Theory

The diffusion of a molecular species in a fluid is described by the following equation (in one dimension)

$$\frac{\partial c}{\partial t} =D_{fi}\frac{ \partial^2 c}{\partial x ^2}.$$ (1)

Here c is the concentration of the molecular species, t is time, x is position, and D_fi is the free molecular diffusivity in fluid i.

At length scales much larger than the typical pore size, diffusion is generally described by the macroscopic diffusion equation [1] (again in one dimension)

$$\frac{\partial C S \phi}{\partial t} =D_{bi} \frac{\partial^2 C}{\partial x^2} .$$ (2)

where C is the concentration of a molecular species in the fluid phase, S is the saturation of fluid i, $\phi$ is the porosity, and D_bi is the bulk (or macroscopic) diffusion coefficient associated with diffusion in fluid phase i and can depend on degree of saturation (here it is assumed the porous medium is uniformily saturated).

Diffusion in a porous medium can be very slow, making measurements of D_bi very time consuming. For the case when the solid is nonconducting and the pore space is fully saturated (S=1) by a conducting fluid with conductivity $\sigma_i$_i , the bulk diffusivity, D'_bi, (the prime denotes the case of S=1) may be obtained by electrical measurements of the bulk conductivity, $\sigma_b$_b, and using the Einstein relation ${D'}_{bi} = D_{fi} \frac{{\sigma'}_b}{\sigma_i}$ [3], where. D_fi is the free diffusivity of the ionic species in fluid i being measured.

Consider, more closely, the case of a porous medium filled with two fluids. Assume one fluid is wetting (energetically favorable to reside near the surface) and the second is non-wetting (energetically favorable not to reside near the pore surface). An example could be concrete filled with water and air where the water preferentially wets the porous concrete surface. If we allow a simple molecule to diffuse in a single fluid component, that molecule is limited to move in a subset of the total pore space which depends on the degree of saturation. We define the relative diffusivity, D_ri=D_bi/D' _bi, where D_bi is the bulk diffusivity in the non-wetting or wetting phase only (labeled i,i=1,2). Since $D_{bi}=D_{fi}\frac{\sigma_{bi}}{\sigma_i}$and ${D'}_{bi}=D_{fi}\frac{\sigma_{b}}{\sigma_i}$, $D_{ri}=D_{bi}/{D'}_{bi}=\frac{\sigma_{bi}}{{\sigma'}_{b}}$. The most widely known empirical relationship between conductivity and fluid saturation S (the fraction of pore space occupied by the designated fluid) is called Archie's second law [8]: _bi / _i =   ⁿS ^m where $\phi$ is the porosity. Note that when S=1 we have Archie's first law [8]. Since it may be viewed that changing the degree of saturation is effectively changing the accessible pore space (or effective porosity) it is often assumed that m=n. However, it is not always clear how conductivity depends on saturation since how and where the pore space is filled should depend on the previous history of the fluids ingress and on the details of the pore space connectivity. Therefore, simply varying the porosity in such semiempirical relations may not be sufficient for an accurate prediction of conductivity.