Etudes in computational rock physics: Alterations and benchmarking

Jack Dvorkin1, Qian Fang2, and Naum Derzhi2

ABSTRACT

We tested computational benchmarking data for theabsolute permeability, electrical formation factor, and elasticmoduli based on the Finney pack, a physical dense randompack of identical spheres digitally rendered into a 3D rectan-gular coordinate system, as the starting digital object. It isaltered by (a) changing the radius of each sphere and (b) geo-metrically inverting these new packs by swapping grains andpores. Porosity, the absolute permeability, electrical forma-tion factor, and elastic moduli are computed for all thesealterations. The direct (grain-based) objects are relevant toclastic rock, and the inverse objects are proxies for carbo-nates with moldic pores. To corroborate these computationalresults, we matched the permeability versus porosity, forma-tion factor versus porosity, and elastic moduli versus poros-ity trends they form by established theoretical rock physicsmodels. These trends persisted when we reduced the scale ofinvestigation by subsampling some of the digital objectsunder examination.

INTRODUCTION

The computational rock physics opens an unprecedented oppor-tunity to experiment on objects that are not amenable to traditionalphysical measurements. These objects include very small rock testsas well as damaged and irregularly shaped fragments. Moreover,once such an object is accurately represented in 3D as a set of in-tegers (e.g., zeros for pores and ones for grains), it can be digitallyaltered to produce its variants to reflect diagenetic and mechanicalprocesses in rock by, e.g., altering the grains and pores to introducecement; placing small particles in the pore-space; replacing oneminerals by another (e.g., feldspar by clay); and introducing or heal-ing natural or stress-induced fractures.

The goal of this work is to use such alteration techniques to createa computational benchmarking data set for the physical propertiesof granular rocks and their geometric opposites, such as carbonateswith moldic porosity. The starting object is the Finney pack — arandom dense pack of identical spheres digitally represented in 3Dspace. It is first altered by uniformly changing (reducing andincreasing) the radii of the spheres. Next, these structures are geo-metrically inverted by swapping grains and pores to make them re-presentative of moldic carbonates or high-porosity volcanic rocks.The computational results for porosity, absolute permeability,

electrical formation factor, and elastic moduli are tabulated and thetrends are plotted. These results form a digital benchmarking dataset for future use which we deem no less significant than publishedphysical data benchmarks, as the data presented here are based onaccurately defined and simple objects that can be used by develo-pers and/or users of digital rock physics laboratory to verify andvalidate new (or newly adopted) algorithms and results.It is important to list two novel points of this work: (a) although

the Finney pack as well as statistically or computationally generatedsphere packs have been used in various computational experiments(e.g., Bosl et al., 1998; Keehm et al., 2001; Knackstedt et al., 2003;Toumelin and Torres-Verdín, 2008; Jin et al., 2009; Sain, 2010),here, for the first time, we systematically alter it and simultaneouslycompute the absolute permeability, electrical formation factor, andelastic moduli for all of its geometrical permutations; and (b) theresults for the geometrically inverse structures form a new bench-mark data set relevant to moldic carbonates.In addition, we subsample selected samples by dividing each into

eight (23) smaller objects. Although the subsampling procedureused here has been employed before, the results are somewhatunexpected and, hence, new. To better understand this novelty, letus recall that when we digitally subsample a microscopic sample ofnatural rock, we often recover an, e.g., permeability-porosity trendformed by several larger samples. This effect was first discoveredby Rothman and Zaleski (2004) and Kameda and Dvorkin (2004)and later elaborated upon by, among others, Kameda et al. (2006)and Richa (2010). In one example (Kameda et al., 2006), subsam-pling a single sandstone sample of fixed porosity first widened the

Manuscript received by the Editor 30 June 2011; revised manuscript received 12 October 2011; published online 22 March 2012.1Stanford University, Department of Geophysics, Stanford, California, USA; Ingrain, Inc., Houston, Texas, USA. E-mail: dvorkin@stanford.edu.2Ingrain, Inc., Houston, Texas, USA. E-mail: fang@ingrainrocks.com; derzhi@ingrainrocks.com.

© 2012 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 77, NO. 3 (MAY-JUNE 2012); P. D45–D52, 10 FIGS., 1 TABLE.10.1190/GEO2011-0236.1

porosity range. Next, when the permeability was computed for eachof the subsamples, the resulting permeability-porosity pairs formeda trend very close to that formed by the physical data from a multi-tude of geologically similar samples in a wide porosity range. Thiseffect still awaits a convincing explanation. One speculation is thatthe pore-space heterogeneity at the microscale duplicates the diag-enesis-induced heterogeneity observed at a much larger scale.However, even if this explanation is plausible, the situation

encountered here with the Finney pack is different: Each digitalobject examined here was formed from the original regular homo-geneous structure by spatially homogeneous alterations. Hence,the above-mentioned speculation, arguably passable for a naturalrock, is irrelevant here. Still, the fact remains that the data pairs ob-tained by subsampling one regular object encroach upon the domainsoccupied by other and different regular objects. The objective of thisspecific exercise is not to offer another explanation to the scale-stability of the trends but rather present a new fact and entice the read-er to contemplate the nature of real and artificial porous systems.Finally, we believe that the results presented here not only can

serve as a benchmark but also illustrate the potential of computa-tional rock physics as applied to natural rock and man-made porousmaterials, such as proppants and foams.

OBJECT OF EXPERIMENT

The Finney pack used here is a dense random pack of 4021nominally identical hard spheres. Finney (1970) recorded the coor-dinates of each center, thus producing a historical data set amenableto analytical and computational experimentation (Figure 1).To accurately represent the spherical geometry on a 3D rectan-

gular grid, we selected a smaller cubical volume around the centerof the original Finney pack (Figure 2, left) to conduct simulations ofviscous fluid flow, electrical current, and linear elastic deformationon this digital object. Resulting from these computational experi-ments are the absolute permeability, electrical formation factor,and the elastic bulk and shear moduli, respectively.The size of the original digital object as well as of all its geome-

trical alterations was 141 × 141 × 141 voxels. The physical size of acubical voxel was 10−3 mm on each side. Hence, the physical sizeof each digital object under examination was 0.141 × 0.141×0.141 mm, and the diameter of each sphere in the original digitalpack was about 44 voxels, which translates into the physical size0.044 mm.

INVERTING AND ALTERING THE PACK

Shown in Figure 2 is a subvolume of the original Finney packwhose porosity is 0.3654. Let us geometrically invert this pack byswapping pores and grains (Figure 2, right). The porosity of thisinverse structure is 0.6346, one minus the porosity of the originalpack. Although the original Finney pack is a proxy for a granularrock, its geometrical inverse can serve as a proxy for moldiccarbonates.We can alter the original pack by changing the radius of each

grain by the same amount and then invert these structures as well.For example, if we reduce the radius of each grain to 0.95 of itsoriginal value (Figure 3), the porosity becomes 0.4560, whereas thatof the inverse structure becomes 0.5440. If we expand the radius ofeach grain to 1.05 of its original value and allow the resulting largerspheres to interpenetrate (Figure 4), the porosity of the resultingstructure becomes 0.2726 and that of the inverse structure becomes0.7274.The porosity symmetry between the direct and inverse structures

is approximately attained for the radius of each sphere 0.92 of theoriginal radius, with the porosity of the direct structure 0.5059 andthat of the inverse structure 0.4941 (Figure 5).

Figure 1. The Finney pack, two view angles. The spheres are light-gray on the black background.

Figure 2. A subset of the Finney pack on a 3D rectangular grid.White is for the solid while black is for the pores. Left: The originalpack. Right: Grains switched to pores and pores switched to grains.The cube has 141 voxels on each side with the size of the digitalsample 2.803221 million ¼ 1413 voxels and the voxel size0.001 mm. The physical size of the cube is 0.141 mm on each side.The diameter of each sphere is approximately 44 voxels or0.044 mm.

Figure 3. Same as Figure 2 but with the grain radius uniformly re-duced to 0.95 of the original radius.

D46 Dvorkin et al.

COMPUTATIONAL ENGINES AND RESULTS

Three sets of physical properties of the packs thus alteredas well as of their geometrically inverse structures were computedas:

• the absolute permeability (k) by means of simulating the in-compressible creeping viscous flow in the pore space usingthe Lattice-Boltzmann numerical method which resolves theappropriate Navier-Stokes equations

• the electrical resistivity assuming that the solid is a perfectresistor while the pore space is a conductor with uniform andfinite resistivity: The resistivity thus obtained is normalizedby that of the pore-filling material. This ratio is called theelectrical formation factor (F).

• the elastic moduli, bulk (K) and shear (G)

The resistivity and elastic properties are computed using thefinite-element method adopted from Garboczi and Day (1995) withthe appropriate field equations numerically resolved.All these computational engines are described in Dvorkin

et al. (2008, 2011) and Tolke et al. (2010). The porosity is computedby dividing the number of voxels that fall into the pore space by thetotal number of voxels in the selected volume.In the elastic simulations, we assumed that the solid was calcite

with the bulk modulus 76.80 GPa and shear modulus 32 GPa,whereas the pore space filling had zero elasticmoduli (infinitely compliant). The reason for se-lecting calcite was to make the elastic propertiesof inverse structures relevant to moldic limestone.These computational results are listed in

Table 1 and displayed in Figures 6, 7, and 8.

CORROBORATION OF THERESULTS

Absolute permeability

Because it is very unlikely to obtain physicalobjects precisely matching the digital geometriesexamined here, the direct experimental validationof our results is hardly feasible. This is whywe resort to implicit corroboration by matching

the results by established theoretical rock physics models. One for-tunate exception is a fundamental value k∕d2 ¼ 680000 mD∕mm2

in which the ratio of the absolute permeability k in mD to the par-ticle diameter squared d2 in mm2 for a random dense pack of iden-tical spheres (see the data plot and references in Bryant et al.[1993]). In the case under examination, only one direct structure,the original Finney pack (Figure 2, left), represents such geometry.

Figure 4. Same as Figure 2 but with the grain radius uniformlyincreased to 1.05 of the original radius.

Figure 5. Porosity versus the grain radius as the grains are uniformly expanded orshrunk. Black symbols: Direct structures. Gray symbols: Inverse structures.

Table 1. Computed properties of altered and inverted Finney packs. From left to right: radii of the spheres (the original radiusis 1.00), porosity of the pack, porosity of the inverse structure; permeability (mD) of the pack, permeability of the inversestructure, formation factor of the pack, formation factor of the inverse structure, bulk modulus (GPa) of the pack, bulkmodulus of the inverse structure, shear modulus of the pack, and shear modulus of the inverse structure. Subscriptd refers to the direct structures and subscript i refers to the inverse structures.

R ϕd ϕi kd ki Fd Fi Kd Ki Gp Gi

0.92 0.5059 0.4941 4147.4 0 2.5667 ∞ 0.000 18.700 0.000 9.710

0.95 0.4560 0.5440 2807.1 0 2.9667 ∞ 0.000 15.480 0.000 8.080

1.00 0.3654 0.6346 1326.3 231 4.1667 6.4000 3.730 9.640 2.760 5.120

1.05 0.2726 0.7274 533.1 4422 6.8000 2.4667 13.680 4.880 9.470 2.490

1.10 0.1895 0.8105 164.3 12,573 13.000 1.6667 26.210 2.030 15.760 0.930

1.15 0.1211 0.8789 35.5 25,895 35.667 1.3000 40.030 0.360 21.260 0.180

1.20 0.0707 0.9293 6.0 51,548 193.33 1.2000 53.010 0.040 25.600 0.030

Etudes in computational rock physics D47

Figure 6. Computed permeability (mD) for the direct and inverse structures. (a) Permeability versus porosity color-coded by the radius of thegrains. The solid curve is from theKozeny-Carman equation as described in the text. An open square is the physicallymeasured permeability of arandom dense pack of identical spheres (see discussion in the text). The dashed curve connects the data points for the inverse-pack structures.(b) Permeability versus the radius of the grains, color-coded by porosity. The dashed curves are connectors between the data points.

Figure 7. Computed electrical formation factor data for the direct and inverse structures. The display is the same as used in Figure 6. Thetheoretical (solid) curve in (a) is from equation 4 with the inputs listed in the text. The dashed curve connects the inverse structure data points.In (b), the dashed curves are connectors between the data points. The arrows indicate that the formation factors of the inverse structuresbecomes infinite (zero conductivity) as its porosity reduces beyond 60%.

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The diameter of the spheres in this digital pack is 0.044 mmand its permeability is 1326.3 mD at porosity 0.3654 (Table 1).Hence, the computed k∕d2 is 685072 mD∕mm2 andclosely matches the experimental value. For d ¼ 0.044 mm,

k∕d2 ¼ 680000 mD∕mm2 translates into k ¼ 1316.5 mD.This value is plotted versus porosity 0.36 in Figure 6a ontop of the permeability-porosity trend for the values listed inTable 1.

Figure 8. Computed elastic bulk (a and c) and shear (b and d) moduli. The theoretical curves in (a and b) are from the stiff-sand model asexplained in the text. The black squares in (c and d) are the elastic moduli of pure calcite (plotted at zero porosity). The dashed curves in (c andd) are the connecting lines. The display is the same as used in Figures 6 and 7.

Etudes in computational rock physics D49

To corroborate the rest of the computational results listed inTable 1, we need to recall the theoretical Kozeny-Carman equation(e.g., Mavko et al., 2009)

k ¼ 109d2

72

ðϕ − ϕpÞ3ð1 − ϕþ ϕpÞ2τ2

; (1)

where, once again, the grain size d is in mm and permeability k is inmD. The other symbols used are τ for the tortuosity and ϕp for thepercolation porosity at which the pore space becomes disconnectedand, hence, k becomes zero. For granular rock, ϕp is usually a smallnumber, between zero and 0.03.In the Kozeny-Carman equation, the tortuosity is defined as the

ratio of the length of a pipe-shaped conduit to the length of the por-ous sample in the direction of the flow. There are no pipe-shapedconduits in a realistic pore-space. This is why τ needs to be treatedas an adjustable parameter that, of course, cannot be smaller thanone. Generally, the tortuosity should be porosity-dependent (e.g.,Zinszner and Pellerin, 2007). Still, because τ is essentially a fittingparameter in a highly idealized model and, as such, can be verydifferent for different samples even at the same porosity; someauthors treat it as a constant for a set of samples of similar nature(see, e.g., Bourbie et al. 1987, for the Fontainebleau sandstone dataset). Because our variants of the Finney pack can also be treated assamples of similar nature, we adopt the latter concept and keep thetortuosity constant.To find this constant, we rearrange equation 1 as

τ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi109

72

1

k∕d2ðϕ − ϕpÞ3

ð1 − ϕþ ϕpÞ2s

(2)

and, by using k∕d2 ¼ 680000 mD∕mm2 and ϕ ¼ 0.36 and alsoassuming ϕp ¼ 0.01, find τ ¼ 1.44.Finally, we use this τ value in equation 1, in which we vary d as a

function of the varying radius R (in units of the original unalteredFinney pack sphere radius) of the expanding or shrinking spheres(as listed in Table 1) by multiplying d ¼ 0.044 mm by R. The re-sulting permeability-porosity curve in Figure 6a closely matchesour computational results.Because the goal of this discussion is to present new computa-

tional results rather than interpret them in terms of yet another idea-lized equation, we do not attempt to match the inverse-packpermeability by an analytical curve. Instead, we connect these datapoints by a line (Figure 6a).In Figure 6b, the same permeability data are plotted versus the

altered radius of the individual spheres. The permeability of the al-tered direct pack equals that of the inverse pack for the sphere radiusabout 1.022 of that of the original unaltered pack. Of course, at thismatching point the porosity of the inverse pack exceeds that of thedirect pack.

Formation factor

The formation factor (F) results are displayed in Figure 7. Thecomputational data are corroborated by the Archie equation (e.g.,Mavko et al., 2009)

F ¼ a∕ϕm: (3)

We modify this equation following the percolation porosityconcept employed in equation 2:

F ¼ a∕ðϕ − ϕpÞm: (4)

Sen et al. (1981) show that F measured on fused glass beads isaccurately matched by equation 4 with a ¼ 1, ϕp ¼ 0.00, andm ¼ 1.5. The curve displayed in Figure 7a to mimic the directstructure data uses a ¼ 1, ϕp ¼ 0.01, and m ¼ 1.5. For the samereason as in the permeability case, we do not attempt to match theinverse-pack formation factor data by an analytical curve but insteadconnect them by a line.It is important to remind the reader that there are theoretical

formation factor models, such as, self-consistent approximationsor differential effective-medium models (e.g., Mavko et al., 2009),that can be possibly forced to mimic these computational data. Suchexercises are beyond the scope of present discussion.

Elastic moduli

The elastic properties listed in Table 1 are displayed in Figure 8.To match the elastic moduli computed for the direct structures weused the stiff-sand model (Mavko et al., 2009) that is the modifiedupper Hashin-Shtrikman bound with the critical porosity 0.45 (asappropriate for a granular structure, see Nur et al., 1998), coordina-tion number 6, and close to zero differential pressure. For theinverse structures, we used the same model and inputs but withthe critical porosity 0.90 (as appropriate for a foam-like structure).This rock-physics model was designed to describe the elastic

behavior of stiff, well-consolidated rock. Although the originalFinney pack represents an unconsolidated granular material, this mod-el still seems appropriate for the case under examination because byextending the radii of the original spheres we essentially “cement” thepack. The inverse structures are representative of a “foam” geometry,relevant to high-porosity carbonates and volcanic pumice, which alsocan be accurately described by the stiff-sand model with appropriatelylarge critical porosity (Nur et al., 1998). In these theoretical modeling,we assumed the solid pure calcite and the pore space empty. The pure-calcite elastic moduli endpoints at zero porosity are also displayed inFigure 8a and 8b for the bulk and shear moduli.

SUBSAMPLING

The trends observed in Figures 6, 7, and 8 are different for thedirect and inverse structures, yet within the structure type, they arestable. The next logical question is whether these trends depend onthe size of the Finney pack sample used. To address this question,we subsample two digital samples of the direct structure and two ofthe inverse structure. This subsampling is done by evenly dividingthe selected cube unto 8 ¼ 23 subcubes.The results of computing the permeability and formation factor of

eight subsamples of the two selected direct structures and two se-lected inverse structures are displayed in Figure 9. The permeabilityand formation factor in the subsamples appeared slightly anisotro-pic. Hence, the results plotted in Figure 9 are the arithmetic averagesof these properties in the three directions. Remarkably, the perme-ability-porosity and formation factor-porosity data of these 32 sub-samples fall on the trends formed by the larger-sample data.We have observed this effect before when subsampling digital

images of natural rock and computing their physical properties(Dvorkin and Nur, 2009). We suggested then that under certain, pos-sibly limited circumstances, pairs of data points obtained on an in-ternally heterogeneous data set form a trend that is valid over a range

D50 Dvorkin et al.

of scales. Such a trend is stationary with respect to position and scale,and so it can be applied to a remotely sensed quantity (e.g., porosity)to arrive at another desired property (e.g., permeability) at the scale ofpractical measurement. We have guardedly avoided making long-

reaching conclusions about the geologic background of this phenom-enon. Still, one plausible speculation might be that small subsamplesfind the internal variability inside the rock that is similar to its macro-scopic variability governed by diagenesis.

Figure 9. Permeability (a) and formation factor (b) versus porosity for the altered direct and inverse Finney pack structures (black circles).These data are the same as displayed in Figures 6 and 7. Smaller gray symbols are for 32 (8 × 4) subsamples of the selected four originalsamples (two for the direct structures and two for the inverse structures). The dashed lines connect the black circles.

Figure 10. Bulk (a) and shear (b) moduli versus porosity for the altered direct and inverse Finney pack structures. These data are the same asdisplayed in Figure 8. Smaller gray symbols are for 32 (8 × 4) subsamples of selected four original samples.

Etudes in computational rock physics D51

However, in the idealized case under examination, this specula-tion is false because the structures we create are very homogeneousbecause they are created by uniformly increasing or reducing theradius of each grain. This scheme mimics very simple diageneticalteration in which the cement is uniformly added to each grain.Natural diagenesis can be much more complex. In spite of this,the trends obtained on the subsamples seem to connect the datapairs belonging to different larger samples. This effect still awaitsan explanation.The subsampling results for the elastic bulk and shear moduli are

displayed in Figure 10. Once again, the subsample data pairs fall onthe trend formed by the original Finney structures, direct and in-verse, used in these computational experiments. Moreover, thesedata pairs fill the space between the data pairs belonging to the ori-ginal structures. Once again, at present, we cannot explain this com-putational-experiment phenomenon. Further elaborate investigationof the scale effect is in order.

CONCLUSION

Simple digital geometry alterations of the original Finney packused in computational engines allow us to generate reference trendsfor the absolute permeability, electrical formation factor, and elasticmoduli versus porosity. Although the alterations of the original packare proxies for natural clastic rock, their geometric inverse struc-tures are proxies for foam-like natural structures, such as volcanicand moldic carbonate rocks. Subsampling of the originally gener-ated samples preserves the trends, counterintuitively, for the caseunder examination.Such experiments, if intended to be carried out in the physical

laboratory, require elaborate sample preparation (e.g., fusing orepoxy-cementing glass beads or creating foam). Yet, these experi-ments are easy and conclusive if carried out in the digital rock-physics laboratory. Such a laboratory may become a massive datagenerator to handle practically any structure, be it a scan of a naturalsample or a sample created by the imagination of a geologist.The results reported here are intended to be used as one of

rock-physics benchmarks, such as physical measurements reportedbefore for glass beads or Fontainebleau and Berea sandstone, butthis time, computationally generated.

REFERENCES

Bosl, W., J. Dvorkin, and A. Nur, 1998, A study of porosity and permeabilityusing a lattice Boltzmann simulation: Geophysical Research Letters, 25,1475–1478, doi: 10.1029/98GL00859.

Bourbie, T., O. Coussy, and B. Zinszner, 1987, Acoustics of porous media:Gulf Publishing Company.

Bryant, S. L., P. R. King, and D. V. Mellor, 1993, Network model evaluationof permeability and spatial correlation in a real random sphere packing:Transport in Porous Media, 11, 53–70, doi: 10.1007/BF00614635.

Dvorkin, J., M. Armbruster, C. Baldwin, Q. Fang, N. Derzhi, C. Gomez, B.Nur, A. Nur, and Y. Mu, 2008, The future of rock physics: Computationalmethods versus lab testing: First Break, 26, 63–68.

Dvorkin, J., N. Derzhi, E. Diaz, and Q. Fang, 2011, Relevance of computa-tional rock physics: Geophysics, 76, no. 5, E141–E153, doi: 10.1190/geo2010-0352.1.

Dvorkin, J., and A. Nur, 2009, Scale of experiment and rock physics trends:The Leading Edge, 28, 110–115, doi: 10.1190/1.3064155.

Finney, J. L., 1970, Random packings and the structure of simple liquids. I.The geometry of random close packing: Proceedings of the Royal Societyof London: Series A, Mathematical and Physical Science, 319, no. 1539,479–493, doi: 10.1098/rspa.1970.0189.

Garboczi, E. J., and A. R. Day, 1995, An algorithm for computing the ef-fective linear elastic properties of heterogeneous materials: Three-dimensional results for composites with equal phase Poisson’s ratios:Journal of the Mechanics and Physics of Solids, 43, 1349–1362, doi:10.1016/0022-5096(95)00050-S.

Jin, G., C. Torres-Verdín, and E. Toumelin, 2009, Comparison of NMRsimulations of porous media derived from analytical and voxelized repre-sentations: Journal of Magnetic Resonance, 200, 313–320, doi: 10.1016/j.jmr.2009.07.021.

Kameda, A., and J. Dvorkin, 2004, To see a rock in a grain of sand: TheLeading Edge, 23, 790–794, doi: 10.1190/1.1786904.

Kameda, A., J. Dvorkin, Y. Keehm, A. Nur, and B. Bosl, 2006, Permeability-porosity transforms from small sandstone fragments: Geophysics, 71, no.1, N11–N19, doi: 10.1190/1.2159054.

Keehm, Y., T. Mukerji, and A. Nur, 2001, Computational rock physics at thepore scale: Transport properties and diagenesis in realistic pore geome-tries: The Leading Edge, 20, 180–183, doi: 10.1190/1.1438904.

Knackstedt, M. A., C. H. Arns, and W. V. Pinczewski, 2003, Velocity-porosity relationships, 1: Accurate velocity model for clean consolidatedsandstones: Geophysics, 68, 1822–1834, doi: 10.1190/1.1635035.

Mavko, G., T. Mukerji, and J. Dvorkin, 2009, The rock physics handbook:Cambridge University Press.

Nur, A., G. Mavko, J. Dvorkin, and D. Galmudi, 1998, Critical porosity: Akey to relating physical properties to porosity in rocks: The Leading Edge,17, 357–362, doi: 10.1190/1.1437977.

Richa, R., 2010, Preservation of transport properties trends: Computationalrock physics approach: Ph.D. thesis, Stanford University.

Rothman, D., and S. Zaleski, 2004, Lattice-gas cellular automata: Simplemodels of complex hydrodynamics: Cambridge University Press.

Sain, R., 2010, Numerical simulation of pore-scale heterogeneity and itseffects on elastic, electrical and transport properties: Ph.D. thesis, Stan-ford University.

Sen, P. N., C. Scala, and M. H. Cohen, 1981, A self-similar model forsedimentary rocks with application to the dielectric constant of fused glassbeads: Geophysics, 46, 781–795, doi: 10.1190/1.1441215.

Tolke, J., C. Baldwin, Y. Mu, Q. Fang, A. Grader, and J. Dvorkin, 2010,Computer simulations of fluid flow in sediment: From images to perme-ability: The Leading Edge, 29, 68–74, doi: 10.1190/1.3284055.

Toumelin, E., and C. Torres-Verdín, 2008, Object-oriented approach for thepore-scale simulation of DC electrical conductivity of two-phase saturatedporous media: Geophysics, 73, no. 2, E67–E79, doi: 10.1190/1.2836675.

Zinszner, B., and F. M. Pellerin, 2007, A geoscientist’s guide to petrophy-sics: IFP Publications.

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