THE ANALTYICAL FORMULATION

The Bergman-Milton (Bergman, 1978, 1980; Milton, 1980, 1981; Bergman, 1982; Korringa and LaTorraca, 1986; Stroud et al., 1986; Berryman, 1992) analytical approach to understanding some generic effective conductivity g^* of two-component inhomogeneous media shows that  

$$g^* = G(g_1,g_2) = g_1G(1,0) + g_2G(0,1) + \int_0^\infty \frac{dx {\cal G}(x)}{\frac{1}{g_1} + \frac{x}{g_2}},$$ (1)

where G(1,0) and G(0,1) are constants depending only on the geometry and ${\cal G}(x) \ge 0$ is a resonance density also depending only on the geometry. The integral in (1) is known as a Stieltjes integral (Baker, 1975). Although the representation (1) has usually been employed to study the behavior of g^* in the complex plane when g₁ and g₂ are themselves complex (corresponding to mixtures of conductors and dielectrics), I will restrict consideration here - as Bergman did in his early work (Bergman, 1978) - to pure conductors so that g₁, g₂, and g^* are all real and nonnegative.

In the limit that one or the other of the two constituents is a perfect insulator (g_i = 0), or in the more common case when one of the constituents is much more strongly conducting than the other, I can define two quantities called formation factors (Archie, 1942) by  

$$\lim_{g_1 \to \infty} \frac{g^*}{g_1} = \lim_{g_1 \to \infty} G(1,g_2/g_1) = G(1,0) = \frac{1}{F_1},$$ (2)

and, similarly, by  

$$\lim_{g_2 \to \infty} \frac{g^*}{g_2} = \lim_{g_2 \to \infty} G(g_1/g_2,1) = G(0,1) = \frac{1}{F_2}.$$ (3)

In a porous material, where solid and pore fluid are each continuously connected throughout the material, both formation factors are finite, and both satisfy $F \ge 1$. The more commonly measured quantity of this type is the electrical formation factor for the continuous fluid component. This measurement has some possible complications due to surface conductance (Johnson et al., 1986; Wildenschild et al., 2000), but it is usually not contaminated by conductance through the bulk solid material because most rock grains can be correctly assumed to be electrically insulating to a very good approximation. Since the formation factor is strictly a measure of the microgeometry of the heterogeneous medium, it is the same number [except for those possible complications already mentioned of surface electrical conduction (Johnson et al., 1986; Wildenschild et al., 2000), which can be eliminated whenever necessary by known experimental methods] for all mathematically equivalent conductivities. For this presentation, I will use F₁ to represent this formation factor associated with the pore space. On the other hand, for thermal conduction the rock grains are the most highly conducting component and the pore fluids tend to be much more poorly conducting - especially so in the case of saturating air. So I will take F₂ to be this formation factor associated with the solid frame of the porous material.