is only 0.8%. Thus, we chooseω = 1.05 for the remaining simulations.

As with all numerical techniques, the accuracy of a lat-tice Boltzmann simulation is limited by the voxel resolu-tion a. At a higher resolution, one can anticipate more ad-equate results. However, the complexity of the proceduredramatically increases with the number of voxels. There-fore, a reasonable compromise between the resolution andthe complexity of computations is needed. To reach suchas compromise, we have performed a series of numerical ex-periments. For the uniform grain packing, shown in Fig. 1,we observed that a = 200 microns can give a reasonableestimation of the permeability for the model samples withdifferent porosities (listed in the next section). For thenon-uniform grain packing (Fig. 2), we have selected thevoxel resolution based on a sensitivity study of the calcu-lated permeability.

Fig. 11 shows the effect of voxel resolution on the cal-culated permeability of the small subsample (x, y, z ∈[2.2, 2.8] mm) of the non-uniform grain packing. Withthe increasing resolution, the calculated permeability de-creases. Our scheme for discretization is not to replace avoxel with several finer voxels of the same phase,23 or tocombine several finer voxels into a supervoxel40 based onthe same pore space image, but to re-discretize the rocksample directly. Thus, with the increasing resolution, thediscretized image resembles more closely the real geome-try of the sample pore space. Therefore, the permeabilitycalculated at a higher resolution should be closer to the“exact” value. We have found that at the resolution a = 5

Tab. 1— The porosities and permeabilities from the LB sim-ulations and experiment for the uniform grain packing. Thepermeability unit is ×10−5 cm2. kx, ky, and kz are the cal-culated permeabilities in the x−, y−, and z−directions. ka isthe average value, (kx +ky +kz)/3.0. The experimental valueske are from Wyllie and Gregory.48 The relative error is definedas e = |kl − ke|/ke × 100%, l = x, y, z.

φ (%) kx ky kz ka ke emin emax

40.1 9.42 9.57 9.88 9.62 9.54 0.31% 3.56%38.0 7.29 7.62 7.93 7.61 7.50 1.60% 5.73%37.0 6.46 6.79 7.20 6.82 6.79 0.00% 6.04%

microns the estimated permeability is ks = 35.9 Darcy,whereas at the resolution a = 10 microns, the permeabil-ity is ks = 37.5 Darcy. The relative error between thesetwo values is 4.5%. Therefore, in the simulations describedin the following sections, we have chosen a = 10 micronsfor the non-uniform grain packing, and its sub-samples.This choice allowed us to save on computational time andinvestigate larger sub-samples of the packing.

Effect of compaction. The initial unconsolidated uni-form grain packing, shown in Fig. 1, is slightly compactedto a porosity in the range of 36 to 42%. Only the effec-tive (or connected) porosity is used in the discussion thatfollows.

Permeabilities at three different porosities during thecompaction are listed in Tab. 1. The experimental dataare from Wyllie and Gregory.48 We find that the max-imum relative error between the calculated permeabilityand the experimental data is 6.04%. The relative erroris less than 1.5%, if the average permeability, defined aska = (kx + ky + kz)/3.0 is used. This good agreementbetween the results of computations and laboratory dataconfirms that the relaxation parameter ω = 1.05 and thespatial resolution a = 200 microns can give a satisfactoryestimate of the absolute permeability for the uniform pack-ing of coarse sand grains, shown in Fig. 1.

Fig. 12 shows the permeability reduction with increas-ing compaction. We find that our numerical results matchwell the Kozeny-Carman equation. This matching is bestfor the average permeability. The Kozeny-Carman equa-tion relates the permeability of a porous medium to itsspecific surface area and porosity as48

kf =φ3

CS20(1 − φ)2

(11)

where C is called the Kozeny-Carman constant, and S0 isthe surface area per unit volume of solid phase. For equalspheres, the specific surface area is S0 = 6/d, thus, in ourexample, S0 = 19.8 cm−1. The constant C is generallyequal to 5 for flow through unconsolidated porous media.48However, we obtained that C = 4.83 yields the best fitof the experimental results. Our calculated permeabilitycan also be approximated as a power law of the porosity,k = AφB , where A = 4.48 × 10−3 cm2 and B = 4.21.


0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.424

5

6

7

8

9

10

11

Porosity

Perm

eability,×

10−

5cm

2

kxkykzkakekf

Fig. 12— The absolute permeability of the uniform grainpacking from our calculations, measurement, and the tunedKozeny-Carman formula with C = 4.83. For the notation, seeTable 1 and Eq. (11).

Our approach is further validated against other pub-lished data. For example, Bryant et al.66 calculated k/d2 =6.80× 10−4 using a network model based on the geometri-cal data of Finney67 for a uniform bead packing with theporosity 36.2%. In our calculations, ka/d2 = 6.60 × 10−4

and ka/d2 = 7.58 × 10−4 for the grain packings with theporosities of 35.8% and 37.0%, respectively. Maier et al.53reported k/d2 = 7.48 × 10−4 for the simulated bead pack-ing with the porosity 37.0%, and our value is 7.58× 10−4.

Sedimentation and compaction may cause anisotropy ofthe absolute rock permeability. Table 1 and Fig. 12 showthat the calculated permeability is always higher in thez−direction than in the x− and y−directions. It is ex-pected that the grain packing resulting from sedimenta-tion under gravity is anisotropic, but the horizontal per-meabilities should be equal to each other. The calcu-lated permeability of the initial spherical grain packing isanisotropic, nearly equal in the x− and y−directions, andslightly higher in the z−direction. With increasing com-paction, the permeability difference in the x−, y−, andz−directions becomes more obvious. This phenomenoncan be explained as follows. The compaction constrictsthe flow channels in the x− and y−directions more thanin the z−direction. The anisotropic permeability causedby compaction was also reported in pore network model-ing.14

Note that over larger distances, the rock permeabil-ity is usually less in the vertical (bedding-perpendicular)z−direction than in the horizontal (bedding) x− andy−directions. At a scale of a single rock layer, the oppositemay be true if the rock grains are close to spherical. Grainsof other shapes, on the other hand, tend to settle in such away that the single layer permeability in the z−directionis less than in the bedding directions (Dr. Pal-Eric Øren,Statoil, private communication).

Tab. 2— The porosities and permeabilities for the naturaland reconstructed Fontainebleau sandstone. φ and fc are theporosity and the amount of cements in the sample. a is thevoxel resolution, micron. The image size in voxels is denotedby D. kx, ky, and kz are the calculated permeabilities in thex−, y−, and z−directions, Darcy. ka is the average value,(kx + ky + kz)/3.0.

SI SC SPB SPU SPT SPT∗ SEφ 0.40 0.29 0.18 0.18 0.18 0.18 0.18fc 0.0 0.0 0.11 0.11 0.11 0.22 -a 10 10 10 10 10 10 nD 1003 1003 1003 1003 1003 1003 903

kx 44.0 10.7 1.49 1.36 1.05 2.41 0.032n2

ky 42.4 10.0 1.34 0.97 0.45 1.27 0.015n2

kz 45.6 12.3 1.81 1.53 1.17 1.89 0.047n2

ka 44.0 11.0 1.55 1.29 0.89 1.86 0.031n2

Effect of the various modes of cementation. In thissection, we begin with an unconsolidated non-uniformgrain packing shown in Fig. 2, and investigate the effectsof various modes of cementation on the absolute perme-ability.

The reference rock chosen for the study is Fontainebleausandstone, which is very pure (> 99% quartz) and wellsorted, with a constant grain size. Only the character ofthe cementation varies. Therefore the Fontainebleau sand-stone model allows us to focus on the effect of cementationon the porosity, permeability, and microstructure, while allother parameters are fixed. In our calculations, we haveused the grain size distribution reported in Ref.1 Visualcomparison17,18 suggests that the pore space of our mod-eled rock closely resembles the target Fontainebleau sand-stone. Here, we compare the modeled and real rock bytheir absolute permeabilities.

The calculated permeabilities are listed in Tab. 2 for thedifferent rock samples. The model sample SI is obtained af-ter gravity-driven sedimentation, i.e., it is the initial grainpacking shown in Fig. 4. The compacted sample SC isshown in Fig. 5. After compaction, the various cementa-tion modes are modeled by setting the growth exponentζ in Eq. (1) to different values: (i) A positive value putsmore cement in the direction of a pore body or a largedistance l(r) within the sample. This type of cementationis called pore body cementing (PB). (ii) If ζ = 0, the sameamount of cement is uniformly deposited on grain surfacesin all directions. This pattern of cementation is called poreuniform cementing (PU). (iii) A negative value of ζ favorspore-throat growth, i.e., more cement is deposited in thenarrow grain contact regions and on the surfaces of smallgrains. This kind of cementation is called pore throat ce-menting (PT).

In this study, the growth exponent ζ is given values of1, 0 and −1, respectively, to model the three different ce-mentation modes. The parameter ξ is equal to 1 to favorcement deposition on the small grain surfaces. The valueof κ is varied to control the amount of cement deposition


0

50

100

0

50

1000

20

40

60

80

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xy

z

(a) SI

0

50

100

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80

100

xy

z(b) SC

0

50

100

0

50

1000

20

40

60

80

100

xy

z

(c) SPT

Fig. 13— Schematic diagram of streamlines at each simulation stage. (a) After sedimentation, SI, (b) after compaction, SC, and(c) after cementation, SPT, favoring pore throat cementing. For clarity, only the major streamlines are shown.

020

4060

80100

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15

x 10−8

u

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(a) SI

020

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80100

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yz

(b) SC

020

4060

80100

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−1

0

1

2

3

x 10−8

uyz

(c) SPT

Fig. 14— Schematic diagram of the fluid velocity profiles across the cross section x = 50 in Fig. 13 at each simulation stage. (a)After sedimentation, SI, (b) after compaction, SC, and (c) after cementation favoring pore throat cementing, SPT.

in the sample, so as to decrease the sample porosity toany prescribed target value. The samples with the finalporosity of 18% are denoted by SPB, SPU, and SPT.

Sample SE is the three-dimensional computer tomo-graphic image of Fontainebleau sandstone∗.18 Its size is90×90×90 voxels. The voxel resolution n is estimated to bebetween 5 and 7.5 microns. Thereafter, the calculated per-meability of SE is in the range of 0.38-2.64 Darcy. One cansee that the calculated permeabilities of the model samplesSPB, SPU and SPT are in good agreement with the targetsample SE. This finding is confirmed by the visual simi-larity between the model samples and the target sample,cf. Fig. 10 and Fig. 11 in Ref.18 Note that some key pa-rameters in the reconstruction procedure, such as the typeand amount of cement and clay in the Fontainebleau sand-stone, and compaction porosity loss during rock formation,are not known exactly; only the final porosity is matched.However, permeabilities of the model samples SPB, SPU,and SPT differ slightly in the magnitude. Apparently, thisdifference is caused by the different cementation patterns.

As one would expect, the permeability of the initial un-consolidated grain packing, SI, decreases upon diagenesis.

∗The tomographic images are courtesy of Schlumberger.

The diagenetic processes modify the pore space geometryand connectivity, resulting in a complicated evolution ofpermeability. In the compaction stage, the decrease in theporosity from 40% to 29% is accompanied by a 4-fold re-duction in the permeability (from 44 to 11 Darcy). How-ever, the permeability is reduced nearly by an order ofmagnitude when the porosity is further reduced from 29%to 18% by cementation. This sharp permeability reductionresults from the fundamental modification of the geome-try and connectivity of the pore spaces shown in Figs 4,5, and 7. Grain interactions and rearrangement due tocompaction reduce the pore sizes and the porosity. Com-paction keeps most of the porosity connected. In contrast,cementation reduces the porosity by depositing cement ongrain surfaces and, therefore, blocking some parts of thepore space. This blocking increases the tortuosity and dis-connect some part of the pore space.

One can clearly observe the microstructures of the flowand connectivity of the pore space from the streamlineplots shown in Fig. 13, and the cross-section velocity pro-files shown in Fig. 14. Fig. 13 illustrates the 3D velocityfield when a single-phase fluid flows through the modelsamples, SI, SC and SPT. The streamlines represent all


0.16 0.18 0.2 0.22 0.24 0.26 0.280

2

4

6

8

10

12

14

Porosity

Perm

eability,D

arc

y

kx

ky

kz

ka

Fig. 15— Effect of the pore throat (PT) cementing on theabsolute permeability of the model sample SC. The porosityis reduced by cementation only. kx, ky, and kz are the calcu-lated permeabilities in the x−, y−, and z−direction. ka is theaverage value, (kx + ky + kz)/3.0.

possible percolation pathways within the model samplesin the x−direction. It can be seen that fluid flow oc-curs through preferential flow channels. After cementa-tion, only few complicated flow channels remain availablefor the fluid flow through sample SPT, Fig. 13c. Fig. 14shows the x-component of the microscopic velocity acrossthe plane x = 50. The magnitude of the velocity is largerin wider channels. Compared with the distribution of theflow channel sizes in the initial grain packing, shown inFig. 14a, the size and number of wide flow channels de-crease significantly during the compaction, see Fig. 14b.Only few small channels remain available for fluid flow af-ter the cementation, Fig. 14c. The negative fluid velocityin some locations implies that the fluid has had to followextremely tortuous microscopic flow paths to get throughthe pore space of the model sample SPT.

Rock microstructure evolves with diagenetic processes.As more cement is deposited on the grain surfaces, theporosity and permeability of the rock continuously de-crease. We assume that the geological environment doesnot change during rock formation, and the newly depositedcements follow the same cementation pattern as the exist-ing ones. Of course, given sufficient knowledge of deposi-tional environment, the cement deposition patterns in thesimulation can be adjusted correspondingly.

Fig. 15 shows permeability reduction of the model sam-ple SC as a function of porosity, when cement is precip-itated on the grain surfaces according to the pore throatcementing pattern. No compaction is involved and theporosity is reduced only by the cementation. One can seethat the permeability reduction in the z−direction is largerthan those in the x− and y−directions. The cementationrole is roughly the same in the permeability alteration inthe x− and y− directions, which is reflected by the nearlyconstant permeability difference between them. A likely

0.16 0.18 0.2 0.22 0.24 0.26 0.280

2

4

6

8

10

12

14

Porosity

Perm

eability,D

arc

y

kx

ky

kz

ka

Fig. 16— Effect of the pore uniform (PU) cementing on theabsolute permeability of the model sample SC. The porosityis reduced by cementation only. kx, ky, and kz are the calcu-lated permeabilities in the x−, y−, and z−direction. ka is theaverage value, (kx + ky + kz)/3.0.

0.16 0.18 0.2 0.22 0.24 0.26 0.280

2

4

6

8

10

12

14

Porosity

Perm

eability,D

arc

ykx

ky

kz

ka

Fig. 17— Effect of the pore body (PB) cementing on the ab-solute permeability of the model sample SC. The porosity isreduced by cementation only. kx, ky, and kz are the calcu-lated permeabilities in the x−, y−, and z−direction. ka is theaverage value, (kx + ky + kz)/3.0.

explanation may be that the pore geometry is similar inthe x− and y−directions in the model sample SC and,therefore, the similar reduction in the size of flow channelsoccurs during the pore throat cementation. The effect ofthe pore uniform and body cementing on the permeabil-ity reduction of the model sample SC is shown in Fig. 16and Fig. 17. These two cementation patterns show simi-lar permeability reduction tendency as in the pore throatcementing.

Fig. 18 displays the average calculated permeability ofthe model sample SC as a function of the porosity, whencement is deposited on the grain surfaces according to the


0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.30

2

4

6

8

10

12

14

Porosity

Perm

eability,D

arc

y

PBPUPTFC

Fig. 18— Effect of different cementation patterns on the av-erage calculated permeability of the model sample SC. Theporosity is reduced by cementation only. The fitting curve(FC) is a power law k = AφB with A = 2.86 × 103 Darcy andB = 4.26.

three different cementation patterns: the pore body, uni-form and throat cementing. The relation between the per-meability and porosity can be expressed as a power lawover almost the entire range of porosities studied here:

k = AφB (12)

where A = 2.86×103 Darcy and B = 4.26. Compared withthe value B = 4.21 for the unconsolidated uniform grainpacking, the larger exponent B = 4.26 indicates that thebehavior of the model samples deviates from the Kozeny-Carman predictions. This deviation may be explained asfollows. Traditionally, the Kozeny-Carman formula is de-rived from a model, in which the connected pore space isreplaced with a multiply connected network of cylindricalcapillaries with a constant cross-sectional area.68 The hy-pothesis of the constant cross-section area (or the so-calledhomogeneity of the pore structure) is crucial in the deriva-tion. However, even the pore space formed by a uniformspherical packing is significantly different from a bundleof capillary tubes. The non-uniform grain sizes and con-sequences of diagenesis (compaction and cementation) canonly enhance the microscopic inhomogeneity of the poregeometry of the model samples.

Fig. 18 shows that in the high porosity range (φ >19.5%), the dependence of the permeability on porosity isabout the same for all three cementation patterns. How-ever, in the low porosity range (φ < 19.5%), the situation isdifferent. Generally, the permeability is controlled by thesizes of the pore throats connecting the pore bodies. In thehigh porosity range, the reduction of permeability by ce-mentation is almost entirely due to the gradual shrinkageof the pore throats, but only very few smallest pore throatscompletely close. Those closures do not decrease much themacroscopic hydraulic conductivity of the sample. The rel-atively large pore throats remain open. With more cement

(a) SPT (b) SPT∗

Fig. 19— The cross-section images cut from the centers ofthe model samples SPT and SPT∗. Only the pore space isshown.

deposited in the sample, even large pore throats graduallyshrink and eventually can be completely blocked. The con-ductive pore throats become narrower and less numerous.Therefore, in the low porosity range the permeability issharply reduced by cementation.

The various modes of cementation have different effectson the permeability change, especially in the low porosityrange. From Tab. 2 and Fig. 18, it seems that the impactof the pore throat cementing on the permeability is muchstronger than those of the pore body and uniform cement-ing. In the pore body cementing, there is a tendency toform sheet-like pores, so that the porosity is reduced dra-matically, but the pore space remains connected even atlow porosity. In contrast, in the pore throat cementing,cement deposits in the grain contact regions, and tendsto close off pore throats. Thus, the pore space becomesdisconnected even at a higher porosity. The influence ofthe pore uniform cementing is between the other two ce-menting patterns, PB and PT. To summarize, the dom-inant mode of cementation determines the magnitude ofpermeability reduction with cementation in the low poros-ity range.

Effect of diagenetic history. The porosity itself doesnot uniquely determine the respective rock-forming pro-cess: the same end-value can be obtained from differentcombinations of compaction and cementation. It turns outthat it is possible for two rocks of same porosity to havesignificantly different permeabilities. As a test, anothermodel sample SPT∗ is constructed as follows. Cement isdeposited directly on the grain surfaces of the initial pack-ing SI, while in the model sample SPT cement is depositedon the grain surfaces of the compacted packing SC. Thesame cementation parameters are used in both samples,i.e., their cementation patterns are similar. In addition,the final effective (connected) porosity is nearly the samefor these two samples: 17.25% for SPT and 17.33% forSPT∗. But, as one can see from Tab. 2, the calculatedpermeability is about twice as large in sample SPT∗ thanin SPT.

The different permeabilities calculated for the modelsamples SPT and SPT∗ result from the different geome-


tries and connectivities of the respective pore spaces, whichin turn result from the different mechanisms of rock for-mation. Fig. 19 displays the thin-section images of thepore space cut from the centers of the model samples SPTand SPT∗. One can observe that the pore space in sampleSPT∗ is well-connected through the relatively large “porethroats.” In contrast, the pore space in sample SPT is con-nected through very narrow “pore throats.” Both sampleshave the same porosity; therefore, the large pores in sampleSPT∗ are also well sorted.

Discussion and conclusionsWe have presented an integrated procedure for the esti-

mation of the absolute permeability of unconsolidated andconsolidated reservoir rock. This procedure is based on di-rect analysis of microtomographic or computer-generated3D images of the pore space. It consists of two majorsteps (1) obtaining an image from CT scanner or simu-lation of formation of sedimentary rock; and (2) the di-rect simulation of viscous fluid flow in the pore space ofthe tomographic or computer-generated rock images. Thisprocedure has been applied to visualize the interplay be-tween the microscopic changes in rock microstructure andthe macroscopic rock properties, such as the porosity andpermeability.

The lattice Boltzmann method has been used to simu-late the single phase flow in the natural and model sam-ples of permeable rock. The absolute sample permeabilityhas been derived directly from the simulated fluid velocityfield. A study of the calculated rock permeability sensitiv-ity with respect to the BGK relaxation parameter ω andthe spatial resolution a has been studied

To test the approximate LB-BGK model and determineits relaxation parameter, we have constructed an unconsol-idated uniform grain packing, and calculated its absolutepermeability. Good agreement was found when the calcu-lated rock permeability was compared with an experimentand a pore network simulation. Also, the calculated per-meability agreed well with the Kozeny-Carman formulawith C = 4.83, when the initial unconsolidated uniformgrain packing was slightly compacted from the porosity of42% to 36%.

The calculated permeability of a model Fontainebleausandstone was found to agree with the estimated perme-ability of the target Fontainebleau sandstone. This agree-ment shows that our depositional model allows successfulreconstruction of sedimentary rock, and provides a reason-ably good prediction of the absolute permeability. Thelimits of predictive capability of our model must be fur-ther tested by comparing the calculated permeabilities ofthe reconstructed rock samples with the target micro-CTrock images, and direct permeability measurements.

There are other significant benefits from using ourmodel. In particular, one can study the evolution of therock microstructure and the absolute permeability withcompaction and cementation. One may start with an un-consolidated non-uniform grain packing, and study thepermeability variation with compaction, different degrees

of cementation, and different cementation patterns.We have found that the relationship between the poros-

ity (or the amount of cement) and permeability can be de-scribed with a simple power law k = AφB over the wholerange of porosity studied here, and for different cemen-tation patterns. However, this relationship could not beextended to the lower porosities (φ < 19%). The perme-ability began to deviate from the power law below a certainvalue of the porosity.

Different permeability reductions resulted from differentcementation patterns, implying that the diagenetic historyof rock formation plays an important role in the develop-ment of rock microstructure and its transport properties.A meaningful rock reconstruction procedure should takeinto account all relevant physical processes of rock forma-tion. In general, it is insufficient just to match the finalporosity during the reconstruction process. This results, inparticular, implies that any permeability-porosity relation-ship, like the Kozeny-Carman equation 11, is not universal.

The results described here are but the first phase of amuch larger research effort aimed at developing a funda-mental and quantitative understanding of rock transportproperties, and their relationship to the diagenetic rockhistory. We conclude by noting that some of the prelimi-nary observations presented here can only be verified by adetailed study of a large number of real rock samples andtheir reconstructed counterparts. One should systemati-cally change the microstructure of the simulated rock sam-ples, to establish the qualitative and quantitative trendsamong the geometric, transport, and mechanical proper-ties of the rock.

NomenclatureRoman lettersa voxel resolution, LA coefficient in the power law, L2

B exponent in the power law, dimensionlessd grain diameter, LD image size, dimensionlesse relative error, dimensionlessfc amount of cements, dimensionlessci lattice velocity, dimensionlesscs lattice speed of sound, dimensionlessC Kozeny-Carman constant, dimensionlessFi particle distribution in the i direction

dimensionlessF e

i local equilibrium distribution, dimensionlessk permeability, L2

ka average permeability, L2

ke experimental permeability, L2

kf permeability from the Kozeny-Carman equation, L2

ks simulated permeability, L2

kx permeability in the x−direction, L2

ky permeability in the y−direction, L2

kz permeability in the z−direction, L2

l distance between the original grain surfaceand the plane of the polyhedron, L

L distance from the grain center to the


pore-cement interface, LL1 sample length, dimensionlessm mass, dimensionlessn number of lattice velocities per lattice

node, dimensionlessN number of pore space nodes, dimensionlessP pressure, dimensionlessq volumetric fluid flux, L/tr vector from the grain centerR mean grain radius, LR0 original grain radius, LS0 specific surface area, L−1

t time, dimensionlessu fluid velocity, dimensionlesswi weight factor associated with i direction,

dimensionlessx location of the lattice node, dimensionlessx space coordinate, Ly space coordinate, Lz space coordinate, L

Greek lettersκ coefficient determining the porosity reduction,

dimensionlessζ exponent controlling the growth direction,

dimensionlessξ parameter controlling the effect of grain size

on the rate of cement overgrowth, dimensionlessφ porosity, dimensionlessω BGK relaxation parameter, dimensionlessρ density, dimensionlessν kinematic viscosity of fluid, dimensionless

Superscript′ dimensional quantity in physical units

AcknowledgementsGratitude is expressed for financial support to the Assis-

tant Secretary for Fossil Energy, Office of Natural Gas andPetroleum Technology, through the National PetroleumTechnology office, Natural Gas and Oil Technology Part-nership under US Department of Energy contract no. DE-AC03-76SF00098 to Lawrence Berkeley National Labora-tory. Partial support was also provided by gifts fromChevronTexaco, ConocoPhillips, and Statoil to UC Oil,Berkeley. We are thankful to Schlumberger for providingthe synchrotron images of Fontainebleau sandstone.

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