272

Homogenization of the ionic transport equations inperiodic porous media
Jason R. Looker

1Introduction

The transport of electrolyte (salt solution) through porous media is a classical multiscale
phenomenon with numerous applications in geophysics and bio-porous materials modeling.
Phenomenological equations relating the transport of ions and solvent to electrical potential,
pressure and concentration gradients have achieved widespread acceptance in the literature.
These equations are assumed to be valid on the scale of the porous body (the macroscopicor Darcy-scale) and represent an average, in some sense, of the pore-scale (or microscopic)
equations. Attempts at deriving macroscopic transport equations from microscopic equations have been successful under various somewhat restrictive assumptions. In this article, I
demonstrate how Darcy-scale transport equations may be derived with minimal assumptions
using homogenization theory. Without any additional assumptions, it can be shown thatthe transport coefficient tensors obey certain fundamental thermodynamic requirements,
namely, Onsager’s reciprocal relations and the positive definiteness of the diagonal coefficient tensors. Onsager’s reciprocal relations state that the tensor relating the independent
thermodynamic forces to their corresponding fluxes, must be symmetric [1].
I consider an N -component electrolyte in a dilute solvent flowing through a rigid porous
body with a periodic microstructure, where the geometry of the porous medium ensuresthat the inertia of the flow may be neglected. The electrolyte flows in response to a static(d.c.) electric field and a constant surface charge density on the pore walls. The magnitudeof the applied field is assumed to be sufficiently small to permit the linearization of the ionictransport (electrokinetic) equations.
Gross and Osterle [2] derived Darcy-scale transport equations for the case of a capillarymodel for charged wide-pore membranes. This simple geometry allowed for a complete analytical solution of the electrokinetic equations including the establishment of the Onsager
relations. By assuming the distance from the pore walls over which electrostatic interactions
are significant to be very small, Edwards [3] employed a volume averaging approach to derive
macroscopic transport equations for a periodic porous body. However, Edwards did not succeed in establishing Onsager’s reciprocal relations. The paper by Marino et al. [4] studies aperiodic porous medium subjected to an external electric field, a pressure gradient and a concentration gradient. The well-known phenomenological equations are assumed to hold and,starting from the electrokinetic equations on the pore-scale, the transport coefficient tensors
are calculated numerically for a number of different microstuctural geometries. Moyne andMurad [5, 6] also considered a periodic porous medium but in addition, they permitted theporous body to swell. No linearization of the equation describing the conservation of ionicspecies on the pore-scale was carried out. However they effectively neglected pore-scale ionicThe author is the winner of the Thomas Cherry Prize for best talk by a PhD student at ANZIAM 2005.
See AustMS Gazette 32 (2005), 103–104.

Homogenization of the ionic transport equations in periodic porous media273

convection. Under the assumption that the ionic concentrations in the pore water are uniform, Revil and Leroy [7] used volume averaging to derive macroscopic transport equations
from linearized microscopic equations. The Onsager relations were established under theseconditions. In all of the aforementioned articles, only a 1-1 electrolyte was considered.
In this article I show that all of the aforementioned assumptions are unnecessary andconsequently, my analysis includes the effects of pore-scale ionic convection. No assumptions
are made regarding the microstructure. Although we assume periodicity of the porous
domain, there exists an equivalence between periodic and statistically homogeneous random
porous media [8].
This work is stimulated by the author’s interest in developing a theory for the electroacoustic characterization of suspensions of porous particles [9]. To determine the electrophoretic motion of a porous particle, the governing equations outside the particle, inside
the particle and the boundary conditions coupling the two domains need to be specified. The
governing equations exterior to the particle are the well-known electrokinetic equations [10].
It is therefore natural to upscale the electrokinetic equations to obtain a Darcy-scale model
inside the particle. The author has used scaling arguments in conjunction with recent resultsin homogenization theory to develop a model for the hydrodynamics of an oscillating porous
particle [11], since electroacoustic experiments generate oscillatory flows. To be consistent
with this work, homogenization theory is employed to upscale the electrokinetic equations.

2Homogenization

Homogenization theory exploits the fact that transport phenomena in porous media occur
on two disparate length scales: a macroscopic scale (the scale of the reservoir, L) anda microscopic scale (the pore scale, `). Transport phenomena can then be characterized
via the small dimensionless parameter  = `/L. In practice we only observe macroscopictransport phenomena and it is sufficient for information regarding the microstructure to beretained purely in the form of averaged quantities, such as porosity and permeability. Thehomogenization process transforms equations on the microscopic scale to effective equations
on the macroscopic scale by determining the limiting behavior of the microscopic equationsas  → 0. For an extensive coverage of homogenization and porous media, see [12].
I briefly describe the construction of a periodic porous domain with period . A complete
description of periodic porous media can be found in [8]. A standard cell consists of a three
dimensional cube C containing a standard obstacle S, where the fluid part of the cell is
Y = C \ S, with boundary ∂C ∪ ∂S. This cell is extended periodically, with period , to allof three dimensional space. A domain Ω is then intersected with the periodically extended
standard cells to give the periodic porous domain. The fluid part of the periodic porous
domain is given the symbol Ω , with boundary ∂Ω \ ∂Ω. The limit  → 0 corresponds tothe size of the cells becoming progressively smaller and greater in number until the fluid
part is annihilated (or, the pore size goes to zero).
Under the assumptions stated in the introduction, the scaled linearized microscopic equations in Ω are, for j = 1, . . . , N [13],nj = n∞j exp (−zj ψ) ,2 ∇2 ψ = −4πNX

zj n j ,(1)
(2)

j=1


λj
∇2 Φj = zj ∇ψ · ∇Φj + E + u ,zj(3)274Jason R. Looker

2 ∇2 u = ∇P +NX

zj ∇nj (Φj + E · x) ,(4)j=1

∇ · u = 0,(5)with boundary conditions on ∂Ω \ ∂Ω,n̂ · ∇ψ = −σ,(6)n̂ · ∇Φj = −E · n̂,(7)u = 0.

(8)

The ionic parameters zj , λj and n∞
j represent the valency, drag and the bulk ionic number
density of the jth ionic species, respectively. The physical constant σ is the surface charge
density and E is the applied electric field. The functions ψ, nj , Φj , u and P are theequilibrium electric potential, equilibrium number density of the jth ionic species, ionic
potential of the jth ionic species, velocity of the solvent, and the partial pressure of thesolvent. The vector n̂ denotes the outward unit normal to ∂Ω \ ∂Ω. Observe that theboundary conditions are applied on the boundary of the pore walls, not on the boundary of
the macroscopic domain ∂Ω. For convenience, all functions and variables in (1)–(8) refer todimensionless quantities. We shall adopt this convention for the remainder of the article.
To study the limit as  → 0 of (1)–(8), we postulate that ψ, u, P and Φj have formal
two-scale asymptotic expansions of the form,f (x) = f0 (x, y) + f1 (x, y) + O2 ,where x ∈ R3 is the macroscopic (or slow) variable, and the fi are Y-periodic with respect
to the microscopic (or fast) variable y = x/. A Y-periodic function has the same value oneach opposite face of ∂C. The two different length scales imply that the derivatives must betransformed according to,1
∇ = ∇x + ∇y ,
21∇2 = ∇2x + ∇x · ∇y + 2 ∇2y .


The asymptotic expansions for ψ, u, P and Φj are substituted into (1)–(8) and terms of thesame order of magnitude are compared, generating a cascade of equations. Solutions aresought in terms of products of macroscopic forcing terms and cell functions. Cell functions
are defined only in Y and are required to be Y-periodic.

3Non-Equilibrium Thermodynamics

After homogenization of the microscopic equations and integration over a cell, it is shownin [14] that the macroscopic fluid velocity and ionic flux of the jth species can be expressedas:ueff (x) = −NXJj · ∇µeff
j (x) − K · ∇peff (x)in Ω,(9)in Ω,(10)j=1seff
jeff
j (x) = −Dj · ∇µj (x) −NX
l=1

Dcjl · ∇µeffl (x) − Lj · ∇peff (x)Homogenization of the ionic transport equations in periodic porous media

275

where µeffand peff are the effective electrochemical potential and pressure, respectively.
jThe electrochemical potential is a thermodynamic measure of how much work is required toreturn a system to equilibrium [15]; it is related to the ionic potential via,∇µj = −zj (∇Φj + E) .
The tensors Jj and Lj are coupling tensors, K is the permeability tensor, Dsj is a self
electrodiffusion tensor and Dcjl accounts for cross electrodiffusion effects. Note that Dcjj ≡ 0.
These transport coefficient tensors are expressed in terms of averages of cell functions thatare solutions to given boundary value problems in Y. For example, the αβ component of
the permeability tensor is given by,1{K}αβ =|Y|

N
X
 11
υ β (y) α dy +|Y|Y
j=1ZZY

 2
υ jβ (y) α dy,for α, β = 1, 2, 3, where |Y| is the volume of Y. The first term is the usual Darcy permeability
tensor [12], but in the presence of ionic motion, K contains extra contributions. The vectors
υ 1β and υ 2jβ represent microscopic solvent velocities: υ 1β is nonzero in the absence of ions,while υ 2jβ arises purely from the pore-scale ionic convection of the jth species.
My aim is to establish the validity of these macroscopic equations. This shall be accomplished by proving, without any additional assumptions, that the above macroscopic
system obeys one of the fundamental laws of non-equilibrium thermodynamics: Onsager’sreciprocal relations. The primary reference for this section is [1].
Onsager’s reciprocal relations are essentially a statement of time reversal invariance. Thatis, if a small perturbation to a system in some reference state is sent back in time, then thesystem will return to that reference state. If the system is expressed in the following format,
J = M · F,

(11)

where J is a vector representing the fluxes and F a vector of the independent thermodynamic forces, Onsager proved that the tensor M must obey M = MT , that is, M must besymmetric. Recasting( 9) and (10) into the form of (11) gives, ∇peffueff
KJ1··· ···JN jeff L1 eff Ds1 Dc12 · · · Dc1N  1  ∇µ1 
 .. 
 ..
.
. .
..
..  ·  .  = − .
 . Dc21
  . .
 . 
 .
..
..   .. ..
 ..  ..
.
.
.   . ceff
LN DN 1 · · · · · · DsN
jN∇µeff
NSymmetry of the coefficient tensor requires that:
K = KT ,TJj = (Lj ) ,Dsj = Dsj

T,

Dcjl = Dclj

T,(12)

must be true for each j, l = 1, . . . , N if the Onsager relations are to hold.
A necessary condition ensuring the entropy production of the system remains positive, isthe positive definiteness of the diagonal tensors in M. Therefore the permeability tensor K
and the self electrodiffusion tensor Dsj are required to be positive definite.
It is rigorously proven in [14] that (12) hold and the permeability and self electrodiffusiontensors are positive definite, for the transport coefficient tensors defined in this article. The
proof follows directly from the ionic and fluid cell problems.

276

4Discussion

Equations (9) and (10) are formally the same as the Darcy-scale equations modelling ionictransport in porous shales derived by Revil and Leroy [7], and are identical to the phenomenological transport equations presented in [1]. Explicit expressions for the transportcoefficient tensors in terms of solutions to cell problems are presented in [14], where it isshown that all of the transport coefficient tensors have been influenced by the inclusion of
pore-scale ionic convection in the present model.
This research constitutes a theoretical framework for a greater understanding of the assumptions underlying the phenomenological equations and similar models describing transport phenomena in porous media, however more work is required. For instance, computing
the transport coefficient tensors numerically by solving the ionic and fluid cell problems will
elucidate the relative importance of the terms in (9) and (10). Furthermore, making the underlying formal mathematical calculations rigorous by establishing the two-scale convergence[12] of the microscopic functions to their homogenized counterparts will further clarify theassumptions upon which (9) and (10) are based, and may be of independent mathematicalinterest.

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[9] R.W. O’Brien, Electro-acoustic effects in a dilute suspension of spherical particles, J. Fluid Mech. 190(1988), 71–86.
[10] R.J. Hunter, Foundations of colloid science, (Oxford University Press Oxford 1987–1989).
[11] J.R. Looker and S.L. Carnie, The hydrodynamics of an oscillating porous sphere, Phys. Fluids 16(2004), 62–72.
[12] U. Hornung (ed.), Homogenization and porous media, Interdisciplinary applied mathematics 6 (SpringerBerlin 1997).
[13] R.W. O’Brien and L.R. White, Electrophoretic mobility of a spherical colloidal particle, Journal of theChemical Society, Faraday Trans. 74 (1978), 1607–1626.
[14] J.R. Looker and S.L. Carnie, Homogenization of the ionic transport equations in periodic porous media,
submitted to Transport in Porous Media (2005).
[15] J. Lyklema, Fundamentals of interface and colloid science (Academic Press London 1991).
E-mail: jrlooker@ms.unimelb.edu.au
Particulate Fluids Processing Centre, Department of Mathematics and Statistics, The University of Melbourne, VIC 3010
Received 2 May 2005, accepted 30 May 2005.