description of non-darcy ﬂows in porous medium systems · 2020. 10. 9. · of darcy’s law is...

PHYSICAL REVIEW E 87, 033012 (2013)

Description of non-Darcy flows in porous medium systems

Amanda L. Dye,1 James E. McClure,2,* Cass T. Miller,1 and William G. Gray11Department of Environmental Sciences and Engineering, University of North Carolina at Chapel Hill,

Chapel Hill, North Carolina 27599-7431, USA2Advanced Research Computing, Virginia Tech, Blacksburg, Virginia 24061-0123, USA

(Received 25 August 2012; published 18 March 2013)

Fluid flow through isotropic and anisotropic porous medium systems is investigated for a range of Reynoldsnumbers corresponding to both Darcy and non-Darcy regimes. A non-dimensional formulation is developed fora Forchheimer approximation of the momentum balance, and lattice Boltzmann simulations are used to elucidatethe effects of porous medium characteristics on macroscale constitutive relation parameters. The geometricorientation tensor of the solid phase is posited as a morphological measure of leading-order importance forthe description of anisotropic flows. Simulation results are presented to confirm this hypothesis, and parametercorrelations are developed to predict closure relation coefficients as a function of porous medium porosity, specificinterfacial area of the solid phase, and the geometric orientation tensor. The developed correlations provideimproved estimates of model coefficients compared to available estimates and extend predictive capabilitiesto fully determine macroscopic momentum parameters for three-dimensional flows in anisotropic porousmedia.

DOI: 10.1103/PhysRevE.87.033012 PACS number(s): 47.56.+r, 81.05.Xj

I. INTRODUCTION

The equations that describe single-fluid-phase flow providea fundamental basis for our understanding of the influenceof of the pore morphology and topology on momentumtransport in porous media. Expressions such as Darcy’s law andForchheimer’s equation describe flow processes in an averagedapproximate sense, such that the complex microscopic struc-ture typical of porous media can be effectively described usinga small number of macroscopic parameters [1,2]. The capacityto describe flow processes for a wide range of pore structuresis contingent on our ability to identify and quantify themorphological and topological properties of a porous mediumthat are of leading-order importance. Existing correlationspredict parameters associated with approximations to theconservation of momentum equation using only the porosityand Sauter-mean diameter as the primary measures of porousmedium morphology [3–5]. These correlations have beendeveloped for restricted ranges of Reynolds numbers (Re)and porous medium morphologies. Furthermore, the availableparametric correlations are insufficient to describe anisotropicsystems where directional dependent effects impact the flowbehavior. Proper mathematical description of anisotropy isessential to predict transport phenomena in many naturallyoccurring porous media.

The current state of knowledge categorizes single-fluid-phase flow into two primary regimes: a creeping flow regime,corresponding to low flow velocities in which viscous forcescompletely dominate inertial forces, and a regime correspond-ing to flow velocities for which inertial forces cannot beneglected [6–13]. These two regimes are most commonlymodeled by two approximations of the macroscale momentumequation for single-fluid-phase flow through a porous media:Darcy’s law and Forchheimer’s equation, respectively. Theone-dimensional forms of these equations, which apply to

*[email protected]

isotropic flows, are familiar and have been studied extensively[1–3,14–19]. More recent work has focused on deriving thesetraditional models for single-fluid-phase flow from first prin-ciples and linking macroscopic phenomena with microscopicflow behavior [6,20–26]. Investigations have also linked themacroscopic permeability to tortuosity in both isotropic andanisotropic porous media [27–30]. While the anisotropic formof Darcy’s law is relatively well established, the proper formof the inertial correction term remains a matter of debate andpredictive correlations for parameters in the anisotropic caseare unavailable at present.

As details of porous medium microstructure become in-creasingly accessible, opportunities exist to use this informa-tion to advance a more complete macroscopic understanding offlow through porous media. Computed microtomography tech-niques can provide three-dimensional, high-resolution imagesthat reveal the pore structure for a wide range of materials[31]. Numerical simulation techniques permit the study ofincreasingly large systems that can approach or exceed thesize required for system properties to become size independentprovided that systems are homogeneous at the macroscale.The emergence of these technologies provides ample oppor-tunities to exploit information collected in the field, such asimaging of core data. Most typically, properties such as thepermeability and porosity are considered [32–34]. However,we suggest that the amount of predictive information that canbe extracted from such data can be extended significantly basedon current technology, particularly for anisotropic porousmedia.

The present state of knowledge is hindered by an in-complete understanding of non-Darcy flow in general atthe macroscale, insufficient links between microscale porousmedium morphology and anisotropic flow characteristics, andthe lack of generally applicable estimations of parametersappearing in common flow equations. The development ofthe thermodynamically constrained averaging theory providesan opportunity to advance such descriptions by explicitlyconnecting spatial scales and identifying critical variables

033012-11539-3755/2013/87(3)/033012(14) ©2013 American Physical Society

http://dx.doi.org/10.1103/PhysRevE.87.033012borregoTypewritten TextCopyright by the American Physical Society. Dye, Amanda L. ; McClure, James E. ; Miller, Cass T. ; et al., Mar 18, 2013. “Description of non-Darcy flows in porous medium systems,” PHYSICAL REVIEW E 87(3): 033012. DOI: 10.1103/PhysRevE.87.033012.

DYE, MCCLURE, MILLER, AND GRAY PHYSICAL REVIEW E 87, 033012 (2013)

expected to affect single-fluid-phase flow [35]. This workcapitalizes on that opportunity.

The primary objectives of this work are(i) to develop a means to generate systematically isotropic

and anisotropic porous medium systems with varying proper-ties;

(ii) to perform highly resolved microscale simulations fora wide variety of porous medium systems at a scale that issufficient to ensure a representative elementary volume (REV)for describing macroscale systems and hence the generality ofthe results;

(iii) to develop a formulation of the macroscale momentumresistance tensor that is consistent with microscale observa-tions of anisotropic, non-Darcy flow behavior;

(iv) to identify porous medium characteristics that are ofleading-order importance for determining parameter valuesappearing in macroscale equations used to describe fluid flow;

(v) to derive an expression that can be used to relate mediaproperties to macroscale flow parameters; and

(vi) to compare parameter correlations developed basedupon this work with existing correlations.

II. BACKGROUND

A. Non-Darcy flow

The three-dimensional momentum equation forsingle-phase flow in anisotropic porous media is approximatedby [35]

−∇pw + ρwgw = R̂w·vw, (1)where pw is the volume-averaged fluid pressure of phasew, ρw is the volume-averaged mass density, gw is themass-averaged gravitational acceleration vector, and vw isthe mass-averaged velocity vector. The momentum resistancetensor R̂

wmust be symmetric and positive semidefinite

to ensure that the entropy production is non-negative. Weconsider Eq. (1) written in the form

ψ |−∇pw + ρwgw| = R̂w·ω|vw|. (2)Unit vectors provide the force orientation,

ψ = −∇pw + ρwgw

|−∇pw + ρwgw| , (3)

and flow orientation,

ω = vw

|vw| . (4)

Since ψ and ω are dimensionless unit vectors, the dimensionedvariables of importance can be identified as

|−∇pw + ρwgw| [m/(l2t2)], (5)

|vw| [l/t], (6)

ρw [m/l3], (7)

μ̂w [m/(lt)], (8)

�ws [1/l], (9)

where μ̂w is the dynamic viscosity and �ws is the surface area ofthe ws interface per unit volume, which is equal to the specificinterfacial area of the solid for this single-fluid-phase system.The surface-to-volume ratio is incorporated by using the Sautermean diameter as the representative length scale for the flow

d = 6 �s

�ws, (10)

where �s is the volume fraction of the solid phase.For convenience, Eq. (2) can be arranged into a dimen-

sionless form to simplify the task of analyzing parametricdependence on medium properties. Multiplying Eq. (2) byρwd3/(μ̂w)2 leads to the expression

ψFc = d2

μ̂wR̂

w · ωRe, (11)

where the dimensionless forcing term is

Fc = ρwd3|−∇pw + ρwgw|

(μ̂w)2(12)

and the Re is

Re = ρwd|vw|μ̂w

. (13)

The momentum resistance tensor is symmetric and positivesemidefinite and can be written in the form

d2

μ̂wR̂

w = Qw·�w·(Qw)−1, (14)

where the column vectors of Qw are a set of orthonormaleigenvectors of R̂

w, denoted qwi , i = 1,2,3. The diagonal

tensor �w contains the associated eigenvalues �wi . Theeigenvectors of R̂

ware assumed to be a property of the porous

medium only.The dependence of the momentum resistance on the flow

velocity has been studied extensively for one-dimensionalflows [2,6,8,21,22]. This dependence is introduced into thethree-dimensional form by assuming that the eigenvalues �wiare a function of Re. For sufficiently small Re, a linearexpansion can be applied to describe the flow behavior

�wi (Re) ≈ �ai + �bi Re. (15)Based on this expansion the momentum resistance tensor takesthe form

d2

μ̂wR̂

w = A + BRe, (16)

where the Darcy momentum resistance tensor is

A = Qw·�a·(Qw)−1, (17)and the inertial momentum resistance tensor is

B = Qw·�b·(Qw)−1. (18)The diagonal tensors �a and �b provide the momentum re-sistance coefficients associated with the Darcy and non-Darcyflow effects. The diagonal values �ai specify the momentumresistance associated with qwi within the Darcy regime andthe diagonal values �bi account for the inertial correction thatbecomes important in the non-Darcy regime. While there are

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DESCRIPTION OF NON-DARCY FLOWS IN POROUS . . . PHYSICAL REVIEW E 87, 033012 (2013)

three independent components of vw, the dependence on Reonly considers the influence of the lone invariant quantity |vw|.The remaining two independent quantities are provided by theflow orientation ω. If A and B are independent of ω, Eq. (16) isa dimensionless form for the momentum resistance proposedby Wang et al. [27]. However, the directional-dependent flowbehavior observed by McClure et al. suggests that the inertialmomentum resistance tensor must also be a function of ω forcertain systems [36].

It would be advantageous to be able to predict A and Bbased on the knowledge of a small set of easily measuredquantities. These tensors provide a macroscopic measure ofthe momentum resistance associated with a particular porousmedium and, therefore, depend on measures of the poremorphology and topology. In the dimensionless formulationthis influence can be incorporated by identifying a set ofdimensionless measures that can be functionally related tothe components of A and B to produce predictive forms.

B. Correlations for flow in isotropic porous media

When the porous medium is isotropic, the tensors appearingin Eq. (16) must reduce to

A = a∗I, (19)

B = b∗I, (20)

where a∗ and b∗ are momentum resistance coefficients thatapply for flows in isotropic porous media. Such flows can bedescribed using the resulting one-dimensional counterpart toEq. (11):

Fc = a∗Re + b∗Re2. (21)The dependence of the coefficients a∗ and b∗ on the

porous medium morphology has been studied extensively[4,5,37–40]. With surface-to-volume effects accounted forby the length scale definition given in Eq. (10), existingcorrelations typically predict the coefficient values a∗ andb∗ as functions of the volume fraction of the w phase �w,or, equivalently, the porosity for this single-fluid-phase case,which is given by

ε = 1 − �s. (22)The most well known of these expressions is Ergun’s equa-tion, which specifies functional forms for the dimensionlesspermeability,

a∗e =ε2

α∗e (1 − ε)2, (23)

and inertial coefficient,

b∗e = β∗e(1 − ε)

ε. (24)

Ergun [3] suggested that α∗e = 150 and β∗e = 1.75, but severalvalues for the coefficients have been proposed. When α∗e =180, Eq. (23) is the widely used Carman-Kozeny relationship[41]. MacDonald et al. [5] determined that 1.8 � β∗e � 4.0served to match experimental data from six different porousmedium systems. For the case of Darcy flow, Rumpf and Gupte

[4] predict the permeability using the relationship

a∗rg = 5.6ε−5.5. (25)Pan et al. [39] found that the Carmen-Kozeny relationship

underestimates the permeability and observed deviations fromthe Rumpf-Gupte relation when systems outside the rangeof experimental support for this expression were considered.They proposed an alternative correlation form in whichan additional dimensionless variable, the relative standarddeviation σ̃D , was also included, whereby

a∗p = α∗p1εα

∗p2(

1 + α∗p3σ̃α∗p4D

) , (26)where α∗p1, α

∗p2, α

∗p3, and α

∗p4 are best-fit coefficients.

III. METHODS

The objectives of this work are accomplished using asimulation approach that relies upon the generation of a largeset of isotropic and anisotropic porous medium systems withspecified variations in size, shape, orientation, and porosity.The factors of leading-order importance for non-Darcy flowin anisotropic systems must first be posited and then evaluatedand compared to simulation results. Simulations must behighly resolved microscale simulations of sufficient sizeto yield a reliable measure of a macroscale system. Eachcomponent of these methods is described in turn.

A. Generation of sphere and ellipsoid packs

Surrogate porous media were constructed by generatingsphere packs with log-normally distributed radii ri . Thesphere centroids, ci , i = 1,2, . . . ,Ns , were determined by acollective rearrangement algorithm designed for applicationsin porous media. The algorithm is a modified version of theone developed by Williams and Philipse to generate packsof sphereocylinders [42]. Our version of the algorithm isconstructed to provide precise control over the final systemporosity ε and to accommodate a lognormal size distributionfor the sphere radii. Initially, a system of spheres is instantiatedinto a domain of fixed size. The initial sphere radii areassigned from a lognormal distribution with variance σ 2 and aninitial mean μ0. The sphere centroids are initially distributedrandomly based upon a uniform distribution throughout thedomain, which is a rectangular prism with sides of lengthLx,Ly , and Lz. The initialization is summarized in thefollowing pseudocode:

for i = 1,2, . . . ,Ns dolog(ri) ← Normal(μ0,σ 2)ci,x ← Uniform(0,Lx)ci,y ← Uniform(0,Ly)ci,z ← Uniform(0,Lz)

end forThe value of μ0 is chosen so that the initial porosity is high

based on the size of the system (in our case the initial porosity isε0 = 0.80). Once the initial system of spheres has been created,a collective rearrangement algorithm is applied to sequentiallyincrease the size of the radii ri using a constant rescalingfactor λ, thereby decreasing ε until the desired porosity isreached. As the sizes of the radii are increased, the value of

033012-3


μ will increase and the system porosity will decrease. Eachtime the radii are rescaled an iterative procedure is applied toshift the sphere centroids so that all overlaps are eliminated.The direction to shift each sphere is determined by the overlapvector xi . The parameter κ was introduced in order to providea way to tune the rate of convergence, which is increasinglyimportant as σ 2 increases. This is due to the fact that sphereoverlaps frequently occur between spheres of widely disparateradii as σ 2 increases. A value of κ = 0.6 was sufficient for thesystems considered in this work. However, it may be necessaryto consider smaller values of κ to generate systems with largervalues of σ 2. The rescaling procedure is summarized in thefollowing pseudocode:

while∑

i

4πr3i3 < (1 − ε)LxLyLz do

ri ← λriμ ← μ + log(λ)while maxi,j (|ci − cj | − ri − rj ) > 1 × 10−10 do

for i = 1,2, . . . ,Ns do

xi =

∑j �=i[|ci − cj | − (ri + rj )](ci − cj )

end forfor i = 1,2, . . . ,Ns do

ci ← ci + κxiend for

end whileend while

Full periodic boundary conditions are enforced, which elimi-nates the potential for boundary effects. For the homogeneouscase σ 2 = 0 the minimum achievable porosity approaches0.367, which is a well-established result for random closepacks of equally sized spheres [43,44]. Lower porosity valuesbecome accessible as σ 2 increases. The variance σ 2 is specifiedas an input parameter and is unchanged by the rescalingprocedure. The final value of μ is determined from the desiredporosity, which can be estimated using the equation

ε = 1 − 4πNsE[r3i

]3LxLyLz

, (27)

where the expected value of the cube of the radius is

E[r3i

] = exp (3μ + 92σ 2). (28)Then insertion of Eq. (28) into Eq. (27) and solution for μyields

μ = 13

log

(3 (1 − ε) LxLyLz

4πNs

)− 3

2σ 2, (29)

which relates the final mean radius for the lognormal distribu-tion to the two independent input parameters ε and σ 2.

The coordination number was used to evaluate the stabilityof each packing in a gravitational field. For a coordinationnumber greater than 6, the average number of spheressupporting each sphere is at least three. This threshold wasused to establish an upper limit on the maximum porosity fora stable packing at a prescribed variance. The porosity rangesobtained for each variance are listed in Table I.

Anisotropic systems were constructed by applying themapping {ηx,γy,ζ z} → {x ′,y ′,z′} to sphere packs. In thiswork the stretch factors fall within the range of 1 � η,γ,ζ �1.7. This procedure stretches the sphere packs to yield a systemof axially aligned ellipsoids. The resulting ellipsoid surfaces

TABLE I. Range of stable, accessible porosity values for log-normal sphere packs.

Variance (σ 2) Porosity range

0 0.37–0.600.1 0.35–0.540.2 0.32–0.480.3 0.30–0.42

are given by the equation(

x ′ − c′xηr

)2+

(y ′ − c′y

γ r

)2+

(z′ − c′z

ζ r

)2= 1, (30)

where {c′x,c′y,c′z} = {ηcx,γ cy,ζ cz}. This mapping preservesboth media porosity and grain contacts and can be usedto generate a sequence of anisotropic media by consideringdifferent stretch factors η, γ , and ζ applied to a givensphere packing. While alternative algorithms exist to generateanisotropic media, this approach is advantageous due to thefact that each anisotropic packing is directly associated withan isotropic counterpart. Changes in the coefficient values cantherefore be directly associated with changes in the systemanisotropy with a minimum number of confounding factors.

B. The orientation tensor as a quantitativemeasure of anisotropy

In order to extend the relationships developed for isotropicporous media to anisotropic systems, a dimensionless mor-phological measure of anisotropy must be identified andfunctionally related to A and B. In this work, we considerthe average geometric orientation tensor [45] for the ws, orsolid, surface as a quantitative measure of anisotropy,

Gws =∫�s

nsns dr∫�s

dr, (31)

where ns is the unit vector outward normal to the solidsurface �s .

The spectral decomposition of Gws provides a straightfor-ward way to determine the anisotropic properties of a porousmedium. The orientation tensor is symmetric and can beexpressed in the form

Gws = Qs·�s·(Qs)−1, (32)where orthonormal eigenvectors of Gws serve as the columnvectors of Qs . The eigenvectors are denoted qsi , i = 1,2,3. Thediagonal matrix �s contains the associated eigenvalues �si .Since ns is a unit vector, the first invariant of the orientationtensor places a constraint on the eigenvalues

tr(Gws) = tr(�s) = 1. (33)This implies that the tensor is uniquely specified according totwo independent quantities and establishes that �s1 = �s2 =�s3 = 1/3 for an isotropic system.

The eigenvectors provide the principal directions ofanisotropy and the eigenvalues quantify the relative surfaceorientation associated with each eigenvector. The significanceof these quantities is illustrated by the simple case of arectangular prism. Consider a rectangular prism for which Ax ,

033012-4


Ay , and Az denote the surface area of the x, y, and z faces,respectively. Based on Eq. (31), the associated orientationtensor can be computed as

Gws = 1Ax + Ay + Az

⎡⎣Ax 0 00 Ay 0

0 0 Az

⎤⎦ . (34)

For this geometry the eigenvectors align with the coordinateaxes and the eigenvalues are

�s0 =Ax

Ax + Ay + Az , (35)

�s1 =Ay

Ax + Ay + Az , and (36)

�s2 =Az

Ax + Ay + Az . (37)

Each eigenvalue �si has the straightforward interpretation asthe relative area of the part of the surface that has a normalvector aligned with qsi .

Since �si are dimensionless and provide a quantitativemeasure of anisotropy, it is natural to incorporate theminto functional forms to predict the coefficients �ai and�bi . Furthermore, if q

si can be used to approximate q

wi the

momentum resistance tensor R̂w

can be fully predicted usinga small set of morphological information: ε, �ws , and Gws . Thisposited leading-order dependency will be investigated througha series of highly resolved simulations.

C. Lattice Boltzmann scheme

Simulations were carried out using a three-dimensional, 19-velocity-vector (D3Q19), multi-relaxation-time lattice Boltz-mann (LB) scheme [46,47]. The LB scheme recovers theNavier-Stokes equations to second order and is valid withinboth the Darcy and the non-Darcy regimes. A solution forthe pore-scale velocity field vw was obtained by simulating asteady-state flow driven by an external force gw. Full periodicboundary conditions were used for all simulations, ensuringthat ∇pw = 0. The density was given by ρw = 1 for allsimulations. Prescribing gw in addition to the fluid viscosity μ̂w

directly determines the value of Fc and ψ for each steady-statesimulation. Based on the microscale steady-state velocity field,the macroscopic velocity was computed according to

vw =∫�w

ρwvw dr∫�w

ρw dr. (38)

The macroscopic velocity can be used to compute the Reaccording to Eq. (13), and ω may be computed from Eq. (4).

D. Determination of representative elementary volumes

To ensure that the macroscopic results are independent ofthe domain size, each simulated system was large enough tobe considered an REV. For the generated packing to be a validREV, the resistance tensor must be independent of both thelattice size of the simulated domain and the packing size ofthe generated media.

Simulations were performed to generate flows for a se-quence of Re < 120. The upper bound on Re was selected toensure that a steady state was achieved. Higher values of Re

TABLE II. Domain sizes and resolution necessary to achieve agrid-independent REV.

σ 2 Ns n3 D (pixels)

0 1500 3603 330.1 1500 3803 300.2 3000 4603 280.3 4000 4903 25

lead to unsteady flows in which vortex shedding may occur. Aleast-squares fit of Eq. (21) was applied to the simulation datato obtain the corresponding values of a∗ and b∗. To develop aporous medium that yielded grid-independent results, a systemcontaining a set number of spheres was packed and the latticesize was increased until the resistance tensor converged to onedefinitive solution. This determined the number of lattice sitesneeded to resolve the mean grain diameter:

D = exp [μ + 12σ 2]. (39)The resulting parameter values are listed in Table II. Since

the relative error for Eq. (21) was of order 1 × 10−2, it wasdetermined that was an appropriate threshold to determine theREV for the purposes of this work. REV size was determinedby incrementally increasing the number of spheres until thevariation in the parameters a∗ and b∗ decreased below 1 ×10−2, relative to the values a∗∞ and b

∗∞ obtained at the highestresolution. The relative variations of these quantities are shownas functions of sphere number for each variance of lognormaldistributed radii in Fig. 1. Table II lists the number of spheresand cubic lattice sizes that define an REV porous medium foreach variance of the log-normally distributed radii.

IV. RESULTS AND DISCUSSION

A. Approximating the eigenvectors of the momentumresistance tensor

In order to produce useful parameter estimates for theanisotropic form of the momentum resistance tensor, theeigenvectors of R̂

wmust be associated with morphological

properties of the solid phase. In an isotropic system every real-valued vector is an eigenvector of the momentum resistancetensor. This is readily apparent from Eq. (20). This is no longerthe case for anisotropic systems. In anisotropic systems theeigenvectors qwi correspond to those directions for which theforce and flow orientations align. Identification of qwi greatlysimplifies the task of generating parameter estimates of R̂

w

since it reduces the number of undetermined parameters. Sincethe eigenvectors of Gws provide a way to identify principaldirections of anisotropy in a given media, it is natural to relatethem to the eigenvectors of R̂

w.

The two-dimensional flows depicted in Fig. 2 providephysical insight into the significance of the eigenvectors innon-Darcy flows. In this case, the symmetry of the solidgeometry ensures that eigenvectors of Gws align with thecoordinate axes. When ψ aligns with one of these eigenvectors,the resulting flow orientation satisfies ω = ψ = qsi , as inFigs. 2(a), 2(b), 2(d), and 2(e). The implication is that for

033012-5


(a) Permeability (b) Inertial

FIG. 1. Coefficient values relative to the associated REV obtained for a∗ and b∗ for a range of sphere packs.

this case the eigenvectors of Gws are also eigenvectors of R̂w

.Figures 2(c) and 2(f) demonstrate that ψ and ω will generallynot be aligned in anisotropic systems.

Specifying the force orientation as ψ = qsi permits us toquantitatively evaluate whether qsi is an eigenvector of R̂

wfor a

particular porous medium. The corresponding flow orientationω can be obtained by simulating the steady-state velocityfield. If qsi is an eigenvector of R̂

w, qsi · ω = 1. The accuracy

of the eigenvector approximation qwi ≈ qsi can therefore beevaluated by computing |1 − ψ · ω|. This error is plotted inFig. 3 as a function of the Re. The properties of the associatedellipsoid packing were ε = 0.38, �s0 = 0.46, �s1 = 0.25, and�s2 = 0.29. These results demonstrate that the eigenvectors ofGws provide an accurate way to identify the eigenvectors ofR̂

wfor ellipsoid packs. The extensibility of this result to more

general anisotropic media requires further study. Based on thisapproximation, we can write

A = Qs·�a·(Qs)−1, (40)

B = Qs·�b·(Qs)−1. (41)

The task is thereby simplified to develop predictive relation-ships laws to approximate �a and �b. Specific forms arepresented in Sec. IV E.

B. Dependence on the flow orientation

It has been known for over a century that the inertialcorrection to Darcy’s law depends on the flow velocity [2,3].The precise form of this dependence has been the subjectof ongoing study [27,48,49]. For one-dimensional flows, thisdependence can be expressed in terms of Re without a lossof generality. For three-dimensional flows Re accounts foronly the invariant of the velocity vector, |vw|, leaving twoindependent components unaccounted for. The remainingtwo independent components specify the flow orientationω. Understanding how the momentum resistance depends

on the flow orientation represents an important challenge tounderstanding flow processes in anisotropic porous media.

The dependence of the momentum resistance on Re isalready established by Eq. (16). If the correction to Darcy’slaw is presumed to depend generally on vw, the implication isthat B is also a function of ω, of which only two componentsare independent. The first problem is that the components ofω are inherently determined by the coordinate system. Sincethe coordinate system is arbitrary, we must develop a way tostudy the impact of the flow orientation that is independent ofthis choice. Since the projection of the flow orientation ontothe eigenvectors qsi does not depend on the coordinate system,we consider how �b depends on the three quantities ω · qs0,ω · qs1, and ω · qs2.

Due to the fact that ω has a fixed unit length, a Taylor seriescannot be applied to approximate the associated impact on theflow behavior. As a consequence, determining the functionaldependence of B on the flow orientation requires empiricism.However, the form of this dependence is subject to restrictions.First, in isotropic porous media Eq. (20) must be recovered forall possible flow orientations. Second, entropy production mustbe positive for all Re. In order to satisfy these constraints, wepropose that the eigenvalues of B can be predicted by the form

�bi =�

b(0)i

∣∣ω · qs0∣∣ + �b(1)i ∣∣ω · qs1∣∣ + �b(2)i ∣∣ω · qs2∣∣∣∣ω · qs0∣∣ + ∣∣ω · qs1∣∣ + ∣∣ω · qs2∣∣ . (42)

The coefficients �b(j )i are functions of the porous mediummorphology that determine the inertial contribution to mo-mentum resistance in the direction of qsi that results from achange in the projection of ω on qsj . A total of nine coefficientsare necessary to specify B in addition to the eigenvectors. Thework of McClure et al. [36] suggests that �b(j )i will also dependon the signed values of ω · qsi for certain porous media. Thispossibility is not explored in this work.

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(a) 0

5 10-6

10-5

1.5 10-5

2 10-5

2.5 10-5

3 10-5

3.5 10-5

qs1qs0

(b) 0

2 10-6

4 10-6

6 10-6

8 10-6

10-5

1.2 10-5

1.4 10-5

1.6 10-5

1.8 10-5

2 10-5

2.2 10-5

2.4 10-5

2.6 10-5

qs1qs0

(c) 0

2 10-6

4 10-6

6 10-6

8 10-6

10-5

1.2 10-5

1.4 10-5

1.6 10-5

1.8 10-5

2 10-5

2.2 10-5

2.4 10-5

2.6 10-5

2.8 10-5

qs1qs0

(d) 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

qs1qs0

(e) 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

qs1qs0

(f) 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

qs1qs0

FIG. 2. (Color online) Streamlines resulting from Darcy (a)–(c) and non-Darcy (d)–(f) flows obtained using various force orientations ina simple periodic two-dimensional flow geometry. Darcy flows correspond to Re = 0.01 and non-Darcy flows correspond to Re = 100. Thegrayscale (color) value represents the magnitude of the flow velocity |vw|.

C. Physical significance of flow coefficients and eigenvectors

Recent work investigating the microscopic origins ofinertial behavior in porous medium flows has linked theformation of eddies with macroscopic deviations from Darcy’slaw [7,8,36,50]. This insight is particularly useful whenattempting to understand the three-dimensional form givenin Eq. (11). Eddies associated with the onset of inertial effectsdistort the flow field such that the resistance to flow increases.In anisotropic systems the size, shape, and location of these ed-dies differ depending on the flow orientation, and correspond-ing differences in the macroscopic flow behavior result. Thiseffect is evident upon considering the simple two-dimensionalflows shown in Fig. 2. Three flow orientations were consideredin a periodic flow domain, providing qualitative insight intothe microscopic phenomena that must be accounted for bymacroscopic coefficient values in anisotropic flows.

In Darcy flows, streamlines are independent of Re andinvariant upon flow reversal, as shown in Figs. 2(a)–2(c). As aresult the eigenvalues �ai are properties of the porous mediumonly. The three eigenvalues can be measured by conducting a

set of three flow experiments in which flow orientations arealigned with each of the eigenvectors.

Inertial flows present a more challenging scenario; stream-lines are no longer independent of Re as inertial effects begin todistort the flow field. For a sufficiently high Re, the formationof eddies will be observed. The size, shape, and position ofthese eddies depend on both the Re and the flow orientation.Macroscopic coefficients must account for the impact that eddyformation has on the resulting flow behavior for all possibleflow orientations. When the direction of ψ is aligned with oneof the eigenvectors of Gws , as shown in Figs. 2(d) and 2(e),the eddies formed do not alter the eigenvectors of R̂

w. As a

consequence, qwi values are presumed to be independent of Re.The coefficients �b(i)i determine the rate that momentum

resistance increases due to the inertial distortions of the flowfield that result when flow is aligned with qsi . The coefficient�

b(j )i , j �= i, contributes to the momentum resistance only if

ω · qsi and ω · qsj are both nonzero. These coefficient valuesprovide a macroscopic measure of the momentum resistancecaused by the varying size, shape, and position of the eddies

033012-7


0.1 1 10 100

10-6

0

1

2

3

4

5

6

7

8

Error for Eigenvector Approximation

Re

|1-

•|

FIG. 3. Error for the approximation qwi ≈ qsi in an axially alignedellipsoid packing.

that form for different flow orientations. This picture isdescribed qualitatively in Fig. 2(f). A comparison of Figs. 2(f)to 2(d) and 2(e) reveals distinct qualitative differences betweenthe flow fields. The necessity of the coefficients �b(j )i , j �= i,can be assessed by considering their capacity to improvethe accuracy of the estimated three-dimensional flow in ananisotropic porous medium.

The three-dimensional analog of Fig. 2 was used to evaluatethe usefulness of Eq. (42). For simplicity, a symmetricgeometry was selected to satisfy �s1 = �s2. If B is independentof ω, only the three eigenvalues are needed to fully specifythe form of the tensor, reducing to the form proposed byWang et al. [27]. Flows aligned with the eigenvectors aresufficient to predict all three coefficients. In Fig. 4, flowsaligned with eigenvectors qs0 and q

s1 are labeled Case A and

Case B, respectively. The relative error of this approximationis determined by how well the quadratic form involving theRe matches the simulated data. In Case C, the flow orientationwas selected to be (qs0 + qs1)/2. Based on the form of Wanget al. [27], Cases A and B provide all of the coefficients needed

Case A (1-D)Case B (1-D)Case C (Wang et. al.)Case C (This Work)

0.01 0.1 1 10 100

0.001

0.01

0.1

Re

|-(

[A+

B R

e]•

Re)

/Fc

|

FIG. 4. (Color online) Relative error associated with variouspredictive forms plotted as a function of Re. Accurate prediction ofthree-dimensional flow behavior requires all coefficients appearingin Eq. (11).

to predict the flow behavior in Case C. However, comparing thepredicted and simulated data demonstrates that while this formpredicts the flow behavior quite well in the Darcy regime, therelative error increases substantially with the onset of inertialeffects at higher Re.

The implication of this result is that differences in thelocation, size, and shape of eddies are dependent on theflow orientation, and these differences are manifested by amacroscopic difference in the flow behavior. To describe thisflow behavior accurately, the inertial tensor B must dependon ω. Flow data from Cases A, B, and C were then usedto determine the associated coefficients, �a0,�

a1 = �a2 and

�b(0)0 ,�

b(1)0 = �b(2)0 , �b(0)1 = �b(0)2 ,�b(1)2 = �b(2)1 , and �b(1)1 =

�b(2)2 , where the equalities are implied by the symmetry of the

flow geometry. The description of the inertial regime improvessignificantly when the effects of flow orientation are accountedfor due to Eq. (42).

D. Parametric estimates for flow in isotropic porous media

Simulations of non-Darcy flow were performed in isotropicsphere packs representing the full range of porosities andvariances listed in Table I and Re < 120. Existing correlationsfor the dimensionless permeability and inertial parameter werethen compared to these simulation results.

In Fig. 5, the simulation results of a specific isotropic caseare shown. Based on the simulated data points, the relative erroris minimized for a∗ and b∗ based on Eq. (21). As a functionof the Re, the relative error exhibits a similar profile for theisotropic case compared to the anisotropic profile plotted inFig. 4. Based on our analysis, the accuracy of Eq. (21) iswithin several percent. Since this is true for both isotropic andanisotropic cases, alternative forms which more accuratelyapproximate the transition region from 1 < Re < 50 wouldbe a logical way to improve the predictive capabilities of thismodel. The consideration of such forms is beyond the scopeof this work.

The functional forms listed in Eqs. (23)–(26) do not fullymatch the simulated data throughout the range of porosityvalues considered. This is unsurprising given that the range ofporosity values considered here is broader than what has beenconsidered in other studies. The Pan et al. [39] permeabilityrelation was based on a set of flow simulations within aporosity range of 0.33 � ε � to 0.45. At the porositiesoutside the experimentally supported range, the Pan et al. [39]relation underestimates the measured permeability values,demonstrated in Fig. 6. The Carmen-Kozeny relation for lowerporosities (ε � 0.42) underestimates the associated coefficient(Fig. 6). The lower porosity deviations are consistent withthe findings of others [14,16,39,51,52]. The Rumpf-Guptepermeability relation deviates the most from the simulateddata, overpredicting the data for ε � 0.38 and underpre-dicting for ε � 0.38. The Rumpf-Gupte data are based onflow experiments using sphere packs with relative standarddeviations of the sphere-size distribution, σ̃D , of 0.0945, 0.32,and 0.327 over a wide range of porosities (0.366 � ε �0.64)and 0 < Re < 100 [4,5]. The cases with σ 2 = 0.3 and 0.36 <ε < 0.42 are the only simulations that fit entirely within theexperimentally supported regime of the Rumpf-Gupte relation.The data within the Rumpf-Gupte experiments correlate with

033012-8


0.1 1 10 100

101

102

103

104

105

Re

Fc

(a) Fc plotted as a function of Re.

0.01 0.1 1 10 100

0.001

0.01

0.1

Re

|([A

+B

Re]

Re)

/Fc|

(b) Relative error associated with the predictive form plottedas a function of Re where a∗ = 452.75 and b∗ = 2.811.

FIG. 5. Simulated data points and best-fit curve for an isotropic sphere pack with a porosity of 0.37.

the porosity range where deviations are at a minimum, anobservation supported by others [39,51,53].

We found that an exponential fit yields more satisfactoryagreement for the full range of the simulation data. Theassociated functional form is

a∗(ε) = α∗1 exp(α∗2ε), (43)where the best-fit coefficient values are α∗1 = 1.95 × 10−5 andα∗2 = 9.85 based on the simulated data points plotted in Fig. 6.

The inertial correction to Darcy’s law is shown in Fig. 7,The coefficient b∗ measured from simulation increases as theporosity decreases. Simulated values for the inertial parameterare shown in comparison with values predicted by the Ergunfunctional form [Eq. (24)] in Fig. 7. Neither of the inertialcoefficients predicted by Ergun or MacDonald provide asatisfactory fit to the simulation data; both relations overpredict

Carman-KozenyRumpf-GumptePan et. al.Proposed Model

2=02=0.12=0.22=0.3

0.30 0.35 0.40 0.45 0.50 0.55 0.60

0.0005

0.001

0.002

0.005

0.01

1 / a

*

FIG. 6. (Color online) Comparison of isotropic constitutive lawsthat predict a∗ with simulated data.

the lower range of porosities (0.40). The Ergun relation wasqualitatively based on a straight tube geometry of the porespace and has been found to be valid only in a range of Revalues, 0 � Re � 75 [3,5]. Based on the packing parametersset by the REV calculations (Table II), the range of Resimulated differs for each variance. The Re range is 0 � Re �140 for the homogeneous packing (σ 2 = 0), 0 � Re � 130 forσ 2 = 0.1, 0 � Re � 115 for σ 2 = 0.2, and 0 � Re � 95 forσ 2 = 0.3. As the Re range for each variance increases past theupper Re limit of the Ergun relation, the difference betweenthe simulated data and the inertial parameters predicted byErgun increases. The Ergun relation has also been found to bevalid only for 0.38 � ε � 0.47 [54].

The Ergun relation matches the simulated data well withinthe restricted range before diverging at a porosity of 0.35. The

2=0.02=0.12=0.22=0.3

ErgunProposed Model

0.30 0.35 0.40 0.45 0.50 0.55 0.601.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

b*

FIG. 7. (Color online) Comparison of isotropic constitutive lawsthat predict b∗ with simulated data.

033012-9


MacDonald et al. relation is the Ergun relation [Eq. (24)] witha modified β̂∗e coefficient based on the comparative analysis ofnumerous experimental results, including the data of Rumpfand Gupte. The analysis took into account data from a widerange of porosities (0.123 � ε � 0.919) and granular shapesand sizes with the goal of deriving an Ergun relation that wasapplicable for packs of nonspherical grains [5]. Much of thelower porosity data (ε � 0.40) came from experiments usingirregularly shaped objects, sand and gravel mixtures, and avariety of undisclosed materials. Since the simulated data onlytake into account porous media composed of smooth spheres,roughness could explain the large deviations between theinertial parameter predicted by the MacDonald et al. relationand the simulated data at ε � 0.40.

To provide a better fit for the full range of simulated data,an alternative functional form is proposed to provide a bettermatch of the simulated data,

b∗(ε) = β∗1(1 − ε)β∗2

εβ∗3

, (44)

where the coefficients β∗1 = 4.21, β∗2 = 1.58, and β∗3 = 0.378were obtained by performing a least-squares analysis tominimize the relative error based on the simulated data pointsshown in Fig. 7.

E. Parametric estimates for flow in anisotropic porous media

The results in Sec. IV D establish functional forms thatpredict the values of flow coefficients in an isotropic system asa function of the media porosity, expanding on the resultsof prior studies. In this section, extended correlations aredeveloped that apply to systems where anisotropic effectscontribute to the flow behavior.

To resolve the flow behavior in anisotropic porous media,six different force orientations were used. These orientationscorresponded to [1,0,0]T , [0,1,0]T ,[0,0,1]T , [

√2,

√2,0]T ,

[√

2,0,√

2]T , and [0,√

2,√

2]T . Between 100 and 150 steady-state simulations were performed for each anisotropic medium.Each steady-state simulation corresponded to values of ψ , ω,Fc, and Re that were used to determine all coefficient valuesfor the media. Based on Eqs. (40) and (41), 12 coefficients arenecessary, in addition to the eigenvectors qsi , to fully specify themacroscopic flow behavior. These coefficients were calculatedby performing a nonlinear least-squares approach to minimize

�i =∑

{Re,F,ω,ψ}

[ψ · qsi

− 1F

(�ai Re +

∑j �

b(j )i

∣∣qsi · ω∣∣∑j

∣∣qsj · ω∣∣ Re2

)ω · qsi

]2, (45)

where the index i = 0,1,2 specifies the eigenvectors associatedwith the principal directions of anisotropy. The full set ofcoefficients was determined by numerical solution of a set ofthe nonlinear equations

∂�i

∂�ai= 0 for i = 0,1,2, (46)

∂�i

∂�b(j )i

= 0 for i,j = 0,1,2. (47)

0.25 0.30 0.35 0.40 0.45 0.50

-0.2

-0.1

0

0.1

0.2

si

s j -

s k

FIG. 8. Domain range for the two independent components of theorientation tensor considered in the constitutive laws.

Simulations were performed using a large number of media toobtain a well-resolved range of anisotropies as shown in Fig. 8.Based on these simulations flow coefficients were determinedfor each medium using Eqs. (46) and (47). Specific functionalforms accounting for anisotropy were then selected to matchthe simulation data.

Since each ellipsoid pack is defined by the mappingprocedure given in Eq. (30), we can understand anisotropiceffects by considering how the introduction of anisotropyalters the flow coefficients for a given sphere pack. In practice,each sphere pack exhibits a small amount of anisotropy asdetermined by the accuracy of the REV. We account forthis by measuring the full set of coefficients for each spherepack, which we denote �a∗i and �

b(j )∗i . The coefficient values

from an ellipsoid packing can then be compared to thesevalues to more accurately determine the associated impacton flow behavior. The components of Gws were determinednumerically for the anisotropic systems considered in thiswork.

Predictive functional forms for �ai and �bi were developed

by constructing separable functions to account for eachindependent variable. Because of the constraint of Eq. (33),two functions are sufficient to fully describe the influence ofGws . Since the indexing of the eigenvectors qsi is nonunique,the constitutive functions must be identical for all permutationsof the index convention. The eigenvalue �si , which provides ameasure of the relative blockage orthogonal to qsi , was selectedas the first of these values. Since the labeling of the remainingaxes is arbitrary, the additional quantity obtained from theorientation tensor must be symmetric with respect to �sj and�sk . This is satisfied by choosing the second independentquantity to be |�j − �k|. We then search for constitutiverelationships of the form

�ai = a∗(ε)a‖(�si

)a⊥

( ∣∣�sj − �sk∣∣ ), i �= j �= k, (48)�

b(i)i = b∗(ε)b‖

(�si

)b⊥

( ∣∣�sj − �sk∣∣ ), i �= j �= k. (49)033012-10


a|| ( si) = 0.369, 2 = 0.0 = 0.38, 2 = 0.0 = 0.42, 2 = 0.0 = 0.45, 2 = 0.0 = 0.35, 2 = 0.1 = 0.38, 2 = 0.1 = 0.38, 2 = 0.2

0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

si

a i / a

*

(a)

b|| ( si) = 0.369, 2 = 0.0 = 0.38, 2 = 0.0 = 0.42, 2 = 0.0 = 0.45, 2 = 0.0 = 0.35, 2 = 0.1 = 0.38, 2 = 0.1 = 0.38, 2 = 0.2

0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

si

b(i)

i /

b*

(b)

FIG. 9. (Color online) Anisotropic simulation data and best-fit functional forms for α‖(�si ) and β‖(�si ).

The functional forms a‖, a⊥, b‖, and b⊥ must equal unity for anisotropic system (i.e., �si = 13 and |�sj − �sk| = 0), which issatisfied by the subsequent expressions given in Eqs. (50)–(53).

Simulation results indicate that �si is responsible for thelargest part of the anisotropic correction to �ai and �

bi . In

order to match the simulation data, the constitutive laws wereformulated as

a‖(�si

) = [1 + α‖1(�si − 13)]α‖2 , (50)

b‖(�si

) = [1 + β‖1(�si − 13)]β‖2 . (51)

The best-fit coefficients are α‖1 = 1.40, β‖1 = 0.351, α‖2 = 1.06,and β‖2 = 11.1. To obtain these coefficients, systems wereconstructed such that �sj = �sk for k �= j �= i. Plots of thesimulation data and resulting functions are shown in Fig. 9.

A smaller contribution is associated with |�sj − �sk|. Alinear functional form was selected to account for this

contribution,

a⊥(∣∣�sj − �sk∣∣) = 1 + α⊥ ∣∣�sj − �sk∣∣ , (52)

and

b⊥(∣∣�sj − �sk∣∣) = 1 + β⊥ ∣∣�sj − �sk∣∣ . (53)

The best-fit coefficients were α⊥ = 0.0173 and β⊥ = 0.229.The ranges of porosity values and variances considered toobtain these results are summarized in Table I and the rangeof orientation tensor eigenvalues is plotted in Fig. 8. Sincea⊥(0) = 1 and b⊥ (0) = 1, the anisotropic contribution disap-pears when an isotropic system is considered. As demonstratedin Fig. 10, the contribution of a⊥ and b⊥ is approximatelythe same order of magnitude as the error contribution. Notcoincidentally, this error is the same order of magnitude asthe error in the quadratic form for the non-Darcy momentumresistance, shown in Fig. 4. Based on this, we conclude that theinclusion of a⊥ and b⊥ is unlikely to improve the descriptionsignificantly. Alternatively, a⊥ and b⊥ could be set to 1, theirvalue for an isotropic system, for simplicity.

a (| sj - sk|)

0 0.05 0.10 0.15 0.20 0.25

0.97

0.98

0.99

1.00

1.01

1.02

1.03

| sj - sk|

a i / [

a* i a

|| (s i)

]

(a)

b (| sj - sk|)

0 0.05 0.10 0.15 0.20 0.250.92

0.94

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

| sj - sk|

b(i)

i /

[ b(

i)*

i b

|| (s i)

]

(b)

FIG. 10. Anisotropic simulation data plotted with best-fit functional forms for a⊥(|�sj − �sk|) and b⊥(|�sj − �sk|).

033012-11


is - j

s ks

0

0.5

1

1.5

2

FIG. 11. (Color online) Anisotropic simulation data with surfacedetermined by best-fit functional forms b†(�si − �sj )b‡(�sk).

The remaining inertial coefficients, �b(j )i , i �= j , contributeonly when the flow is not aligned with one of the eigenvectorsqsi . Parameter values for these coefficients presumed that thesecoefficients could be described by

�b(j )i = b∗(ε)b†

(�si − �sj

)b‡

(�sk

). (54)

The coefficient �b(j )i provides the relative change to themomentum resistance in the direction of qsi that results from achange in

b†(�si − �sj

) = exp [β†(�si − �sj )], (55)b‡

(�sk

) = exp [β‡(�sk − 13)], (56)where β† = 0.742 and β‡ = −2.05. A plot of the resultingfunctional form is shown in Fig. 11.

The anisotropic parameter correlations given in Eqs. (50)–(56) combine with the results in Sec. IV D to providea complete set of predictive coefficients for Eq. (11) forany system where the porosity, specific surface area, andorientation tensor are known. The functional forms proposedto account for these various contributions are summarized inTable III. Several basic principles were relied upon to producethese relationships, which are summarized as follows.

(1) The eigenvalues of tensors A and B are each non-negative, to ensure that the second law of thermodynamicsis satisfied for all Re values.

(2) By constructing the formulation in terms of the eigen-decomposition of the momentum resistance tensor and iden-tifying the associated eigenvectors, the description is fullyindependent of the coordinate system choice.

(3) The tensor form of the momentum transport equationreduces exactly to the one-dimensional form given in Eq. (21)if the solid surface is isotropic (�si = 13 ).

Specific functional forms were selected to satisfy theseconstraints while matching the simulated data as closely aspossible.

Microscopic imaging techniques can be exploited to obtaindirect measurements of both surface area and orientationtensor values. When working with such data, a spectraldecomposition must be applied to the numerically computedorientation tensor Gws to obtain qsi and �

si . Our results suggest

that the eigenvectors of Gws provide a good approximationfor the eigenvectors of A and B. Since our results wereobtained for systems of axially aligned ellipsoids, futurestudies should consider the extensibility of these results tomore general anisotropic porous media. However, the useof dimensionless variables should facilitate the extension ofthese results to a wide range of systems. Since the surfacearea and orientation tensor can be constructed easily basedon computed microtomography data, this would seem topresent a natural opportunity to examine the extensibility ofthese results to more general anisotropic porous media. Itis possible that other dimensionless morphological measuresexert an influence on momentum transport for certain systems,and the identification of such measures presents additionalopportunities to expand the predictive capabilities for non-Darcy flows beyond the leading-order factors identifiedherein.

V. SUMMARY AND CONCLUSIONS

In this work, a large set of LB simulations has beenperformed to produce constitutive relationships that fullypredict all coefficients required to close a tensorial form of themomentum transport equations for single-phase flow in porousmedia. Essential closure information for both isotropic andanisotropic flows in porous media is provided as a function ofnondimensional measures of the porous medium morphology.In the isotropic case, simulated data were compared to existing

TABLE III. Summary of functional forms and coefficient values proposed to predict momentum equation parameters (i �= j �= k). TensorsA and B can be determined by combining these expressions with Eqs. (48) and (49), Eq. (54), and Eqs. (40) and (41).

Parameter Dependence Functional form Coefficient values

�ai a∗(ε) α∗1 exp(α

∗2ε) α

∗1 = 1.95 × 10−5, α∗2 = 9.85

�ai a‖(�si ) [1 + α‖1(�si − 13 )]α

‖2 α

‖1 = 1.40, α‖2 = 1.06

�ai a⊥(|�sj − �sk|) 1 + α⊥|�sj − �sk| α⊥ = 0.0173

�b(i)i , �

b(j )i b

∗(ε) β∗1(1−ε)β∗2

εβ∗3

β∗1 = 4.21, β∗2 = 1.58, β∗3 = 0.378.�

b(i)i b

‖(�si ) [1 + β‖1 (�si − 13 )]β‖2 β

‖1 = 0.351, β‖2 = 11.1

�b(i)i b

⊥(|�sj − �sk|) 1 + β⊥|�sj − �sk|. β⊥ = 0.229�

b(j )i b

†(�si − �sj ) exp[β†(�si − �sj )] β† = 0.742�

b(j )i b

‡(�sk) exp[β‡(�sk − 13 )] β‡ = −2.05

033012-12


relationships including those due to Ergun, Rumpf and Gupte,MacDonald et al., Carmen-Kozeny, and Pan et al. As shownin our work, these equations have varying degrees of accuracyin their application, depending on the quantity and quality ofthe data used to derive them. Our work improves upon theseexisting correlations by considering data for a wider rangeof morphological and flow characteristics than previously ex-amined. We also studied the dependence of single-fluid-phaseflow through anisotropic media on morphological character-istics beyond porosity and postulated a quantitative measureof anisotropy that we demonstrated to be predictive of themacroscopic flow behavior. This result is significant becauseit concretely establishes a connection between the macroscopicflow coefficients and an averaged measure of the anisotropyof underlying solid phase morphology: the orientation tensor.As a consequence, the predictive capabilities for anisotropicflows are extended significantly. A summary of the majorcontributions of this work is as follows.

(1) A collective rearrangement algorithm and mappingprocedure is developed to generate packed periodic systems ofspheres and ellipsoids that match a specified porosity and sizedistribution.

(2) An approach is derived to simulate highly resolvedflows that are essentially grid independent for a representativeelementary volume of porous medium systems, and guidanceis given on how such systems can be created.

(3) The momentum resistance of non-Darcy flows is demon-strated to depend linearly on the Re and nonlinearly on the floworientation, accounting for all three independent componentsof the flow velocity. An empirical form is introduced to accountfor the influence of the flow orientation in anisotropic flows.

(4) Porosity, specific interfacial area, and geometric orien-tation of the solid surface are shown to be the variables ofleading-order importance in assessing Darcy and non-Darcysingle-fluid-phase flow through anisotropic porous media.

(5) The spectral decomposition of the momentum resistancetensor is studied and an approach is proposed to predict boththe eigenvectors and the eigenvalues. An empirical functionalform is posited to represent the momentum resistance tensorand correlations are developed to predict parameter valuesbased upon a large number of highly resolved simulations.

(6) The functional form developed in this work to predict theresistance tensor is compared to available estimates and shownto be significantly more accurate than previous estimates whenthe entire range of simulations performed in this work isexamined.

ACKNOWLEDGMENT

This work was supported by National Science FoundationGrant No. ATM-0941235 and Department of Energy GrantNo. DE-SC0002163x.

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