Research ArticleComputing and Comparing Effective Properties for Flow andTransport in Computer-Generated Porous Media

Rebecca Allen and Shuyu Sun

King Abdullah University of Science and Technology Thuwal 23955-6900 Saudi Arabia

Correspondence should be addressed to Shuyu Sun shuyusunkaustedusa

Received 18 October 2016 Accepted 7 December 2016 Published 13 February 2017

Academic Editor Micol Todesco

Copyright copy 2017 Rebecca Allen and Shuyu Sun This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We compute effective properties (ie permeability hydraulic tortuosity and diffusive tortuosity) of three different digital porousmedia samples including in-line array of uniform shapes staggered-array of squares and randomly distributed squares Thepermeability and hydraulic tortuosity are computed by solving a set of rescaled Stokes equations obtained by homogenizationand the diffusive tortuosity is computed by solving a homogenization problem given for the effective diffusion coefficient thatis inversely related to diffusive tortuosity We find that hydraulic and diffusive tortuosity can be quantitatively different by up toa factor of ten in the same pore geometry which indicates that these tortuosity terms cannot be used interchangeably We alsofind that when a pore geometry is characterized by an anisotropic permeability the diffusive tortuosity (and correspondingly theeffective diffusion coefficient) can also be anisotropic This finding has important implications for reservoir-scale modeling of flowand transport as it is more realistic to account for the anisotropy of both the permeability and the effective diffusion coefficient

1 Introduction

Whenmodeling subsurface flow and transport in a reservoirit is common practice to represent the geological formationwith effective properties instead of resolving the preciselocation of fluid and solid phase Effective properties suchas permeability diffusivity and tortuosity can be computedfrompore-scalemodeling of fluid flow and transport througha porous media sample (ie a core sample) taken from ageological formation Permeability and effective diffusivityare the well-known and important properties used in Darcy-and reservoir-scale flow and transport equations whiletortuosity is given less importance and subject to multipledefinitions Tortuosity is generally defined as a ratio betweenthe effective path traveled by a species and the unit lengthof the domain As noted in [1] different types of tortuosity(geometric hydraulic diffusive dispersive and electrical)are distinguished by the type of species transport underconsiderationHydraulic tortuosity refers to the effective pathtraveled by a fluid particle driven by a force while diffusivetortuosity refers to the effective path traveled by a speciesdriven by molecular diffusion [1]

Pore-scale modeling is performed on digital samples ofporous media which can be obtained using imaging andconversion techniques as those described in [2 3] or byusing an algorithm that generates samples which possess thesame statistical characteristics of a real porous media sample[4ndash6] Table 1 presents a summary of studies which usedeither real or computer-generated porous media samples realrefers to a digital sample obtained by imaging and conversionto binary form and computer-generated refers to a digitalsample obtained by an algorithm or random reconstructionWe note that Table 1 is only a short list of recent works onthe topic and that pore-scale simulation to obtain effectiveproperties dates back to Cancelliere et al [7] and possiblyearlier

In regard to obtaining effective properties at the pore-scale we are motivated to distinguish between two differenttypes of tortuosity (namely hydraulic and diffusive) that havebeen identifiedWhile a few studies have compared tortuositydefinitions and have shown they are indeed quantitativelydifferent within the same pore structure [8 9] other studiescontinue to use these definitions interchangeably or donot distinguish the type of tortuosity they are computing

HindawiGeofluidsVolume 2017 Article ID 4517259 24 pageshttpsdoiorg10115520174517259

2 Geofluids

Table 1 Recent work using real or computer-generated porous media samples comparison of sample types porosities and computedproperties

Study Real sample Computer-generatedsample Porosity range Effective properties

computed

[16] 3D fibrous material(membrane) 045ndash09

permeabilityhydraulictortuosity

[15] 3D samples Fontainbleau (F) Berea (B)Carbonate (C) Sphere pack F 0147 B 0184 C

0247 Spheres 0343

Permeabilityelectrical

resistivity amongothers

[14]

2D staggeredcylinders 3D

body-centered-cubicspheres

025ndash098Permeabilityhydraulictortuosity

[53] 3D vuggy limestone 016ndash081 Permeability

[21]3D samples of volcanic tuff glass bead

column sandpack sandstonescarbonates

3D samples generatedby level-setpercolation

Real 043 06Generated 035 06

Permeabilityhydraulictortuosity

[12] 3D carbonate sandpacks sandstone 017ndash038 Permeability

[22]

3D samples generatedby spheri-

calnonsphericalgrain models

005ndash055Permeabilityhydraulictortuosity

[13]2D randomly

distributed freelyoverlapping squares

0367ndash099 Hydraulictortuosity

The usefulness of hydraulic tortuosity becomes apparentwhen formulating a prediction for permeability such as theKozeny-Carman equation Also tortuosity can be seen as amore intuitive quantity for fluid flow and transport througha pore geometry than permeability or the effective diffusioncoefficient

The outline of this work is as follows first we willconduct a literature review on the methods commonly usedto compute the effective properties of digital porous mediathenwewill present the details of thesemethodswe employedherein followed by three example geometries (in-line arrayof uniform shapes staggered-array of squares and randomlydistributed squares) and our obtained trends between poros-ity and the effective properties We will conclude with themain findings of this work

2 Literature Review on Effective Properties

21 Permeability From the list given in Table 1 an importantand commonly computed effective property is permeabil-ity The notion of a porous mediarsquos permeability can beunderstood in terms of Darcyrsquos equation which computes amacroscale fluid velocity u by

u = minusk120583 sdot (nabla119875 + 120588g) (1)

where k is the permeability tensor The other quantities arefluid viscosity and density 120583 and 120588 macroscale pressure 119875and gravitational acceleration g Eqn (1) can be obtained

mathematically by the method of homogenization [10 11]which is a multiscale expansion of the fluid flow equationsat the pore-scale (ie Stokes equations) see Section 3

Out of the papers listed in Table 1 the general approachto compute the permeability field of the porousmedia sampleis as follows

(1) Obtain pore-scale geometries converted from realmedia sample images or generated by an algorithmthat may or may not use media parameters such asgrain size distribution and so on

(2) Solve Stokes flow in pore space of media differentsolvers have been used (finite difference [12] finiteelement lattice Boltzmann [13ndash16] explicit jumpmethod [15] etc) from academically developedcodes commercial software and open-source soft-ware

(3) Compute permeability assuming Darcyrsquos equation isvalid for the pore structure and flow field resultspermeability is computed based on volume averagesof velocity and pressure field

A further step is to fit the permeability data to a for-mula which proposes a relationship between permeabilityand other porous media properties known as the Kozeny-Carman (KC) equation [17 18]

22 Kozeny-Carman Equation An idealized pore geometryof parallel cylindrical channels was used to formulate the

Geofluids 3

Kozeny-Carman (KC) equation [17] This equation proposesa relationship between porosity 120601 and permeability 119896 aswell as other parameters such as hydraulic tortuosity 120591ℎ andspecific surface area 119878 and is

119896 = 12060131205731198961205912ℎ1198782 (2)

where 120573119896 is a fitting parameter (sometimes referred to as theshape factor or theKCconstant though inconsistently)whichis used to account for different channel configurations Arange of shape factors have beenpresented in literature [17 18]which correspond to cylindrical elliptical rectangular andtriangular cross-sectional channel shapes The parameter 120591ℎin (2) is the ratio between effective path traveled by thefluid particle and the length of the sample 120591ℎ = 119871119890119871 119904 Inorder to be consistent with the literature we are referencingand comparing our results against we call this the hydraulictortuosity although we recognize that other work refers to 1205912ℎas the hydraulic tortuosity as similarly pointed out in [19]Thespecific surface area is the ratio of fluid-solid interface area119860119891minus119904 to the total volume in the domain

119878 = 119860119891minus119904pores119881tot

(3)

Notice that (2) is not written as direction-specific In factCarrier III [20] mentions that a limitation of the KC equationis that it does not explicitly account for anisotropy (adirection-specific property) even though the permeabilityof a real geological formation is typically greater in thehorizontal than in the vertical direction Despite this lim-itation the Kozeny-Carman equation could still apply todirection-specific flows where 119896 = 119896119894119894 and 120591ℎ = 120591ℎ119894119894 arethe permeability and hydraulic tortuosity in the principaldirections respectively

Various studies [14 21 22] include attempts at establishinga modification to the KC equation or propose a shape factor120573119896 for a specific class of porous media Xu and Yu [23] com-piled a list of 11 studies that proposed amodification to theKCequation and each work corresponded to a different mediatype (textile glass and fiber square particles sandstone etc)These proposedKCmodifications included different parame-ters like effective porosity percolation threshold grain radiusfractal dimension interconnectivity parameter and others

23 Tortuosity

231 Different Types of Tortuosity As stated in the introduc-tion different types of tortuosities have been defined accord-ing to the type of species transport under considerationPrevious works have recognized this and some provided acomparison between specific tortuosity types [1 8 13] Whatis important is that these forms of tortuosity should not beexpected to be the same in the same porous structure [24]unless the pore size distribution is very narrow as pointedout in Ghanbarian et al [8] Perhaps unintentionally a fewprevious studies have failed to distinguish the differencebetween tortuosity types For example Ohkubo [25] mea-sured diffusive tortuosity through porous media that was

represented by plate-like obstacles and used the diffusivetortuosity results to compute fitting coefficients present inKoponen et alrsquos [26] tortuosity-porosity trend equationHowever Koponen et alrsquos [26] trend was empirically derivedfrom simulation results that computed hydraulic tortuosityAlso Sun et al [27] applied the method of homogenizationto the diffusive transport equation in order to compute theeffective diffusion coefficient of periodic unit cells and thecorresponding tortuosity What they did not point out wasthat their approach gave them the diffusive tortuosity whilesome of the trends they compared their results to were withrespect to hydraulic tortuosity Unless tortuosity types arequantitatively identical in the same class of porousmedia useof a trend that is specific to one type of tortuosity should notimmediately be used to describe the trend of another type oftortuosity

A few studies [8 9 28] have focused on quantifyingthe difference between two specific tortuosities in the samestructure For example hydraulic and electrical tortuositywere compared in bothDavid [28] andZhang andKnackstedt[9]David [28] used a network as an analog for themedia porespace and the study focused on quantifying the tortuosityof the critical or preferential pathway not the tortuosity ofthe entire flow field Zhang and Knackstedt [9] computedtortuosity using the entire flow field which they obtainedby numerical simulations for fluid flow (via lattice GasAutomata) and electrical current (via finite difference) Bothof these works found that the hydraulic tortuosity was higherthan electrical tortuosity in the same structure (up to an orderofmagnitude in low porosity configurations [9]) Once againit was emphasized in Ghanbarian et alrsquos [8] review that due tothe difference between tortuosity types their models are notinterchangeable

Regarding diffusive tortuosity both Quintard et al [29]and Valdes-Parada et al [1] stated that its quantificationbecomes more complex when more than just passive dif-fusion is occurring in the system Valdes-Parada et al [1]showed how the quantification of diffusive tortuosity can bedifferent depending on the consideration of different masstransport terms (ie passive diffusion only or diffusion withconvection reaction or hydrodynamic dispersion) Theyconcluded that the consideration of passive diffusion leadsto the only appropriate definition of tortuosity Their workfocused on mass transport of species through fluid and didnot measure hydraulic tortuosity

232 Various Tortuosity-Porosity Trends Many authors havetheoretically or empirically derived tortuosity as a functionof porosity (see Tables 2 and 3 where the tortuosity cor-responds to fluid flow (hydraulic) and diffusive transportresp) Similar tables that summarize the theoretically andempirically derived trends proposed in literature appear inShen and Chen [30] Boudreau [31] and Ahmadi et al[32] The most recent and complete comparison of tortuositytrends (or models) that we have seen to date was made byGhanbarian et al [8] which includes an explicit comparisonbetween many different tortuosity types namely geometrichydraulic electrical diffusive and even tortuositymodels forunsaturated porous media

4 Geofluids

Table 2 Past work on hydraulic tortuosity

Study Samples considered Tortuosityndashporosity fit Notes

[54] Packed beds 120591 = 1 + 119901 ln( 1120601) 119901 is function of particle shape

[19] 2D random overlapping monosized squares 05 lt 120601 lt 1 120591 = 1 + 08 (1 minus 120601)[26] 2D random overlapping monosized squares 04 lt 120601 lt 1 120591 = 1 + 065 1 minus 120601

(120601 minus 120601119888)019 120601119888 is critical porosity[24] 2D random overlapping monosized squares 120591 prop 1 + 119877 119878120601 119877 is hydraulic radius 119878 is specific surface area[13] 2D freely overlapping squares 120591 prop 1 + (1 minus 120601)12

Table 3 Past work on diffusive tortuosity

Study Samples considered Tortuosityndashporosity fit Notes[48]lowast Array of cylinders (2D) 120591 = 2 minus 120601[50]lowastlowast Diffusion of electrolytes in membrane 120591 = (2 minus 120601120601 )2[52] Bed of uniform spheres (applicable to overlapping nonuniform spheres) 120591 = 1 minus 12 ln120601[43] Isotropic system 0 lt 120601 lt 05 120591 = 120601minus04[31] Fine-grained uncemented sediments 120591 = (1 minus ln (1206012))12[55] Isotropic representative unit cell 120591 = 120601

1 minus (1 minus 120601)119898 2D119898 = 12 3D119898 = 23[56] Random partial overlapping shapes 120591 = 11 minus 120572 (1 minus 120601) 120572 = shape factor

[46] Voronoi channel geometries 02 lt 120601 lt 06 120591 = (075120601minus10819)12lowastAs referenced in [57]lowastlowastAs referenced in [30 31]

In the same way that the KC equation is specific to a givenclass of porous media any porosity-tortuosity trend that isempirically derived depends on the porousmedia consideredSome authors rely on synthetic porous media such as 2Drandomly distributed squares [13 19 24 26 33 34] 3D highporosity plate-like obstacles [25] or unit cells of centereduniform shapes while others rely on real samples of mediataken by imaging (via scanning electron microscopy (SEM)or X-ray microtomography) and digitization

Additionally it is important to note that empiricallyderived trends are applicable to a limited range of porosityonly as the trend was developed given simulated data of acertain porosity range The porosity limit could be a resultof the idealized geometry under consideration Ghanbarianet al [8] noticed that most tortuosity models were derived byfocusing on the higher porosity structures and recommendedthat research moves more to focus on the lower porositystructure where the porosity approaches the percolationthreshold (ie the minimum porosity for which flow is stillpossible through structure)

While many trends (or models) have been proposedto compute tortuosity as a function of porosity and otherfitting parameters Valdes-Parada et al [1] stated that anydefinition of tortuosity should not be considered as a functionof porosity but rather a function of the pore geometryonly The idea behind this statement seems to rest in the

fact that there is no universally agreed upon trend betweentortuosity and porosity as pointed out by Matyka et al [24]and that the geometrical features of a pore structure shouldhave more influence on the overall tortuosity of flow ortransport through the geometry than the value of porosityHowever since it is possible that a unique relationshipbetween tortuosity and porosity can indeed exist for specialclasses of porous media [24] the development of tortuosity-porosity trends remains to be an insightful topic of research

Also Liu and Kitanidis [35] stated that tortuosity is atensorial property Despite this statement the majority ofwork does not report directional-specific quantities whichmay leave the reader with the impression that tortuosity is ascalar quantity A scalar quantity could be appropriate in poregeometries that exhibit an anisotropic microstructure whilestill exhibiting an isotropic macrostructure For examplepore geometry composed of randomly distributed squaresis geometrically anisotropic at the microscale however theeffective properties computed for the REV can be isotropic innature In this work we demonstrate this point by example

3 Methods to Compute Effective Properties

31 Permeability The method of homogenization has beenused to mathematically derive Darcyrsquos equation from theStokes equations Through this derivation the macroscopic

Geofluids 5

D120576

120597D120576

120576

120576x2

x1

y2

y1

Y-cell

(a) (b)

Figure 1 Macroscale (a) and microscale (b) domains used in thehomogenization problemThe solid structure in the119884-cell illustratesthe geometry of our first example in-line array of solid shapes (iecircles or squares)

property known as permeability 119896119894119895 is defined as the spatialaverage of a rescaled pore space velocity119873119894119895(119910) In the follow-ing we present the basic steps that lead to such a definitionand the reader is referred to [11] for more derivation details

The problem is first defined by the macroscale (a)and microscale (b) systems shown in Figure 1 where themacroscale domain is comprised of repeating unit cells ofsize 120576 which is comprised of both fluid and solid spaceDue to periodicity the macroscale domain of porosity 120601 canbe represented by the unit cell with periodic boundariesAssuming slow and steady flow of incompressible fluidthrough the pore space of themedia the governing equationsare given by the Stokes equations as follows

minus 120597120597119909119894119901120576 + 120583 120597120597119909119895120597120597119909119895 V120576119894 = minus120588119892119895 (4)

120597120597119909119894 V120576119894 = 0 (5)

where 119894 = 1 2 3 and 119895 = 1 2 3 and repeated indices implysummation 119901 is pressure 120583 is fluid viscosity V119894 is the porespace fluid velocity 120588 is fluid density and 119892119895 is gravitationalaccelerationThe boundary condition at the fluid-solid inter-face is given by

V119894 = 0 (6)

which comes from the no-flow and no-slip boundary condi-tions Although the slip boundary condition is necessary incertain contexts (such as gas flow and non-Newtonian fluid)the no-slip boundary condition is a valid assumption forviscous flow in porous media and is widely used in modelingstudies of pore-scale flow [12 15 24]

In the Stokes equations 119901120576 and V120576119894 can vary within thepore space and thus are functions of the macroscale (119909) andmicroscale (120576) however 120583 119892 and 120588 are treated as constantsand are thus functions of themacroscale (119909) only By defininga fast variable 119910 = 119909120576 the cell of unit length 120576 is scaledinto the 119884-cell (shown in Figure 1(b)) where 0 le 119910 le 1 A

multiscale expansion is used to write 119901120576 and V120576119894 in terms oftheir leading and higher order terms as follows

119901120576 = 119901(119909 119909120576 ) = 119901(0) (119909 119910) + 120576119901(1) (119909 119910) + sdot sdot sdot (7)

V120576119894 = V119894 (119909 119909120576 )= V(0)119894 (119909 119910) + 120576V(1)119894 (119909 119910) + 1205762V(2)119894 (119909 119910) + sdot sdot sdot

(8)

Upon substitution of the expanded variables into (4)ndash(6)and then the collection of like order terms the second-orderpressure term and the third-order velocity term are

119901(1) (119909 119910) = 119876119895 (119910) (minus120588119892119895 + 120597120597119909119895119901(0) (119909))

V(2) (119909 119910) = 119873119894119895 (119910)120583 (minus120588119892119895 + 120597120597119909119895119901(0) (119909)) (9)

such that 119876119895(119910) and119873119894119895(119910) satisfyminus 120597120597119910119894119876119895 (119910) + 120597120597119910119896

120597120597119910119896119873119894119895 (119910) = 120575119894119895 (10)

120597120597119910119894119873119894119895 (119910) = 0 (11)

with the fluid-solid boundary condition given by

119873119894119895 (119910) = 0 (12)

In the above formulation 119876119895(119910) and 119873119894119895(119910) are functionsof the microscale only and are independent of macroscalequantities 120583 120588 and 119892119895 They can be thought of as rescaledpressure and pore space velocity respectively and (10)-(11)can be thought of as rescaled Stokes equations to be solvedwithin the 119884-cell In (10) 120575119894119895 is the Kronecker delta function(120575119894119895 = 1 for 119894 = 119895 and 120575119894119895 = 0 for 119894 = 119895)

Upon further mathematical steps (collecting of like orderterms taking the spatial average over the cell and applyingthe divergence theorem and the periodicity of the cell)the spatial average of V(2)(119909 119910) in the cell is found to bedivergence free nabla sdot ⟨V(2)(119909 119910)⟩119910 = 0 This implies that themacroscopic flow equation is satisfied byDarcyrsquos equation (1)that is

120597120597119909119894 (119896119894119895120583 ( 120597120597119909119895119901 (119909) minus 120588119892119895)) = 0 997888rarr

120597120597119909119894 119906119894 (119909) = 0(13)

where 119901(119909) is the macroscale pressure and 119906119894(119909) is themacroscale velocity when the full permeability tensor for an119899-dimensional problem is defined as

119896119894119895 fl ⟨119873119894119895 (119910)⟩119910 = 1119884 int119884119873119894119895 (119910) 119889119884 (14)

6 Geofluids

where 119884 is the volume of the 119884-cell (solid and fluid spacecombined)

In this work we use the staggered grid finite differencemethod (ie the marker and cell or MAC scheme [36]) forthe numerical solution of the rescaled Stokes problemdefinedby (10)ndash(12) in the 119884-cell Our numerical implementationusing a matrix-oriented approach is explained in [37] Toillustrate the numerical scheme let us consider the followingStokes problem (in its original form before being rescaledby homogenization) in two-dimensional space find the porespace velocity V = (V119909 V119910)119879 and the pressure 119901 such that

minus120583ΔV + nabla119901 = 119891 in Ω fl (0 119871119909) times (0 119871119910) (15)

nabla sdot V = 0 in Ω (16)

V = 0 on 120597Ω (17)

where the velocity has a unique solution while the pressure isunique up to an additive constant Here 119871119909 and 119871119910 denotethe length of the rectangular domain Ω in the 119909- and 119910-directions respectively

To construct a staggered grid finite difference scheme forthe Stokes problem we introduce a possibly nonuniform gridofΩ as follows

0 = 1199090 lt 1199091 lt sdot sdot sdot lt 119909119872 = 1198711199090 = 1199100 lt 1199101 lt sdot sdot sdot lt 119910119873 = 119871119910 (18)

The grid is comprised of fluid and solid cells (each cell beingfully occupied by one phase) and the computational schemeacts uniquely on the computational cells belonging to thefluid The pressure at the (fluid) cell center (119909119894+12 119910119895+12)is 119901(119909119894+12 119910119895+12) approximated by 119901119894+12119895+12 Similarly weapproximate the 119909-velocity V119909 at each 119909-edge (fluid) centerV119909(119909119894 119910119895+12) by V119909119894119895+12 and the 119910-velocity V119910 at each 119910-edge(fluid) center V119910(119909119894minus12 119910119895) by V119910119894minus12119895

By integrating the119909-component of (15) over cellΩ119894119895+12 =(119909119894minus12 119909119894+12) times (119910119895 119910119895+1) we haveminus [int119910119895+1

119910119895(120597V119909120597119909 (119909119894+12 119910) minus 120597V119909120597119909 (119909119894minus12 119910)) 119889119910

+ int119909119894+12119909119894minus12

(120597V119909120597119910 (119909 119910119895+1) minus 120597V119909120597119910 (119909 119910119895)) 119889119909]+ int119910119895+1

119910119895(119901 (119909119894+12 119910) minus 119901 (119909119894minus12 119910)) 119889119910

= intΩ119894119895+12

119891119909 (119909 119910) 119889119909 119889119910

(19)

We then approximate those edge integrals by midpointvalues and the volume integral by its center value and divideboth sides by |Ω119894119895+12| = (119909119894+12 minus 119909119894minus12)(119910119895+1 minus 119910119895)

By introducing the following forward and backwarddifference operators

Δ+119909V119909119894119895+12 = V119909119894+1119895+12 minus V119909119894119895+12119909119894+1 minus 119909119894 Δminus119909119901119894+12119895+12 = 119901119894+12119895+12 minus 119901119894minus12119895+12119909119894+12 minus 119909119894minus12

(20)

and similar notations for operations in the 119910-direction wecan express the results as

minus Δ+119909V119909119894119895+12 minus Δminus119909V119909119894119895+12119909119894+12 minus 119909119894minus12 minus Δ+119910V119909119894119895+12 minus Δminus119910V119909119894119895+12119910119895+1 minus 119910119895+ Δminus119909119901119894+12119895+12 = 119891119909119894119895+12

(21)

Similarly by integrating the 119910-component of (15) overcell Ω119894minus12119895 = (119909119894minus1 119909119894) times (119910119895minus12 119895119895+12) and then applyingmidpoint quadrature rule we can obtain

minus Δ+119909V119910119894minus12119895 minus Δminus119909V119910119894minus12119895119909119894 minus 119909119894minus1 minus Δ+119910V119910119894minus12119895 minus Δminus119910V119910119894minus12119895119910119895+12 minus 119910119895minus12+ Δminus119910119901119894minus12119895+12 = 119891119910119894minus12119895

(22)

By integrating (16) over cell Ω119894minus12119895+12 = (119909119894 minus 119909119894minus1) times(119910119895 119895119895+1) we obtain its discrete form

Δminus119909V119909119894119895+12 + Δ+119910V119910119894minus12119895 = 0 (23)

Eqns (21)ndash(23) form a linear equation system for thepressure and velocity and the equation system defines theMAC scheme for the Stokes problem of (15)ndash(17) Note thatthe no-flow boundary condition supplies the zero velocitycomponent that acts normal to any fluid-solid interface andperiodic conditions are applied along the domainrsquos externalboundaries

We apply this numerical scheme to solve the set ofrescaled Stokes equations in Eqns (10)ndash(12) Our computer-generatedmedia samples are 2-dimensional (2D) thus the setis solved with different forcings 120575119894119895 where 119894 = 1 2 and 119895 = 1 2Upon obtaining the rescaled pore space velocity 119873119894119895(119910) inthe 119884-cell we then compute the full permeability tensor by(14) If our computer-generated media are characterized by anondiagonal tensor we determine the principal componentsof the permeability tensor by diagonalization [11 38] Thenotation used in our results section is k and klowast for thepermeability tensors in coordinate systems x and xlowastrespectively that is

k = [11989611 1198961211989621 11989622] in x 997904rArr

klowast = [119896max 00 119896119905] in xlowast

(24)

32 Hydraulic Tortuosity The approach commonly used inliterature to compute hydraulic tortuosity is presented inKoponen et al [19 26] and further validation of this ap-proach is given in Duda et al [13] In this approach the

Geofluids 7

hydraulic tortuosity 120591ℎ or a particular direction is computedbased on the pore space fluid velocities driven by a force Forexample the hydraulic tortuosity of flow that is driven in the119909 direction is

120591ℎ119909 = ⟨Vmag⟩Ω⟨V119909⟩Ω (25)

where ⟨sdot⟩Ω is the spatial average in the unit cell domain ΩVmag is the magnitude of the fluid velocities (ie the fluidspeed) and V119909 is the velocity component in the 119909 directionThe spatial average of V119909 inΩ is

⟨V119909⟩Ω = 1|119881| int119881119891 V119891119909119889119881119891 = 1|119881|sum119894119895 V

119891119909 (119894 119895) Δ119881119891 (26)

where119881 is the volume ofΩ (pore and solid space combined)119881119891 is the fluid space and the integral is taken over the fluidspace only The spatial average for Vmag is taken in the samewayThe tortuosity in the 119910 direction is computed in a similarmanner as follows

120591ℎ119910 = ⟨Vmag⟩Ω⟨V119910⟩Ω (27)

where the velocity is the solution for flow driven in the119910 direction We note that (25) and (27) correspond to thehydraulic tortuosity as we have defined it in Section 22 thatis 120591ℎ = 119871119890119871 119904 If the term 1205912ℎ which appears in the Kozeny-Carmen equation is labelled as the hydraulic tortuosity then(25) and (27) should be squared

While this method has been used in many works (ie[26 33 39 40]) other approaches to compute hydraulictortuosity have been proposed For example Matyka et al[24] took measurements of streamlines that passed through agiven cross-sectional area in the domain with a constant flux(ie not all streamlines in the domain would be measured)The aim of this approach is to capture the hydraulic tortu-osity of the main conducting channels in a pore geometryAlso Ahmadi et al [32] formulated analytical functionsfor hydraulic tortuosity based on volume averaging of massbalance equations

In our work we follow Koponen et alrsquos [19 26] approachas given by (25)ndash(27) however we do not compute the porespace velocity fields V119909 and V119910 explicitly Instead during thepermeability computation for our 2D pore geometries weobtain the rescaled pore space velocity fields 119873119894119895 in the 119884-cell where 119894 = 1 2 and 119895 = 1 2 It can be seen that 11987311and 11987321 are related to V119909 and V119910 respectively for flow thatis driven by g = ( 0) and that 11987312 and 11987322 are relatedto V119909 and V119910 respectively for flow that is driven by g =(0 ) Thus it is mathematically equivalent to compute thehydraulic tortuosities by

120591ℎ119909 = ⟨radic11987311 + 11987321⟩Ω⟨11987311⟩Ω 120591ℎ119910 = ⟨radic11987312 + 11987322⟩Ω⟨11987322⟩Ω

(28)

where Ω refers to the domain of the 119884-cell and the spatialaverages of 119873119894119895 are computed in the same way as shown by(26) Since the numerical scheme computes 119873119894119895 on the fluidedge-centers we compute the fluid cell-center quantities byaveraging across the grid cells where 119873119894119895 = 0 on the fluid-solid interfaces as per the boundary condition in (12)

As mentioned we diagonalize the permeability tensorto obtain the principal permeability components and theprincipal directions 120579 and 120579+1205872 fl Thus in order tomakean appropriate comparison between direction-specific per-meability and hydraulic tortuosity we compute the hydraulictortuosity in these principal directions For example 120591ℎ in theprincipal direction 120579 is

120591ℎ120579 = ⟨1003816100381610038161003816100381611987312057911989410038161003816100381610038161003816⟩Ω⟨119873120579⟩Ω (29)

where |119873120579119894 | = radic(119873120579

1 )2 + (1198731205792 )2 and where119873120579

1 1198731205792 and119873120579 are

obtained by vector transformations from 119873119894119895 120591ℎ in the otherprincipal direction is similarly obtained The notation weuse in our results section is 120591ℎmax = 120591ℎ120579 and 120591ℎ119905 = 120591ℎ33 Effective Diffusion Coefficient and Diffusive TortuosityThe problem is similarly defined by the two scales illustratedin Figure 1 Within the pore space the diffusive and convec-tive transport of a species of concentration 119888 is given by

120597120597119905119888120576 + 1120576 V120576119894 120597120597119909119894 119888120576 =120597120597119909119894119863119894119895

120597120597119909119895 119888120576119899119894119863119894119895

120597120597119909119895 119888120576 = 0V120576119894 = 0

(30)

where V120576119894 is divergence free (5) 120576 implies the variables arefunctions of the microscale and 119863119894119895 is the molecular diffu-sion coefficient The last two equations give the boundaryconditions on the fluid-solid interface Using similar homog-enization steps as presented in Section 31 the multiscaleexpansion

119888120576 = 119888 (119909 119909120576 )= 119888(0) (119909 119910) + 120576119888(1) (119909 119910) + 1205762119888(2) (119909 119910) + sdot sdot sdot

(31)

and (8) are substituted into the governing transport equations(30) Upon collecting like order terms the second-order termof 119888 is

119888(1) (119909 119910) = 119875119896 (119909 119910) 120597120597119909119896 119888(0) (119909) (32)

where 119888(0)(119909) is the macroscale concentration and 119875119896(119909 119910)satisfies 120597120597119910119894

120597120597119910119894119875119896 (119909 119910) = 0 (33)

119899119894 120597120597119910119894119875119896 (119909 119910) = minus119899119896 (34)

8 Geofluids

where (34) gives the boundary condition and 119899119894 is a unitvector normal to the fluid-solid interface (positive in thedirection away from the fluid space) Eqns (33) and (34)correspond to the assumption of an isotropic moleculardiffusion coefficient that is119863119894119895 = 119863120575119894119895 Upon further algebraand averaging over the 119884-cell the macroscale transportequation is obtained as follows

120601 120597120597119905119888 (119909) = 120597120597119909119894119863efflowast119894119896

120597120597119909119896 119888 (119909) (35)

where the effective diffusion coefficient119863efflowast119894119896 is given by

119863efflowast119894119896119863 = 1119884 int

119884(120575119894119896 + 120597120597119910119894119875119896 (119909 119910)) 119889119884 (36)

Intuitively the more tortuous the porous media the slowerthe rate of effective diffusion thus a commonly used relation-ship between effective and molecular diffusivity is

119863efflowast119894119896 = 119863120601120591119889119894119896 (37)

where 120591119889119894119896 is the diffusive tortuosity [41]The above approach to compute the effective diffusion

coefficient and the diffusive tortuosity has been used inseveral previous studies which formulate (33)ndash(36) using themethod of volume averaging [1 42ndash45] or by the method ofhomogenization [27] Other approaches to compute diffusivetortuosity include simulation of the diffusion process usinga random walk [25] or numerically solving the microscalediffusion equation using lattice Boltzmann modeling [46]Typically the effective diffusivity is computed and then usedto calculate diffusive tortuosity by the inverse relationshipshown in (37)

By (36) and (37) diffusive tortuosity is computed by

120591119889119894119896 = (119868119894119896 + 11003816100381610038161003816100381611988411989110038161003816100381610038161003816 int119884119891 (120597120597119910119894119875119896 (119909 119910)) 119889119910119891)

minus1 (38)

where 119884119891 = 120601119884 and where 119875119896(119909 119910) is the solution to(33) with the boundary condition in (34) This is the sameformulation that was used in Sun et al [27] by the method ofhomogenization and in Kim et al [43] and Valdes-Parada etal [1] by the method of volume averaging

The full diffusive tortuosity can also be diagonalized usingthe same procedure outlined in Section 31 to obtain thecomponents 120591119889max and 120591119889119905 corresponding to the principaldirections The rotation required to obtain the principalcomponents of klowast is not necessarily identical to the rotationrequired to obtain the principal components of 120591lowast119889 as willbe shown by one of the following pore geometry examples(randomly distributed squares)

4 Results

While real geological formations are typically of low porosity(120601 lt 03) we have generated several idealized geometrieswithin the porosity range of 033 lt 120601 lt 1 These idealized

Simulated result (Stokes) circleAnalytical result (Gebart eqn)

10minus4

10minus3

10minus2

10minus1

100

101

102

Nor

mal

ized

per

mea

bilit

y

03 04 05 06 07 08 09 102

Porosity

Figure 2 Permeability-porosity trend for in-line array of circles(ie infinitely long cylinders) Permeability units are nondimen-sional (ie unit length times unit length) The permeability is isotropicand is normalized by 1199032 where 119903 is the solid circle radius Gebartrsquos[47] analytical solution was given in (A2)

geometries and porosities have been used in literature (seeTable 1) and thus provide ground for comparison and oppor-tunity to make useful comparisons of the effective propertiesin high porosity representations of porous media

Permeability hydraulic tortuosity anddiffusive tortuosityare computed for each pore geometry using the methodsoutlined in Sections 31 32 and 33 respectively Each poregeometry is generated by defining certain input parameters(ie radius of solid circle side length of square and den-sity of randomly distributed squares) and these inputs aresummarized in Table 4 By using a range of values for theinput parameters we obtain unit cells which we treat asrepresentative elementary volumes over a range of porositiesAfter computing the permeability and tortuosity we plotthe data to observe permeability-porosity and tortuosity-porosity trends

41 In-Line Array of Uniform Shapes The unit cell or 119884-cellin Figure 1(b) illustrates our first type of generated poregeometry and the shape in the center of the cell is eithera circle or square A range of porosities of the unit cell areobtained by changing the radius of the solid circle or thelength of the solid square centered in the cell

411 Permeability and Hydraulic Tortuosity The full perme-ability tensor was computed for 033 lt 120601 lt 1 and resultsare plotted in Figure 2 Following [14] we computed andplotted the analytical solution from Gebart [47] which isexplained in Appendix A Permeability is isotropic (ie 119896119909119909 =119896119910119910) for this pore geometry The range of our simulateddata fits well against the analytical solution We computed

Geofluids 9

Table 4 Input parameters used to generate different pore-geometries (or different 119884-cells) and their corresponding porosity ranges In thistable cell refers to a cell in the mesh used to discretize the 119884-cellGeometry Initial mesh dimensions Inputs Porosity rangeIn-line array of shapes(see Figure 1) 50 times 50 cells (119884-cell length 1 unit) (a) Circle diameter 44 to 8 cells

(b) Square length 48 to 2 cells033 lt 120601 lt 1007 lt 120601 lt 1

Staggered array ofsquares (see Figure 7)

100 times 50 cells (119884-cell length 119871119887 = 05units 119886119887 = 1 119871119886119871119887 = 1 as per Figure 8) Square length 40 to 8 cells 036 le 120601 lt 1

Randomly distributedsquares (see Figure 14) 100 times 100 cells (119884-cell length 1 unit) Increasing density of squares

(with constant square length 001unit = 1 cell)

045 le 120601 lt 1

0005

001

0015

002

0025

003

0035

Figure 3 Fluid flow past a solid circle 120601 = 071 driven by 119892119909 Non-dimensional fluid speed (ie magnitude of velocities) is indicatedby color and flow path is indicated by flow lines (direction is left toright) Speed profile and flow path driven by 119892119910 are same as shownbut rotated 90 degrees

our permeability data on increasingly finer meshes until thepermeability converged within a 1 relative error To reachthis convergence criteria after starting with a mesh size of50 times 50 the lowest porosity cell required a refined mesh of500 times 500 and the highest porosity cell required 300 times 300The shape of the circular edge was refined at each meshrefinement in order to represent the circle as realistically aspossible and to minimize artifacts caused by discretization

We obtained the hydraulic tortuosity in the 119909 and 119910directions for porosities within the range 033 lt 120601 lt 1 withthe same convergence criteria used to obtain permeabilityFigure 3 illustrates the fluid speed and direction for a unitcell of 120601 = 071 and the hydraulic tortuosity in both 119909 and119910 directions were computed to be 120591ℎ119909 = 120591ℎ119910 = 10185 Thefluid speed is highest in the pore space that is uninterruptedby the solid and the flow lines indicate the direction is parallelto the macroscopic flow The flow lines diverge in the regionthat is interrupted by the solid however the fluid speed isessentially zero in that pore space So while these flow lines

120601 = 033

120601 = 039

120601 = 050

120601 = 059

120601 = 072

120601 = 087

120601 = 098

1

12

14

16

18

2

22

Com

pute

d di

ffusiv

e 120591

1 2 3 4 50

Number of cells times105

Figure 4Number of grid cells required to reach converged 120591119889 (usinga convergence criteria of 01 relative error) for select porosities ofin-line array of circles

appear to illustrate a tortuous pathway through this geometrythe tortuosity in either direction is in fact calculated to beasymp1 because the highest speeds are acting along a nontortuouspathway In the pore space that is not restricted or interruptedin the direction parallel to the driving force the fluid velocityis able to develop into the parabolic profile that characterizeschannel flow

412 Diffusive Tortuosity We obtained the full diffusive tor-tuosity tensor on increasingly refined meshes until the tortu-osity converged within a 01 relative error Starting with amesh of 50 times 50 the lowest porosity required a refined meshof 700 times 700 while the highest porosity required 250 times 250 tomeet the convergence criteria Solid edges were also refinedwith mesh refinement as mentioned previously In Figure 4we plot the convergence behavior of 120591119889 for a select numberof porosity cases in order to illustrate that each porosity caseconvergences with similar behavior Figure 5(a) illustrates thefields of 119875119896 (119896 = 1 2 or 119909 119910) for a unit cell of 120601 = 071which were obtained by solving the homogenized problem

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 3

Kozeny-Carman (KC) equation [17] This equation proposesa relationship between porosity 120601 and permeability 119896 aswell as other parameters such as hydraulic tortuosity 120591ℎ andspecific surface area 119878 and is

119896 = 12060131205731198961205912ℎ1198782 (2)

where 120573119896 is a fitting parameter (sometimes referred to as theshape factor or theKCconstant though inconsistently)whichis used to account for different channel configurations Arange of shape factors have beenpresented in literature [17 18]which correspond to cylindrical elliptical rectangular andtriangular cross-sectional channel shapes The parameter 120591ℎin (2) is the ratio between effective path traveled by thefluid particle and the length of the sample 120591ℎ = 119871119890119871 119904 Inorder to be consistent with the literature we are referencingand comparing our results against we call this the hydraulictortuosity although we recognize that other work refers to 1205912ℎas the hydraulic tortuosity as similarly pointed out in [19]Thespecific surface area is the ratio of fluid-solid interface area119860119891minus119904 to the total volume in the domain

119878 = 119860119891minus119904pores119881tot

(3)

Notice that (2) is not written as direction-specific In factCarrier III [20] mentions that a limitation of the KC equationis that it does not explicitly account for anisotropy (adirection-specific property) even though the permeabilityof a real geological formation is typically greater in thehorizontal than in the vertical direction Despite this lim-itation the Kozeny-Carman equation could still apply todirection-specific flows where 119896 = 119896119894119894 and 120591ℎ = 120591ℎ119894119894 arethe permeability and hydraulic tortuosity in the principaldirections respectively

Various studies [14 21 22] include attempts at establishinga modification to the KC equation or propose a shape factor120573119896 for a specific class of porous media Xu and Yu [23] com-piled a list of 11 studies that proposed amodification to theKCequation and each work corresponded to a different mediatype (textile glass and fiber square particles sandstone etc)These proposedKCmodifications included different parame-ters like effective porosity percolation threshold grain radiusfractal dimension interconnectivity parameter and others

23 Tortuosity

231 Different Types of Tortuosity As stated in the introduc-tion different types of tortuosities have been defined accord-ing to the type of species transport under considerationPrevious works have recognized this and some provided acomparison between specific tortuosity types [1 8 13] Whatis important is that these forms of tortuosity should not beexpected to be the same in the same porous structure [24]unless the pore size distribution is very narrow as pointedout in Ghanbarian et al [8] Perhaps unintentionally a fewprevious studies have failed to distinguish the differencebetween tortuosity types For example Ohkubo [25] mea-sured diffusive tortuosity through porous media that was

represented by plate-like obstacles and used the diffusivetortuosity results to compute fitting coefficients present inKoponen et alrsquos [26] tortuosity-porosity trend equationHowever Koponen et alrsquos [26] trend was empirically derivedfrom simulation results that computed hydraulic tortuosityAlso Sun et al [27] applied the method of homogenizationto the diffusive transport equation in order to compute theeffective diffusion coefficient of periodic unit cells and thecorresponding tortuosity What they did not point out wasthat their approach gave them the diffusive tortuosity whilesome of the trends they compared their results to were withrespect to hydraulic tortuosity Unless tortuosity types arequantitatively identical in the same class of porousmedia useof a trend that is specific to one type of tortuosity should notimmediately be used to describe the trend of another type oftortuosity

A few studies [8 9 28] have focused on quantifyingthe difference between two specific tortuosities in the samestructure For example hydraulic and electrical tortuositywere compared in bothDavid [28] andZhang andKnackstedt[9]David [28] used a network as an analog for themedia porespace and the study focused on quantifying the tortuosityof the critical or preferential pathway not the tortuosity ofthe entire flow field Zhang and Knackstedt [9] computedtortuosity using the entire flow field which they obtainedby numerical simulations for fluid flow (via lattice GasAutomata) and electrical current (via finite difference) Bothof these works found that the hydraulic tortuosity was higherthan electrical tortuosity in the same structure (up to an orderofmagnitude in low porosity configurations [9]) Once againit was emphasized in Ghanbarian et alrsquos [8] review that due tothe difference between tortuosity types their models are notinterchangeable

Regarding diffusive tortuosity both Quintard et al [29]and Valdes-Parada et al [1] stated that its quantificationbecomes more complex when more than just passive dif-fusion is occurring in the system Valdes-Parada et al [1]showed how the quantification of diffusive tortuosity can bedifferent depending on the consideration of different masstransport terms (ie passive diffusion only or diffusion withconvection reaction or hydrodynamic dispersion) Theyconcluded that the consideration of passive diffusion leadsto the only appropriate definition of tortuosity Their workfocused on mass transport of species through fluid and didnot measure hydraulic tortuosity

232 Various Tortuosity-Porosity Trends Many authors havetheoretically or empirically derived tortuosity as a functionof porosity (see Tables 2 and 3 where the tortuosity cor-responds to fluid flow (hydraulic) and diffusive transportresp) Similar tables that summarize the theoretically andempirically derived trends proposed in literature appear inShen and Chen [30] Boudreau [31] and Ahmadi et al[32] The most recent and complete comparison of tortuositytrends (or models) that we have seen to date was made byGhanbarian et al [8] which includes an explicit comparisonbetween many different tortuosity types namely geometrichydraulic electrical diffusive and even tortuositymodels forunsaturated porous media

4 Geofluids

Table 2 Past work on hydraulic tortuosity

Study Samples considered Tortuosityndashporosity fit Notes

[54] Packed beds 120591 = 1 + 119901 ln( 1120601) 119901 is function of particle shape

[19] 2D random overlapping monosized squares 05 lt 120601 lt 1 120591 = 1 + 08 (1 minus 120601)[26] 2D random overlapping monosized squares 04 lt 120601 lt 1 120591 = 1 + 065 1 minus 120601

(120601 minus 120601119888)019 120601119888 is critical porosity[24] 2D random overlapping monosized squares 120591 prop 1 + 119877 119878120601 119877 is hydraulic radius 119878 is specific surface area[13] 2D freely overlapping squares 120591 prop 1 + (1 minus 120601)12

Table 3 Past work on diffusive tortuosity

Study Samples considered Tortuosityndashporosity fit Notes[48]lowast Array of cylinders (2D) 120591 = 2 minus 120601[50]lowastlowast Diffusion of electrolytes in membrane 120591 = (2 minus 120601120601 )2[52] Bed of uniform spheres (applicable to overlapping nonuniform spheres) 120591 = 1 minus 12 ln120601[43] Isotropic system 0 lt 120601 lt 05 120591 = 120601minus04[31] Fine-grained uncemented sediments 120591 = (1 minus ln (1206012))12[55] Isotropic representative unit cell 120591 = 120601

1 minus (1 minus 120601)119898 2D119898 = 12 3D119898 = 23[56] Random partial overlapping shapes 120591 = 11 minus 120572 (1 minus 120601) 120572 = shape factor

[46] Voronoi channel geometries 02 lt 120601 lt 06 120591 = (075120601minus10819)12lowastAs referenced in [57]lowastlowastAs referenced in [30 31]

In the same way that the KC equation is specific to a givenclass of porous media any porosity-tortuosity trend that isempirically derived depends on the porousmedia consideredSome authors rely on synthetic porous media such as 2Drandomly distributed squares [13 19 24 26 33 34] 3D highporosity plate-like obstacles [25] or unit cells of centereduniform shapes while others rely on real samples of mediataken by imaging (via scanning electron microscopy (SEM)or X-ray microtomography) and digitization

Additionally it is important to note that empiricallyderived trends are applicable to a limited range of porosityonly as the trend was developed given simulated data of acertain porosity range The porosity limit could be a resultof the idealized geometry under consideration Ghanbarianet al [8] noticed that most tortuosity models were derived byfocusing on the higher porosity structures and recommendedthat research moves more to focus on the lower porositystructure where the porosity approaches the percolationthreshold (ie the minimum porosity for which flow is stillpossible through structure)

While many trends (or models) have been proposedto compute tortuosity as a function of porosity and otherfitting parameters Valdes-Parada et al [1] stated that anydefinition of tortuosity should not be considered as a functionof porosity but rather a function of the pore geometryonly The idea behind this statement seems to rest in the

fact that there is no universally agreed upon trend betweentortuosity and porosity as pointed out by Matyka et al [24]and that the geometrical features of a pore structure shouldhave more influence on the overall tortuosity of flow ortransport through the geometry than the value of porosityHowever since it is possible that a unique relationshipbetween tortuosity and porosity can indeed exist for specialclasses of porous media [24] the development of tortuosity-porosity trends remains to be an insightful topic of research

Also Liu and Kitanidis [35] stated that tortuosity is atensorial property Despite this statement the majority ofwork does not report directional-specific quantities whichmay leave the reader with the impression that tortuosity is ascalar quantity A scalar quantity could be appropriate in poregeometries that exhibit an anisotropic microstructure whilestill exhibiting an isotropic macrostructure For examplepore geometry composed of randomly distributed squaresis geometrically anisotropic at the microscale however theeffective properties computed for the REV can be isotropic innature In this work we demonstrate this point by example

3 Methods to Compute Effective Properties

31 Permeability The method of homogenization has beenused to mathematically derive Darcyrsquos equation from theStokes equations Through this derivation the macroscopic

Geofluids 5

D120576

120597D120576

120576

120576x2

x1

y2

y1

Y-cell

(a) (b)

Figure 1 Macroscale (a) and microscale (b) domains used in thehomogenization problemThe solid structure in the119884-cell illustratesthe geometry of our first example in-line array of solid shapes (iecircles or squares)

property known as permeability 119896119894119895 is defined as the spatialaverage of a rescaled pore space velocity119873119894119895(119910) In the follow-ing we present the basic steps that lead to such a definitionand the reader is referred to [11] for more derivation details

The problem is first defined by the macroscale (a)and microscale (b) systems shown in Figure 1 where themacroscale domain is comprised of repeating unit cells ofsize 120576 which is comprised of both fluid and solid spaceDue to periodicity the macroscale domain of porosity 120601 canbe represented by the unit cell with periodic boundariesAssuming slow and steady flow of incompressible fluidthrough the pore space of themedia the governing equationsare given by the Stokes equations as follows

minus 120597120597119909119894119901120576 + 120583 120597120597119909119895120597120597119909119895 V120576119894 = minus120588119892119895 (4)

120597120597119909119894 V120576119894 = 0 (5)

where 119894 = 1 2 3 and 119895 = 1 2 3 and repeated indices implysummation 119901 is pressure 120583 is fluid viscosity V119894 is the porespace fluid velocity 120588 is fluid density and 119892119895 is gravitationalaccelerationThe boundary condition at the fluid-solid inter-face is given by

V119894 = 0 (6)

which comes from the no-flow and no-slip boundary condi-tions Although the slip boundary condition is necessary incertain contexts (such as gas flow and non-Newtonian fluid)the no-slip boundary condition is a valid assumption forviscous flow in porous media and is widely used in modelingstudies of pore-scale flow [12 15 24]

In the Stokes equations 119901120576 and V120576119894 can vary within thepore space and thus are functions of the macroscale (119909) andmicroscale (120576) however 120583 119892 and 120588 are treated as constantsand are thus functions of themacroscale (119909) only By defininga fast variable 119910 = 119909120576 the cell of unit length 120576 is scaledinto the 119884-cell (shown in Figure 1(b)) where 0 le 119910 le 1 A

multiscale expansion is used to write 119901120576 and V120576119894 in terms oftheir leading and higher order terms as follows

119901120576 = 119901(119909 119909120576 ) = 119901(0) (119909 119910) + 120576119901(1) (119909 119910) + sdot sdot sdot (7)

V120576119894 = V119894 (119909 119909120576 )= V(0)119894 (119909 119910) + 120576V(1)119894 (119909 119910) + 1205762V(2)119894 (119909 119910) + sdot sdot sdot

(8)

Upon substitution of the expanded variables into (4)ndash(6)and then the collection of like order terms the second-orderpressure term and the third-order velocity term are

119901(1) (119909 119910) = 119876119895 (119910) (minus120588119892119895 + 120597120597119909119895119901(0) (119909))

V(2) (119909 119910) = 119873119894119895 (119910)120583 (minus120588119892119895 + 120597120597119909119895119901(0) (119909)) (9)

such that 119876119895(119910) and119873119894119895(119910) satisfyminus 120597120597119910119894119876119895 (119910) + 120597120597119910119896

120597120597119910119896119873119894119895 (119910) = 120575119894119895 (10)

120597120597119910119894119873119894119895 (119910) = 0 (11)

with the fluid-solid boundary condition given by

119873119894119895 (119910) = 0 (12)

In the above formulation 119876119895(119910) and 119873119894119895(119910) are functionsof the microscale only and are independent of macroscalequantities 120583 120588 and 119892119895 They can be thought of as rescaledpressure and pore space velocity respectively and (10)-(11)can be thought of as rescaled Stokes equations to be solvedwithin the 119884-cell In (10) 120575119894119895 is the Kronecker delta function(120575119894119895 = 1 for 119894 = 119895 and 120575119894119895 = 0 for 119894 = 119895)

Upon further mathematical steps (collecting of like orderterms taking the spatial average over the cell and applyingthe divergence theorem and the periodicity of the cell)the spatial average of V(2)(119909 119910) in the cell is found to bedivergence free nabla sdot ⟨V(2)(119909 119910)⟩119910 = 0 This implies that themacroscopic flow equation is satisfied byDarcyrsquos equation (1)that is

120597120597119909119894 (119896119894119895120583 ( 120597120597119909119895119901 (119909) minus 120588119892119895)) = 0 997888rarr

120597120597119909119894 119906119894 (119909) = 0(13)

where 119901(119909) is the macroscale pressure and 119906119894(119909) is themacroscale velocity when the full permeability tensor for an119899-dimensional problem is defined as

119896119894119895 fl ⟨119873119894119895 (119910)⟩119910 = 1119884 int119884119873119894119895 (119910) 119889119884 (14)

6 Geofluids

where 119884 is the volume of the 119884-cell (solid and fluid spacecombined)

In this work we use the staggered grid finite differencemethod (ie the marker and cell or MAC scheme [36]) forthe numerical solution of the rescaled Stokes problemdefinedby (10)ndash(12) in the 119884-cell Our numerical implementationusing a matrix-oriented approach is explained in [37] Toillustrate the numerical scheme let us consider the followingStokes problem (in its original form before being rescaledby homogenization) in two-dimensional space find the porespace velocity V = (V119909 V119910)119879 and the pressure 119901 such that

minus120583ΔV + nabla119901 = 119891 in Ω fl (0 119871119909) times (0 119871119910) (15)

nabla sdot V = 0 in Ω (16)

V = 0 on 120597Ω (17)

where the velocity has a unique solution while the pressure isunique up to an additive constant Here 119871119909 and 119871119910 denotethe length of the rectangular domain Ω in the 119909- and 119910-directions respectively

To construct a staggered grid finite difference scheme forthe Stokes problem we introduce a possibly nonuniform gridofΩ as follows

0 = 1199090 lt 1199091 lt sdot sdot sdot lt 119909119872 = 1198711199090 = 1199100 lt 1199101 lt sdot sdot sdot lt 119910119873 = 119871119910 (18)

The grid is comprised of fluid and solid cells (each cell beingfully occupied by one phase) and the computational schemeacts uniquely on the computational cells belonging to thefluid The pressure at the (fluid) cell center (119909119894+12 119910119895+12)is 119901(119909119894+12 119910119895+12) approximated by 119901119894+12119895+12 Similarly weapproximate the 119909-velocity V119909 at each 119909-edge (fluid) centerV119909(119909119894 119910119895+12) by V119909119894119895+12 and the 119910-velocity V119910 at each 119910-edge(fluid) center V119910(119909119894minus12 119910119895) by V119910119894minus12119895

By integrating the119909-component of (15) over cellΩ119894119895+12 =(119909119894minus12 119909119894+12) times (119910119895 119910119895+1) we haveminus [int119910119895+1

119910119895(120597V119909120597119909 (119909119894+12 119910) minus 120597V119909120597119909 (119909119894minus12 119910)) 119889119910

+ int119909119894+12119909119894minus12

(120597V119909120597119910 (119909 119910119895+1) minus 120597V119909120597119910 (119909 119910119895)) 119889119909]+ int119910119895+1

119910119895(119901 (119909119894+12 119910) minus 119901 (119909119894minus12 119910)) 119889119910

= intΩ119894119895+12

119891119909 (119909 119910) 119889119909 119889119910

(19)

We then approximate those edge integrals by midpointvalues and the volume integral by its center value and divideboth sides by |Ω119894119895+12| = (119909119894+12 minus 119909119894minus12)(119910119895+1 minus 119910119895)

By introducing the following forward and backwarddifference operators

Δ+119909V119909119894119895+12 = V119909119894+1119895+12 minus V119909119894119895+12119909119894+1 minus 119909119894 Δminus119909119901119894+12119895+12 = 119901119894+12119895+12 minus 119901119894minus12119895+12119909119894+12 minus 119909119894minus12

(20)

and similar notations for operations in the 119910-direction wecan express the results as

minus Δ+119909V119909119894119895+12 minus Δminus119909V119909119894119895+12119909119894+12 minus 119909119894minus12 minus Δ+119910V119909119894119895+12 minus Δminus119910V119909119894119895+12119910119895+1 minus 119910119895+ Δminus119909119901119894+12119895+12 = 119891119909119894119895+12

(21)

Similarly by integrating the 119910-component of (15) overcell Ω119894minus12119895 = (119909119894minus1 119909119894) times (119910119895minus12 119895119895+12) and then applyingmidpoint quadrature rule we can obtain

minus Δ+119909V119910119894minus12119895 minus Δminus119909V119910119894minus12119895119909119894 minus 119909119894minus1 minus Δ+119910V119910119894minus12119895 minus Δminus119910V119910119894minus12119895119910119895+12 minus 119910119895minus12+ Δminus119910119901119894minus12119895+12 = 119891119910119894minus12119895

(22)

By integrating (16) over cell Ω119894minus12119895+12 = (119909119894 minus 119909119894minus1) times(119910119895 119895119895+1) we obtain its discrete form

Δminus119909V119909119894119895+12 + Δ+119910V119910119894minus12119895 = 0 (23)

Eqns (21)ndash(23) form a linear equation system for thepressure and velocity and the equation system defines theMAC scheme for the Stokes problem of (15)ndash(17) Note thatthe no-flow boundary condition supplies the zero velocitycomponent that acts normal to any fluid-solid interface andperiodic conditions are applied along the domainrsquos externalboundaries

We apply this numerical scheme to solve the set ofrescaled Stokes equations in Eqns (10)ndash(12) Our computer-generatedmedia samples are 2-dimensional (2D) thus the setis solved with different forcings 120575119894119895 where 119894 = 1 2 and 119895 = 1 2Upon obtaining the rescaled pore space velocity 119873119894119895(119910) inthe 119884-cell we then compute the full permeability tensor by(14) If our computer-generated media are characterized by anondiagonal tensor we determine the principal componentsof the permeability tensor by diagonalization [11 38] Thenotation used in our results section is k and klowast for thepermeability tensors in coordinate systems x and xlowastrespectively that is

k = [11989611 1198961211989621 11989622] in x 997904rArr

klowast = [119896max 00 119896119905] in xlowast

(24)

32 Hydraulic Tortuosity The approach commonly used inliterature to compute hydraulic tortuosity is presented inKoponen et al [19 26] and further validation of this ap-proach is given in Duda et al [13] In this approach the

Geofluids 7

hydraulic tortuosity 120591ℎ or a particular direction is computedbased on the pore space fluid velocities driven by a force Forexample the hydraulic tortuosity of flow that is driven in the119909 direction is

120591ℎ119909 = ⟨Vmag⟩Ω⟨V119909⟩Ω (25)

where ⟨sdot⟩Ω is the spatial average in the unit cell domain ΩVmag is the magnitude of the fluid velocities (ie the fluidspeed) and V119909 is the velocity component in the 119909 directionThe spatial average of V119909 inΩ is

⟨V119909⟩Ω = 1|119881| int119881119891 V119891119909119889119881119891 = 1|119881|sum119894119895 V

119891119909 (119894 119895) Δ119881119891 (26)

where119881 is the volume ofΩ (pore and solid space combined)119881119891 is the fluid space and the integral is taken over the fluidspace only The spatial average for Vmag is taken in the samewayThe tortuosity in the 119910 direction is computed in a similarmanner as follows

120591ℎ119910 = ⟨Vmag⟩Ω⟨V119910⟩Ω (27)

where the velocity is the solution for flow driven in the119910 direction We note that (25) and (27) correspond to thehydraulic tortuosity as we have defined it in Section 22 thatis 120591ℎ = 119871119890119871 119904 If the term 1205912ℎ which appears in the Kozeny-Carmen equation is labelled as the hydraulic tortuosity then(25) and (27) should be squared

While this method has been used in many works (ie[26 33 39 40]) other approaches to compute hydraulictortuosity have been proposed For example Matyka et al[24] took measurements of streamlines that passed through agiven cross-sectional area in the domain with a constant flux(ie not all streamlines in the domain would be measured)The aim of this approach is to capture the hydraulic tortu-osity of the main conducting channels in a pore geometryAlso Ahmadi et al [32] formulated analytical functionsfor hydraulic tortuosity based on volume averaging of massbalance equations

In our work we follow Koponen et alrsquos [19 26] approachas given by (25)ndash(27) however we do not compute the porespace velocity fields V119909 and V119910 explicitly Instead during thepermeability computation for our 2D pore geometries weobtain the rescaled pore space velocity fields 119873119894119895 in the 119884-cell where 119894 = 1 2 and 119895 = 1 2 It can be seen that 11987311and 11987321 are related to V119909 and V119910 respectively for flow thatis driven by g = ( 0) and that 11987312 and 11987322 are relatedto V119909 and V119910 respectively for flow that is driven by g =(0 ) Thus it is mathematically equivalent to compute thehydraulic tortuosities by

120591ℎ119909 = ⟨radic11987311 + 11987321⟩Ω⟨11987311⟩Ω 120591ℎ119910 = ⟨radic11987312 + 11987322⟩Ω⟨11987322⟩Ω

(28)

where Ω refers to the domain of the 119884-cell and the spatialaverages of 119873119894119895 are computed in the same way as shown by(26) Since the numerical scheme computes 119873119894119895 on the fluidedge-centers we compute the fluid cell-center quantities byaveraging across the grid cells where 119873119894119895 = 0 on the fluid-solid interfaces as per the boundary condition in (12)

As mentioned we diagonalize the permeability tensorto obtain the principal permeability components and theprincipal directions 120579 and 120579+1205872 fl Thus in order tomakean appropriate comparison between direction-specific per-meability and hydraulic tortuosity we compute the hydraulictortuosity in these principal directions For example 120591ℎ in theprincipal direction 120579 is

120591ℎ120579 = ⟨1003816100381610038161003816100381611987312057911989410038161003816100381610038161003816⟩Ω⟨119873120579⟩Ω (29)

where |119873120579119894 | = radic(119873120579

1 )2 + (1198731205792 )2 and where119873120579

1 1198731205792 and119873120579 are

obtained by vector transformations from 119873119894119895 120591ℎ in the otherprincipal direction is similarly obtained The notation weuse in our results section is 120591ℎmax = 120591ℎ120579 and 120591ℎ119905 = 120591ℎ33 Effective Diffusion Coefficient and Diffusive TortuosityThe problem is similarly defined by the two scales illustratedin Figure 1 Within the pore space the diffusive and convec-tive transport of a species of concentration 119888 is given by

120597120597119905119888120576 + 1120576 V120576119894 120597120597119909119894 119888120576 =120597120597119909119894119863119894119895

120597120597119909119895 119888120576119899119894119863119894119895

120597120597119909119895 119888120576 = 0V120576119894 = 0

(30)

where V120576119894 is divergence free (5) 120576 implies the variables arefunctions of the microscale and 119863119894119895 is the molecular diffu-sion coefficient The last two equations give the boundaryconditions on the fluid-solid interface Using similar homog-enization steps as presented in Section 31 the multiscaleexpansion

119888120576 = 119888 (119909 119909120576 )= 119888(0) (119909 119910) + 120576119888(1) (119909 119910) + 1205762119888(2) (119909 119910) + sdot sdot sdot

(31)

and (8) are substituted into the governing transport equations(30) Upon collecting like order terms the second-order termof 119888 is

119888(1) (119909 119910) = 119875119896 (119909 119910) 120597120597119909119896 119888(0) (119909) (32)

where 119888(0)(119909) is the macroscale concentration and 119875119896(119909 119910)satisfies 120597120597119910119894

120597120597119910119894119875119896 (119909 119910) = 0 (33)

119899119894 120597120597119910119894119875119896 (119909 119910) = minus119899119896 (34)

8 Geofluids

where (34) gives the boundary condition and 119899119894 is a unitvector normal to the fluid-solid interface (positive in thedirection away from the fluid space) Eqns (33) and (34)correspond to the assumption of an isotropic moleculardiffusion coefficient that is119863119894119895 = 119863120575119894119895 Upon further algebraand averaging over the 119884-cell the macroscale transportequation is obtained as follows

120601 120597120597119905119888 (119909) = 120597120597119909119894119863efflowast119894119896

120597120597119909119896 119888 (119909) (35)

where the effective diffusion coefficient119863efflowast119894119896 is given by

119863efflowast119894119896119863 = 1119884 int

119884(120575119894119896 + 120597120597119910119894119875119896 (119909 119910)) 119889119884 (36)

Intuitively the more tortuous the porous media the slowerthe rate of effective diffusion thus a commonly used relation-ship between effective and molecular diffusivity is

119863efflowast119894119896 = 119863120601120591119889119894119896 (37)

where 120591119889119894119896 is the diffusive tortuosity [41]The above approach to compute the effective diffusion

coefficient and the diffusive tortuosity has been used inseveral previous studies which formulate (33)ndash(36) using themethod of volume averaging [1 42ndash45] or by the method ofhomogenization [27] Other approaches to compute diffusivetortuosity include simulation of the diffusion process usinga random walk [25] or numerically solving the microscalediffusion equation using lattice Boltzmann modeling [46]Typically the effective diffusivity is computed and then usedto calculate diffusive tortuosity by the inverse relationshipshown in (37)

By (36) and (37) diffusive tortuosity is computed by

120591119889119894119896 = (119868119894119896 + 11003816100381610038161003816100381611988411989110038161003816100381610038161003816 int119884119891 (120597120597119910119894119875119896 (119909 119910)) 119889119910119891)

minus1 (38)

where 119884119891 = 120601119884 and where 119875119896(119909 119910) is the solution to(33) with the boundary condition in (34) This is the sameformulation that was used in Sun et al [27] by the method ofhomogenization and in Kim et al [43] and Valdes-Parada etal [1] by the method of volume averaging

The full diffusive tortuosity can also be diagonalized usingthe same procedure outlined in Section 31 to obtain thecomponents 120591119889max and 120591119889119905 corresponding to the principaldirections The rotation required to obtain the principalcomponents of klowast is not necessarily identical to the rotationrequired to obtain the principal components of 120591lowast119889 as willbe shown by one of the following pore geometry examples(randomly distributed squares)

4 Results

While real geological formations are typically of low porosity(120601 lt 03) we have generated several idealized geometrieswithin the porosity range of 033 lt 120601 lt 1 These idealized

Simulated result (Stokes) circleAnalytical result (Gebart eqn)

10minus4

10minus3

10minus2

10minus1

100

101

102

Nor

mal

ized

per

mea

bilit

y

03 04 05 06 07 08 09 102

Porosity

Figure 2 Permeability-porosity trend for in-line array of circles(ie infinitely long cylinders) Permeability units are nondimen-sional (ie unit length times unit length) The permeability is isotropicand is normalized by 1199032 where 119903 is the solid circle radius Gebartrsquos[47] analytical solution was given in (A2)

geometries and porosities have been used in literature (seeTable 1) and thus provide ground for comparison and oppor-tunity to make useful comparisons of the effective propertiesin high porosity representations of porous media

Permeability hydraulic tortuosity anddiffusive tortuosityare computed for each pore geometry using the methodsoutlined in Sections 31 32 and 33 respectively Each poregeometry is generated by defining certain input parameters(ie radius of solid circle side length of square and den-sity of randomly distributed squares) and these inputs aresummarized in Table 4 By using a range of values for theinput parameters we obtain unit cells which we treat asrepresentative elementary volumes over a range of porositiesAfter computing the permeability and tortuosity we plotthe data to observe permeability-porosity and tortuosity-porosity trends

41 In-Line Array of Uniform Shapes The unit cell or 119884-cellin Figure 1(b) illustrates our first type of generated poregeometry and the shape in the center of the cell is eithera circle or square A range of porosities of the unit cell areobtained by changing the radius of the solid circle or thelength of the solid square centered in the cell

411 Permeability and Hydraulic Tortuosity The full perme-ability tensor was computed for 033 lt 120601 lt 1 and resultsare plotted in Figure 2 Following [14] we computed andplotted the analytical solution from Gebart [47] which isexplained in Appendix A Permeability is isotropic (ie 119896119909119909 =119896119910119910) for this pore geometry The range of our simulateddata fits well against the analytical solution We computed

Geofluids 9

Table 4 Input parameters used to generate different pore-geometries (or different 119884-cells) and their corresponding porosity ranges In thistable cell refers to a cell in the mesh used to discretize the 119884-cellGeometry Initial mesh dimensions Inputs Porosity rangeIn-line array of shapes(see Figure 1) 50 times 50 cells (119884-cell length 1 unit) (a) Circle diameter 44 to 8 cells

(b) Square length 48 to 2 cells033 lt 120601 lt 1007 lt 120601 lt 1

Staggered array ofsquares (see Figure 7)

100 times 50 cells (119884-cell length 119871119887 = 05units 119886119887 = 1 119871119886119871119887 = 1 as per Figure 8) Square length 40 to 8 cells 036 le 120601 lt 1

Randomly distributedsquares (see Figure 14) 100 times 100 cells (119884-cell length 1 unit) Increasing density of squares

(with constant square length 001unit = 1 cell)

045 le 120601 lt 1

0005

001

0015

002

0025

003

0035

Figure 3 Fluid flow past a solid circle 120601 = 071 driven by 119892119909 Non-dimensional fluid speed (ie magnitude of velocities) is indicatedby color and flow path is indicated by flow lines (direction is left toright) Speed profile and flow path driven by 119892119910 are same as shownbut rotated 90 degrees

our permeability data on increasingly finer meshes until thepermeability converged within a 1 relative error To reachthis convergence criteria after starting with a mesh size of50 times 50 the lowest porosity cell required a refined mesh of500 times 500 and the highest porosity cell required 300 times 300The shape of the circular edge was refined at each meshrefinement in order to represent the circle as realistically aspossible and to minimize artifacts caused by discretization

We obtained the hydraulic tortuosity in the 119909 and 119910directions for porosities within the range 033 lt 120601 lt 1 withthe same convergence criteria used to obtain permeabilityFigure 3 illustrates the fluid speed and direction for a unitcell of 120601 = 071 and the hydraulic tortuosity in both 119909 and119910 directions were computed to be 120591ℎ119909 = 120591ℎ119910 = 10185 Thefluid speed is highest in the pore space that is uninterruptedby the solid and the flow lines indicate the direction is parallelto the macroscopic flow The flow lines diverge in the regionthat is interrupted by the solid however the fluid speed isessentially zero in that pore space So while these flow lines

120601 = 033

120601 = 039

120601 = 050

120601 = 059

120601 = 072

120601 = 087

120601 = 098

1

12

14

16

18

2

22

Com

pute

d di

ffusiv

e 120591

1 2 3 4 50

Number of cells times105

Figure 4Number of grid cells required to reach converged 120591119889 (usinga convergence criteria of 01 relative error) for select porosities ofin-line array of circles

appear to illustrate a tortuous pathway through this geometrythe tortuosity in either direction is in fact calculated to beasymp1 because the highest speeds are acting along a nontortuouspathway In the pore space that is not restricted or interruptedin the direction parallel to the driving force the fluid velocityis able to develop into the parabolic profile that characterizeschannel flow

412 Diffusive Tortuosity We obtained the full diffusive tor-tuosity tensor on increasingly refined meshes until the tortu-osity converged within a 01 relative error Starting with amesh of 50 times 50 the lowest porosity required a refined meshof 700 times 700 while the highest porosity required 250 times 250 tomeet the convergence criteria Solid edges were also refinedwith mesh refinement as mentioned previously In Figure 4we plot the convergence behavior of 120591119889 for a select numberof porosity cases in order to illustrate that each porosity caseconvergences with similar behavior Figure 5(a) illustrates thefields of 119875119896 (119896 = 1 2 or 119909 119910) for a unit cell of 120601 = 071which were obtained by solving the homogenized problem

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

4 Geofluids

Table 2 Past work on hydraulic tortuosity

Study Samples considered Tortuosityndashporosity fit Notes

[54] Packed beds 120591 = 1 + 119901 ln( 1120601) 119901 is function of particle shape

[19] 2D random overlapping monosized squares 05 lt 120601 lt 1 120591 = 1 + 08 (1 minus 120601)[26] 2D random overlapping monosized squares 04 lt 120601 lt 1 120591 = 1 + 065 1 minus 120601

(120601 minus 120601119888)019 120601119888 is critical porosity[24] 2D random overlapping monosized squares 120591 prop 1 + 119877 119878120601 119877 is hydraulic radius 119878 is specific surface area[13] 2D freely overlapping squares 120591 prop 1 + (1 minus 120601)12

Table 3 Past work on diffusive tortuosity

Study Samples considered Tortuosityndashporosity fit Notes[48]lowast Array of cylinders (2D) 120591 = 2 minus 120601[50]lowastlowast Diffusion of electrolytes in membrane 120591 = (2 minus 120601120601 )2[52] Bed of uniform spheres (applicable to overlapping nonuniform spheres) 120591 = 1 minus 12 ln120601[43] Isotropic system 0 lt 120601 lt 05 120591 = 120601minus04[31] Fine-grained uncemented sediments 120591 = (1 minus ln (1206012))12[55] Isotropic representative unit cell 120591 = 120601

1 minus (1 minus 120601)119898 2D119898 = 12 3D119898 = 23[56] Random partial overlapping shapes 120591 = 11 minus 120572 (1 minus 120601) 120572 = shape factor

[46] Voronoi channel geometries 02 lt 120601 lt 06 120591 = (075120601minus10819)12lowastAs referenced in [57]lowastlowastAs referenced in [30 31]

In the same way that the KC equation is specific to a givenclass of porous media any porosity-tortuosity trend that isempirically derived depends on the porousmedia consideredSome authors rely on synthetic porous media such as 2Drandomly distributed squares [13 19 24 26 33 34] 3D highporosity plate-like obstacles [25] or unit cells of centereduniform shapes while others rely on real samples of mediataken by imaging (via scanning electron microscopy (SEM)or X-ray microtomography) and digitization

Additionally it is important to note that empiricallyderived trends are applicable to a limited range of porosityonly as the trend was developed given simulated data of acertain porosity range The porosity limit could be a resultof the idealized geometry under consideration Ghanbarianet al [8] noticed that most tortuosity models were derived byfocusing on the higher porosity structures and recommendedthat research moves more to focus on the lower porositystructure where the porosity approaches the percolationthreshold (ie the minimum porosity for which flow is stillpossible through structure)

While many trends (or models) have been proposedto compute tortuosity as a function of porosity and otherfitting parameters Valdes-Parada et al [1] stated that anydefinition of tortuosity should not be considered as a functionof porosity but rather a function of the pore geometryonly The idea behind this statement seems to rest in the

fact that there is no universally agreed upon trend betweentortuosity and porosity as pointed out by Matyka et al [24]and that the geometrical features of a pore structure shouldhave more influence on the overall tortuosity of flow ortransport through the geometry than the value of porosityHowever since it is possible that a unique relationshipbetween tortuosity and porosity can indeed exist for specialclasses of porous media [24] the development of tortuosity-porosity trends remains to be an insightful topic of research

Also Liu and Kitanidis [35] stated that tortuosity is atensorial property Despite this statement the majority ofwork does not report directional-specific quantities whichmay leave the reader with the impression that tortuosity is ascalar quantity A scalar quantity could be appropriate in poregeometries that exhibit an anisotropic microstructure whilestill exhibiting an isotropic macrostructure For examplepore geometry composed of randomly distributed squaresis geometrically anisotropic at the microscale however theeffective properties computed for the REV can be isotropic innature In this work we demonstrate this point by example

3 Methods to Compute Effective Properties

31 Permeability The method of homogenization has beenused to mathematically derive Darcyrsquos equation from theStokes equations Through this derivation the macroscopic

Geofluids 5

D120576

120597D120576

120576

120576x2

x1

y2

y1

Y-cell

(a) (b)

Figure 1 Macroscale (a) and microscale (b) domains used in thehomogenization problemThe solid structure in the119884-cell illustratesthe geometry of our first example in-line array of solid shapes (iecircles or squares)

property known as permeability 119896119894119895 is defined as the spatialaverage of a rescaled pore space velocity119873119894119895(119910) In the follow-ing we present the basic steps that lead to such a definitionand the reader is referred to [11] for more derivation details

The problem is first defined by the macroscale (a)and microscale (b) systems shown in Figure 1 where themacroscale domain is comprised of repeating unit cells ofsize 120576 which is comprised of both fluid and solid spaceDue to periodicity the macroscale domain of porosity 120601 canbe represented by the unit cell with periodic boundariesAssuming slow and steady flow of incompressible fluidthrough the pore space of themedia the governing equationsare given by the Stokes equations as follows

minus 120597120597119909119894119901120576 + 120583 120597120597119909119895120597120597119909119895 V120576119894 = minus120588119892119895 (4)

120597120597119909119894 V120576119894 = 0 (5)

where 119894 = 1 2 3 and 119895 = 1 2 3 and repeated indices implysummation 119901 is pressure 120583 is fluid viscosity V119894 is the porespace fluid velocity 120588 is fluid density and 119892119895 is gravitationalaccelerationThe boundary condition at the fluid-solid inter-face is given by

V119894 = 0 (6)

which comes from the no-flow and no-slip boundary condi-tions Although the slip boundary condition is necessary incertain contexts (such as gas flow and non-Newtonian fluid)the no-slip boundary condition is a valid assumption forviscous flow in porous media and is widely used in modelingstudies of pore-scale flow [12 15 24]

In the Stokes equations 119901120576 and V120576119894 can vary within thepore space and thus are functions of the macroscale (119909) andmicroscale (120576) however 120583 119892 and 120588 are treated as constantsand are thus functions of themacroscale (119909) only By defininga fast variable 119910 = 119909120576 the cell of unit length 120576 is scaledinto the 119884-cell (shown in Figure 1(b)) where 0 le 119910 le 1 A

multiscale expansion is used to write 119901120576 and V120576119894 in terms oftheir leading and higher order terms as follows

119901120576 = 119901(119909 119909120576 ) = 119901(0) (119909 119910) + 120576119901(1) (119909 119910) + sdot sdot sdot (7)

V120576119894 = V119894 (119909 119909120576 )= V(0)119894 (119909 119910) + 120576V(1)119894 (119909 119910) + 1205762V(2)119894 (119909 119910) + sdot sdot sdot

(8)

Upon substitution of the expanded variables into (4)ndash(6)and then the collection of like order terms the second-orderpressure term and the third-order velocity term are

119901(1) (119909 119910) = 119876119895 (119910) (minus120588119892119895 + 120597120597119909119895119901(0) (119909))

V(2) (119909 119910) = 119873119894119895 (119910)120583 (minus120588119892119895 + 120597120597119909119895119901(0) (119909)) (9)

such that 119876119895(119910) and119873119894119895(119910) satisfyminus 120597120597119910119894119876119895 (119910) + 120597120597119910119896

120597120597119910119896119873119894119895 (119910) = 120575119894119895 (10)

120597120597119910119894119873119894119895 (119910) = 0 (11)

with the fluid-solid boundary condition given by

119873119894119895 (119910) = 0 (12)

In the above formulation 119876119895(119910) and 119873119894119895(119910) are functionsof the microscale only and are independent of macroscalequantities 120583 120588 and 119892119895 They can be thought of as rescaledpressure and pore space velocity respectively and (10)-(11)can be thought of as rescaled Stokes equations to be solvedwithin the 119884-cell In (10) 120575119894119895 is the Kronecker delta function(120575119894119895 = 1 for 119894 = 119895 and 120575119894119895 = 0 for 119894 = 119895)

Upon further mathematical steps (collecting of like orderterms taking the spatial average over the cell and applyingthe divergence theorem and the periodicity of the cell)the spatial average of V(2)(119909 119910) in the cell is found to bedivergence free nabla sdot ⟨V(2)(119909 119910)⟩119910 = 0 This implies that themacroscopic flow equation is satisfied byDarcyrsquos equation (1)that is

120597120597119909119894 (119896119894119895120583 ( 120597120597119909119895119901 (119909) minus 120588119892119895)) = 0 997888rarr

120597120597119909119894 119906119894 (119909) = 0(13)

where 119901(119909) is the macroscale pressure and 119906119894(119909) is themacroscale velocity when the full permeability tensor for an119899-dimensional problem is defined as

119896119894119895 fl ⟨119873119894119895 (119910)⟩119910 = 1119884 int119884119873119894119895 (119910) 119889119884 (14)

6 Geofluids

where 119884 is the volume of the 119884-cell (solid and fluid spacecombined)

In this work we use the staggered grid finite differencemethod (ie the marker and cell or MAC scheme [36]) forthe numerical solution of the rescaled Stokes problemdefinedby (10)ndash(12) in the 119884-cell Our numerical implementationusing a matrix-oriented approach is explained in [37] Toillustrate the numerical scheme let us consider the followingStokes problem (in its original form before being rescaledby homogenization) in two-dimensional space find the porespace velocity V = (V119909 V119910)119879 and the pressure 119901 such that

minus120583ΔV + nabla119901 = 119891 in Ω fl (0 119871119909) times (0 119871119910) (15)

nabla sdot V = 0 in Ω (16)

V = 0 on 120597Ω (17)

where the velocity has a unique solution while the pressure isunique up to an additive constant Here 119871119909 and 119871119910 denotethe length of the rectangular domain Ω in the 119909- and 119910-directions respectively

To construct a staggered grid finite difference scheme forthe Stokes problem we introduce a possibly nonuniform gridofΩ as follows

0 = 1199090 lt 1199091 lt sdot sdot sdot lt 119909119872 = 1198711199090 = 1199100 lt 1199101 lt sdot sdot sdot lt 119910119873 = 119871119910 (18)

The grid is comprised of fluid and solid cells (each cell beingfully occupied by one phase) and the computational schemeacts uniquely on the computational cells belonging to thefluid The pressure at the (fluid) cell center (119909119894+12 119910119895+12)is 119901(119909119894+12 119910119895+12) approximated by 119901119894+12119895+12 Similarly weapproximate the 119909-velocity V119909 at each 119909-edge (fluid) centerV119909(119909119894 119910119895+12) by V119909119894119895+12 and the 119910-velocity V119910 at each 119910-edge(fluid) center V119910(119909119894minus12 119910119895) by V119910119894minus12119895

By integrating the119909-component of (15) over cellΩ119894119895+12 =(119909119894minus12 119909119894+12) times (119910119895 119910119895+1) we haveminus [int119910119895+1

119910119895(120597V119909120597119909 (119909119894+12 119910) minus 120597V119909120597119909 (119909119894minus12 119910)) 119889119910

+ int119909119894+12119909119894minus12

(120597V119909120597119910 (119909 119910119895+1) minus 120597V119909120597119910 (119909 119910119895)) 119889119909]+ int119910119895+1

119910119895(119901 (119909119894+12 119910) minus 119901 (119909119894minus12 119910)) 119889119910

= intΩ119894119895+12

119891119909 (119909 119910) 119889119909 119889119910

(19)

We then approximate those edge integrals by midpointvalues and the volume integral by its center value and divideboth sides by |Ω119894119895+12| = (119909119894+12 minus 119909119894minus12)(119910119895+1 minus 119910119895)

By introducing the following forward and backwarddifference operators

Δ+119909V119909119894119895+12 = V119909119894+1119895+12 minus V119909119894119895+12119909119894+1 minus 119909119894 Δminus119909119901119894+12119895+12 = 119901119894+12119895+12 minus 119901119894minus12119895+12119909119894+12 minus 119909119894minus12

(20)

and similar notations for operations in the 119910-direction wecan express the results as

minus Δ+119909V119909119894119895+12 minus Δminus119909V119909119894119895+12119909119894+12 minus 119909119894minus12 minus Δ+119910V119909119894119895+12 minus Δminus119910V119909119894119895+12119910119895+1 minus 119910119895+ Δminus119909119901119894+12119895+12 = 119891119909119894119895+12

(21)

Similarly by integrating the 119910-component of (15) overcell Ω119894minus12119895 = (119909119894minus1 119909119894) times (119910119895minus12 119895119895+12) and then applyingmidpoint quadrature rule we can obtain

minus Δ+119909V119910119894minus12119895 minus Δminus119909V119910119894minus12119895119909119894 minus 119909119894minus1 minus Δ+119910V119910119894minus12119895 minus Δminus119910V119910119894minus12119895119910119895+12 minus 119910119895minus12+ Δminus119910119901119894minus12119895+12 = 119891119910119894minus12119895

(22)

By integrating (16) over cell Ω119894minus12119895+12 = (119909119894 minus 119909119894minus1) times(119910119895 119895119895+1) we obtain its discrete form

Δminus119909V119909119894119895+12 + Δ+119910V119910119894minus12119895 = 0 (23)

Eqns (21)ndash(23) form a linear equation system for thepressure and velocity and the equation system defines theMAC scheme for the Stokes problem of (15)ndash(17) Note thatthe no-flow boundary condition supplies the zero velocitycomponent that acts normal to any fluid-solid interface andperiodic conditions are applied along the domainrsquos externalboundaries

We apply this numerical scheme to solve the set ofrescaled Stokes equations in Eqns (10)ndash(12) Our computer-generatedmedia samples are 2-dimensional (2D) thus the setis solved with different forcings 120575119894119895 where 119894 = 1 2 and 119895 = 1 2Upon obtaining the rescaled pore space velocity 119873119894119895(119910) inthe 119884-cell we then compute the full permeability tensor by(14) If our computer-generated media are characterized by anondiagonal tensor we determine the principal componentsof the permeability tensor by diagonalization [11 38] Thenotation used in our results section is k and klowast for thepermeability tensors in coordinate systems x and xlowastrespectively that is

k = [11989611 1198961211989621 11989622] in x 997904rArr

klowast = [119896max 00 119896119905] in xlowast

(24)

32 Hydraulic Tortuosity The approach commonly used inliterature to compute hydraulic tortuosity is presented inKoponen et al [19 26] and further validation of this ap-proach is given in Duda et al [13] In this approach the

Geofluids 7

hydraulic tortuosity 120591ℎ or a particular direction is computedbased on the pore space fluid velocities driven by a force Forexample the hydraulic tortuosity of flow that is driven in the119909 direction is

120591ℎ119909 = ⟨Vmag⟩Ω⟨V119909⟩Ω (25)

where ⟨sdot⟩Ω is the spatial average in the unit cell domain ΩVmag is the magnitude of the fluid velocities (ie the fluidspeed) and V119909 is the velocity component in the 119909 directionThe spatial average of V119909 inΩ is

⟨V119909⟩Ω = 1|119881| int119881119891 V119891119909119889119881119891 = 1|119881|sum119894119895 V

119891119909 (119894 119895) Δ119881119891 (26)

where119881 is the volume ofΩ (pore and solid space combined)119881119891 is the fluid space and the integral is taken over the fluidspace only The spatial average for Vmag is taken in the samewayThe tortuosity in the 119910 direction is computed in a similarmanner as follows

120591ℎ119910 = ⟨Vmag⟩Ω⟨V119910⟩Ω (27)

where the velocity is the solution for flow driven in the119910 direction We note that (25) and (27) correspond to thehydraulic tortuosity as we have defined it in Section 22 thatis 120591ℎ = 119871119890119871 119904 If the term 1205912ℎ which appears in the Kozeny-Carmen equation is labelled as the hydraulic tortuosity then(25) and (27) should be squared

While this method has been used in many works (ie[26 33 39 40]) other approaches to compute hydraulictortuosity have been proposed For example Matyka et al[24] took measurements of streamlines that passed through agiven cross-sectional area in the domain with a constant flux(ie not all streamlines in the domain would be measured)The aim of this approach is to capture the hydraulic tortu-osity of the main conducting channels in a pore geometryAlso Ahmadi et al [32] formulated analytical functionsfor hydraulic tortuosity based on volume averaging of massbalance equations

In our work we follow Koponen et alrsquos [19 26] approachas given by (25)ndash(27) however we do not compute the porespace velocity fields V119909 and V119910 explicitly Instead during thepermeability computation for our 2D pore geometries weobtain the rescaled pore space velocity fields 119873119894119895 in the 119884-cell where 119894 = 1 2 and 119895 = 1 2 It can be seen that 11987311and 11987321 are related to V119909 and V119910 respectively for flow thatis driven by g = ( 0) and that 11987312 and 11987322 are relatedto V119909 and V119910 respectively for flow that is driven by g =(0 ) Thus it is mathematically equivalent to compute thehydraulic tortuosities by

120591ℎ119909 = ⟨radic11987311 + 11987321⟩Ω⟨11987311⟩Ω 120591ℎ119910 = ⟨radic11987312 + 11987322⟩Ω⟨11987322⟩Ω

(28)

where Ω refers to the domain of the 119884-cell and the spatialaverages of 119873119894119895 are computed in the same way as shown by(26) Since the numerical scheme computes 119873119894119895 on the fluidedge-centers we compute the fluid cell-center quantities byaveraging across the grid cells where 119873119894119895 = 0 on the fluid-solid interfaces as per the boundary condition in (12)

As mentioned we diagonalize the permeability tensorto obtain the principal permeability components and theprincipal directions 120579 and 120579+1205872 fl Thus in order tomakean appropriate comparison between direction-specific per-meability and hydraulic tortuosity we compute the hydraulictortuosity in these principal directions For example 120591ℎ in theprincipal direction 120579 is

120591ℎ120579 = ⟨1003816100381610038161003816100381611987312057911989410038161003816100381610038161003816⟩Ω⟨119873120579⟩Ω (29)

where |119873120579119894 | = radic(119873120579

1 )2 + (1198731205792 )2 and where119873120579

1 1198731205792 and119873120579 are

obtained by vector transformations from 119873119894119895 120591ℎ in the otherprincipal direction is similarly obtained The notation weuse in our results section is 120591ℎmax = 120591ℎ120579 and 120591ℎ119905 = 120591ℎ33 Effective Diffusion Coefficient and Diffusive TortuosityThe problem is similarly defined by the two scales illustratedin Figure 1 Within the pore space the diffusive and convec-tive transport of a species of concentration 119888 is given by

120597120597119905119888120576 + 1120576 V120576119894 120597120597119909119894 119888120576 =120597120597119909119894119863119894119895

120597120597119909119895 119888120576119899119894119863119894119895

120597120597119909119895 119888120576 = 0V120576119894 = 0

(30)

where V120576119894 is divergence free (5) 120576 implies the variables arefunctions of the microscale and 119863119894119895 is the molecular diffu-sion coefficient The last two equations give the boundaryconditions on the fluid-solid interface Using similar homog-enization steps as presented in Section 31 the multiscaleexpansion

119888120576 = 119888 (119909 119909120576 )= 119888(0) (119909 119910) + 120576119888(1) (119909 119910) + 1205762119888(2) (119909 119910) + sdot sdot sdot

(31)

and (8) are substituted into the governing transport equations(30) Upon collecting like order terms the second-order termof 119888 is

119888(1) (119909 119910) = 119875119896 (119909 119910) 120597120597119909119896 119888(0) (119909) (32)

where 119888(0)(119909) is the macroscale concentration and 119875119896(119909 119910)satisfies 120597120597119910119894

120597120597119910119894119875119896 (119909 119910) = 0 (33)

119899119894 120597120597119910119894119875119896 (119909 119910) = minus119899119896 (34)

8 Geofluids

where (34) gives the boundary condition and 119899119894 is a unitvector normal to the fluid-solid interface (positive in thedirection away from the fluid space) Eqns (33) and (34)correspond to the assumption of an isotropic moleculardiffusion coefficient that is119863119894119895 = 119863120575119894119895 Upon further algebraand averaging over the 119884-cell the macroscale transportequation is obtained as follows

120601 120597120597119905119888 (119909) = 120597120597119909119894119863efflowast119894119896

120597120597119909119896 119888 (119909) (35)

where the effective diffusion coefficient119863efflowast119894119896 is given by

119863efflowast119894119896119863 = 1119884 int

119884(120575119894119896 + 120597120597119910119894119875119896 (119909 119910)) 119889119884 (36)

Intuitively the more tortuous the porous media the slowerthe rate of effective diffusion thus a commonly used relation-ship between effective and molecular diffusivity is

119863efflowast119894119896 = 119863120601120591119889119894119896 (37)

where 120591119889119894119896 is the diffusive tortuosity [41]The above approach to compute the effective diffusion

coefficient and the diffusive tortuosity has been used inseveral previous studies which formulate (33)ndash(36) using themethod of volume averaging [1 42ndash45] or by the method ofhomogenization [27] Other approaches to compute diffusivetortuosity include simulation of the diffusion process usinga random walk [25] or numerically solving the microscalediffusion equation using lattice Boltzmann modeling [46]Typically the effective diffusivity is computed and then usedto calculate diffusive tortuosity by the inverse relationshipshown in (37)

By (36) and (37) diffusive tortuosity is computed by

120591119889119894119896 = (119868119894119896 + 11003816100381610038161003816100381611988411989110038161003816100381610038161003816 int119884119891 (120597120597119910119894119875119896 (119909 119910)) 119889119910119891)

minus1 (38)

where 119884119891 = 120601119884 and where 119875119896(119909 119910) is the solution to(33) with the boundary condition in (34) This is the sameformulation that was used in Sun et al [27] by the method ofhomogenization and in Kim et al [43] and Valdes-Parada etal [1] by the method of volume averaging

The full diffusive tortuosity can also be diagonalized usingthe same procedure outlined in Section 31 to obtain thecomponents 120591119889max and 120591119889119905 corresponding to the principaldirections The rotation required to obtain the principalcomponents of klowast is not necessarily identical to the rotationrequired to obtain the principal components of 120591lowast119889 as willbe shown by one of the following pore geometry examples(randomly distributed squares)

4 Results

While real geological formations are typically of low porosity(120601 lt 03) we have generated several idealized geometrieswithin the porosity range of 033 lt 120601 lt 1 These idealized

Simulated result (Stokes) circleAnalytical result (Gebart eqn)

10minus4

10minus3

10minus2

10minus1

100

101

102

Nor

mal

ized

per

mea

bilit

y

03 04 05 06 07 08 09 102

Porosity

Figure 2 Permeability-porosity trend for in-line array of circles(ie infinitely long cylinders) Permeability units are nondimen-sional (ie unit length times unit length) The permeability is isotropicand is normalized by 1199032 where 119903 is the solid circle radius Gebartrsquos[47] analytical solution was given in (A2)

geometries and porosities have been used in literature (seeTable 1) and thus provide ground for comparison and oppor-tunity to make useful comparisons of the effective propertiesin high porosity representations of porous media

Permeability hydraulic tortuosity anddiffusive tortuosityare computed for each pore geometry using the methodsoutlined in Sections 31 32 and 33 respectively Each poregeometry is generated by defining certain input parameters(ie radius of solid circle side length of square and den-sity of randomly distributed squares) and these inputs aresummarized in Table 4 By using a range of values for theinput parameters we obtain unit cells which we treat asrepresentative elementary volumes over a range of porositiesAfter computing the permeability and tortuosity we plotthe data to observe permeability-porosity and tortuosity-porosity trends

41 In-Line Array of Uniform Shapes The unit cell or 119884-cellin Figure 1(b) illustrates our first type of generated poregeometry and the shape in the center of the cell is eithera circle or square A range of porosities of the unit cell areobtained by changing the radius of the solid circle or thelength of the solid square centered in the cell

411 Permeability and Hydraulic Tortuosity The full perme-ability tensor was computed for 033 lt 120601 lt 1 and resultsare plotted in Figure 2 Following [14] we computed andplotted the analytical solution from Gebart [47] which isexplained in Appendix A Permeability is isotropic (ie 119896119909119909 =119896119910119910) for this pore geometry The range of our simulateddata fits well against the analytical solution We computed

Geofluids 9

Table 4 Input parameters used to generate different pore-geometries (or different 119884-cells) and their corresponding porosity ranges In thistable cell refers to a cell in the mesh used to discretize the 119884-cellGeometry Initial mesh dimensions Inputs Porosity rangeIn-line array of shapes(see Figure 1) 50 times 50 cells (119884-cell length 1 unit) (a) Circle diameter 44 to 8 cells

(b) Square length 48 to 2 cells033 lt 120601 lt 1007 lt 120601 lt 1

Staggered array ofsquares (see Figure 7)

100 times 50 cells (119884-cell length 119871119887 = 05units 119886119887 = 1 119871119886119871119887 = 1 as per Figure 8) Square length 40 to 8 cells 036 le 120601 lt 1

Randomly distributedsquares (see Figure 14) 100 times 100 cells (119884-cell length 1 unit) Increasing density of squares

(with constant square length 001unit = 1 cell)

045 le 120601 lt 1

0005

001

0015

002

0025

003

0035

Figure 3 Fluid flow past a solid circle 120601 = 071 driven by 119892119909 Non-dimensional fluid speed (ie magnitude of velocities) is indicatedby color and flow path is indicated by flow lines (direction is left toright) Speed profile and flow path driven by 119892119910 are same as shownbut rotated 90 degrees

our permeability data on increasingly finer meshes until thepermeability converged within a 1 relative error To reachthis convergence criteria after starting with a mesh size of50 times 50 the lowest porosity cell required a refined mesh of500 times 500 and the highest porosity cell required 300 times 300The shape of the circular edge was refined at each meshrefinement in order to represent the circle as realistically aspossible and to minimize artifacts caused by discretization

We obtained the hydraulic tortuosity in the 119909 and 119910directions for porosities within the range 033 lt 120601 lt 1 withthe same convergence criteria used to obtain permeabilityFigure 3 illustrates the fluid speed and direction for a unitcell of 120601 = 071 and the hydraulic tortuosity in both 119909 and119910 directions were computed to be 120591ℎ119909 = 120591ℎ119910 = 10185 Thefluid speed is highest in the pore space that is uninterruptedby the solid and the flow lines indicate the direction is parallelto the macroscopic flow The flow lines diverge in the regionthat is interrupted by the solid however the fluid speed isessentially zero in that pore space So while these flow lines

120601 = 033

120601 = 039

120601 = 050

120601 = 059

120601 = 072

120601 = 087

120601 = 098

1

12

14

16

18

2

22

Com

pute

d di

ffusiv

e 120591

1 2 3 4 50

Number of cells times105

Figure 4Number of grid cells required to reach converged 120591119889 (usinga convergence criteria of 01 relative error) for select porosities ofin-line array of circles

appear to illustrate a tortuous pathway through this geometrythe tortuosity in either direction is in fact calculated to beasymp1 because the highest speeds are acting along a nontortuouspathway In the pore space that is not restricted or interruptedin the direction parallel to the driving force the fluid velocityis able to develop into the parabolic profile that characterizeschannel flow

412 Diffusive Tortuosity We obtained the full diffusive tor-tuosity tensor on increasingly refined meshes until the tortu-osity converged within a 01 relative error Starting with amesh of 50 times 50 the lowest porosity required a refined meshof 700 times 700 while the highest porosity required 250 times 250 tomeet the convergence criteria Solid edges were also refinedwith mesh refinement as mentioned previously In Figure 4we plot the convergence behavior of 120591119889 for a select numberof porosity cases in order to illustrate that each porosity caseconvergences with similar behavior Figure 5(a) illustrates thefields of 119875119896 (119896 = 1 2 or 119909 119910) for a unit cell of 120601 = 071which were obtained by solving the homogenized problem

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 5

D120576

120597D120576

120576

120576x2

x1

y2

y1

Y-cell

(a) (b)

Figure 1 Macroscale (a) and microscale (b) domains used in thehomogenization problemThe solid structure in the119884-cell illustratesthe geometry of our first example in-line array of solid shapes (iecircles or squares)

property known as permeability 119896119894119895 is defined as the spatialaverage of a rescaled pore space velocity119873119894119895(119910) In the follow-ing we present the basic steps that lead to such a definitionand the reader is referred to [11] for more derivation details

The problem is first defined by the macroscale (a)and microscale (b) systems shown in Figure 1 where themacroscale domain is comprised of repeating unit cells ofsize 120576 which is comprised of both fluid and solid spaceDue to periodicity the macroscale domain of porosity 120601 canbe represented by the unit cell with periodic boundariesAssuming slow and steady flow of incompressible fluidthrough the pore space of themedia the governing equationsare given by the Stokes equations as follows

minus 120597120597119909119894119901120576 + 120583 120597120597119909119895120597120597119909119895 V120576119894 = minus120588119892119895 (4)

120597120597119909119894 V120576119894 = 0 (5)

where 119894 = 1 2 3 and 119895 = 1 2 3 and repeated indices implysummation 119901 is pressure 120583 is fluid viscosity V119894 is the porespace fluid velocity 120588 is fluid density and 119892119895 is gravitationalaccelerationThe boundary condition at the fluid-solid inter-face is given by

V119894 = 0 (6)

which comes from the no-flow and no-slip boundary condi-tions Although the slip boundary condition is necessary incertain contexts (such as gas flow and non-Newtonian fluid)the no-slip boundary condition is a valid assumption forviscous flow in porous media and is widely used in modelingstudies of pore-scale flow [12 15 24]

In the Stokes equations 119901120576 and V120576119894 can vary within thepore space and thus are functions of the macroscale (119909) andmicroscale (120576) however 120583 119892 and 120588 are treated as constantsand are thus functions of themacroscale (119909) only By defininga fast variable 119910 = 119909120576 the cell of unit length 120576 is scaledinto the 119884-cell (shown in Figure 1(b)) where 0 le 119910 le 1 A

multiscale expansion is used to write 119901120576 and V120576119894 in terms oftheir leading and higher order terms as follows

119901120576 = 119901(119909 119909120576 ) = 119901(0) (119909 119910) + 120576119901(1) (119909 119910) + sdot sdot sdot (7)

V120576119894 = V119894 (119909 119909120576 )= V(0)119894 (119909 119910) + 120576V(1)119894 (119909 119910) + 1205762V(2)119894 (119909 119910) + sdot sdot sdot

(8)

Upon substitution of the expanded variables into (4)ndash(6)and then the collection of like order terms the second-orderpressure term and the third-order velocity term are

119901(1) (119909 119910) = 119876119895 (119910) (minus120588119892119895 + 120597120597119909119895119901(0) (119909))

V(2) (119909 119910) = 119873119894119895 (119910)120583 (minus120588119892119895 + 120597120597119909119895119901(0) (119909)) (9)

such that 119876119895(119910) and119873119894119895(119910) satisfyminus 120597120597119910119894119876119895 (119910) + 120597120597119910119896

120597120597119910119896119873119894119895 (119910) = 120575119894119895 (10)

120597120597119910119894119873119894119895 (119910) = 0 (11)

with the fluid-solid boundary condition given by

119873119894119895 (119910) = 0 (12)

In the above formulation 119876119895(119910) and 119873119894119895(119910) are functionsof the microscale only and are independent of macroscalequantities 120583 120588 and 119892119895 They can be thought of as rescaledpressure and pore space velocity respectively and (10)-(11)can be thought of as rescaled Stokes equations to be solvedwithin the 119884-cell In (10) 120575119894119895 is the Kronecker delta function(120575119894119895 = 1 for 119894 = 119895 and 120575119894119895 = 0 for 119894 = 119895)

Upon further mathematical steps (collecting of like orderterms taking the spatial average over the cell and applyingthe divergence theorem and the periodicity of the cell)the spatial average of V(2)(119909 119910) in the cell is found to bedivergence free nabla sdot ⟨V(2)(119909 119910)⟩119910 = 0 This implies that themacroscopic flow equation is satisfied byDarcyrsquos equation (1)that is

120597120597119909119894 (119896119894119895120583 ( 120597120597119909119895119901 (119909) minus 120588119892119895)) = 0 997888rarr

120597120597119909119894 119906119894 (119909) = 0(13)

where 119901(119909) is the macroscale pressure and 119906119894(119909) is themacroscale velocity when the full permeability tensor for an119899-dimensional problem is defined as

119896119894119895 fl ⟨119873119894119895 (119910)⟩119910 = 1119884 int119884119873119894119895 (119910) 119889119884 (14)

6 Geofluids

where 119884 is the volume of the 119884-cell (solid and fluid spacecombined)

In this work we use the staggered grid finite differencemethod (ie the marker and cell or MAC scheme [36]) forthe numerical solution of the rescaled Stokes problemdefinedby (10)ndash(12) in the 119884-cell Our numerical implementationusing a matrix-oriented approach is explained in [37] Toillustrate the numerical scheme let us consider the followingStokes problem (in its original form before being rescaledby homogenization) in two-dimensional space find the porespace velocity V = (V119909 V119910)119879 and the pressure 119901 such that

minus120583ΔV + nabla119901 = 119891 in Ω fl (0 119871119909) times (0 119871119910) (15)

nabla sdot V = 0 in Ω (16)

V = 0 on 120597Ω (17)

where the velocity has a unique solution while the pressure isunique up to an additive constant Here 119871119909 and 119871119910 denotethe length of the rectangular domain Ω in the 119909- and 119910-directions respectively

To construct a staggered grid finite difference scheme forthe Stokes problem we introduce a possibly nonuniform gridofΩ as follows

0 = 1199090 lt 1199091 lt sdot sdot sdot lt 119909119872 = 1198711199090 = 1199100 lt 1199101 lt sdot sdot sdot lt 119910119873 = 119871119910 (18)

The grid is comprised of fluid and solid cells (each cell beingfully occupied by one phase) and the computational schemeacts uniquely on the computational cells belonging to thefluid The pressure at the (fluid) cell center (119909119894+12 119910119895+12)is 119901(119909119894+12 119910119895+12) approximated by 119901119894+12119895+12 Similarly weapproximate the 119909-velocity V119909 at each 119909-edge (fluid) centerV119909(119909119894 119910119895+12) by V119909119894119895+12 and the 119910-velocity V119910 at each 119910-edge(fluid) center V119910(119909119894minus12 119910119895) by V119910119894minus12119895

By integrating the119909-component of (15) over cellΩ119894119895+12 =(119909119894minus12 119909119894+12) times (119910119895 119910119895+1) we haveminus [int119910119895+1

119910119895(120597V119909120597119909 (119909119894+12 119910) minus 120597V119909120597119909 (119909119894minus12 119910)) 119889119910

+ int119909119894+12119909119894minus12

(120597V119909120597119910 (119909 119910119895+1) minus 120597V119909120597119910 (119909 119910119895)) 119889119909]+ int119910119895+1

119910119895(119901 (119909119894+12 119910) minus 119901 (119909119894minus12 119910)) 119889119910

= intΩ119894119895+12

119891119909 (119909 119910) 119889119909 119889119910

(19)

We then approximate those edge integrals by midpointvalues and the volume integral by its center value and divideboth sides by |Ω119894119895+12| = (119909119894+12 minus 119909119894minus12)(119910119895+1 minus 119910119895)

By introducing the following forward and backwarddifference operators

Δ+119909V119909119894119895+12 = V119909119894+1119895+12 minus V119909119894119895+12119909119894+1 minus 119909119894 Δminus119909119901119894+12119895+12 = 119901119894+12119895+12 minus 119901119894minus12119895+12119909119894+12 minus 119909119894minus12

(20)

and similar notations for operations in the 119910-direction wecan express the results as

minus Δ+119909V119909119894119895+12 minus Δminus119909V119909119894119895+12119909119894+12 minus 119909119894minus12 minus Δ+119910V119909119894119895+12 minus Δminus119910V119909119894119895+12119910119895+1 minus 119910119895+ Δminus119909119901119894+12119895+12 = 119891119909119894119895+12

(21)

Similarly by integrating the 119910-component of (15) overcell Ω119894minus12119895 = (119909119894minus1 119909119894) times (119910119895minus12 119895119895+12) and then applyingmidpoint quadrature rule we can obtain

minus Δ+119909V119910119894minus12119895 minus Δminus119909V119910119894minus12119895119909119894 minus 119909119894minus1 minus Δ+119910V119910119894minus12119895 minus Δminus119910V119910119894minus12119895119910119895+12 minus 119910119895minus12+ Δminus119910119901119894minus12119895+12 = 119891119910119894minus12119895

(22)

By integrating (16) over cell Ω119894minus12119895+12 = (119909119894 minus 119909119894minus1) times(119910119895 119895119895+1) we obtain its discrete form

Δminus119909V119909119894119895+12 + Δ+119910V119910119894minus12119895 = 0 (23)

Eqns (21)ndash(23) form a linear equation system for thepressure and velocity and the equation system defines theMAC scheme for the Stokes problem of (15)ndash(17) Note thatthe no-flow boundary condition supplies the zero velocitycomponent that acts normal to any fluid-solid interface andperiodic conditions are applied along the domainrsquos externalboundaries

We apply this numerical scheme to solve the set ofrescaled Stokes equations in Eqns (10)ndash(12) Our computer-generatedmedia samples are 2-dimensional (2D) thus the setis solved with different forcings 120575119894119895 where 119894 = 1 2 and 119895 = 1 2Upon obtaining the rescaled pore space velocity 119873119894119895(119910) inthe 119884-cell we then compute the full permeability tensor by(14) If our computer-generated media are characterized by anondiagonal tensor we determine the principal componentsof the permeability tensor by diagonalization [11 38] Thenotation used in our results section is k and klowast for thepermeability tensors in coordinate systems x and xlowastrespectively that is

k = [11989611 1198961211989621 11989622] in x 997904rArr

klowast = [119896max 00 119896119905] in xlowast

(24)

32 Hydraulic Tortuosity The approach commonly used inliterature to compute hydraulic tortuosity is presented inKoponen et al [19 26] and further validation of this ap-proach is given in Duda et al [13] In this approach the

Geofluids 7

hydraulic tortuosity 120591ℎ or a particular direction is computedbased on the pore space fluid velocities driven by a force Forexample the hydraulic tortuosity of flow that is driven in the119909 direction is

120591ℎ119909 = ⟨Vmag⟩Ω⟨V119909⟩Ω (25)

where ⟨sdot⟩Ω is the spatial average in the unit cell domain ΩVmag is the magnitude of the fluid velocities (ie the fluidspeed) and V119909 is the velocity component in the 119909 directionThe spatial average of V119909 inΩ is

⟨V119909⟩Ω = 1|119881| int119881119891 V119891119909119889119881119891 = 1|119881|sum119894119895 V

119891119909 (119894 119895) Δ119881119891 (26)

where119881 is the volume ofΩ (pore and solid space combined)119881119891 is the fluid space and the integral is taken over the fluidspace only The spatial average for Vmag is taken in the samewayThe tortuosity in the 119910 direction is computed in a similarmanner as follows

120591ℎ119910 = ⟨Vmag⟩Ω⟨V119910⟩Ω (27)

where the velocity is the solution for flow driven in the119910 direction We note that (25) and (27) correspond to thehydraulic tortuosity as we have defined it in Section 22 thatis 120591ℎ = 119871119890119871 119904 If the term 1205912ℎ which appears in the Kozeny-Carmen equation is labelled as the hydraulic tortuosity then(25) and (27) should be squared

While this method has been used in many works (ie[26 33 39 40]) other approaches to compute hydraulictortuosity have been proposed For example Matyka et al[24] took measurements of streamlines that passed through agiven cross-sectional area in the domain with a constant flux(ie not all streamlines in the domain would be measured)The aim of this approach is to capture the hydraulic tortu-osity of the main conducting channels in a pore geometryAlso Ahmadi et al [32] formulated analytical functionsfor hydraulic tortuosity based on volume averaging of massbalance equations

In our work we follow Koponen et alrsquos [19 26] approachas given by (25)ndash(27) however we do not compute the porespace velocity fields V119909 and V119910 explicitly Instead during thepermeability computation for our 2D pore geometries weobtain the rescaled pore space velocity fields 119873119894119895 in the 119884-cell where 119894 = 1 2 and 119895 = 1 2 It can be seen that 11987311and 11987321 are related to V119909 and V119910 respectively for flow thatis driven by g = ( 0) and that 11987312 and 11987322 are relatedto V119909 and V119910 respectively for flow that is driven by g =(0 ) Thus it is mathematically equivalent to compute thehydraulic tortuosities by

120591ℎ119909 = ⟨radic11987311 + 11987321⟩Ω⟨11987311⟩Ω 120591ℎ119910 = ⟨radic11987312 + 11987322⟩Ω⟨11987322⟩Ω

(28)

where Ω refers to the domain of the 119884-cell and the spatialaverages of 119873119894119895 are computed in the same way as shown by(26) Since the numerical scheme computes 119873119894119895 on the fluidedge-centers we compute the fluid cell-center quantities byaveraging across the grid cells where 119873119894119895 = 0 on the fluid-solid interfaces as per the boundary condition in (12)

As mentioned we diagonalize the permeability tensorto obtain the principal permeability components and theprincipal directions 120579 and 120579+1205872 fl Thus in order tomakean appropriate comparison between direction-specific per-meability and hydraulic tortuosity we compute the hydraulictortuosity in these principal directions For example 120591ℎ in theprincipal direction 120579 is

120591ℎ120579 = ⟨1003816100381610038161003816100381611987312057911989410038161003816100381610038161003816⟩Ω⟨119873120579⟩Ω (29)

where |119873120579119894 | = radic(119873120579

1 )2 + (1198731205792 )2 and where119873120579

1 1198731205792 and119873120579 are

obtained by vector transformations from 119873119894119895 120591ℎ in the otherprincipal direction is similarly obtained The notation weuse in our results section is 120591ℎmax = 120591ℎ120579 and 120591ℎ119905 = 120591ℎ33 Effective Diffusion Coefficient and Diffusive TortuosityThe problem is similarly defined by the two scales illustratedin Figure 1 Within the pore space the diffusive and convec-tive transport of a species of concentration 119888 is given by

120597120597119905119888120576 + 1120576 V120576119894 120597120597119909119894 119888120576 =120597120597119909119894119863119894119895

120597120597119909119895 119888120576119899119894119863119894119895

120597120597119909119895 119888120576 = 0V120576119894 = 0

(30)

where V120576119894 is divergence free (5) 120576 implies the variables arefunctions of the microscale and 119863119894119895 is the molecular diffu-sion coefficient The last two equations give the boundaryconditions on the fluid-solid interface Using similar homog-enization steps as presented in Section 31 the multiscaleexpansion

119888120576 = 119888 (119909 119909120576 )= 119888(0) (119909 119910) + 120576119888(1) (119909 119910) + 1205762119888(2) (119909 119910) + sdot sdot sdot

(31)

and (8) are substituted into the governing transport equations(30) Upon collecting like order terms the second-order termof 119888 is

119888(1) (119909 119910) = 119875119896 (119909 119910) 120597120597119909119896 119888(0) (119909) (32)

where 119888(0)(119909) is the macroscale concentration and 119875119896(119909 119910)satisfies 120597120597119910119894

120597120597119910119894119875119896 (119909 119910) = 0 (33)

119899119894 120597120597119910119894119875119896 (119909 119910) = minus119899119896 (34)

8 Geofluids

where (34) gives the boundary condition and 119899119894 is a unitvector normal to the fluid-solid interface (positive in thedirection away from the fluid space) Eqns (33) and (34)correspond to the assumption of an isotropic moleculardiffusion coefficient that is119863119894119895 = 119863120575119894119895 Upon further algebraand averaging over the 119884-cell the macroscale transportequation is obtained as follows

120601 120597120597119905119888 (119909) = 120597120597119909119894119863efflowast119894119896

120597120597119909119896 119888 (119909) (35)

where the effective diffusion coefficient119863efflowast119894119896 is given by

119863efflowast119894119896119863 = 1119884 int

119884(120575119894119896 + 120597120597119910119894119875119896 (119909 119910)) 119889119884 (36)

Intuitively the more tortuous the porous media the slowerthe rate of effective diffusion thus a commonly used relation-ship between effective and molecular diffusivity is

119863efflowast119894119896 = 119863120601120591119889119894119896 (37)

where 120591119889119894119896 is the diffusive tortuosity [41]The above approach to compute the effective diffusion

coefficient and the diffusive tortuosity has been used inseveral previous studies which formulate (33)ndash(36) using themethod of volume averaging [1 42ndash45] or by the method ofhomogenization [27] Other approaches to compute diffusivetortuosity include simulation of the diffusion process usinga random walk [25] or numerically solving the microscalediffusion equation using lattice Boltzmann modeling [46]Typically the effective diffusivity is computed and then usedto calculate diffusive tortuosity by the inverse relationshipshown in (37)

By (36) and (37) diffusive tortuosity is computed by

120591119889119894119896 = (119868119894119896 + 11003816100381610038161003816100381611988411989110038161003816100381610038161003816 int119884119891 (120597120597119910119894119875119896 (119909 119910)) 119889119910119891)

minus1 (38)

where 119884119891 = 120601119884 and where 119875119896(119909 119910) is the solution to(33) with the boundary condition in (34) This is the sameformulation that was used in Sun et al [27] by the method ofhomogenization and in Kim et al [43] and Valdes-Parada etal [1] by the method of volume averaging

The full diffusive tortuosity can also be diagonalized usingthe same procedure outlined in Section 31 to obtain thecomponents 120591119889max and 120591119889119905 corresponding to the principaldirections The rotation required to obtain the principalcomponents of klowast is not necessarily identical to the rotationrequired to obtain the principal components of 120591lowast119889 as willbe shown by one of the following pore geometry examples(randomly distributed squares)

4 Results

While real geological formations are typically of low porosity(120601 lt 03) we have generated several idealized geometrieswithin the porosity range of 033 lt 120601 lt 1 These idealized

Simulated result (Stokes) circleAnalytical result (Gebart eqn)

10minus4

10minus3

10minus2

10minus1

100

101

102

Nor

mal

ized

per

mea

bilit

y

03 04 05 06 07 08 09 102

Porosity

Figure 2 Permeability-porosity trend for in-line array of circles(ie infinitely long cylinders) Permeability units are nondimen-sional (ie unit length times unit length) The permeability is isotropicand is normalized by 1199032 where 119903 is the solid circle radius Gebartrsquos[47] analytical solution was given in (A2)

geometries and porosities have been used in literature (seeTable 1) and thus provide ground for comparison and oppor-tunity to make useful comparisons of the effective propertiesin high porosity representations of porous media

Permeability hydraulic tortuosity anddiffusive tortuosityare computed for each pore geometry using the methodsoutlined in Sections 31 32 and 33 respectively Each poregeometry is generated by defining certain input parameters(ie radius of solid circle side length of square and den-sity of randomly distributed squares) and these inputs aresummarized in Table 4 By using a range of values for theinput parameters we obtain unit cells which we treat asrepresentative elementary volumes over a range of porositiesAfter computing the permeability and tortuosity we plotthe data to observe permeability-porosity and tortuosity-porosity trends

41 In-Line Array of Uniform Shapes The unit cell or 119884-cellin Figure 1(b) illustrates our first type of generated poregeometry and the shape in the center of the cell is eithera circle or square A range of porosities of the unit cell areobtained by changing the radius of the solid circle or thelength of the solid square centered in the cell

411 Permeability and Hydraulic Tortuosity The full perme-ability tensor was computed for 033 lt 120601 lt 1 and resultsare plotted in Figure 2 Following [14] we computed andplotted the analytical solution from Gebart [47] which isexplained in Appendix A Permeability is isotropic (ie 119896119909119909 =119896119910119910) for this pore geometry The range of our simulateddata fits well against the analytical solution We computed

Geofluids 9

Table 4 Input parameters used to generate different pore-geometries (or different 119884-cells) and their corresponding porosity ranges In thistable cell refers to a cell in the mesh used to discretize the 119884-cellGeometry Initial mesh dimensions Inputs Porosity rangeIn-line array of shapes(see Figure 1) 50 times 50 cells (119884-cell length 1 unit) (a) Circle diameter 44 to 8 cells

(b) Square length 48 to 2 cells033 lt 120601 lt 1007 lt 120601 lt 1

Staggered array ofsquares (see Figure 7)

100 times 50 cells (119884-cell length 119871119887 = 05units 119886119887 = 1 119871119886119871119887 = 1 as per Figure 8) Square length 40 to 8 cells 036 le 120601 lt 1

Randomly distributedsquares (see Figure 14) 100 times 100 cells (119884-cell length 1 unit) Increasing density of squares

(with constant square length 001unit = 1 cell)

045 le 120601 lt 1

0005

001

0015

002

0025

003

0035

Figure 3 Fluid flow past a solid circle 120601 = 071 driven by 119892119909 Non-dimensional fluid speed (ie magnitude of velocities) is indicatedby color and flow path is indicated by flow lines (direction is left toright) Speed profile and flow path driven by 119892119910 are same as shownbut rotated 90 degrees

our permeability data on increasingly finer meshes until thepermeability converged within a 1 relative error To reachthis convergence criteria after starting with a mesh size of50 times 50 the lowest porosity cell required a refined mesh of500 times 500 and the highest porosity cell required 300 times 300The shape of the circular edge was refined at each meshrefinement in order to represent the circle as realistically aspossible and to minimize artifacts caused by discretization

We obtained the hydraulic tortuosity in the 119909 and 119910directions for porosities within the range 033 lt 120601 lt 1 withthe same convergence criteria used to obtain permeabilityFigure 3 illustrates the fluid speed and direction for a unitcell of 120601 = 071 and the hydraulic tortuosity in both 119909 and119910 directions were computed to be 120591ℎ119909 = 120591ℎ119910 = 10185 Thefluid speed is highest in the pore space that is uninterruptedby the solid and the flow lines indicate the direction is parallelto the macroscopic flow The flow lines diverge in the regionthat is interrupted by the solid however the fluid speed isessentially zero in that pore space So while these flow lines

120601 = 033

120601 = 039

120601 = 050

120601 = 059

120601 = 072

120601 = 087

120601 = 098

1

12

14

16

18

2

22

Com

pute

d di

ffusiv

e 120591

1 2 3 4 50

Number of cells times105

Figure 4Number of grid cells required to reach converged 120591119889 (usinga convergence criteria of 01 relative error) for select porosities ofin-line array of circles

appear to illustrate a tortuous pathway through this geometrythe tortuosity in either direction is in fact calculated to beasymp1 because the highest speeds are acting along a nontortuouspathway In the pore space that is not restricted or interruptedin the direction parallel to the driving force the fluid velocityis able to develop into the parabolic profile that characterizeschannel flow

412 Diffusive Tortuosity We obtained the full diffusive tor-tuosity tensor on increasingly refined meshes until the tortu-osity converged within a 01 relative error Starting with amesh of 50 times 50 the lowest porosity required a refined meshof 700 times 700 while the highest porosity required 250 times 250 tomeet the convergence criteria Solid edges were also refinedwith mesh refinement as mentioned previously In Figure 4we plot the convergence behavior of 120591119889 for a select numberof porosity cases in order to illustrate that each porosity caseconvergences with similar behavior Figure 5(a) illustrates thefields of 119875119896 (119896 = 1 2 or 119909 119910) for a unit cell of 120601 = 071which were obtained by solving the homogenized problem

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

6 Geofluids

where 119884 is the volume of the 119884-cell (solid and fluid spacecombined)

In this work we use the staggered grid finite differencemethod (ie the marker and cell or MAC scheme [36]) forthe numerical solution of the rescaled Stokes problemdefinedby (10)ndash(12) in the 119884-cell Our numerical implementationusing a matrix-oriented approach is explained in [37] Toillustrate the numerical scheme let us consider the followingStokes problem (in its original form before being rescaledby homogenization) in two-dimensional space find the porespace velocity V = (V119909 V119910)119879 and the pressure 119901 such that

minus120583ΔV + nabla119901 = 119891 in Ω fl (0 119871119909) times (0 119871119910) (15)

nabla sdot V = 0 in Ω (16)

V = 0 on 120597Ω (17)

where the velocity has a unique solution while the pressure isunique up to an additive constant Here 119871119909 and 119871119910 denotethe length of the rectangular domain Ω in the 119909- and 119910-directions respectively

To construct a staggered grid finite difference scheme forthe Stokes problem we introduce a possibly nonuniform gridofΩ as follows

0 = 1199090 lt 1199091 lt sdot sdot sdot lt 119909119872 = 1198711199090 = 1199100 lt 1199101 lt sdot sdot sdot lt 119910119873 = 119871119910 (18)

The grid is comprised of fluid and solid cells (each cell beingfully occupied by one phase) and the computational schemeacts uniquely on the computational cells belonging to thefluid The pressure at the (fluid) cell center (119909119894+12 119910119895+12)is 119901(119909119894+12 119910119895+12) approximated by 119901119894+12119895+12 Similarly weapproximate the 119909-velocity V119909 at each 119909-edge (fluid) centerV119909(119909119894 119910119895+12) by V119909119894119895+12 and the 119910-velocity V119910 at each 119910-edge(fluid) center V119910(119909119894minus12 119910119895) by V119910119894minus12119895

By integrating the119909-component of (15) over cellΩ119894119895+12 =(119909119894minus12 119909119894+12) times (119910119895 119910119895+1) we haveminus [int119910119895+1

119910119895(120597V119909120597119909 (119909119894+12 119910) minus 120597V119909120597119909 (119909119894minus12 119910)) 119889119910

+ int119909119894+12119909119894minus12

(120597V119909120597119910 (119909 119910119895+1) minus 120597V119909120597119910 (119909 119910119895)) 119889119909]+ int119910119895+1

119910119895(119901 (119909119894+12 119910) minus 119901 (119909119894minus12 119910)) 119889119910

= intΩ119894119895+12

119891119909 (119909 119910) 119889119909 119889119910

(19)

We then approximate those edge integrals by midpointvalues and the volume integral by its center value and divideboth sides by |Ω119894119895+12| = (119909119894+12 minus 119909119894minus12)(119910119895+1 minus 119910119895)

By introducing the following forward and backwarddifference operators

Δ+119909V119909119894119895+12 = V119909119894+1119895+12 minus V119909119894119895+12119909119894+1 minus 119909119894 Δminus119909119901119894+12119895+12 = 119901119894+12119895+12 minus 119901119894minus12119895+12119909119894+12 minus 119909119894minus12

(20)

and similar notations for operations in the 119910-direction wecan express the results as

minus Δ+119909V119909119894119895+12 minus Δminus119909V119909119894119895+12119909119894+12 minus 119909119894minus12 minus Δ+119910V119909119894119895+12 minus Δminus119910V119909119894119895+12119910119895+1 minus 119910119895+ Δminus119909119901119894+12119895+12 = 119891119909119894119895+12

(21)

Similarly by integrating the 119910-component of (15) overcell Ω119894minus12119895 = (119909119894minus1 119909119894) times (119910119895minus12 119895119895+12) and then applyingmidpoint quadrature rule we can obtain

minus Δ+119909V119910119894minus12119895 minus Δminus119909V119910119894minus12119895119909119894 minus 119909119894minus1 minus Δ+119910V119910119894minus12119895 minus Δminus119910V119910119894minus12119895119910119895+12 minus 119910119895minus12+ Δminus119910119901119894minus12119895+12 = 119891119910119894minus12119895

(22)

By integrating (16) over cell Ω119894minus12119895+12 = (119909119894 minus 119909119894minus1) times(119910119895 119895119895+1) we obtain its discrete form

Δminus119909V119909119894119895+12 + Δ+119910V119910119894minus12119895 = 0 (23)

Eqns (21)ndash(23) form a linear equation system for thepressure and velocity and the equation system defines theMAC scheme for the Stokes problem of (15)ndash(17) Note thatthe no-flow boundary condition supplies the zero velocitycomponent that acts normal to any fluid-solid interface andperiodic conditions are applied along the domainrsquos externalboundaries

We apply this numerical scheme to solve the set ofrescaled Stokes equations in Eqns (10)ndash(12) Our computer-generatedmedia samples are 2-dimensional (2D) thus the setis solved with different forcings 120575119894119895 where 119894 = 1 2 and 119895 = 1 2Upon obtaining the rescaled pore space velocity 119873119894119895(119910) inthe 119884-cell we then compute the full permeability tensor by(14) If our computer-generated media are characterized by anondiagonal tensor we determine the principal componentsof the permeability tensor by diagonalization [11 38] Thenotation used in our results section is k and klowast for thepermeability tensors in coordinate systems x and xlowastrespectively that is

k = [11989611 1198961211989621 11989622] in x 997904rArr

klowast = [119896max 00 119896119905] in xlowast

(24)

32 Hydraulic Tortuosity The approach commonly used inliterature to compute hydraulic tortuosity is presented inKoponen et al [19 26] and further validation of this ap-proach is given in Duda et al [13] In this approach the

Geofluids 7

hydraulic tortuosity 120591ℎ or a particular direction is computedbased on the pore space fluid velocities driven by a force Forexample the hydraulic tortuosity of flow that is driven in the119909 direction is

120591ℎ119909 = ⟨Vmag⟩Ω⟨V119909⟩Ω (25)

where ⟨sdot⟩Ω is the spatial average in the unit cell domain ΩVmag is the magnitude of the fluid velocities (ie the fluidspeed) and V119909 is the velocity component in the 119909 directionThe spatial average of V119909 inΩ is

⟨V119909⟩Ω = 1|119881| int119881119891 V119891119909119889119881119891 = 1|119881|sum119894119895 V

119891119909 (119894 119895) Δ119881119891 (26)

where119881 is the volume ofΩ (pore and solid space combined)119881119891 is the fluid space and the integral is taken over the fluidspace only The spatial average for Vmag is taken in the samewayThe tortuosity in the 119910 direction is computed in a similarmanner as follows

120591ℎ119910 = ⟨Vmag⟩Ω⟨V119910⟩Ω (27)

where the velocity is the solution for flow driven in the119910 direction We note that (25) and (27) correspond to thehydraulic tortuosity as we have defined it in Section 22 thatis 120591ℎ = 119871119890119871 119904 If the term 1205912ℎ which appears in the Kozeny-Carmen equation is labelled as the hydraulic tortuosity then(25) and (27) should be squared

While this method has been used in many works (ie[26 33 39 40]) other approaches to compute hydraulictortuosity have been proposed For example Matyka et al[24] took measurements of streamlines that passed through agiven cross-sectional area in the domain with a constant flux(ie not all streamlines in the domain would be measured)The aim of this approach is to capture the hydraulic tortu-osity of the main conducting channels in a pore geometryAlso Ahmadi et al [32] formulated analytical functionsfor hydraulic tortuosity based on volume averaging of massbalance equations

In our work we follow Koponen et alrsquos [19 26] approachas given by (25)ndash(27) however we do not compute the porespace velocity fields V119909 and V119910 explicitly Instead during thepermeability computation for our 2D pore geometries weobtain the rescaled pore space velocity fields 119873119894119895 in the 119884-cell where 119894 = 1 2 and 119895 = 1 2 It can be seen that 11987311and 11987321 are related to V119909 and V119910 respectively for flow thatis driven by g = ( 0) and that 11987312 and 11987322 are relatedto V119909 and V119910 respectively for flow that is driven by g =(0 ) Thus it is mathematically equivalent to compute thehydraulic tortuosities by

120591ℎ119909 = ⟨radic11987311 + 11987321⟩Ω⟨11987311⟩Ω 120591ℎ119910 = ⟨radic11987312 + 11987322⟩Ω⟨11987322⟩Ω

(28)

where Ω refers to the domain of the 119884-cell and the spatialaverages of 119873119894119895 are computed in the same way as shown by(26) Since the numerical scheme computes 119873119894119895 on the fluidedge-centers we compute the fluid cell-center quantities byaveraging across the grid cells where 119873119894119895 = 0 on the fluid-solid interfaces as per the boundary condition in (12)

As mentioned we diagonalize the permeability tensorto obtain the principal permeability components and theprincipal directions 120579 and 120579+1205872 fl Thus in order tomakean appropriate comparison between direction-specific per-meability and hydraulic tortuosity we compute the hydraulictortuosity in these principal directions For example 120591ℎ in theprincipal direction 120579 is

120591ℎ120579 = ⟨1003816100381610038161003816100381611987312057911989410038161003816100381610038161003816⟩Ω⟨119873120579⟩Ω (29)

where |119873120579119894 | = radic(119873120579

1 )2 + (1198731205792 )2 and where119873120579

1 1198731205792 and119873120579 are

obtained by vector transformations from 119873119894119895 120591ℎ in the otherprincipal direction is similarly obtained The notation weuse in our results section is 120591ℎmax = 120591ℎ120579 and 120591ℎ119905 = 120591ℎ33 Effective Diffusion Coefficient and Diffusive TortuosityThe problem is similarly defined by the two scales illustratedin Figure 1 Within the pore space the diffusive and convec-tive transport of a species of concentration 119888 is given by

120597120597119905119888120576 + 1120576 V120576119894 120597120597119909119894 119888120576 =120597120597119909119894119863119894119895

120597120597119909119895 119888120576119899119894119863119894119895

120597120597119909119895 119888120576 = 0V120576119894 = 0

(30)

where V120576119894 is divergence free (5) 120576 implies the variables arefunctions of the microscale and 119863119894119895 is the molecular diffu-sion coefficient The last two equations give the boundaryconditions on the fluid-solid interface Using similar homog-enization steps as presented in Section 31 the multiscaleexpansion

119888120576 = 119888 (119909 119909120576 )= 119888(0) (119909 119910) + 120576119888(1) (119909 119910) + 1205762119888(2) (119909 119910) + sdot sdot sdot

(31)

and (8) are substituted into the governing transport equations(30) Upon collecting like order terms the second-order termof 119888 is

119888(1) (119909 119910) = 119875119896 (119909 119910) 120597120597119909119896 119888(0) (119909) (32)

where 119888(0)(119909) is the macroscale concentration and 119875119896(119909 119910)satisfies 120597120597119910119894

120597120597119910119894119875119896 (119909 119910) = 0 (33)

119899119894 120597120597119910119894119875119896 (119909 119910) = minus119899119896 (34)

8 Geofluids

where (34) gives the boundary condition and 119899119894 is a unitvector normal to the fluid-solid interface (positive in thedirection away from the fluid space) Eqns (33) and (34)correspond to the assumption of an isotropic moleculardiffusion coefficient that is119863119894119895 = 119863120575119894119895 Upon further algebraand averaging over the 119884-cell the macroscale transportequation is obtained as follows

120601 120597120597119905119888 (119909) = 120597120597119909119894119863efflowast119894119896

120597120597119909119896 119888 (119909) (35)

where the effective diffusion coefficient119863efflowast119894119896 is given by

119863efflowast119894119896119863 = 1119884 int

119884(120575119894119896 + 120597120597119910119894119875119896 (119909 119910)) 119889119884 (36)

Intuitively the more tortuous the porous media the slowerthe rate of effective diffusion thus a commonly used relation-ship between effective and molecular diffusivity is

119863efflowast119894119896 = 119863120601120591119889119894119896 (37)

where 120591119889119894119896 is the diffusive tortuosity [41]The above approach to compute the effective diffusion

coefficient and the diffusive tortuosity has been used inseveral previous studies which formulate (33)ndash(36) using themethod of volume averaging [1 42ndash45] or by the method ofhomogenization [27] Other approaches to compute diffusivetortuosity include simulation of the diffusion process usinga random walk [25] or numerically solving the microscalediffusion equation using lattice Boltzmann modeling [46]Typically the effective diffusivity is computed and then usedto calculate diffusive tortuosity by the inverse relationshipshown in (37)

By (36) and (37) diffusive tortuosity is computed by

120591119889119894119896 = (119868119894119896 + 11003816100381610038161003816100381611988411989110038161003816100381610038161003816 int119884119891 (120597120597119910119894119875119896 (119909 119910)) 119889119910119891)

minus1 (38)

where 119884119891 = 120601119884 and where 119875119896(119909 119910) is the solution to(33) with the boundary condition in (34) This is the sameformulation that was used in Sun et al [27] by the method ofhomogenization and in Kim et al [43] and Valdes-Parada etal [1] by the method of volume averaging

The full diffusive tortuosity can also be diagonalized usingthe same procedure outlined in Section 31 to obtain thecomponents 120591119889max and 120591119889119905 corresponding to the principaldirections The rotation required to obtain the principalcomponents of klowast is not necessarily identical to the rotationrequired to obtain the principal components of 120591lowast119889 as willbe shown by one of the following pore geometry examples(randomly distributed squares)

4 Results

While real geological formations are typically of low porosity(120601 lt 03) we have generated several idealized geometrieswithin the porosity range of 033 lt 120601 lt 1 These idealized

Simulated result (Stokes) circleAnalytical result (Gebart eqn)

10minus4

10minus3

10minus2

10minus1

100

101

102

Nor

mal

ized

per

mea

bilit

y

03 04 05 06 07 08 09 102

Porosity

Figure 2 Permeability-porosity trend for in-line array of circles(ie infinitely long cylinders) Permeability units are nondimen-sional (ie unit length times unit length) The permeability is isotropicand is normalized by 1199032 where 119903 is the solid circle radius Gebartrsquos[47] analytical solution was given in (A2)

geometries and porosities have been used in literature (seeTable 1) and thus provide ground for comparison and oppor-tunity to make useful comparisons of the effective propertiesin high porosity representations of porous media

Permeability hydraulic tortuosity anddiffusive tortuosityare computed for each pore geometry using the methodsoutlined in Sections 31 32 and 33 respectively Each poregeometry is generated by defining certain input parameters(ie radius of solid circle side length of square and den-sity of randomly distributed squares) and these inputs aresummarized in Table 4 By using a range of values for theinput parameters we obtain unit cells which we treat asrepresentative elementary volumes over a range of porositiesAfter computing the permeability and tortuosity we plotthe data to observe permeability-porosity and tortuosity-porosity trends

41 In-Line Array of Uniform Shapes The unit cell or 119884-cellin Figure 1(b) illustrates our first type of generated poregeometry and the shape in the center of the cell is eithera circle or square A range of porosities of the unit cell areobtained by changing the radius of the solid circle or thelength of the solid square centered in the cell

411 Permeability and Hydraulic Tortuosity The full perme-ability tensor was computed for 033 lt 120601 lt 1 and resultsare plotted in Figure 2 Following [14] we computed andplotted the analytical solution from Gebart [47] which isexplained in Appendix A Permeability is isotropic (ie 119896119909119909 =119896119910119910) for this pore geometry The range of our simulateddata fits well against the analytical solution We computed

Geofluids 9

Table 4 Input parameters used to generate different pore-geometries (or different 119884-cells) and their corresponding porosity ranges In thistable cell refers to a cell in the mesh used to discretize the 119884-cellGeometry Initial mesh dimensions Inputs Porosity rangeIn-line array of shapes(see Figure 1) 50 times 50 cells (119884-cell length 1 unit) (a) Circle diameter 44 to 8 cells

(b) Square length 48 to 2 cells033 lt 120601 lt 1007 lt 120601 lt 1

Staggered array ofsquares (see Figure 7)

100 times 50 cells (119884-cell length 119871119887 = 05units 119886119887 = 1 119871119886119871119887 = 1 as per Figure 8) Square length 40 to 8 cells 036 le 120601 lt 1

Randomly distributedsquares (see Figure 14) 100 times 100 cells (119884-cell length 1 unit) Increasing density of squares

(with constant square length 001unit = 1 cell)

045 le 120601 lt 1

0005

001

0015

002

0025

003

0035

Figure 3 Fluid flow past a solid circle 120601 = 071 driven by 119892119909 Non-dimensional fluid speed (ie magnitude of velocities) is indicatedby color and flow path is indicated by flow lines (direction is left toright) Speed profile and flow path driven by 119892119910 are same as shownbut rotated 90 degrees

our permeability data on increasingly finer meshes until thepermeability converged within a 1 relative error To reachthis convergence criteria after starting with a mesh size of50 times 50 the lowest porosity cell required a refined mesh of500 times 500 and the highest porosity cell required 300 times 300The shape of the circular edge was refined at each meshrefinement in order to represent the circle as realistically aspossible and to minimize artifacts caused by discretization

We obtained the hydraulic tortuosity in the 119909 and 119910directions for porosities within the range 033 lt 120601 lt 1 withthe same convergence criteria used to obtain permeabilityFigure 3 illustrates the fluid speed and direction for a unitcell of 120601 = 071 and the hydraulic tortuosity in both 119909 and119910 directions were computed to be 120591ℎ119909 = 120591ℎ119910 = 10185 Thefluid speed is highest in the pore space that is uninterruptedby the solid and the flow lines indicate the direction is parallelto the macroscopic flow The flow lines diverge in the regionthat is interrupted by the solid however the fluid speed isessentially zero in that pore space So while these flow lines

120601 = 033

120601 = 039

120601 = 050

120601 = 059

120601 = 072

120601 = 087

120601 = 098

1

12

14

16

18

2

22

Com

pute

d di

ffusiv

e 120591

1 2 3 4 50

Number of cells times105

Figure 4Number of grid cells required to reach converged 120591119889 (usinga convergence criteria of 01 relative error) for select porosities ofin-line array of circles

appear to illustrate a tortuous pathway through this geometrythe tortuosity in either direction is in fact calculated to beasymp1 because the highest speeds are acting along a nontortuouspathway In the pore space that is not restricted or interruptedin the direction parallel to the driving force the fluid velocityis able to develop into the parabolic profile that characterizeschannel flow

412 Diffusive Tortuosity We obtained the full diffusive tor-tuosity tensor on increasingly refined meshes until the tortu-osity converged within a 01 relative error Starting with amesh of 50 times 50 the lowest porosity required a refined meshof 700 times 700 while the highest porosity required 250 times 250 tomeet the convergence criteria Solid edges were also refinedwith mesh refinement as mentioned previously In Figure 4we plot the convergence behavior of 120591119889 for a select numberof porosity cases in order to illustrate that each porosity caseconvergences with similar behavior Figure 5(a) illustrates thefields of 119875119896 (119896 = 1 2 or 119909 119910) for a unit cell of 120601 = 071which were obtained by solving the homogenized problem

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 7

hydraulic tortuosity 120591ℎ or a particular direction is computedbased on the pore space fluid velocities driven by a force Forexample the hydraulic tortuosity of flow that is driven in the119909 direction is

120591ℎ119909 = ⟨Vmag⟩Ω⟨V119909⟩Ω (25)

where ⟨sdot⟩Ω is the spatial average in the unit cell domain ΩVmag is the magnitude of the fluid velocities (ie the fluidspeed) and V119909 is the velocity component in the 119909 directionThe spatial average of V119909 inΩ is

⟨V119909⟩Ω = 1|119881| int119881119891 V119891119909119889119881119891 = 1|119881|sum119894119895 V

119891119909 (119894 119895) Δ119881119891 (26)

where119881 is the volume ofΩ (pore and solid space combined)119881119891 is the fluid space and the integral is taken over the fluidspace only The spatial average for Vmag is taken in the samewayThe tortuosity in the 119910 direction is computed in a similarmanner as follows

120591ℎ119910 = ⟨Vmag⟩Ω⟨V119910⟩Ω (27)

where the velocity is the solution for flow driven in the119910 direction We note that (25) and (27) correspond to thehydraulic tortuosity as we have defined it in Section 22 thatis 120591ℎ = 119871119890119871 119904 If the term 1205912ℎ which appears in the Kozeny-Carmen equation is labelled as the hydraulic tortuosity then(25) and (27) should be squared

While this method has been used in many works (ie[26 33 39 40]) other approaches to compute hydraulictortuosity have been proposed For example Matyka et al[24] took measurements of streamlines that passed through agiven cross-sectional area in the domain with a constant flux(ie not all streamlines in the domain would be measured)The aim of this approach is to capture the hydraulic tortu-osity of the main conducting channels in a pore geometryAlso Ahmadi et al [32] formulated analytical functionsfor hydraulic tortuosity based on volume averaging of massbalance equations

In our work we follow Koponen et alrsquos [19 26] approachas given by (25)ndash(27) however we do not compute the porespace velocity fields V119909 and V119910 explicitly Instead during thepermeability computation for our 2D pore geometries weobtain the rescaled pore space velocity fields 119873119894119895 in the 119884-cell where 119894 = 1 2 and 119895 = 1 2 It can be seen that 11987311and 11987321 are related to V119909 and V119910 respectively for flow thatis driven by g = ( 0) and that 11987312 and 11987322 are relatedto V119909 and V119910 respectively for flow that is driven by g =(0 ) Thus it is mathematically equivalent to compute thehydraulic tortuosities by

120591ℎ119909 = ⟨radic11987311 + 11987321⟩Ω⟨11987311⟩Ω 120591ℎ119910 = ⟨radic11987312 + 11987322⟩Ω⟨11987322⟩Ω

(28)

where Ω refers to the domain of the 119884-cell and the spatialaverages of 119873119894119895 are computed in the same way as shown by(26) Since the numerical scheme computes 119873119894119895 on the fluidedge-centers we compute the fluid cell-center quantities byaveraging across the grid cells where 119873119894119895 = 0 on the fluid-solid interfaces as per the boundary condition in (12)

As mentioned we diagonalize the permeability tensorto obtain the principal permeability components and theprincipal directions 120579 and 120579+1205872 fl Thus in order tomakean appropriate comparison between direction-specific per-meability and hydraulic tortuosity we compute the hydraulictortuosity in these principal directions For example 120591ℎ in theprincipal direction 120579 is

120591ℎ120579 = ⟨1003816100381610038161003816100381611987312057911989410038161003816100381610038161003816⟩Ω⟨119873120579⟩Ω (29)

where |119873120579119894 | = radic(119873120579

1 )2 + (1198731205792 )2 and where119873120579

1 1198731205792 and119873120579 are

obtained by vector transformations from 119873119894119895 120591ℎ in the otherprincipal direction is similarly obtained The notation weuse in our results section is 120591ℎmax = 120591ℎ120579 and 120591ℎ119905 = 120591ℎ33 Effective Diffusion Coefficient and Diffusive TortuosityThe problem is similarly defined by the two scales illustratedin Figure 1 Within the pore space the diffusive and convec-tive transport of a species of concentration 119888 is given by

120597120597119905119888120576 + 1120576 V120576119894 120597120597119909119894 119888120576 =120597120597119909119894119863119894119895

120597120597119909119895 119888120576119899119894119863119894119895

120597120597119909119895 119888120576 = 0V120576119894 = 0

(30)

where V120576119894 is divergence free (5) 120576 implies the variables arefunctions of the microscale and 119863119894119895 is the molecular diffu-sion coefficient The last two equations give the boundaryconditions on the fluid-solid interface Using similar homog-enization steps as presented in Section 31 the multiscaleexpansion

119888120576 = 119888 (119909 119909120576 )= 119888(0) (119909 119910) + 120576119888(1) (119909 119910) + 1205762119888(2) (119909 119910) + sdot sdot sdot

(31)

and (8) are substituted into the governing transport equations(30) Upon collecting like order terms the second-order termof 119888 is

119888(1) (119909 119910) = 119875119896 (119909 119910) 120597120597119909119896 119888(0) (119909) (32)

where 119888(0)(119909) is the macroscale concentration and 119875119896(119909 119910)satisfies 120597120597119910119894

120597120597119910119894119875119896 (119909 119910) = 0 (33)

119899119894 120597120597119910119894119875119896 (119909 119910) = minus119899119896 (34)

8 Geofluids

where (34) gives the boundary condition and 119899119894 is a unitvector normal to the fluid-solid interface (positive in thedirection away from the fluid space) Eqns (33) and (34)correspond to the assumption of an isotropic moleculardiffusion coefficient that is119863119894119895 = 119863120575119894119895 Upon further algebraand averaging over the 119884-cell the macroscale transportequation is obtained as follows

120601 120597120597119905119888 (119909) = 120597120597119909119894119863efflowast119894119896

120597120597119909119896 119888 (119909) (35)

where the effective diffusion coefficient119863efflowast119894119896 is given by

119863efflowast119894119896119863 = 1119884 int

119884(120575119894119896 + 120597120597119910119894119875119896 (119909 119910)) 119889119884 (36)

Intuitively the more tortuous the porous media the slowerthe rate of effective diffusion thus a commonly used relation-ship between effective and molecular diffusivity is

119863efflowast119894119896 = 119863120601120591119889119894119896 (37)

where 120591119889119894119896 is the diffusive tortuosity [41]The above approach to compute the effective diffusion

coefficient and the diffusive tortuosity has been used inseveral previous studies which formulate (33)ndash(36) using themethod of volume averaging [1 42ndash45] or by the method ofhomogenization [27] Other approaches to compute diffusivetortuosity include simulation of the diffusion process usinga random walk [25] or numerically solving the microscalediffusion equation using lattice Boltzmann modeling [46]Typically the effective diffusivity is computed and then usedto calculate diffusive tortuosity by the inverse relationshipshown in (37)

By (36) and (37) diffusive tortuosity is computed by

120591119889119894119896 = (119868119894119896 + 11003816100381610038161003816100381611988411989110038161003816100381610038161003816 int119884119891 (120597120597119910119894119875119896 (119909 119910)) 119889119910119891)

minus1 (38)

where 119884119891 = 120601119884 and where 119875119896(119909 119910) is the solution to(33) with the boundary condition in (34) This is the sameformulation that was used in Sun et al [27] by the method ofhomogenization and in Kim et al [43] and Valdes-Parada etal [1] by the method of volume averaging

The full diffusive tortuosity can also be diagonalized usingthe same procedure outlined in Section 31 to obtain thecomponents 120591119889max and 120591119889119905 corresponding to the principaldirections The rotation required to obtain the principalcomponents of klowast is not necessarily identical to the rotationrequired to obtain the principal components of 120591lowast119889 as willbe shown by one of the following pore geometry examples(randomly distributed squares)

4 Results

While real geological formations are typically of low porosity(120601 lt 03) we have generated several idealized geometrieswithin the porosity range of 033 lt 120601 lt 1 These idealized

Simulated result (Stokes) circleAnalytical result (Gebart eqn)

10minus4

10minus3

10minus2

10minus1

100

101

102

Nor

mal

ized

per

mea

bilit

y

03 04 05 06 07 08 09 102

Porosity

Figure 2 Permeability-porosity trend for in-line array of circles(ie infinitely long cylinders) Permeability units are nondimen-sional (ie unit length times unit length) The permeability is isotropicand is normalized by 1199032 where 119903 is the solid circle radius Gebartrsquos[47] analytical solution was given in (A2)

geometries and porosities have been used in literature (seeTable 1) and thus provide ground for comparison and oppor-tunity to make useful comparisons of the effective propertiesin high porosity representations of porous media

Permeability hydraulic tortuosity anddiffusive tortuosityare computed for each pore geometry using the methodsoutlined in Sections 31 32 and 33 respectively Each poregeometry is generated by defining certain input parameters(ie radius of solid circle side length of square and den-sity of randomly distributed squares) and these inputs aresummarized in Table 4 By using a range of values for theinput parameters we obtain unit cells which we treat asrepresentative elementary volumes over a range of porositiesAfter computing the permeability and tortuosity we plotthe data to observe permeability-porosity and tortuosity-porosity trends

41 In-Line Array of Uniform Shapes The unit cell or 119884-cellin Figure 1(b) illustrates our first type of generated poregeometry and the shape in the center of the cell is eithera circle or square A range of porosities of the unit cell areobtained by changing the radius of the solid circle or thelength of the solid square centered in the cell

411 Permeability and Hydraulic Tortuosity The full perme-ability tensor was computed for 033 lt 120601 lt 1 and resultsare plotted in Figure 2 Following [14] we computed andplotted the analytical solution from Gebart [47] which isexplained in Appendix A Permeability is isotropic (ie 119896119909119909 =119896119910119910) for this pore geometry The range of our simulateddata fits well against the analytical solution We computed

Geofluids 9

Table 4 Input parameters used to generate different pore-geometries (or different 119884-cells) and their corresponding porosity ranges In thistable cell refers to a cell in the mesh used to discretize the 119884-cellGeometry Initial mesh dimensions Inputs Porosity rangeIn-line array of shapes(see Figure 1) 50 times 50 cells (119884-cell length 1 unit) (a) Circle diameter 44 to 8 cells

(b) Square length 48 to 2 cells033 lt 120601 lt 1007 lt 120601 lt 1

Staggered array ofsquares (see Figure 7)

100 times 50 cells (119884-cell length 119871119887 = 05units 119886119887 = 1 119871119886119871119887 = 1 as per Figure 8) Square length 40 to 8 cells 036 le 120601 lt 1

Randomly distributedsquares (see Figure 14) 100 times 100 cells (119884-cell length 1 unit) Increasing density of squares

(with constant square length 001unit = 1 cell)

045 le 120601 lt 1

0005

001

0015

002

0025

003

0035

Figure 3 Fluid flow past a solid circle 120601 = 071 driven by 119892119909 Non-dimensional fluid speed (ie magnitude of velocities) is indicatedby color and flow path is indicated by flow lines (direction is left toright) Speed profile and flow path driven by 119892119910 are same as shownbut rotated 90 degrees

our permeability data on increasingly finer meshes until thepermeability converged within a 1 relative error To reachthis convergence criteria after starting with a mesh size of50 times 50 the lowest porosity cell required a refined mesh of500 times 500 and the highest porosity cell required 300 times 300The shape of the circular edge was refined at each meshrefinement in order to represent the circle as realistically aspossible and to minimize artifacts caused by discretization

We obtained the hydraulic tortuosity in the 119909 and 119910directions for porosities within the range 033 lt 120601 lt 1 withthe same convergence criteria used to obtain permeabilityFigure 3 illustrates the fluid speed and direction for a unitcell of 120601 = 071 and the hydraulic tortuosity in both 119909 and119910 directions were computed to be 120591ℎ119909 = 120591ℎ119910 = 10185 Thefluid speed is highest in the pore space that is uninterruptedby the solid and the flow lines indicate the direction is parallelto the macroscopic flow The flow lines diverge in the regionthat is interrupted by the solid however the fluid speed isessentially zero in that pore space So while these flow lines

120601 = 033

120601 = 039

120601 = 050

120601 = 059

120601 = 072

120601 = 087

120601 = 098

1

12

14

16

18

2

22

Com

pute

d di

ffusiv

e 120591

1 2 3 4 50

Number of cells times105

Figure 4Number of grid cells required to reach converged 120591119889 (usinga convergence criteria of 01 relative error) for select porosities ofin-line array of circles

appear to illustrate a tortuous pathway through this geometrythe tortuosity in either direction is in fact calculated to beasymp1 because the highest speeds are acting along a nontortuouspathway In the pore space that is not restricted or interruptedin the direction parallel to the driving force the fluid velocityis able to develop into the parabolic profile that characterizeschannel flow

412 Diffusive Tortuosity We obtained the full diffusive tor-tuosity tensor on increasingly refined meshes until the tortu-osity converged within a 01 relative error Starting with amesh of 50 times 50 the lowest porosity required a refined meshof 700 times 700 while the highest porosity required 250 times 250 tomeet the convergence criteria Solid edges were also refinedwith mesh refinement as mentioned previously In Figure 4we plot the convergence behavior of 120591119889 for a select numberof porosity cases in order to illustrate that each porosity caseconvergences with similar behavior Figure 5(a) illustrates thefields of 119875119896 (119896 = 1 2 or 119909 119910) for a unit cell of 120601 = 071which were obtained by solving the homogenized problem

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

8 Geofluids

where (34) gives the boundary condition and 119899119894 is a unitvector normal to the fluid-solid interface (positive in thedirection away from the fluid space) Eqns (33) and (34)correspond to the assumption of an isotropic moleculardiffusion coefficient that is119863119894119895 = 119863120575119894119895 Upon further algebraand averaging over the 119884-cell the macroscale transportequation is obtained as follows

120601 120597120597119905119888 (119909) = 120597120597119909119894119863efflowast119894119896

120597120597119909119896 119888 (119909) (35)

where the effective diffusion coefficient119863efflowast119894119896 is given by

119863efflowast119894119896119863 = 1119884 int

119884(120575119894119896 + 120597120597119910119894119875119896 (119909 119910)) 119889119884 (36)

Intuitively the more tortuous the porous media the slowerthe rate of effective diffusion thus a commonly used relation-ship between effective and molecular diffusivity is

119863efflowast119894119896 = 119863120601120591119889119894119896 (37)

where 120591119889119894119896 is the diffusive tortuosity [41]The above approach to compute the effective diffusion

coefficient and the diffusive tortuosity has been used inseveral previous studies which formulate (33)ndash(36) using themethod of volume averaging [1 42ndash45] or by the method ofhomogenization [27] Other approaches to compute diffusivetortuosity include simulation of the diffusion process usinga random walk [25] or numerically solving the microscalediffusion equation using lattice Boltzmann modeling [46]Typically the effective diffusivity is computed and then usedto calculate diffusive tortuosity by the inverse relationshipshown in (37)

By (36) and (37) diffusive tortuosity is computed by

120591119889119894119896 = (119868119894119896 + 11003816100381610038161003816100381611988411989110038161003816100381610038161003816 int119884119891 (120597120597119910119894119875119896 (119909 119910)) 119889119910119891)

minus1 (38)

where 119884119891 = 120601119884 and where 119875119896(119909 119910) is the solution to(33) with the boundary condition in (34) This is the sameformulation that was used in Sun et al [27] by the method ofhomogenization and in Kim et al [43] and Valdes-Parada etal [1] by the method of volume averaging

The full diffusive tortuosity can also be diagonalized usingthe same procedure outlined in Section 31 to obtain thecomponents 120591119889max and 120591119889119905 corresponding to the principaldirections The rotation required to obtain the principalcomponents of klowast is not necessarily identical to the rotationrequired to obtain the principal components of 120591lowast119889 as willbe shown by one of the following pore geometry examples(randomly distributed squares)

4 Results

While real geological formations are typically of low porosity(120601 lt 03) we have generated several idealized geometrieswithin the porosity range of 033 lt 120601 lt 1 These idealized

Simulated result (Stokes) circleAnalytical result (Gebart eqn)

10minus4

10minus3

10minus2

10minus1

100

101

102

Nor

mal

ized

per

mea

bilit

y

03 04 05 06 07 08 09 102

Porosity

Figure 2 Permeability-porosity trend for in-line array of circles(ie infinitely long cylinders) Permeability units are nondimen-sional (ie unit length times unit length) The permeability is isotropicand is normalized by 1199032 where 119903 is the solid circle radius Gebartrsquos[47] analytical solution was given in (A2)

geometries and porosities have been used in literature (seeTable 1) and thus provide ground for comparison and oppor-tunity to make useful comparisons of the effective propertiesin high porosity representations of porous media

Permeability hydraulic tortuosity anddiffusive tortuosityare computed for each pore geometry using the methodsoutlined in Sections 31 32 and 33 respectively Each poregeometry is generated by defining certain input parameters(ie radius of solid circle side length of square and den-sity of randomly distributed squares) and these inputs aresummarized in Table 4 By using a range of values for theinput parameters we obtain unit cells which we treat asrepresentative elementary volumes over a range of porositiesAfter computing the permeability and tortuosity we plotthe data to observe permeability-porosity and tortuosity-porosity trends

41 In-Line Array of Uniform Shapes The unit cell or 119884-cellin Figure 1(b) illustrates our first type of generated poregeometry and the shape in the center of the cell is eithera circle or square A range of porosities of the unit cell areobtained by changing the radius of the solid circle or thelength of the solid square centered in the cell

411 Permeability and Hydraulic Tortuosity The full perme-ability tensor was computed for 033 lt 120601 lt 1 and resultsare plotted in Figure 2 Following [14] we computed andplotted the analytical solution from Gebart [47] which isexplained in Appendix A Permeability is isotropic (ie 119896119909119909 =119896119910119910) for this pore geometry The range of our simulateddata fits well against the analytical solution We computed

Geofluids 9

Table 4 Input parameters used to generate different pore-geometries (or different 119884-cells) and their corresponding porosity ranges In thistable cell refers to a cell in the mesh used to discretize the 119884-cellGeometry Initial mesh dimensions Inputs Porosity rangeIn-line array of shapes(see Figure 1) 50 times 50 cells (119884-cell length 1 unit) (a) Circle diameter 44 to 8 cells

(b) Square length 48 to 2 cells033 lt 120601 lt 1007 lt 120601 lt 1

Staggered array ofsquares (see Figure 7)

100 times 50 cells (119884-cell length 119871119887 = 05units 119886119887 = 1 119871119886119871119887 = 1 as per Figure 8) Square length 40 to 8 cells 036 le 120601 lt 1

Randomly distributedsquares (see Figure 14) 100 times 100 cells (119884-cell length 1 unit) Increasing density of squares

(with constant square length 001unit = 1 cell)

045 le 120601 lt 1

0005

001

0015

002

0025

003

0035

Figure 3 Fluid flow past a solid circle 120601 = 071 driven by 119892119909 Non-dimensional fluid speed (ie magnitude of velocities) is indicatedby color and flow path is indicated by flow lines (direction is left toright) Speed profile and flow path driven by 119892119910 are same as shownbut rotated 90 degrees

our permeability data on increasingly finer meshes until thepermeability converged within a 1 relative error To reachthis convergence criteria after starting with a mesh size of50 times 50 the lowest porosity cell required a refined mesh of500 times 500 and the highest porosity cell required 300 times 300The shape of the circular edge was refined at each meshrefinement in order to represent the circle as realistically aspossible and to minimize artifacts caused by discretization

We obtained the hydraulic tortuosity in the 119909 and 119910directions for porosities within the range 033 lt 120601 lt 1 withthe same convergence criteria used to obtain permeabilityFigure 3 illustrates the fluid speed and direction for a unitcell of 120601 = 071 and the hydraulic tortuosity in both 119909 and119910 directions were computed to be 120591ℎ119909 = 120591ℎ119910 = 10185 Thefluid speed is highest in the pore space that is uninterruptedby the solid and the flow lines indicate the direction is parallelto the macroscopic flow The flow lines diverge in the regionthat is interrupted by the solid however the fluid speed isessentially zero in that pore space So while these flow lines

120601 = 033

120601 = 039

120601 = 050

120601 = 059

120601 = 072

120601 = 087

120601 = 098

1

12

14

16

18

2

22

Com

pute

d di

ffusiv

e 120591

1 2 3 4 50

Number of cells times105

Figure 4Number of grid cells required to reach converged 120591119889 (usinga convergence criteria of 01 relative error) for select porosities ofin-line array of circles

appear to illustrate a tortuous pathway through this geometrythe tortuosity in either direction is in fact calculated to beasymp1 because the highest speeds are acting along a nontortuouspathway In the pore space that is not restricted or interruptedin the direction parallel to the driving force the fluid velocityis able to develop into the parabolic profile that characterizeschannel flow

412 Diffusive Tortuosity We obtained the full diffusive tor-tuosity tensor on increasingly refined meshes until the tortu-osity converged within a 01 relative error Starting with amesh of 50 times 50 the lowest porosity required a refined meshof 700 times 700 while the highest porosity required 250 times 250 tomeet the convergence criteria Solid edges were also refinedwith mesh refinement as mentioned previously In Figure 4we plot the convergence behavior of 120591119889 for a select numberof porosity cases in order to illustrate that each porosity caseconvergences with similar behavior Figure 5(a) illustrates thefields of 119875119896 (119896 = 1 2 or 119909 119910) for a unit cell of 120601 = 071which were obtained by solving the homogenized problem

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 9

Table 4 Input parameters used to generate different pore-geometries (or different 119884-cells) and their corresponding porosity ranges In thistable cell refers to a cell in the mesh used to discretize the 119884-cellGeometry Initial mesh dimensions Inputs Porosity rangeIn-line array of shapes(see Figure 1) 50 times 50 cells (119884-cell length 1 unit) (a) Circle diameter 44 to 8 cells

(b) Square length 48 to 2 cells033 lt 120601 lt 1007 lt 120601 lt 1

Staggered array ofsquares (see Figure 7)

100 times 50 cells (119884-cell length 119871119887 = 05units 119886119887 = 1 119871119886119871119887 = 1 as per Figure 8) Square length 40 to 8 cells 036 le 120601 lt 1

Randomly distributedsquares (see Figure 14) 100 times 100 cells (119884-cell length 1 unit) Increasing density of squares

(with constant square length 001unit = 1 cell)

045 le 120601 lt 1

0005

001

0015

002

0025

003

0035

Figure 3 Fluid flow past a solid circle 120601 = 071 driven by 119892119909 Non-dimensional fluid speed (ie magnitude of velocities) is indicatedby color and flow path is indicated by flow lines (direction is left toright) Speed profile and flow path driven by 119892119910 are same as shownbut rotated 90 degrees

our permeability data on increasingly finer meshes until thepermeability converged within a 1 relative error To reachthis convergence criteria after starting with a mesh size of50 times 50 the lowest porosity cell required a refined mesh of500 times 500 and the highest porosity cell required 300 times 300The shape of the circular edge was refined at each meshrefinement in order to represent the circle as realistically aspossible and to minimize artifacts caused by discretization

We obtained the hydraulic tortuosity in the 119909 and 119910directions for porosities within the range 033 lt 120601 lt 1 withthe same convergence criteria used to obtain permeabilityFigure 3 illustrates the fluid speed and direction for a unitcell of 120601 = 071 and the hydraulic tortuosity in both 119909 and119910 directions were computed to be 120591ℎ119909 = 120591ℎ119910 = 10185 Thefluid speed is highest in the pore space that is uninterruptedby the solid and the flow lines indicate the direction is parallelto the macroscopic flow The flow lines diverge in the regionthat is interrupted by the solid however the fluid speed isessentially zero in that pore space So while these flow lines

120601 = 033

120601 = 039

120601 = 050

120601 = 059

120601 = 072

120601 = 087

120601 = 098

1

12

14

16

18

2

22

Com

pute

d di

ffusiv

e 120591

1 2 3 4 50

Number of cells times105

Figure 4Number of grid cells required to reach converged 120591119889 (usinga convergence criteria of 01 relative error) for select porosities ofin-line array of circles

appear to illustrate a tortuous pathway through this geometrythe tortuosity in either direction is in fact calculated to beasymp1 because the highest speeds are acting along a nontortuouspathway In the pore space that is not restricted or interruptedin the direction parallel to the driving force the fluid velocityis able to develop into the parabolic profile that characterizeschannel flow

412 Diffusive Tortuosity We obtained the full diffusive tor-tuosity tensor on increasingly refined meshes until the tortu-osity converged within a 01 relative error Starting with amesh of 50 times 50 the lowest porosity required a refined meshof 700 times 700 while the highest porosity required 250 times 250 tomeet the convergence criteria Solid edges were also refinedwith mesh refinement as mentioned previously In Figure 4we plot the convergence behavior of 120591119889 for a select numberof porosity cases in order to illustrate that each porosity caseconvergences with similar behavior Figure 5(a) illustrates thefields of 119875119896 (119896 = 1 2 or 119909 119910) for a unit cell of 120601 = 071which were obtained by solving the homogenized problem

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

10 Geofluids

015

01

005

0

minus005

minus01

minus015

(a) 1198751199090

01

02

03

04

05

06

07

08

09

(b) 119862119909 = 119875119909 + 119909119904

02

04

06

08

1

12

14

16

18

(c) Diffusion speed and direction

Figure 5 Fields for a unit cell of 120601 = 071 comprised of a solid circle (a) solution to the homogenized problem (b) concentration fieldtransformed from 119875119896 and (c) magnitude of concentration gradient radic(nabla119909119862119909)2 + (nabla119910119862119909)2 (diffusion speed) and direction The fields withrespect to 119910 are the same as shown above but rotated by 90 degrees

This work circleThis work square

1

11

12

13

14

15

16

17

18

19

2

Hyd

raul

ic to

rtuo

sity

02 04 06 08 10Porosity

(a)

This work circleThis work squareRayleigh (

1

11

12

13

14

15

16

17

18

19

2

Diff

usiv

e tor

tuos

ity

02 04 06 08 10Porosity

120591 = 2 minus 120601)

Vald es-Parada circleVald es-Parada square

(b)

Figure 6 Tortuosity-porosity trends for in-line array of uniform shapes (circle or square) (a) Hydraulic tortuosity is a very weak functionof porosity (b) Diffusive tortuosity for in-line array of squares follows the analytical solution (120591 = 2 minus 120601) from [48] closely and both in-linearray of squares and in-line array of circles match closely results by F J Valdes-Parada (provided via personal communication December 32016)

described in Section 33 The resulting diffusive tortuositytensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 12887 minus1 times 10minus16minus1 times 10minus16 12887 ] (39)

where 120591119889 = (1205911015840119889)minus1 Notice the tensor is symmetric and diago-nal (ie 120591119889119909119910 = 120591119889119910119909 = 0 to machine precision) and isotropic(ie 120591119889119909119909 = 120591119889119910119910) To illustrate the passive diffusive transportthrough this geometry we have illustrated the diffusionspeed and direction in Figure 5(c) which was computedfrom the concentration field illustrated in Figure 5(b) Thisconcentration field satisfiesnabla2119862 = 0with Dirichlet boundary

conditions on the right and left boundaries such that theconcentration gradient across the unit cell is one

413 Comparison of Tortuosities for In-Line Array Unit CellsThe comparison between our isotropic results for 120591ℎ and120591119889 illustrated in Figure 6 allows us to emphasize that thesetortuosity types are quantitatively different This may besurprising at first since the streamlines in Figures 3 and 5(c)appear to indicate that the flow fields are relatively analogousHowever the fluid speeds shown in Figure 3 are such thatthe fastest moving flow exists along the nontortuous path-way while the fastest passive diffusive transport (or highestconcentration gradient) shown in Figure 5(c) exists along

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 11

Table 5 Hydraulic and diffusive streamline comparison in-linearray of circles 120601 = 071 Flow is from left to right (0 119910119904119895) indicatethe starting location of the 119895th streamline

Hydraulic streamlines Diffusive streamlines(0 119910119904119895) 120582(119910119895) in Figure 3 120582(119910119895) in Figure 5(c)005 10011 10010015 10107 10103025 10348 10348035 10827 10852045 11665 118140495 12444 127320505 12444 12732055 11665 11814065 10827 10852075 10348 10348085 10107 10103095 10011 10010

the circlersquos rounded fluid-solid boundary This differencein speeds (or streamline weights) leads to a quantitativedifference between these two tortuosity types

To further make our point we quantitatively compare thehydraulic and diffusive streamlines in Figures 3 and 5(c) Thelength 120582(119910) of 12 streamlines with starting positions given by(0 119910119904119895) is tabulated in Table 5 These lengths are computedby 120582119895 = 119897119895119871 where 119897119895 is the length of the 119895th streamline and119871 = 1 for the length of the pore geometry This table revealsthat the length (or tortuosity) of a hydraulic streamline isrelatively similar to the diffusive streamline with the samestarting location However the hydraulic speeds are such thatthe fastest moving flow exists along the nontortuous pathwaywhile the fastest diffusive flow (or highest concentration gra-dient) exists along the circlersquos rounded fluid-solid boundaryThese speeds mean that individual streamlines contributeto the effective tortuosity with different weights since theweight of an individual hydraulic streamline is different fromthe diffusive streamline the overall hydraulic and diffusivetortuosity of the unit cell are expected to be quantitativelydifferent

In Figure 6 we plot the approximate analytical solutionfrom Rayleigh [48] which is 120591 = 2 minus 120601 (see Table 3) Given asystem comprised of a periodic array of cylinders one canimagine that there is a limit as to how small the porositycan become before the pore space is no longer conductingWe find that the diffusive tortuosity data for in-line array ofsquares fits to this solution with an expected convex shape(comparison between Rayleighrsquos [48] analytical solution andsimulated data for in-line array of squares was shown inFigure 4 of Valdes-Parada et al [1]) but the data for in-linearray of circles deviates as its critical porosity (120601119888 = 1 minus1205874 asymp 021) is approached At first glance one might thinkthe reason for this discrepancy is due to an inadequate meshrefinement as the solid circle becomes larger the fluid spacebetween adjacent solids becomes smaller and a finer meshmay be required to obtain an accurate solution However

the convergence of our in-line array of circles results wasdemonstrated in Figure 4 Furthermore our results matchquite closely those obtained by F J Valdes-Parada (providedvia personal communication December 3 2016) using thefinite elementmethodAs such our diffusive tortuosity ratheris exhibiting validated convex and concave trends

In terms of the hydraulic tortuosity we observe thatit is independent of porosity (or a very weak function of120601) for this isotropic (symmetrical) geometry Regardless ofthe solid circlersquos radius the fluid speeds are highest in thepore space that lies parallel to the driving force direction asnoted in Figure 3 Since the fluid speeds are highest along anontortuous pathway the tortuosity of the macroscopic flowthrough the unit cell is computed to be close to one Wecan expect the scenario to be different for a geometry thatcontains solids which interrupt the main flow path as seen inthe next example of staggered solids

42 Staggered-Array of Uniform Shapes Figure 7 illustratesour second type of pore geometry Unlike the previous poregeometry staggered-array of uniform shapes is an anisotropicgeometry Input parameters to define this geometry areillustrated in Appendix B (Figure 8) and for the followingstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 By varyingthe solid square length a range of porosities are obtained

421 Permeability and Hydraulic Tortuosity The permeabil-ity for our second pore geometry was computed for a range ofporosities Startingwith an initial mesh resolution of 100times50themeshes required refinement up to amaximumof 700times350in order to meet the convergence criteria of 119896rel error lt 1

This pore geometry configuration is anisotropic (ie119896119909119909 = 119896119910119910) and the trend of the principal permeabilitiesover the porosity range tested is shown in Figure 9 The ana-lytical solution for an in-line array of cylinders (ie Gebart[47]) is plotted against our staggered-array permeability forcomparison while a fit is not expected The permeability ischaracterized by an anisotropic ratio as follows

120574 fl119896119910119910119896119909119909 (40)

where 119896119910119910 gt 119896119909119909 for all porosities studied as shown inFigure 10

The hydraulic tortuosity in the 119909 and 119910 directions wascomputed using the same convergence criteria used to obtainpermeability Our results for a staggered-array unit cell of120601 = 073 are shown in Figure 11 and the resulting hydraulictortuosities were 120591ℎ119909 = 13539 and 120591ℎ119910 = 10567 Inthis example the pore geometry is such that there is nota straight (or uninterrupted) flow path when the flow isdriven in the 119909 direction However there is a straight flowpath when the flow is driven in the 119910 direction Due to thischaracteristic we observe that the hydraulic flow is moretortuous in the 119909 direction than in the 119910 direction whichis confirmed by the anisotropic ratio for hydraulic tortuosityplotted in Figure 10 By observing Figure 11 we notice that thefluid speeds are highest in the pore regions located betweentwo square corners We also notice in Figure 11(a) that the

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

12 Geofluids

(a) 120601 = 036 (b) 120601 = 064 (c) 120601 = 084

Figure 7 Unit cell for staggered-array of uniform shapes (ie squares) represented by cells with periodic boundaries and porosity 120601 For allstaggered-array examples 119886119887 = 1 and 119871119886119871119887 = 1 (refer to Figure 8 for dimension details)

a

a

2

b

b

2

Lb

2La

Figure 8 A staggered (anisotropic) unit cell with dimensions 120601 =06 119886119887 = 3 and 119871119886119871119887 = 1

03 04 05 06 07 08 09 1

Porosity

10minus4

10minus3

10minus2

10minus1

Non

norm

aliz

ed P

erm

eabi

lity

(unit2

)

Analytical Result (Gebart eqn nonnormalized)

Simulated Kxx

Simulated Kyy

Figure 9 Permeability-porosity trend for staggered-array ofsquares

fluid speed is close to being constant along a sinusoidal flowpath which appears to occupy most of the pore space excepttowards the fluid-solid boundaries The horizontally drivenflow is interrupted by the center square and causes the flowpath to diverge However in Figure 11(b) the vertically driven

03 04 05 06 07 08 09 1

Porosity

05

06

07

08

09

1

11

12

13

14

15

Ani

sotro

pic r

atio

Permeability KyyKxx

Hydraulic tortuosity 120591y120591xDiffusive tortuosity 120591yy120591xx

Figure 10 Anisotropic ratios in staggered-array of squares Forall porosities the anisotropic ratios for hydraulic and diffusivetortuosity are less than one indicating that the hydraulic flow andpassive diffusive transport are more tortuous in the 119909 direction thanin the 119910 direction The hydraulic flow is more permeable in the 119910direction

flow is not interrupted to the extent that would cause the flowpath to become sinusoidal Instead the geometry of the porespace allows for a straight flow path to develop We do notnotice this sinusoidal flow path of constant speed rather wenotice that the speed is more localized (ie speed is highestbetween adjacent solid corners and lowest along solid squaresides)

422 Diffusive Tortuosity We obtained the full diffusivetortuosity tensor for this staggered geometry using a conver-gence criteria of 120591119889rel error lt 01 Starting with a uniformmesh of 100 times 50 the lowest porosity required a refinedmesh of 1000 times 500 while the highest porosity required 200 times100 in order to meet the convergence criteria Results for a

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 13

8

6

4

2

times10minus3

(a) Flow in 119909 direction

8

12

10

6

4

2

times10minus3

(b) Flow in 119910 direction

Figure 11 Fluid flow through a staggered-array of squares 120601 = 073 Fluid speed (ie magnitude of velocities) is indicated by color anddirection is indicated by flow lines Fluid speed is nondimensional

01

0

minus01

minus02

02

(a) 119875119909

05

1

15

(b) 119862119909 = 119875119909 + 119909119904

051152253

(c) Diffusion speed and direction

minus015minus01minus005000501015

(d) 119875119910

02

04

06

08

(e) 119862119910 = 119875119910 + 119910119904

051152253

(f) Diffusion speed and direction

Figure 12 Fields for a staggered-array unit cell of 120601 = 073 (a d) solution to the homogenization problem (b e) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c f) magnitude of concentration gradient (diffusionspeed) and direction

staggered-array unit cell of 120601 = 073 are shown in Figure 12and the diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 13514 minus1 times 10minus151 times 10minus17 13092 ] (41)

where the tensor is symmetric and diagonal (ie 120591119889119909119910 =120591119889119910119909 = 0 to machine precision) and anisotropic (ie 120591119889119909119909 =120591119889119910119910) In this geometry of staggered squares 120591119889119909119909 gt 120591119889119910119910and thus passive diffusive transport is more tortuous in the 119909direction due to an interrupted flow path

423 Comparison of Tortuosities for Staggered-Array UnitCells Our results for hydraulic and diffusive tortuosity overthe range of porosities are plotted in Figure 13 Diffusivetortuosity data from Ryan et al [49] and Kim et al [43](both obtained from Kim et al [43]) are also plotted forcomparison While the works of Ryan et al [49] and Kimet al [43] were based on using the method of volume aver-aging to obtain the boundary-value problem for diffusivitypresented in Section 33 we are able to compare our resultsto theirs since both homogenization and volume averaging

approaches lead to the same boundary-value problem Tosolve the boundary-value problem presented in Section 33Ryan et al [49] employed the finite difference method whileKim et al [43] employed the boundary element methodWe note that one of Kim et alrsquos [43] data points shown inFigure 13(b) unexpectedly corresponds to a diffusive tortu-osity of less than 1 After confirming this value from Table 6of their work (where 119886119887 = 1 119871119886119871119887 = 1) we do nothave an explanation as to how 120591119889 could be smaller than 1other than a possible typographical error Since they reportedvalues for the parameter 120601Deff119863 rather than 120591119889 explicitly itmay not have been easily apparent to them that one of theirvalues corresponded to 120591119889 lt 1 (Refer to Appendix B wherewe explain more about the notation as well as compare ournumerical results againstmore of their results) Regarding thediffusive tortuosity our simulated data fits closely to thesetwo literature data however in Figure 13(a) we observe our120591119889119909119909 data diverges at lower porosities (120601 lt 06) While theexact reason for this discrepancy is not clear possible causesmay be attributed to the difference in numerical methodsemployed or the geometry of the staggered-array (especiallyat lower porosities that contain very narrow flow channels)

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

14 Geofluids

Ryan et al (1981) FDM dataKim et al (1987) BEM data

Tort

uosit

y (120591

)

Simulated hydraulic 120591xSimulated diffusive 120591xx

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(a) Tortuosity in horizontal direction

Kim et al (1987) BEM dataSimulated hydraulic 120591ySimulated diffusive 120591yy

Tort

uosit

y (120591

)

08

1

12

14

16

18

2

02 04 06 08 10Porosity

(b) Tortuosity in vertical direction

Figure 13 Tortuosity-porosity trends for staggered-array of squares Simulated data is plotted against (diffusive) data from literature thatemployed finite difference (FD) and boundary element (BE) methods In this geometry hydraulic tortuosity in the 119910 direction is a weakfunction of porosity

Table 6 Data comparison for diffusive tortuosity

Unit cell specs Kim et alrsquos [43] results Our results120601 119886119887 119871119886119871119887 120601119863eff

119909119909 119863 120601119863eff119910119910 119863 1206011205911015840119889119909119909 1206011205911015840119889119910119910

074 15 1 0741 0028 07396 00234080 15 1 0787 0116 07959 01107088 15 1 0870 0225 08675 02197091 15 1 0902 0301 09112 0302206 1 1 0427 0412 03863 04125065 1 1 0478 0464 04397 0463007 1 1 0535 0519 04940 05140075 1 1 0596 0581 05625 0578208 1 1 0664 0648 06439 06547085 1 1 0738 0722 07221 0728909 1 1 0819 0805 08064 08097095 1 1 0907 0897 08976 0898506 3 1 0494 0161 04814 01620065 3 1 0562 0209 05465 0208507 3 1 0627 0270 06187 02693075 3 1 0689 0345 06747 0345308 3 1 0748 0434 07350 04341085 3 1 0807 0541 08012 0540509 3 1 0868 0667 08627 06637095 3 1 0933 0817 09282 08201

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 15

(a) 120601 = 053 (b) 120601 = 061 (c) 120601 = 070 (d) 120601 = 080 (e) 120601 = 090

Figure 14 Various pore structures for randomly distributed squares geometry Any fluid area isolated from flow path is converted to solidthus 120601 is effective porosity of geometry

Figure 13(b) shows a closer fit between our 120591119889119910119910 data tothe literature data in comparison with the fit shown inFigure 13(a) Since the narrowest channels exist at the lowestporosity and are oriented in the 119909 direction this observationimplies our diffusive tortuosity data may be sensitive to themesh resolution between adjacent solid boundaries Despitethe difference between our data to that reported in literaturethe anisotropic nature of this geometry is evident from ourresults Over the range of porosities we observe that thehydraulic flow is predominantly parallel to the driving forceunless an obstacle prevents a straight flow path Thus 120591ℎ119909 gt120591ℎ119910 for all porosities as shown in Figure 13 and 120591ℎ119910 is avery weak function of porosity The anisotropic ratios of thetortuosities are 120591ℎ119910120591ℎ119909 120591119889119910119910120591119889119909119909

(42)

and are plotted in Figure 10 along with the permeabilityratio previously defined Over the whole porosity rangeboth anisotropic ratios of the tortuosities are less than oneindicating the hydraulic flow and passive diffusive transportare more tortuous in the 119909 direction than in the 119910 directionThe hydraulic tortuosity shows a greater degree of anisotropythan diffusive tortuosity because 120591ℎ119909 asymp 1 for all porositieswhile 120591ℎ119910 increases towards the lower porosities43 Randomly Distributed Squares Our third pore geometrygenerated comprised of randomly distributed squares whichis a type commonly used in literature (eg [13 19 26]) Inthose works squares are freely overlapping however in ourwork our square size is equivalent to the cell size of theinitial mesh resolution and thus no squares overlap In thisgeometry periodic boundary conditions were implementedby checking for the existence of any solid-fluid interfacesalong the external boundaries of the domain and if detectedno-flow boundary conditions were applied at the interface(ie the velocity component acting normal to the fluid-solid interface was assigned a value of zero in the systemof equations) We imposed the constraint given by Koza etal [34] (ie the ratio of obstacle length to domain length lt001) which was recommended to avoid anisotropic-relatedstatistical errors This same constraint was also respected

in Duda et al [13] Our geometries ranged from (effective)porosities of 045 lt 120601 lt 099 Example pore geometriesare shown in Figure 14 Our algorithm to generate these poregeometries included a step to check for any fluid sites thatmaybe isolated from the flow path which if found were changedto be solid sites By filling in the isolated fluid sites we wereable to represent the effective porosity of the geometry asnoted in Koponen et al [26] Since lower porosity geometriescontained more solid sites than higher porosities they wereespecially susceptible to isolated fluid sites Filling in theisolated sites created large solids within the pore geometry(especially noted in Figure 14(a)) and the impact of thesesolids on results will be discussed Due to the nature of thisrandomly generated geometry we used between 8 and 11realizations of a given porosity between 046 and 095 and5 realizations for the porosity of 097 Thus the followingresults are presented as averaged quantities with error barsto indicate the spread between the maximum and minimumquantities computed

431 Permeability and Hydraulic Tortuosity We computedpermeability using a convergence criteria of 119896rel error lt 1Starting with an initial mesh of 100times100 the lowest porosityrequired a refined mesh of 400times400 and the highest porosityrequired 200 times 200 to meet the convergence criteria

The full permeability tensors (in coordinate system x)for these pore geometries were symmetric and nondiagonalthus a transformation was done to obtain the diagonal tensorin the principal coordinate system xlowast The full permeabil-ity components and diagonal permeability components areplotted in Figure 15 The nature of the anisotropic ratio ofthe diagonal permeability tensor will be discussed and shownlater in Figure 16(b)

We computed hydraulic tortuosity from the flow fieldresults which were obtained using the convergence criteriafor permeability previously mentioned For a configurationcase of 120601 = 07 results for hydraulic flow are shown inFigure 17 and the computed hydraulic tortuosities are 120591ℎ119909 =13727 and 120591ℎ119910 = 13393 From Figure 17 the tortuous natureof the fluid pathways is evident We note that a few ldquohotspotsrdquo of fastest moving fluid (with speeds Vmag gt 5 times 10minus5)exist in the geometry while the rest of the pore space ischaracterized by moderate (1 times 10minus5 lt Vmag lt 5 times 10minus5)to slow (Vmag lt 1 times 10minus5) moving fluid The high speeds

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

16 Geofluids

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3Pe

rmea

bilit

y

Av sim kxxAv sim kyy

Av sim kxy = kyxAv sim minuskxy = minuskyx

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

10minus8

10minus7

10minus6

10minus5

10minus4

10minus3

Perm

eabi

lity

Av sim kmaxAv sim kt(b) Diagonalized tensor components in xlowast

Figure 15 Permeability-porosity trend for randomly distributed squares The full permeability tensor contains nonzero off-diagonalcomponents (absolute values are plotted however negative and positive quantities are indicated by different symbols) and is diagonalizedto obtain the principal permeability components 119896max and 119896119905 x and xlowast are related by rotation angle 120579119896 which is plotted in Figure 16(a)The anisotropic ratio of the principal components is plotted in Figure 16(b)

Coo

rdin

ate R

otat

ion

xrarrxlowast

Permeability 120579kDiffusive tortuosity 120579d

minus90

minus60

minus30

0

30

60

90

04 05 06 07 08 09 1

Porosity

(a) Rotation angle 120579

Permeability kmaxktHydraulic tortuosity 120591max120591tDiffusive tortuosity 120591max120591t

04 05 06 07 08 09 1

Porosity

1

15

2

25

3

35

Ani

sotro

pic R

atio

(b) Anisotropic ratios

Figure 16 Anisotropic ratios in geometries comprised of randomly distributed squares (a) rotation to obtain diagonal tensors (klowast and 120591lowast119889 ) isindependent of porosity (b) anisotropic ratios of diagonal tensors and anisotropic ratio of hydraulic tortuosity At lower porosities (120601 lt 06)anisotropic properties have been induced

exist within the pore geometryrsquos narrow channels and uponmagnifying these regions we observed a parabolic velocityprofile indicative of channel flow While Figure 17 illustratesfluid flow in the 119910 direction only similar characteristics werefound for flow in the 119909 direction thus we have omitted

those illustrations The computed hydraulic tortuosity at thisporosity is essentially isotropic that is 120591ℎ119909 asymp 120591ℎ119910 So whilethe geometry is comprised of randomly distributed squaresthe overall geometry exhibits an isotropic hydraulic tor-tuosity (However this is not the case at lower porosity

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 17

04

04

02

02

06

06

08

08

1

1

1

2

3

4

5

6

00

0

times10minus5

(a) Fluid speed

085

08

075

07

06

065

06505505045

1

2

3

4

5

6

0

times10minus5

(b) Close-up of boxed region in (a)

Figure 17 Flow (driven by 119892119910) through randomly distributed squares 120601 = 070 Nondimensional fluid speed is indicated by color Zero speedat solid sites is kept as dark blue for clarity of illustration (while these fluid speeds do not actually exist)

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

15

2

25

3

35

4

45

Hydraulic 120591maxHydraulic 120591t

Figure 18 Hydraulic tortuosity-porosity trend for randomly dis-tributed squares The flow pathways are more tortuous as geometryis comprised of more solid squares (ie lower porosity)

configurations as the anisotropic ratios which wewill discusslater using Figure 16(b) will indicate) In agreement withresults presented in [13] the trend shown in Figure 18indicates that the flow pathways are more tortuous as thegeometry is comprised of more solid squares

432 Fit to Kozeny-Carman Equation As presented inSection 22 the Kozeny-Carman equation relates permeabil-ity 119896 to porosity 120601 hydraulic tortuosity 120591ℎ specific surfacearea 119878 and a fitting coefficient 120573119896 (which we refer to as theshape factor) Specific surface area for the range of porositieswas computed by (3) and is shown in Figure 19 The specific

04 05 06 07 08 09 1

Porosity

0

10

20

30

40

50

60

70

80

90Sp

ecifi

c sur

face

area

(S)

1(u

nit)

Figure 19 Specific surface area for randomly distributed squaresover the range of porosities studied Specific surface area increaseslinearly as porosity decreases however this trend deviates at lowerporosities due to the large nonuniform shapes introduced into thelower porosity geometries

surface area increases as the porosity decreases because thegeometry is comprised of more solid squares However oncethe porosity is lowered to the point where isolated fluidsites must be converted to large solid areas the fluid-solidinterfacial area is reduced along with the specific surface areaWe computed the shape factor 120573119896 by (2) for this particularpore geometry using our permeability hydraulic tortuosityand specific surface area data presented in Figures 15(a) 18and 19 respectively The trend for 120573119896 over the porosities isplotted in Figure 20 and we observe that the shape factordoes not change significantly within a porosity interval of07 lt 120601 lt 090 which was similarly reported in Duda et al[13]

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

18 Geofluids

04 05 06 07 08 09 1

Porosity

fit for kmax and 120591hmax datafit for kt and 120591ht data

0

10

20

30

40

50

60

70

Shap

e fac

tor (

120573k)

unitl

ess

Figure 20 Shape factor 120573119896 used to fit simulated permeabilityand hydraulic tortuosity data to Kozeny-Carman equation forgeometries comprised of randomly distributed squares 120573119896 appearsto be constant between 07 lt 120601 lt 09

433 Diffusive Tortuosity We obtained diffusive tortuosityfor this geometry using different convergence criteria forthree ranges of porosities due to high mesh refinementrequirements Convergence criteria of 120591rel error lt 3 2 and1 were used for porosity ranges of 045 lt 120601 lt 058 058 lt120601 lt 07 and 07 lt 120601 lt 099 respectively The geometrieswere generated using a mesh of 100 times 100 and the conver-gence criteria required a refinement of 2000 times 2000 for thelowest porosity and 200times200 for the highest porosity Resultsfor a geometry of 120601 = 07 are shown in Figure 21 and theresulting diffusive tortuosity tensor is

120591119889 = [120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910] = [ 27584 minus02914minus02914 26175 ] (43)

Notice the tensor is symmetric (ie 120591119889119909119910 = 120591119889119910119909) but notdiagonal (ie the off-diagonal components are nonzero) Assuch we diagonalized the full diffusive tortuosity tensor andresults for the whole porosity range studied are presented inFigure 22

434 Comparison of Tortuosities for Randomly DistributedSquares Our hydraulic and diffusive tortuosity data areplotted in Figure 23 for a range of porosity values along withsome of the trends from literature that we reported in Tables2 and 3 The trend from Koponen et al [26] was obtainedfor hydraulic tortuosity for randomly distributed squaresThe trend from Mackie and Meares [50] was for diffusivetransport of electrolytes through a membrane and theirexpression for the tortuosity-porosity trend was credited inthe past as one of themost successfully employed correlationsfor membrane systems [51] The trend from Weissberg [52]was also for diffusive transport through a geometry com-prised of uniform spheres It was given as an upper bound

for effective diffusion and thus could be considered as thelower bound for diffusive tortuosity since they are inverselyrelated Our hydraulic data fits moderately close with theKoponen et alrsquos [26] trend Our diffusion data follows asimilar trend to Mackie and Meares [50] until 120601 lt 06 andremains above Weissbergrsquos [52] lower bound In general ourdata demonstrates that diffusive tortuosity is not equivalentto hydraulic tortuosity and can be up to ten times greaterespecially at low porosities

435 InducedAnisotropic Properties at Lower PorosityGeome-tries Due to the method we used to generate the ran-domly distributed squares geometry anisotropic propertieswere introduced into the pore structure especially at lowerporosities This is evident in Figure 24 where low andmedium porosities are compared The lower porosity haslarge nonuniform solid shapes within the geometry Torepresent the connected pore space or effective pore space[26] we filled in the isolated fluid sites which resulted fromthe random distribution of squares in the domain The resultis a nonuniform distribution of solids in the geometry and aninduced degree of anisotropy that is shown in Figure 16(b)(Note that Figure 16(a) shows that the rotation required toobtain the diagonal tensors klowast and 120591lowast119889 is independent ofporosity) At the higher porosities the degree of anisotropyof all three properties is moderately close to one and thuswe could consider those geometries as exhibiting isotropicbehavior As the porosity is lowered the permeability anddiffusive tortuosity become anisotropic and the degree ofanisotropy of the permeability is higher than that of thediffusive tortuosity The degree of anisotropy of hydraulictortuosity is less than diffusive tortuosity and is close to beingisotropic for most of the porosity range A possible reason forthis finding is that hydraulic flow develops preferential path-ways where the fluid speed is highest along a pathway that isunhindered by solids as shown in Figure 25(b) If vertical andhorizontal preferential pathways are geometrically similar anisotropic hydraulic tortuosity is exhibited On the other handthe diffusive speeds are highest adjacent to the fluid-solidboundaries as seen in Figure 25(d) thus the geometry of thepore structure appears to impact the diffusive tortuositymorethan it impacts the hydraulic tortuosity (This point is alsoillustrated by comparing the flows in a higher porosity inFigures 17(b) and 21(d))

5 Conclusion

In this study we computed the effective properties knownas permeability hydraulic tortuosity and diffusive tortuosityusing well-known and commonly employed formulationsbased on themethod of homogenizationWhilewe noted thatpast work has recognized the difference between tortuositytypes we emphasized that a few studies have failed todistinguish the difference and have mistakenly used theminterchangeably In this work hydraulic tortuosity was com-puted based on the approach given in Koponen et al [19 26]Their approach uses pore space velocity fields k howeverwe used the rescaled velocity fields 119873119894119895 which we obtainedby solving a set of rescaled Stokes equations formulated

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 19

02

minus02

015

minus015

01

005

minus005

0

minus01

(a) 119875119909

0

02

04

06

08

1

(b) 119862119909 = 119875119909 + 119909119904

0

1

2

3

4

5

0

02

04

06

08

1

0 02 04 06 08 1

(c) radic(nabla119909119862119909)2 + (nabla119910119862119909)2

085

08

065

0650605505045

07

075

0

1

2

3

4

5

(d) Close-up of boxed region in (c)

Figure 21 Fields for geometry of randomly distributed squares 120601 = 07 (a) solution to the homogenization problem (b) concentration fieldtransformed from 119875119896 (assuming concentration gradient across unit cell is one) and (c d) magnitude of concentration gradient (diffusionspeed)

by homogenization ((10)ndash(12)) Using 119873119894119895 in (28) is math-ematically equivalent to using k in (25) and (27) Diffusivetortuosity was computed by solving the homogenized orboundary-value problem given for the effective diffusioncoefficient as employed in Kim et al [43] Valdes-Parada etal [1] and Sun et al [27]

We generated several different pore geometries includingboth in-line array and staggered-array of uniform shapesand randomly distributed squares For the in-line arrayof circles geometry our simulated permeability data wasvalidated against the analytical solution from Gebart [47]and our simulated diffusive tortuosity data was validatedagainst Rayleighrsquos [48] trend We studied the anisotropy ofthe computed properties and fit data from one of the poregeometries to the Kozeny-Carman equation to obtain theshape factor 120573119896

The main findings from the geometries studied are asfollows

(1) Hydraulic tortuosity is not equal to diffusive tor-tuosity in the same pore geometry In the in-linearray of uniform shapes (either circle or square)

hydraulic tortuosity was weakly dependent on poros-ity while diffusive tortuosity was almost linearlyrelated to porosity (recall Figure 6) In the randomlydistributed squares geometry the diffusive tortuositywas greater (up to a factor of ten) than hydraulictortuosity as porosity decreased (recall Figure 23)In all examples hydraulic speeds were highest alongthe mid-channel space between adjacent solids andformed a parabolic velocity profile while diffusivespeeds were highest along the fluid-solid boundarywhich could be extremely irregular or complex (asseen in the randomly distributed squares geome-try) we suspected this boundary impact on flowto be the main reason why these two tortuositytypes were quantitatively different in the same poregeometry

(2) Results from the three different pore geometry con-figurations used in this work indicate that 120591119889 gt 120591ℎfor themajority of the porosity range tested Howeverwe do not claim that this relationship is necessarilytrue for all types of porousmedia For example a pore

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

20 Geofluids

004 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

1

5

10

15

20

25

Diffusive 120591maxDiffusive 120591t

(a) Full tensor components in x

04 05 06 07 08 09 1

Porosity

Tort

uosit

y (120591

)

Diffusive 120591xxDiffusive 120591yyDiffusive 120591xy = 120591yx

minus5

0

5

10

15

20

25

(b) Diagonalized tensor components in xlowast

Figure 22 Diffusive tortuosity-porosity trend for randomly distributed squares The full diffusive tortuosity tensor contains nonzero off-diagonal components and is diagonalized to obtain the principal diffusive tortuosity components 120591119889max and 120591119889119905 x and xlowast are related byrotation angle 120579119889 which is plotted in Figure 16(a) The anisotropic ratio of the principal components is plotted in Figure 16(b)

04 05 06 07 08 09 1

Tort

uosit

y (120591

)

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

Porosity 120601

0

5

10

15

20

25

(a) Tortuosity-porosity trends

Tort

uosit

y (120591

)

04 05 06 07 08 09 1

Porosity 120601

1

15

2

25

3

35

4

45

5

Simulated hydraulic 120591maxSimulated hydraulic 120591tSimulated diffusive 120591maxSimulated diffusive 120591t

Analy hydraulic 120591 Koponen et al (1997)Analy diffusive 120591 Mackie amp Meares (1955)Analy diffusive 120591 Weissberg (1963)

(b) Close-up of data

Figure 23 Comparison of hydraulic (120591ℎ119909 and 120591ℎ119910) and diffusive (120591119889119909119909 and 120591119889119910119910) tortuosity trends for pore structures of randomly distributedsquares against trends reported in literature

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 21

(a) 120601 = 04525 (b) 120601 = 07005

Figure 24 Induced anisotropy in lower porosity geometries comprised of randomly distributed squares

02 04 06 08 10

0

02

04

06

08

1 times10minus5

0

05

1

15

2

25

(a) Fluid speed055 06 065 07 075

times10minus5

0

05

1

15

2

25

015

02

025

03

035

(b) Close-up of boxed region in (a)

0 02 04 06 08 10

02

04

06

08

1

0

05

1

15

2

25

3

35

4

(c) Diffusion speed055 06 065 07 075

015

02

025

03

035

0

05

1

15

2

25

3

35

4

(d) Close-up of boxed region in (c)

Figure 25 Comparison of hydraulic and diffusive flows through a low porosity geometry comprised of randomly distributed squares 120601 =045 (a b) Fluid flow driven by 119892119909 (c d) Diffusive flow driven by nabla119862 = 1 from west to east ldquoHot spotsrdquo of fastest moving hydraulic flow areevident in channel centers while diffusive speeds are fastest along fluid-solid boundaries

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

22 Geofluids

network model that was considered in Ghanbarianet al [8] led to the conclusion that 120591119889 le 120591ℎ Thus wesimply note an equality held true for the three typesof synthetic porous media tested in our work

(3) Related to the first point hydraulic and diffusive tor-tuosity (as well as permeability) can exhibit differentanisotropic behavior in the same pore geometry Inthe staggered-array of uniform shapes (square) weobserved that the hydraulic tortuosity had a greaterdegree of anisotropy compared to diffusive tortuos-ityrsquos anisotropy (recall Figure 10) This behavior wasdue to the nature of the geometry which allowed fora nontortuous hydraulic flow in one of the principaldirections In the randomly distributed squares thelarge nonuniform shapes that were introduced intothe geometry as a result of isolated fluid sites led toan induced anisotropic behavior at higher porositiesall properties exhibited essentially isotropic behaviorbut at lower porosities k and 120591119889 became anisotropicby varying degrees It is interesting to note that120591ℎ displays the least degree of anisotropic behav-ior this finding may be related to the existence ofpreferential pathways or regions of high-speed flowthroughminimally tortuous pathwayswithin the poregeometry

(4) Two geometries (ie staggered-array of squares ran-domly distributed squares) demonstrated that whenthe permeability is anisotropic the effective diffu-sion coefficient (which is inversely proportional todiffusive tortuosity) can also be anisotropic howevernot necessarily by the same degree In general thedegree of anisotropic permeability was greater thanthe degree of anisotropic diffusion in the same poregeometry (recall Figures 10 and 16(b)) This find-ing has important implications for flow and trans-port modeling at reservoir-scale because the use ofDarcyrsquos equation and a transport equation (with adiffusive flux term) requires permeability and theeffective diffusion coefficient as input parametersrespectively

(5) A few qualitative statements can be made regardingthe flow behavior demonstrated in some of the exam-ples (ie staggered-array of squares and randomlydistributed squares) Hydraulic tortuosity and perme-ability are generally related the more tortuous theflow pathway is in a direction the slower the fluidspeed is and the less permeable it is in that direc-tion Lower porosity geometries are characterized byslower fluid speeds andmore tortuous hydraulic path-ways in addition tomore tortuous diffusive pathwaysBy (37) which shows the effective diffusion coefficientis inversely proportional to diffusive tortuosity we candeduce that the more tortuous the diffusing flow ina direction the slower the rate of (effective) diffusivetransport

Appendix

A Validation of Permeability Calculation

To validate our numerical implementation and computationof permeability we compared our results with the analyticalsolution given by Gebart [47] This analytical solution ofpermeability corresponds to an in-line array of infinitely longcylinders The porosity is computed as a function of thecylindrical radius 119903 as follows

120601 = 119881119891119881119905 = 119881119905 minus 119881119904119881119905 = 1 minus 119881119904119881119905 = 1 minus 12058711990321198712 (A1)

where119881119891 119881119904 are the fluid and solid volumes respectively119881119905 =119881119891 + 119881119904 is the total volume and the unit cell is 119871 times 119871 Thus byvarying 119903 the porosity is also varied Gebartrsquos [47] analyticalsolution for permeability (normalized by 119903) as a function ofporosity is

1198701199032 = 119888119892(radic1 minus 1206011198881 minus 120601 minus 1)52

(A2)

where 119888119892 is the geometric factor which depends on thefiber arrangement and 120601119888 is the critical porosity (percolationthreshold) For a quadratic fiber arrangement the geometricfactor is 169120587radic2 and the percolation threshold is 120601119888 =1 minus 1205874 for an array of cylinders

B Validation of DiffusiveTortuosity Calculation

For benchmarking purposes we computed the diffusivetortuosity for staggered unit cells and compared our resultsto those reported by Kim et al [43] who used the boundaryelement method (BEM) In our work we used finite differ-ence The method to compute the diffusive tortuosity tensorwas presented Section 33 and for a 2D problem the tensorcomponents are computed by

[120591119889119909119909 120591119889119909119910120591119889119910119909 120591119889119910119910]

= [[[[[

1 + 1119881119891 int119881119891

120597119875119909120597119909 119889119881119891 1119881119891 int119881119891

120597119875119910120597119909 1198891198811198911119881119891 int

119881119891

120597119875119909120597119910 119889119881119891 1 + 1119881119891 int119881119891

120597119875119910120597119910 119889119881119891]]]]]

minus1

(B1)

where119881119891 = 119884119891 Note that diffusive tortuosity is the inverse ofthe diffusive tortuosity factor that is 120591119889 = (1205911015840119889)minus1 Recalling(37) the diffusive tortuosity factor is related to the effectivediffusion coefficient by

1206011205911015840119889 = 120601Deff

119863 (B2)

where 120601Deff = Defflowast The form of (B2) helps to make appro-priate comparisons in Table 6The input parameters required

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Geofluids 23

to construct the staggered unit cell are porosity 120601 119886119887 ratioand 119871119886119871119887 ratio and it is constructed to meet the config-uration presented in Figure 8 Kim et al [43] performedmeasurements on many staggered unit cell cases and Table 6presents a comprehensive comparison of our results againsttheir data

Disclosure

Rebecca Allen is now at SINTEF Digital Mathematics andCybernetics Forskningsveien 1 0314 Oslo Norway

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge that the researchreported in this publication was supported by fundingfrom King Abdullah University of Science and Technology(KAUST) The authors gratefully acknowledge and thankDr Francisco J Valdes-Parada for providing the diffusivetortuosity results for in-line array of circle and squaregeometries which were used for comparison in Figure 6with permission

References

[1] F J Valdes-Parada M L Porter and B D Wood ldquoThe role oftortuosity in upscalingrdquo Transport in Porous Media vol 88 no1 pp 1ndash30 2011

[2] M A Knackstedt S Latham M Madadi A Sheppard TVarslot and C Arns ldquoDigital rock physics 3D imaging ofcore material and correlations to acoustic and flow propertiesrdquoLeading Edge vol 28 no 1 pp 28ndash33 2009

[3] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart I imaging and segmentationrdquo Computers ampGeosciences vol 50 pp 25ndash32 2013

[4] S Torquato and Y Jiao ldquoRobust algorithm to generate a diverseclass of dense disordered and ordered sphere packings via linearprogrammingrdquo Physical Review E Statistical Nonlinear andSoft Matter Physics vol 82 no 6 article 061302 2010

[5] C L Y Yeong and S Torquato ldquoReconstructing randommediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 57 no 1 pp 495ndash506 1998

[6] A Yang C T Miller and L D Turcoliver ldquoSimulation ofcorrelated and uncorrelated packing of random size spheresrdquoPhysical Review E - Statistical Physics Plasmas Fluids andRelated Interdisciplinary Topics vol 53 no 2 article 1516 1996

[7] A Cancelliere C Chang E Foti D H Rothman and SSucci ldquoThe permeability of a random medium comparison ofsimulationwith theoryrdquoPhysics of Fluids A FluidDynamics vol2 no 12 pp 2085ndash2088 1990

[8] B Ghanbarian A G Hunt R P Ewing and M SahimildquoTortuosity in porous media a critical reviewrdquo Soil ScienceSociety of America Journal vol 77 no 5 pp 1461ndash1477 2013

[9] X Zhang andM A Knackstedt ldquoDirect simulation of electricaland hydraulic tortuosity in porous solidsrdquoGeophysical ResearchLetters vol 22 no 17 pp 2333ndash2336 1995

[10] E Sanchez-Palencia ldquoHomogenization method for the studyof composite mediardquo in Asymptotic Analysis II pp 192ndash214Springer 1983

[11] J Bear and A H Cheng Modeling Groundwater Flow andContaminant Transport vol 23 of Theory and Applications ofTransport in Porous Media Springer 2010

[12] P Mostaghimi M J Blunt and B Bijeljic ldquoComputationsof absolute permeability on micro-CT imagerdquo MathematicalGeosciences vol 45 no 1 pp 103ndash125 2013

[13] A Duda Z Koza and M Matyka ldquoHydraulic tortuosity inarbitrary porous media flowrdquo Physical Review E vol 84 no 3Article ID 036319 pp 1ndash8 2011

[14] A Ebrahimi Khabbazi J S Ellis and A Bazylak ldquoDevelopinga new form of the Kozeny-Carman parameter for structuredporous media through lattice-Boltzmann modelingrdquo Comput-ers amp Fluids vol 75 pp 35ndash41 2013

[15] H Andra N Combaret J Dvorkin et al ldquoDigital rock physicsbenchmarksmdashpart II computing effective propertiesrdquoComput-ers amp Geosciences vol 50 pp 33ndash43 2013

[16] A Nabovati J Hinebaugh A Bazylak and C H Amon ldquoEffectof porosity heterogeneity on the permeability and tortuosity ofgas diffusion layers in polymer electrolyte membrane fuel cellsrdquoJournal of Power Sources vol 248 pp 83ndash90 2014

[17] P Carman ldquoFluid flow through granular bedsrdquo Transactions-Institution of Chemical Engineers vol 15 1937

[18] P C Carman ldquoPermeability of saturated sands soils and claysrdquoThe Journal of Agricultural Science vol 29 no 2 pp 262ndash2731939

[19] A Koponen M Kataja and J Timonen ldquoTortuous flow inporous mediardquo Physical Review E vol 54 no 1 pp 406ndash4101996

[20] W D Carrier III ldquoGoodbye Hazen hello Kozeny-CarmanrdquoJournal of Geotechnical and Geoenvironmental Engineering vol129 no 11 pp 1054ndash1056 2003

[21] J D Hyman P K Smolarkiewicz and C Larrabee WinterldquoPedotransfer functions for permeability a computationalstudy at pore scalesrdquo Water Resources Research vol 49 no 4pp 2080ndash2092 2013

[22] F D E Latief and U Fauzi ldquoKozeny-Carman and empiricalformula for the permeability of computer rock modelsrdquo Inter-national Journal of RockMechanics andMining Sciences vol 50pp 117ndash123 2012

[23] P Xu and B Yu ldquoDeveloping a new form of permeability andKozenyndashCarman constant for homogeneous porous media bymeans of fractal geometryrdquo Advances in Water Resources vol31 no 1 pp 74ndash81 2008

[24] M Matyka A Khalili and Z Koza ldquoTortuosity-porosityrelation in porous media flowrdquo Physical Review E vol 78 no2 Article ID 026306 2008

[25] T Ohkubo ldquoTortuosity based on anisotropic diffusion processin structured plate-like obstacles by Monte Carlo simulationrdquoTransport in Porous Media vol 72 no 3 pp 339ndash350 2008

[26] A Koponen M Kataja and J Timonen ldquoPermeability andeffective porosity of porous mediardquo Physical Review E vol 56no 3 pp 3319ndash3325 1997

[27] Z Sun X Tang and G Cheng ldquoNumerical simulation fortortuosity of porous mediardquo Microporous and MesoporousMaterials vol 173 pp 37ndash42 2013

[28] C David ldquoGeometry of flow paths for fluid transport in rocksrdquoJournal of Geophysical Research vol 98 no 7 pp 12ndash278 1993

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

24 Geofluids

[29] M Quintard L Bletzacker D Chenu and S Whitaker ldquoNon-linear multicomponent mass transport in porous mediardquoChemical Engineering Science vol 61 no 8 pp 2643ndash26692006

[30] L Shen and Z Chen ldquoCritical review of the impact of tortuosityon diffusionrdquo Chemical Engineering Science vol 62 no 14 pp3748ndash3755 2007

[31] B P Boudreau ldquoThe diffusive tortuosity of fine-grained unlithi-fied sedimentsrdquo Geochimica et Cosmochimica Acta vol 60 no16 pp 3139ndash3142 1996

[32] M M Ahmadi S Mohammadi and A N Hayati ldquoAnalyticalderivation of tortuosity and permeability of monosized spheresa volume averaging approachrdquo Physical Review E - StatisticalNonlinear and Soft Matter Physics vol 83 no 2 article 0263122011

[33] A Nabovati and A C M Sousa ldquoFluid flow simulation inrandom porous media at pore level using the lattice Boltzmannmethodrdquo Journal of Engineering Science and Technology vol 2no 3 pp 226ndash237 2007

[34] Z Koza M Matyka and A Khalili ldquoFinite-size anisotropy instatistically uniform porous mediardquo Physical Review E vol 79no 6 Article ID 066306 pp 1ndash7 2009

[35] Y Liu and P K Kitanidis ldquoTortuosity and Archiersquos lawrdquo inAdvances in Hydrogeology P K Mishra and K L KuhlmanEds chapter 6 pp 115ndash126 Springer New York NY USA 2013

[36] F H Harlow and J E Welch ldquoNumerical calculation oftime-dependent viscous incompressible flow of fluid with freesurfacerdquoThe Physics of Fluids vol 8 no 12 pp 2182ndash2189 1965

[37] S Sun A Salama and M F El Amin ldquoMatrix-oriented imple-mentation for the numerical solution of the partial differentialequations governing flows and transport in porous mediardquoComputers and Fluids vol 68 pp 38ndash46 2012

[38] J R Fanchi ldquoMathematics of fluid flowrdquo in Petroleum Engi-neering Handbook L W Lake Ed chapter 2 pp 45ndash76Society of Petroleum Engineers 2007 httppetrowikiorgPEH3AMathematics of Fluid Flow

[39] A Ghassemi and A Pak ldquoPore scale study of permeabilityand tortuosity for flow through particulate media using LatticeBoltzmann methodrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 35 no 8 pp 886ndash9012011

[40] M Icardi G Boccardo D L Marchisio T Tosco and RSethi ldquoPore-scale simulation of fluid flow and solute disper-sion in three-dimensional porous mediardquo Physical Review EmdashStatistical Nonlinear and Soft Matter Physics vol 90 no 1Article ID 013032 2014

[41] M M Clark Transport Modeling for Environmental Engineersand Scientists John Wiley amp Sons 2nd edition 2009

[42] J A Ochoa S Whitaker and P Stroeve ldquoDetermination ofcell membrane permeability in concentrated cell ensemblesrdquoBiophysical Journal vol 52 no 5 pp 763ndash774 1987

[43] J Kim J Ochoa and S Whitaker ldquoDiffusion in anisotropicporousmediardquo Transport in PorousMedia vol 2 no 4 pp 327ndash356 1987

[44] A E Saez J C Perfetti and I Rusinek ldquoPrediction of effectivediffusivities in porous media using spatially periodic modelsrdquoTransport in Porous Media vol 6 no 2 pp 143ndash157 1991

[45] S Beyhaghi andKM Pillai ldquoEstimation of tortuosity and effec-tive diffusivity tensors using closure formulation in a sinteredpolymer wick during transport of a nondilute multicomponentliquidmixturerdquo Special Topics and Reviews in PorousMedia vol2 no 4 pp 267ndash282 2011

[46] Y Yong X Lou S Li C Yang and X Yin ldquoDirect simulationof the influence of the pore structure on the diffusion processin porous mediardquo Computers amp Mathematics with Applicationsvol 67 no 2 pp 412ndash423 2014

[47] B R Gebart ldquoPermeability of unidirectional reinforcements forRTMrdquo Journal of Composite Materials vol 26 no 8 pp 1100ndash1133 1992

[48] L Rayleigh ldquoLVIOn the influence of obstacles arranged in rect-angular order upon the properties of a mediumrdquo PhilosophicalMagazine Series 5 vol 34 no 211 Article ID 481502 pp 481ndash502 1892

[49] D Ryan R Carbonell and S Whitaker ldquoA theory of diffusionand reaction in porous mediardquo in Proceedings of the AmericanInstitute of Chemical Engineers vol 71 of AIChE SymposiumSeries 202 pp 46ndash62 January 1981

[50] J S Mackie and P Meares ldquoThe diffusion of electrolytes in acation-exchange resin membrane I Theoreticalrdquo Proceedingsof the Royal Society A Mathematical Physical and EngineeringSciences vol 232 no 1191 pp 498ndash509 1955

[51] S B Iversen V K Bhatia K Dam-Johansen and G JonssonldquoCharacterization of microporous membranes for use in mem-brane contactorsrdquo Journal ofMembrane Science vol 130 no 1-2pp 205ndash217 1997

[52] H L Weissberg ldquoEffective diffusion coefficient in porousmediardquo Journal of Applied Physics vol 34 no 9 pp 2636ndash26391963

[53] M C Sukop H Huang P F Alvarez E A Variano and K JCunningham ldquoEvaluation of permeability and non-Darcy flowin vuggy macroporous limestone aquifer samples with latticeBoltzmann methodsrdquo Water Resources Research vol 49 no 1pp 216ndash230 2013

[54] J Comiti and M Renaud ldquoA new model for determiningmean structure parameters of fixed beds from pressure dropmeasurements application to beds packedwith parallelepipedalparticlesrdquo Chemical Engineering Science vol 44 no 7 pp 1539ndash1545 1989

[55] E du Plessis S Woudberg and J Prieur du Plessis ldquoPore-scalemodelling of diffusion in unconsolidated porous structuresrdquoChemical Engineering Science vol 65 no 8 pp 2541ndash2551 2010

[56] L Pisani ldquoSimple expression for the tortuosity of porousmediardquoTransport in Porous Media vol 88 no 2 pp 193ndash203 2011

[57] J A Ochoa-Tapia P Stroeve and S Whitaker ldquoDiffusivetransport in two-phase media spatially periodic models andmaxwellrsquos theory for isotropic and anisotropic systemsrdquo Chem-ical Engineering Science vol 49 no 5 pp 709ndash726 1994

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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