In numerous studies on the electrolyte flows in rocks, the pore pressure and the streaming (electric) potential interplay through the equation, where is the current density, is the saturated rock conductivity, and is the electrokinetic cross-coupling term. Hydrogeological applications concern the study of water leakage from dams [1], groundwater flows in geothermal fields and volcanoes [2], estimation of water resources [3]. In electrochemistry, the above equation form a basis for managing microchip separations of analytes in nano-channels [4]; there is also an evidence that this equation find applications in hydrocarbon recovery [5,6].

By the Helmholtz-Smoluchowski theory [7], the term is given by the formula where is the porosity, is the dielectric permittivity of the saturating fluid, is the viscosity, and is the so-called -potential, the electric potential across the diffuse part of the interfacial double layer. In [8,9,10], the above formula is substituted by (or more sophisticated formulas), with being a dimensionless formation factor.

The goal of the present paper is to give more mathematical insight into the physico-chemical nature of the cross coupling coefficient. Restricting ourselves to one-dimensional flows, we derive a representation formula for by the two-scale homogenization technique [11,12], starting from the equations of the ions transport through a layered membrane with a periodical structure. On this way we arrive at electro-osmotic macro-equations, whereas electrokinetic coupling coefficients can be determined from micro-equations defined on the periodicity cell.

Homogenization is a process in which the composite material with microscopic structure is replaced by an equivalent material with macroscopic, homogeneous properties. There are two methods of up-scaling coupled equations at the microscale to equations valid at macroscale for fluid-saturated porous media. The first is the volume averaging and the second is the two-scale and multiscale homogenization. Volume averaging has been applied successfully to derive the form of Biot's equations of poroelasticity [13], and a wide variety of other up-scaling problems in double-porosity poroelasticity [14]. The averaging theorem used by all these authors is due to J. C. Slattery (1967) [15] and is based on well-known Green's theorem together with the idea that in relatively small regions volume averages of spatial gradient in statistically homogeneous media are presumably closely related to gradient of volume averages.

The two-scale homogenization method requires that the heterogeneous microstructure of a rock sample is described by spatially periodic parameters and the microscale of the heterogeneous porous medium is much smaller than the macroscale of most interest. The approach involves assuming that any quantity can be treated as a function of a macroscale variable and a microscale variable. The two-scale homogenization is a well established method in the theory of partial differential equations with rapidly oscillating periodic coefficients. This method has a lot of important applications in various branches of physics, mechanics and modern technology: porous media, composite and perforated materials, thermal conduction, acoustics, electromagnetism. For general references on the homogenization theory we refer to [12,16,17,18].

The two-scale homogenization method can give formulas for coefficients in the up-scaled equations, whereas volume averaging methods give the form of the up-scaled equations but generally must be supplemented with physical arguments and/or data in order to determine the coefficients. A more detailed comparison of two up-scaling methods can be found in [19].

The present study is applicable to sandstones if surface conductivity can be neglected. When passing to claycontaining rocks one should also take into account bound charges concentrating on the interface surfaces. Such rocks are not considered here.

Within the frame of the nonequilibrium thermodynamics, the fluxes (the Darcy's volume fluid velocity and the electric current density) are derived as a linear combination of thermodynamical forces (the pressure gradient and the electric potential gradient):

where is the permeability. Our goal is to show that these equations, specified for one-dimensional flows through a layered membrane, can be derived by the two-scale homogenization technique starting from the equations valid at microscale. While deriving the up-scaled equations (1) and (2), (which can trace back to Helmholtz and von Smoluchowski) we obtain a formula for the cross-coupling coefficient.

  In this section, we summarize equations that govern the flows of a binary electrolyte solution through the pore space of a solid dielectric. To make clear our hypotheses on physical parameters, we use the Gaussian system of units. Clearly, while comparing final calculations with experiments, we apply the SI units. The electric field E obeys the charge conservation law

where is the fluid dielectric permittivity, is the charge of a positive ion, is the charge of the negative ion, is the ion concentration. Viscous incompressible flows of the electrolyte solution is governed by the Navier-Stokes equations [7]

with the inertial terms being neglected in the first momentum equation. Here, is the velocity of the bulk fluid. The motion of both the ionic species satisfies the transport equation

(5)

with the flux given by the Nernst-Plank relation [7]

where is the diffusion coefficient, is the Boltzmann constant, is the absolute temperature.

Inside the solid dielectric, the electrical field obeys the equation

where is the solid dielectric permittivity. In what follows, stands for the electric potential,.

The solid-fluid boundary conditions will be formulated below for one-dimensional flows.

To motivate our further study we keep in mind a vertical membrane of thickness (see Figure 1) when the inflow pressure (on the left) is grater than the outflow pressure. It is the pressure gradient which mainly controls the flow. It is also possible that the flow is due to the external electrical field. Commonly, an inflow concentration of the -th ion is prescribed on the left.

Now, to perform analytical study of the flow equations, we consider a vertical "membrane" of an infinite thickness. We study electrolyte steady flows through the horizontal layer of thickness consisting of horizontal thin slits of the same thickness separated by layers of a solid

dielectric of the same thickness. The central points of the liquid intervals are the points of reference where the ion inflow densities take the prescribed values.

Let and stand for fluid and solid domains

In the domain, we look for steady solutions of the fluid Equations (3-5) in the form

where and are given constants. Under these assumptions, Equations (3-5) in each fluid interval become

We study horizontal flows along the -axis, hence. The latter equality is equivalent to

Integrating between and, we exclude the concentration functions from consideration by the formula

In the solid intervals, the potential satisfies the equatio

In what follows we assume that the dielectric permittivity function and the fluid indicator function

are extended periodically on the real line. Given a function continuous everywhere except a point, we introduce the jump as follows

In some sandstones, surface conductivity can be neglected depending on the pore water salinity and the cation exchange capacity of the mineral surface. For such sandstones, the “electric” boundary conditions reduce to the conditions of continuity of the potential and the normal component the electric induction vector:

where k = 1,…, n-1 and l = 0,…, n-1.

The velocity satisfies the no-slip conditions

We assume that satisfies the external boundary conditions

with the prescribed -potentials and. We introduce a function, which takes the value of the integer part of the number. Then the functions

take constant values for. Thus to define on the whole interval, one should solve the non-local Poisson-Boltzmann equation

jointly with the conditions (8) and (10). Observe that the function is periodic, and on the interval of periodicity.

With the function at hand, one can find a velocity from Equations (6) and the boundary conditions (9).

We look for an asymptotic solution of problem (11), (8), (10), (6), (9) for the functions and, assuming that the ratio is a small parameter for some positive entire number. We argue by the homogenization approach [24], so the entire interval is fixed and varies in. In that case and

Here, is the porosity.

We call a slow variable and we introduce the fast variable. With being small, the periodic functions and oscillate strongly and they can be represented as functions of the fast variable:

where

are periodic functions with the period equal to 1. In what follows the functions

and

are extended periodically for all. The functions, , can be written as

and

In the notations used, the function on the interval is a solution of the problem

where is equal to

It follows from Equations (6) that the bulk velocity satisfies the equation

Let us perform scaling, using the symbol for a reference value of the dimensional quantity and the symbol for a dimensionless quantity of, i. e.. We use the following notations:

The quantity

is known as the Debye length. In terms of dimensionless variables Equations (13) and (11) in the fluid domain take the form

Here,

In the solid domain Equation (7) becomes

Assuming that the dimensionless quantities satisfy the equalities

we obtain a hierarchy of problems to study. In this paper we restrict ourselves to the case when all the powers are equal to zero, i. e.. The meaning of these hypotheses is the following. The relation implies that electroosmotic force and thermal force are of the same order. Observe that the relation holds, for example, for the symmetric electrolyte (where and) in water at, with the valency and with the -potential equal to [4]. When is not small, the Debye-Hückel linearization of the Poisson-Boltzmann equation does not work. Under the condition the Debye length can be longer compared to electrical double layer, moreover the double layer overlapping could occur. Indeed, it is a useful rule of thumb that [4] where is the valency. For the above mentioned electrolyte with the counterion molar concentration we have, whereas the double electric layer is normally only a few nanometers thick [4] and the nanocapillary membrane may have the pore diameter of  15 [20]. For such cases the hypothesis is natural. Hypothesis amounts to the effect that the horizontal pressure gradient and the applied horizontal electrical field are of the same order. The relation means that viscous response is of the same order as the applied horizontal pressure gradient.

There is one more assumption that we impose on the Péclet number:

The hypothesis implies that convection  and diffusion are of the same order.

We close this section by reminding the Debye-Hückel approach to the Poisson-Boltzmann Equation (11) in the single layer with the boundary conditions and as and. In the case of symmetric electrolyte, the linearized equation (11), in the SI system of units where is substituted by 1, becomes  , since the nonlocal term vanishes as. Clearly, is a solution. This explains the notion (14).

We proceed by returning to the dimensional variables. Using the method of the two-scale expansions [12], we look for the solution of Equation (12) in the form of an expansion series

where the functions are periodic in the variable, , with a period equal to 1 for each. We introduce the flux

Clearly,

We present this flux as a series

where the functions are 1-periodic in for all.

Using the formula

and substituting the series (16) and (19) into equality (17), we obtain an equality which looks like

Thus for all. In particular, the three first equalities can be written as

and

Substituting the series (16) and (19) into equality (18) and paying attention to the powers and, we obtain the equations

Equations (20) and (21) allow one to determine the functions, and uniquely. Indeed, with a function independent of the variable, we look for by the method of separation of variables assuming that there exists a -periodic function such that

Substituting this presentation into equation (20), we find that the function solves the following problem on the interval:

The latter integral condition serves for uniqueness. We integrate and arrive at the formulas

Next, we use periodicity and integrate equation (21) with respect to to obtain the following macro-equation for:

As for the function, we look it in the form

Substituting this presentation into equation (21), we find that  is a periodic solution of the problem

This problem has a unique solution provided

Thus, we have established the following asymptotic equality for the electric potential:

Integrating Equation (13), we obtain the following formula for velocity in each fluid domain:

where

We extent the function by zero to the solid intervals and denote such an extension by. Now, with standing for, we have for all that

With given by the expansion series (16), we look for in the form

where the functions are -periodic in and for. After simple calculations, we find that

Using the properties of functions, , , we obtain that is equal to

for

As for, we find that it is equal to

for

By virtue of the multiplier in the right side of equation (11), we can assume that

Then, the variables belong to the interval

also. As is between and, the inequalities

hold and the second derivatives of in Equation (25) are meaningful. In addition, it follows from Equations (23), (24) and (25) that, for, the functions satisfy the equations

Thus, we obtain

Substituting equations (28) and (29) into equation (27) and considering only the power, one can show that the function does not depend on the variable and has the form

Integrating equation (30) over the periodicity cell, we obtain the macroscopic equation

where the hydrodynamic and electrochemical mobilities are defined by the formulas

Thus, we have established the following asymptotic equality for the velocity field:

We introduce the total electric current whose horizontal component in the fluid phase is equal to

We extent the function by zero to the solid intervals and denote such an extension by. Due to hypothesis (15), we have that  . This is why we look for in the form of the expansion series

It follows from (32) and (28) that

By integration, we arrive at the macroscopic equation

where and

Thus, we have established the following asymptotic equality for the electric current:

The asymptotic equalities (26), (31) and (33) are valid in the sense of weak or two-scale convergences; mathematical aspects of these asymptotic expansions are extensively investigated in [21,22,23].

We introduce the Darcy volumetric flow rate and the current density. By the above asymptotic analysis we have derived the macroequations (which are valid up to terms,)

which describe electrolyte flow and distribution of the electric potential across a layered membrane under the assumption that

are prescribed data. For such a membrane, the effective dielectric permittivity and the electrokinetic coupling coefficients are given by the formulas

Formula (35) stating that the effective permittivity of the layered membrane is the harmonic mean of and was first derived by Maxwell (Maxwell 1881) in a different way [24]. Observe, that the Onsager reciprocity relation [25] is not imposed but derived in the above calculations as a consequence of the homogenization procedure. Moreover, the inequality

providing nonnegativity of the entropy production rate is also satisfied automatically [26] due to the representation formulas (36) and (37). The inequality (38) becomes equality if both the diffusion coefficients are negligible. Observe that for some free solutions [27].

We emphasize that the coupling coefficients in the macro-equations (34) are given by the representation formulas (36) and (37) as a result of an extensive analysis of the micro-equations (22) and (25) for the functions and defined on the periodicity cell.

Clearly, the electroosmosis Equations (1) and (2) should coincide with the system (34) for onedimensional flows. Whereas the formula

for the cross-coupling coefficient have a drawback of measuring the -potential, the kinetic coefficients derived by homogenization for the ideal (layered) porous medium do not depend on. One can exploit this advantage in calculation of the coupling coefficient for general porous media.

Applying the general Equations (1) and (2) to the ideal (layered) porous medium, we find that

Now, we have

and inequality (38) gives the following estimate for the electrokinetic cross-coupling coefficient:

As for real rocks, formulas (39) and (40) suggest to take in the form

where is a dimensionless geometrical factor. In applications, the above formula can be of use if no data are available for the diffusion coefficients and the potential. We emphasize that formula (41) is not a physical law but rather an engineering formula which can be of help for some sandstones when surface conductivity can be neglected.

Firstly, we evaluate for a rock sample on the basis of the F.F. Reuss experiment [28]. Such an experiment reveals that a difference in the electric potentials applied to water in a U-tube results in a change of water levels when the low part of tube is plugged with a sandstone sample (Figure 2).

We calculate the weight of salt water which fills the cylinder of height with cross section area (Figure 2). We have, where is the

gravitational acceleration and is the water density. The pressure drop across the sandstone plug is equal to

. On the other hand it follows from Equations (34) that at equilibrium, when, we have.

In [29,30], a mathematical model (jointly with a computer code) is developed for calculation of the electric conductivity of a saturated rock. The model allows one to find an optimal Archie-like law

where is the conductivity of the saturating fluid, is the percolation threshold porosity, is the cementation factor. For sandstones, it was calculated in [29] that 0.03, 1.5. Thus, for sandstones, formula (41) becomes

Now, the factor can be evaluated from the formula

We perform calculation assuming that, as in [31], the applied potential difference results in the water level difference. The rock data are taken from [6]:, , , ,. With these data at hand, we find that. It is the cross-coupling coefficient that can be measured in applications [32]. With the factor given above, we find that in agreement with the data in [6].

Next, we calculate the factor for the rock sample composed of Berea sandstone 500 starting from experimental measurements of streaming potential when a fluid, with a prescribed NaCl concentration (500 ppm), flows through the sample [5]. Given data, , , , we find from formula (41) that.

We have proposed a two-scale model for one-dimensional horizontal electroosmotic flows in a number of thin horizontal slits, with a horizontal pressure gradient and a horizontal electrical field being driving forces. The model is derived within the framework of homogenization in the up-scaling of the pore-scale description consisting of Stokes equation for bulk fluid flow and the Nernst-Plunk equation for the ion transport. The homogenized model is a generalization both of the Darcy law and the Ohm low. According to this model, both the fluid flux and the electric flux depend linearly on the horizontal pressure gradient and the horizontal electrical field, with the coupling coefficients obeying the Onsager symmetry condition and not depending on the - potential.

As for three-dimensional general flows in sandstones in the case when surface current is negligible, the cross-coupling coefficient is obtained in the approximate form, where is the fluid saturated rock electric conductivity, is the rock permeability, is the fluid viscosity, and is dimensionless geometrical factor which depends on the sample. We evaluated that for Berea sandstones.

The authors were supported by Russian Fund of Fundamental Researches (grant 10-05-00835-a) and State Contract No. 14.740.11.0355 of Federal Special-Purpose Program “Personal”.