Effective medium theory and network theory applied to the transport properties of rock

The effective medium theory (EMT) proposed by Kirkpatrick (1973) is known to be a useful tool to describe the conducting properties of heterogeneous media. We check the validity of the theory on several probability distribution functions. The accuracy of the theory is inferred by comparing the results given by the EMT to those obtained when using an exact method based on the resolution of two‐dimensional conducting networks. It turns out that the EMT approximation is quite good for quasi‐uniform distributions of conducting elements, whereas large discrepancies appear for highly contrasted distributions such as decreasing exponential functions. The reason is that ensemble averaging and spatial averaging are not equivalent for broad, nonuniform distributions. Therefore the effective medium theory may not be appropriate for the calculation of the transport properties of rocks which exhibit this type of distribution. The nature of the distribution varies, however, depending on the rock type and the length scale of interest. Critical path analysis can be a complementary tool to deal with highly heterogeneous conducting media. The accuracy of such a technique appears to be better than that of the EMT approximation. But it depends also on the representativity of the input data. Finally, an example of data relative to Fontainebleau Sandstone is developed in this paper. It is concluded that for this rock, network theory is the most accurate tool for calculating transport properties from statistically averaged observations on two‐dimensional sections.