On the size of representative volume element for Darcy law in ...martinos.mechanical.illinois.edu/files/0000/79dfe38efb55...for Darcy law in random media BY X. DUANDM. OSTOJA-STARZEWSKI*

On the size of representative volume elementfor Darcy law in random media

BY X. DU AND M. OSTOJA-STARZEWSKI*

Department of Mechanical Engineering, McGill Institute for AdvancedMaterials, McGill University, Montréal, Quebec H3A 2K6, Canada

Most studies of effective properties of random heterogeneous materials are based on theassumption of the existence of a representative volume element (RVE), withoutquantitatively specifying its size L relative to that of the micro-heterogeneity d. In thispaper, we study the finite-size scaling trend to RVE of the Darcy law for Stokesian flowin random porous media, without invoking any periodic structure assumptions, but onlyassuming the microstructure’s statistics to be spatially homogeneous and ergodic. Byanalogy to the existing methodology in thermomechanics of random materials, we firstformulate a Hill–Mandel condition for the Darcy flow velocity and pressure gradientfields. This dictates uniform Neumann and Dirichlet boundary conditions, which, withthe help of two variational principles, lead to scale-dependent hierarchies on effective(RVE level) permeability. To quantitatively assess the scaling trend towards the RVE,these hierarchies are computed for various porosities of random disc systems, where thedisc centres are generated by a planar hard-core Poisson point field. Overall, it turns outthat the higher is the density of random discs—or, equivalently, the narrower are themicro-channels in the system—the smaller is the size of RVE pertaining to the Darcylaw.

Keywords: random media; representative volume element; mesoscale bounds;Darcy law; Stokes flow

*A

RecAcc

1. Introduction

Most studies of random heterogeneous materials are based on the assumption ofthe existence of a representative volume element (RVE), and are focused on theproperties of the RVE without quantitatively specifying its size L relative to thatof the micro-heterogeneity d (e.g. Berryman & Milton 1985; Nemat-Nasser &Hori 1999; Milton 2002; Torquato 2002). More specifically, the RVE is taken tosatisfy the so-called ‘separation of scales’,

d!L/LMacro; ð1:1Þwhere d is the microscale (average size of heterogeneity, figure 1a), LMacro is themacroscale of the continuum problem considered. Thus, the right-hand sideinequality in (1.1) ensures the applicability of continuum mechanics with Darcy’slaw as the constitutive relation governing the response of the RVE. However, the

Proc. R. Soc. A (2006) 462, 2949–2963

doi:10.1098/rspa.2006.1704

Published online 18 April 2006

uthor for correspondence ([email protected]).

eived 18 October 2005epted 27 February 2006 2949 q 2006 The Royal Society

d .d

d. d

(a) (b)

Figure 1. (a) A planar model of a random medium, formed from a non-overlapping distribution ofcircular discs; one deterministic configuration at a nominal porosity of 50% is shown. (b) DomainBdðuÞ and its partitioning into four subdomains Bsd0 ðuÞ.

X. Du and M. Ostoja-Starzewski2950

left-hand side inequality in (1.1) is postulated rather vaguely (and sometimesinvolving d/L), without clearly specifying what is meant by d!L, or,sometimes, postulating a periodic unit cell. In this paper, we study thisinequality for flow in porous media within a general framework of scale-dependent homogenization of random media (Ostoja-Starzewski 2001) withoutany periodicity assumption, in the vein of Hill’s (1963) definition of RVE.

In order to attain the said homogenization, two conditions must be satisfied:(i) the microstructure’s statistics must be spatially homogeneous and ergodic;(ii) the material’s effective constitutive response must be the same under uniformboundary conditions of Dirichlet (essential) and Neumann (natural) types. Thisapproach has been applied in studies of various effective continuum properties insolid mechanics—examples are: classical elasticity (e.g. Huet 1990; Sab 1992;Kanit et al. 2003), thermal conductivity (Ostoja-Starzewski & Schulte 1996),viscoelasticity (Huet 1995), physically nonlinear elasticity (Hazanov 1999),elasto-plasticity and rigid-plasticity (Jiang et al. 2001; Ostoja-Starzewski 2005;Li & Ostoja-Starzewski in press), linear thermoelasticity (Du & Ostoja-Star-zewski in press) and finite (thermo)elasticity (Khisaeva & Ostoja-Starzewski2006).

In this paper, we consider a steady, single-phase Stokesian fluid flow in randomporous media with explicit account of a spatially disordered microstructure. Inphenomenological, deterministic continuum mechanics, such a flow is describedby Darcy’s law (1856). With the slow flow in a porous medium, also consideredas incompressible and viscous, Darcy’s law is governed by the equations(Dullien 1979)

U ZKK

m$Vp; ð1:2Þ

V$U Z gðxÞ; ð1:3Þwhere U is the Darcy (volume averaged) velocity, Vp is the applied pressuregradient driving the flow, m is the fluid viscosity and K is the permeability,

Proc. R. Soc. A (2006)

2951RVE of Darcy law in random media

a second-rank tensor that depends on the microstructure of the porous medium.Equation (1.3) is the continuity equation, wherein g(x) is the source/sink term.The tensor K is usually considered a material property, since it definitely dependson the porous microstructure. (In this paper, we interchangeably use thesymbolic notation for first- and second-rank tensors a and b or indicial notationsai and bij , whichever is more convenient.)

Clearly, equations (1.2) and (1.3) show that Darcy’s law shares the samemathematical formulation as conductivity problems. However, as stated byTorquato (2002, p. 348), ‘As in the trapping problem, there is no localconstitutive relation in flow problem. Furthermore, both fluid permeability andtrapping constant are scale-dependent properties, in contrast to both theeffective conductivity and elastic moduli, which are scale-invariant properties.’As is well known, the local governing equation of fluid in pore spaces is the Stokesequation.

At this point, let us briefly review prior work on permeability problems inporous media. Evidently, due to the complex, randomly structured nature ofsuch media, deterministic solutions are not possible. Statistical methods aretherefore often used for the determination of rigorous (upper/lower) bounds onthe effective (RVE level) permeability. In the very early work done by Prager(1961), a variational principle of energy dissipation rate was applied and, with aproposed trial stress field, the upper bound was obtained. More rigorous boundshave been proposed by Berryman & Milton (1985) and a different trial stresswith two point bounds was used. They were able to reduce the upper bound by afactor from 9/10 to 2/3. Another important approach has been invented anddeveloped in Rubinstein & Torquato (1989). In their derivation of Darcy’s Lawfrom the Stokes equation, an energy-formed functional of permeability wasformulated with the ensemble of a ‘tensor velocity field’ and a ‘vector pressurefield’. The variational principle was applied on this energy-formed functional,and both upper and lower bounds were obtained with properly selected trailfunctions. While all the aforementioned works gave bounds on effectivepermeability, the size of RVE was only required to be much larger than themicroscale.

In our work, we introduce a mesoscale domain of size L, recall equation (1.1),and use a non-dimensional parameter

dZL

d: ð1:4Þ

The response of that domain is therefore a function of (i) mesoscale d, (ii) theboundary conditions and (iii) the actual microstructure, such as that shown infigure 1a. Since the microstructure, in the ensemble sense, is random, themesoscale response is statistical, but tends to become deterministic as d grows. Itis this tendency in function of d which is the focus of our study.

2. Mathematical formulation

(a ) Variational principles

In the general studies on property bounds, the variational principles usuallyprovide the basic starting point (Prager 1961; Huet 1990; Jeulin &



Ostoja-Starzewski 2001; Torquato 2002). Now, as mentioned before, Darcy’s law(1.2) and (1.3) is not a local constitutive relation, since the slow flow in voids isgoverned by the Stokes equation:

mV2v ZVp;

V$v Z 0;

(ð2:1Þ

where v is the actual fluid velocity. In any chosen element dV of the porousmedium, the energy dissipation rate f in a fluid of viscosity m is usuallycalculated as

fZ1

2mV

ðVf

sijsij dV ; ð2:2Þ

where Vf is void region, sij is a symmetric, zero-trace stress tensor, which can beobtained by solving the Stokes equation directly.

Let us now consider a trial stress field, ~sij , satisfying

~sij;j Z ~p;i in Vf ; ð2:3Þwhere ~p is a scalar function, a trial pressure field, and

~sij ZKp0dij on vB; ð2:4Þ

where p0 is a prescribed pressure on vB, the surface of the domain B of the porousmedium. Prager (1961) and Berryman & Milton (1985) stated the variationalprinciple for an element as follows:

~fZ1

2mV

ðVf

~sij ~sij dVRKijm

p;ip;j ; ð2:5Þ

where ~f is the trial dissipation rate calculated with the trial stress function. Aswe shall see in §2c, this gives the upper bound on the effective permeability Kijaccording to Prager (1961). The equality fZ ~f occurs only when ~sij is equal tothe true stress, and this is equivalent to the law of energy conservation. This factwill be used in the subsequent numerical simulations.

On the other hand, the variational principle based on Darcy’s law (1.2) and(1.3) can be expressed in terms of the pressure gradient and the Darcy velocity.Hence, the energy dissipation rate in the porous domain can also be written as

fZ1

V

ððKp;i$UiÞdV : ð2:6Þ

Here, we use another form of Darcy’s law, in which the pressure gradient is afunction of Darcy’s velocity:

p;i ZKmRij$Uj ; or VpZKmR$U ; ð2:7Þwith Rij ðhRÞ being the resistance tensor of the porous medium.

For the effective tensors—that is, on the scale L where the separation of scales(1.1) holds—we have the following relation:

K eff$Reff Z I; ð2:8Þwhere I is the second-rank identity tensor. However, when the element scale issmaller than the RVE size, the foregoing relation is generally not true.



According to the preceding discussion, the energy dissipation can be expressedin two different quadratic forms:

fP Z1

V

ðKijm

p;ip;j

� �dV ; ð2:9Þ

fC Z1

V

ððmRijUiUjÞdV : ð2:10Þ

If we look at Darcy’s law (1.2) and (1.3), the following equation can easily bederived in the absence of sources and sinks inside the material domain:

V$ðK$VpÞZ 0: ð2:11ÞOf course, when considering homogeneous and isotropic media, the foregoingequation becomes the Laplace equation.

Clearly, we see an analogy of the ansatz of flow through random porous mediato that of mechanics of random solids (Jeulin & Ostoja-Starzewski 2001). That is,the pair (pressure field, Darcy velocity) corresponds to the pair (strain, stress) inanti-plane elasticity, or the pair (temperature, heat flux) in heat conductivity.Then, fP is treated as the potential energy, while fC is the complementaryenergy.

Now, recall the following two variational principles in terms of potential andcomplementary energies.

Principle 1. Among the admissible solutions for the pressure field in a porousmedium domain, the true solution possesses the minimum potentialenergy.

Principle 2. Among the admissible solutions for the Darcy velocity field in aporous medium domain, the true solution possesses the minimumcomplementary energy.

As noted earlier, there is another energy formulation with associatedvariational principles (Torquato 2002). However, as they do not involveboundary conditions, they will not be used in the present study.

(b ) Hill condition in porous media

When considering a mesoscale domain of the heterogeneous medium, thedistribution of surface value will be used to evaluate the constitutive response. Inthe context of elasticity as defined by Huet (1990, 1995), this results in the so-called apparent properties. As a first step, the Hill condition must be satisfied. Inan elastic heterogeneous material, that condition can be expressed as follows:

s : 3 Z �s : �35

ðvBðtK�s$nÞ$ðuK�3$xÞdS Z 0; ð2:12Þ

where n is the unit normal vector of a surface element dS, x is the coordinate ofdS. Combining this with the average strain and stress theorems in elasticity, thisensures that the energetic definition is equivalent to the mechanical definition ofHooke’s law. Hence, the variational principles based on energy can be applied toevaluate the constitutive response of heterogeneous material in terms of theboundary values.



Analogous to this, a similar theory must also be established in permeability ofporous media. Thus, in a given porous medium domain V, the pressure gradientand the Darcy velocity can be expressed in the form of average values andfluctuations,

p;i Z p;i Cp0;i; ð2:13Þ

Ui ZUi CU0i : ð2:14Þ

The energy dissipation is calculated as

KfZ p;iUi Z p;iUi Cp0;iU

0i : ð2:15Þ

From the foregoing equation it appears that, in general, the energy dissipation

cannot be calculated from surface values, except only when p0;iU0i Z0. By

inspiration of Hill’s condition in elasticity (Hill 1963), the following derivationapplies:

p0;iU0i Z p;iUiKp;iUi Z

1

V

ðVðp;iKp;iÞðUiKUiÞdV : ð2:16Þ

Owing to the continuity equation Ui;iZ0, we get

p0;iU0i Z

1

V

ðV½ðpKp;j xjÞðUiKUiÞ�;i dV : ð2:17Þ

Applying the Green–Gauss theorem, we arrive at

p0;iU0i Z

1

V

ðvBðpKp;j xjÞðUiniKUiniÞdS: ð2:18Þ

Therefore, in order to satisfy p;iUiZp;iUi , the term (2.18) must vanish. Weconclude that the Hill condition for Darcy’s law is

p;iUi Z p;iUi5

ðvBðpKp;j xjÞðUiniKUiniÞdS Z 0: ð2:19Þ

To satisfy (2.19), there are three possible boundary conditions:

pZVp$x or pZ p;i xi on vB; ð2:20ÞU$n ZU 0$n or Uini ZU0ini on vB; ð2:21Þ

ðpKVp$xÞ$ðU$nKU 0$nÞZ0 or ðpKp;i xiÞðUiniKU0inÞZ0 on vB: ð2:22ÞFollowing the terminology of thermomechanics of solid random media, (2.20) iscalled a uniform Dirichlet (or essential) boundary condition, (2.21) is called auniform Neumann (or natural) boundary condition, while (2.22) is called auniform orthogonal-mixed boundary condition. The Hill condition (2.19) ensuresthat the energy dissipation calculated from the surface values of the mesoscaleelement under either of these conditions is equal to the energy dissipationobtained from the volume integration over that element.

At this point, we introduce apparent dissipation functionals on the mesoscalefPd ðVpÞ and fCd ð �UÞ, which are homogeneous functions of degree 2 in the volumeaverage pressure gradient Vp and the volume average Darcy velocity �U ,respectively. Thus, we have (Ziegler 1983)

�Ui Z1

2

vfP

vp;i; p;i Z

1

2

vfC

vUi: ð2:23Þ



With this we have a full analogy between the Stokesian flow in porous media andthe anti-plane elasticity in a matrix of finite stiffness, containing infinitely stiffinclusions.

Note that there is no body force considered in the above derivations. Also, theabove derivations use the tensor index notation for clarity, while furtherderivations will mostly use the symbolic notation.

(c ) Scale-dependent hierarchies of bounds

We take the porous medium as a set BZfBðuÞ;u2Ug of realizations BðuÞ,defined over the sample space U. Thus, for an elementary event u2U, we have arealization BðuÞ, theoretically of infinite extent, from which we cut out amesoscale specimen BdðuÞ, of finite size LZdd (figure 1a). As we shall see below,that specimen is characterized by apparent (or mesoscale) permeability, KDd , andresistance, RNd , tensors, corresponding to (2.20) and (2.21), respectively,according to

U ZKDd ðuÞ

m$Vp; ð2:24Þ

VpZmRNd ðuÞ$U : ð2:25ÞWe require the statistics of material properties to be statistically homogeneous

and mean ergodic. The latter property means that the spatial average equals theensemble average

KðuÞh limV/N

1

V

ðKðx;uÞdv Z

ðUKðx;uÞdPðuÞhhKðxÞi; ð2:26Þ

where h i denotes the ensemble average and PðuÞ is the probability function ofthe sample u. Note, of course, that KeffZðReffÞK1 is not given by the aboveequality.

As a particular case, one may assume the statistics of microstructure to beisotropic, which would yield

KðuÞ Z hKðxÞiZK I; ð2:27Þi.e. the isotropy of the effective response on the RVE level.

In the following, we partition a square-shaped BdðuÞ into 2m subdomainsBsd0 ðuÞ (in two-dimensional, mZ2; in three-dimensional, mZ3), sZ1, 2,., 2

m.Thus, each Bsd0 ðuÞ has the mesoscale d0Zd=2. Clearly,

BdðuÞZ[2msZ1

Bsd0 ðuÞ: ð2:28Þ

Henceforth, for simplicity, we work in two dimensions as shown by figure 1b; theprocedure is totally analogous in three dimensions.

We now define two types of boundary conditions here: namely, the‘unrestricted’ and ‘restricted’ boundary conditions for BdðuÞ and all fourBsd0 ðuÞ’s, respectively.

First, let us consider an ‘unrestricted’ Dirichlet boundary condition: theprescribed pressure on the boundary of BdðuÞ:

pZVp$x on vBdðuÞ: ð2:29Þ



In view of the Hill condition, the potential energy can be expressed as

fPd ZVp$KDdm

$Vp; ð2:30Þ

where hKDd i is the permeability tensor first appearing in (2.24).On the other hand, we can also set up a ‘restricted’ Dirichlet boundary

condition: the prescribed pressure on the boundary of all four vBsd0 ðuÞ in (2.28):pZVp$x; on vBsd0 ðuÞ; sZ 1;.; 4; ð2:31Þ

which, given the Hill condition, leads to

fDrd ZVp$KDrdm

$Vp: ð2:32Þ

Here, KDrd is different from KDd of (2.30), whereby ‘r’ denotes the restriction.

We now observe that the solution under the restricted condition (2.31) on allfour subdomains Bsd0 ðuÞ is an admissible solution with respect to the unrestrictedcondition (2.29) on BdðuÞ, but not vice versa. Thus, in view of the minimumpotential energy principle, we have the inequality: fPd %f

Pr

d ZP4

sZ1 fPs

d0 , wherefP

s

d0 are the energies of particular subdomains in (2.28). Henceforth, given (2.28),this inequality relationship becomes

VdVp$KDdm

$Vp%X4sZ1

Vsd0Vp$KDsd0

m$Vp; ð2:33Þ

where KDsd0 pertain to particular subdomains in (2.28). Note that VdZ4Vd0 here.Consequently, given the medium’s statistical homogeneity and ergodicity, thefollowing relationship is obtained upon taking the ensemble average over the Uspace:

hKDd i%hKDsd0 i; cd0 Z d=2: ð2:34ÞThe inequality relation between tensors in (2.30) means that, when t$A$t%t$B$t for any t, then A%B. By induction, (2.34) leads to a scale-dependenthierarchy (upper bound) of the permeability of porous media flow

Keff%/%hKD2di%hKDd i%/%hKD1 i: ð2:35ÞNote that hKD1 i pertains to the smallest scale where Darcy’s law still applies; byanalogy to elasticity of random media, that tensor may be called a Voigt bound.

Now, let us consider an ‘unrestricted’ Neumann boundary condition: theprescribed velocity on the boundary of BdðuÞ,

U$n ZU 0$n; on vBdðuÞ; ð2:36Þand a ‘restricted’ Neumann boundary condition: the prescribed velocity on theboundary of all four vBsd0 ðuÞ in (2.28),

U$n ZU 0$n; on vBsd0 ðuÞ; sZ 1;.; 4: ð2:37Þ

Next, consider the energy dissipation rate under (2.36) and (2.37), respectively,via the analogy to complementary energy. Note that the Hill condition isapplicable in both cases, and, corresponding to (2.36) and (2.37), thecomplementary energies are

fCd ZU 0$mRNd $U 0; f

Crd ZU 0$mR

Nrd $U 0: ð2:38Þ

Here, RNd is the resistance tensor first appearing in (2.25).



Now, in view of the minimum complementary energy principle (of §2a), thesolution under the restricted boundary condition (2.37) is admissible with respectto the unrestricted boundary condition (2.36), but not vice versa. Hence,

VdU 0$mRNd $U 0%

X4sZ1

Vsd0U 0$mRNsd0 $U 0; ð2:39Þ

where RNsd0 pertain to particular subdomains in (2.28). As before, upon taking theensemble average over the space U of the assumed random porous media, wearrive at the inequality for the resistance tensor on two mesoscales, d and d0,

hRNd i%hRNd0 i; cd0 Z d=2: ð2:40ÞClearly, this indicates the upper bound character of resistance tensors withrespect to ReffZðKeffÞK1. Since d is arbitrary, from (2.40), we obtain a scale-dependent hierarchy (of lower bounds) of the inverse of the resistance tensor

ReffK1

R/RhRN2diK1RhRNd iK1R/RhRN1 iK1: ð2:41ÞAgain by analogy to elasticity of random media, tensor hRD1 i pertains to thesmallest scale where Darcy’s law still applies; and it may be called a Reussbound.

Noting (2.8), the hierarchies (2.35) and (2.41) can be combined into a single,two-sided hierarchy:

hKD1 iR/RhKDd iRhKD2diR/RKeff Z ðReffÞK1R/RhRN2diK1

RhRNd iK1R/RhRN1 iK1; cd0 Z d=2:ð2:42Þ

This relation states that for random heterogeneous porous media, which arestatistically homogeneous and ergodic:

(i) the effective permeability tensor is bounded from above by thepermeability tensor obtained under the Dirichlet boundary condition,and from below by the permeability tensor obtained under the Neumannboundary condition;

(ii) the hierarchy (2.42) shows the scale dependence of apparent permeabilitytensors under Dirichlet or Neumann boundary condition, which with theincreasing mesoscale d tends to converge towards the effectivepermeability.

Note that, just like in the case of elastic materials, the hierarchy (2.42) may beextended to arbitrary scales d0!d, not only those satisfying d0Zd=2(Ostoja-Starzewski 1999).

3. The RVE size via computational fluid mechanics

(a ) Direct simulation

The foregoing discussion also indicates a procedure to quantitatively homogenizethe heterogeneous medium and determine the RVE size. In the following, weconduct numerical simulations to quantitatively demonstrate the convergenttendency of the hierarchy (2.42). Starting from the energy conservation law in



(2.5), we solve Stokes’ equation (2.1) numerically in the void of a random modelfor a given boundary condition, either Dirichlet or Neumann. In order to dealwith the geometric complexity of random porous media, we employ a controlvolume finite-difference method, and to avoid the enhancement of incompressi-bility by Stokes flow, which causes an unrealistic oscillation of the pressure fieldin computation, a so-called semi-implicit method for pressure-linked equationsrevised with collocated equal-order discretization technique is used (Patankar1980; Chung 2000).

The porous media model investigated in the present research is a two-dimensional flow past a bed of round discs randomly distributed by a planarhard-core Poisson point field (figure 1a). The latter condition means that no twodiscs may touch. In fact, in order to avoid the narrow-neck effects in fluid field,the minimum distance between disc centres is set at 1.1 diameter of the disc.Such effects may be introduced as an extension of our model in future work.

We set up boundary conditions for Darcy’s law. First, for the Dirichletboundary condition with Vp prescribed by (2.20) for a mesoscale specimen at agiven d, we must also pose a proper boundary condition for the Stokes flow in thevoid region. Beside the pressure distribution (2.20), and according to (2.3) and(2.4), we set v1;2Z0 on the top and bottom of the element, and v2;1Z0 on theleft- and right-hand side of the element for the direct numerical simulation of thevoid region. This leads to a calculation of the mesoscale permeability tensor Kij .

On the other hand, for the Neumann boundary condition, given U in (2.21),also according to (2.3) and (2.4), we set v2Z0 on the top and bottom boundariesof the element. This allows us to calculate the mesoscale resistance tensor Rij .

Figure 2 illustrates the comparison between both kinds of boundary conditions.We can clearly see in figure 2b that the pressure distributes nonlinearly on sideboundaries in order to maintain the flow within the top and bottom boundaryaccording to the requirement of Neumann boundary condition. Nevertheless, for asufficiently large mesoscale, the fluctuation of pressure becomes so negligible thatthe results of the Neumann boundary-value problem begin to coincide with thoseof the Dirichlet boundary-value problem (except for the obvious non-uniquenessof the Neumann problem). This is the RVE limit we want to assess.

(b ) Numerical results and discussion

To assess the hierarchy (2.42), we proceed with the following steps:

(i) choose a specific mesoscale d and a nominal porosity;(ii) generate a mesoscale realization BdðuÞ of the disordered medium, as

described in §3a;(iii) separately solve the Dirichlet and Neumann boundary-value problems for

each BdðuÞ, and store the apparent permeability and resistance tensors:KdðuÞ and RdðuÞ;

(iv) proceed from (i) to (iii) in a Monte Carlo sense over the sample space U togenerate an ensemble of results at a fixed mesoscale d : fKdðuÞ;u2Ugand fRdðuÞ;u2Ug;

(v) change the mesoscale and repeat steps (i)–(iv) to find the first momentshKDd i and hRNd i;

(vi) change the nominal porosity and repeat steps (i)–(v).


Figure 2. Pressure fields in a realization of the random porous medium on mesoscale dZ4 underthe uniform (a) Dirichlet and (b) Neumann boundary conditions.


In this procedure, we use dZ4, 8, 16, 32, 64 and 80, and nominal porosities 50, 60,70 and 80%. The corresponding numbers of Monte Carlo realizations are,respectively, 160, 80, 50, 36, 25 and 16.

Figure 3 displays the simulation results of the random medium at 60%porosity in terms of the ensemble average half-traces of the permeability andresistance tensors. The bounding character of these tensors and theirconvergence to the RVE (effective medium) with the increasing mesoscale areclearly seen. In fact, at dZ80, the ensemble averages of both tensors areapproximately equal within 0.4%.

Another viewpoint of that convergence to RVE may be offered in terms of thedeparture of one-half of the ensemble-averaged scalar product of KDd ðuÞ andRNd ðuÞ from unity, for we must have at dZN,

1

2Keff$Reff Z 1: ð3:1Þ

The statistical isotropy of the underlying Poisson point field with exclusiondictates the isotropy of not only the effective (RVE level) properties, but also theisotropy of hKDd i and hRNd i. Thus, we can work with

D ZKDd1jRNd1jK1; ð3:2Þ

to characterize the above-mentioned departure from the RVE properties; here 1jare the indices of particular components of the KDd and R

Nd tensors. The said

departure is shown in figure 4, both for the preceding case of porosity, as well asfor the entire range: 50–80%. Clearly, the scaling effect of permeability in porousmedia depends more strongly on the size of micro-channels than on the discdiameter d itself. The rate of convergence to RVE decreases with increasingporosity. For example, at dZ32, the RVE size can be considered to be attainedat porosity 50%. However, for a higher porosity, say 80%, the RVE size is stillnot achieved even at dZ80.


0 16 32 48 64 80 9620

25

30

35

40

45

d (L /d )

Tr(

Kij)/

2 (×

10–9

m2 )

two-dimensional round discs60% porosity

Dirichlet B.C.

Neumann B.C.

Figure 3. Effect of increasing window scale on the convergence of the hierarchy (2.42) of mesoscalebounds in terms of the trace of K, obtained by computational fluid mechanics.


Of course, the rate of convergence to RVE also depends on the acceptableerror. Therefore, in figure 5, we show d with respect to the scale change andporosities. We can see that, for a 10% error, in the medium with 80% porosity,the RVE can be reached at the mesoscale dZ56. However, for a 1% error, itcannot be reached even at dy100.

As a final issue, we assess the rate of convergence to the isotropic behaviour.That property is measured using the second invariant of either KdðuÞ or RdðuÞ,i.e. the radius of the Mohr circle of the second-rank tensor,

r Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK11KK22

2

� �2CK212

s: ð3:3Þ

Also, the variance of r converges to zero as well. Figure 6 shows the coefficient ofvariation of r (i.e. its standard deviation divided by the mean) in function of themesoscale. We see that the variance converges to zero faster than the ensembleaverage in the present study.

4. Conclusion

We summarize this study as follows.

(i) We set up the mathematical basis for finding effective permeability andRVE size for porous media with spatially homogeneous and ergodicstatistics. By analogy to the existing methodology in thermomechanics ofsolid random media, we first formulate a Hill–Mandel condition for theDarcy flow velocity and pressure gradient fields. This dictates uniformNeumann and Dirichlet boundary conditions, which, with the help of theminimum potential and complementary energy principles for the Stokesflow, lead to scale-dependent hierarchies on effective (RVE level)permeability.


0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

d (L

/d)

f (porosity)

〈K1j〉 〈Rj1〉–1:

10%

5%

1%

D N

Figure 5. The effects of porosity and mesoscale on the departure D of permeability from that of theeffective medium (RVE level).

0 16 32 48 64 80 961.0

1.5

2.0

2.5

〈Kij

〉〈 R

jk 〉

(i =

k =

1, 2

)N

D

two dimensional round discs—porosity:

80%70%60%50%

d (L /d )

Figure 4. Effect of increasing window scale on the convergence of (2.42) in function of mesoscale d,obtained by computational fluid mechanics.


(ii) The scale-dependent hierarchy of mesoscale bounds on the effective (RVElevel) permeability tensor is derived: the ensemble average of thepermeability tensor is bounded from above by Dirichlet boundary-valueproblems, and from below by Neumann boundary-value problems. Theupper and lower bounds converge towards the RVE response with thewindow scale increasing.

(iii) The hierarchy of bounds is computed for a range of volume fractions ofrandom media made up of planar discs, where the disc centres aregenerated by a planar hard-core Poisson point field. Overall, it turns out


0

0.2

0.4

20 40 60 80 100d (L /d)

r'/r

ave

60% porosity

Figure 6. The standard deviation of r over the ensemble average of r for the 60% porosity medium.


that the higher is the density of random discs—or, equivalently, thenarrower are the micro-channels in the system—the smaller is the size ofthe RVE pertaining to the Darcy law. In particular, the condition d/Lin the separation of scales (1.1) is not always necessary, and it is possiblethat the RVE is attained at d!L.

This paper sets the stage for computation of the RVE size in more generalsettings: permeability of non-Newtonian fluids; permeability in three-dimen-sional microstructures; poroelasticity, where the fluid flows in a porous elasticmicrostructure.

This work has been made possible through the support by the Canada Research Chairs programand the NSERC.

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http://dx.doi.org/doi:10.1016/S0022-5096(00)00034-Xhttp://dx.doi.org/doi:10.1016/S0022-5096(00)00034-Xhttp://dx.doi.org/doi:10.1016/S0020-7683(03)00143-4http://dx.doi.org/doi:10.1098/rspa.2005.1614http://dx.doi.org/doi:10.1098/rspa.2005.1614http://dx.doi.org/doi:10.1016/S0167-6636(99)00039-3http://dx.doi.org/doi:10.1016/j.ijplas.2004.06.008http://dx.doi.org/doi:10.1103/PhysRevB.54.278http://dx.doi.org/doi:10.1103/PhysRevB.54.278http://dx.doi.org/doi:10.1063/1.1706246http://dx.doi.org/doi:10.1063/1.1706246

On the size of representative volume element for Darcy law in random mediaIntroductionMathematical formulationVariational principlesHill condition in porous mediaScale-dependent hierarchies of bounds

The RVE size via computational fluid mechanicsDirect simulationNumerical results and discussion

ConclusionThis work has been made possible through the support by the Canada Research Chairs program and the NSERC.References

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On the size of representative volume element for Darcy law in ...martinos.mechanical.illinois.edu/files/0000/79dfe38efb55...for Darcy law in random media BY X. DUANDM. OSTOJA-STARZEWSKI*