Next: Pore-structure-defined length scales
There are a wide variety of random, porous materials, whose transport properties are of scientific and technological interest [1,2]. Important examples include sedimentary rocks, soils, catalysts, ceramic filters, and cement-based materials such as concrete. Transport properties include the rate of either diffusive, electrical, or fluid movement through fluid-filled pore space. The technological reasons for interest in these transport properties varies from material to material. For example, the rate of flow of water and oil in sedimentary rock is important to the petroleum industry, while the water-aided movement of ions is more important for cement-based materials and plays a central role in their degradation, and in their use as containment barriers for toxic and radioactive wastes. In general, however, a common scientific goal for all such materials is to develop a quantitative understanding of the relationship betwen pore structure and transport properties.
Recently, there has also been considerable interest [3,4,5,6,7,8,9] in relationships between transport properties. In particular, the question of the existence of a useful relationship betwen the fluid permeability k of a fluid-saturated porous material, and its bulk electrical conductivity σ has been studied. The fluid permeability is defined via Darcy's law :
where v is the fluid velocity averaged over the total cross section of the porous sample, η is the fluid viscosity, and ΔP is the pressure drop over the sample length L. The analogous defining equation for the bulk conductivity of the same porous material, where the saturating fluid is a conductor and the solid phase is an insulator, is Ohm's law:
where ΔV is the potential drop across a sample of length L and cross-sectional area A, and I is the total current passing through the sample. We define Γ ≡ σ/σo, where σo is the conductivity of the saturating fluid, as the dimensionless conductivity. Since k has dimensions of length squared, a length scale l, based on the pore structure, must be defined in order to relate k and Γ.
To show heuristically how k and Γ separately depend on pore structure, it is useful to write them as [1,3,4]
|Γ = βφ, k = β'φl 2,||(3)|
where φ is the total porosity, which corrects for the reduced cross-sectional area available for flow, l is some length scale defined by the pore structure, and β and β' are factors that take into account pore shape, tortuosity, connectivity, etc. Given Eq. (3), it is easy to see that an estimate of k can be made by using Γ = βφ as an estimate of the β'φ product, and then defining a pore structure length scale l. The prediction for k would then be
|k = cΓl 2,||(4)|
where c = β'/β is an unknown, but hopefully calculable, constant scale factor. Different predictions of the appropriate length scale l have been made in the literature. We consider the three following length scales: h, the pore volume divided by the pore surface area, sometimes called the hydraulic radius ; Λ, defined by electrical conductivity measurements ; and dc, defined by a nonwetting fluid injection experiment . A fourth length scale defined in terms of a difffusion-limited trapping length, which we do not consider, has also been examined [5,6] and shown to have limited success in predicting permeability.
The physical assumptions that underlie the usage of the above-mentioned length scales are quite different. It is therefore important to test and compare each length scale in order to understand what properties of a material's pore structure most strongly determine its permeability. It is also of interest to try to determine how and if, at a fundamental level, the electrical and fluid-flow problems are related. In this paper we carry out this program, using computer simulation, applied to a simple two-dimensional (2D) model of a continuum random porous material. Such an experimental program has been suggested by Le Doussal , and computer simulation studies with similar goals [5,8], using mainly periodic models, appeard as this manuscript was in preparation.
An outline of this paper is as follows. In Sec. 2 the various length scales are defined and in Sec. 3 we discuss the structural models and algorithms used to compute k, Γ, and the three length scales. In Sec. 4 we analyze the example of a simple periodic porous medium, and in Sec. 5 results on random porous material models are presented. In Sec. 6, correlation functions are used to quantitatively compare and contrast the electrical and fluid problems. Our findings are discussed and summarized in Sec. 7.