Effective Properties of Rocks

Effective Medium Theory (EMT) is a method to estimate the macroscopic properties of a composite materialComposite material based upon the relative fractions and specific properties of its constituent parts. Its history, and the mathematical treatment of its different versions (Maxwell’s ApproximationMaxwell’s Approximation, the Self Consistent MethodSelf Consistent Method, and Differential Effective MethodDifferential Effective Method) are given in Sect. 9.1. Section 9.1.1 illustrates the use of EMT through[aut]Kirkpatrick, S. (Kirkpatrick 1973) determination of the effective conductivity of a network of random conductorsEffective conductivity of a network of random conductors. Differential Effective Medium (DEM) approximation (Bruggeman 1935)[aut]Bruggeman, D.A.G. is treated in Sect. 9.2, in the context of electric conductivityElectric conductivity and elasticityElasticity. In Sect. 9.3 we[aut]Hashin, Z. discuss[aut]Shtrikman, S. variational inequalitiesVariational inequalities (Hashin and Shtrikman in Mech Phys Solids 1963, and in Phys Rev 1963), and a generalized mean value theorem (Korvin 1978, 1982), for effective rock properties, and present an example (Korvin 2012) for the anisotropic case. Section 9.4 reviews attempts to theoretically deriveArchie’s law Archie’s law \({\sigma }_{rock}={\sigma }_{fluid}{\Phi }^{m}\) from basic principles, and to determine the cementation exponentCementation exponent m.