Effect of micro-inhomogeneity on the effective stress coefficients and undrained bulk modulus of a poroelastic medium: A double spherical shell model

Abstract

Although most rocks are complex multi-mineralic aggregates, quantitative interpretation workflows usually ignore this complexity and employ Gassmann equation and effective stress laws that assume a micro-homogeneous (mono-mineralic) rock. Even though the Gassmann theory and effective stress concepts have been generalized to micro-inhomogeneous rocks, they are seldom if at all used in practice because they require a greater number of parameters, which are difficult to measure or infer from data. Furthermore, the magnitude of the effect of micro-heterogeneity on fluid substitution and on effective stress coefficients is poorly understood. In particular, it is an open question whether deviations of the experimentally measurements of the effective stress coefficients for drained and undrained elastic moduli from theoretical predictions can be explained by the effect of micro-heterogeneity. In an attempt to bridge this gap, we consider an idealized model of a micro-inhomogeneous medium: a Hashin assemblage of double spherical shells. Each shell consists of a spherical pore surrounded by two concentric spherical layers of two different isotropic minerals. By analyzing the exact solution of this problem, we show that the results are exactly consistent with the equations of Brown and Korringa (which represent an extension of Gassmann's equation to micro-inhomogeneous media). We also show that the effective stress coefficients for bulk volume α, for porosity nϕ and for drained and undrained moduli are quite sensitive to the degree of heterogeneity (contrast between the moduli of the two mineral components).For instance, while for micro-homogeneous rocks the theory gives nϕ = 1, for strongly micro-inhomogenous rocks, nϕ may span a range of values from –∞ to ∞ (depending on the contrast between moduli of inner and outer shells). Furthermore, the effective stress coefficient for pore volume (Biot–Willis coefficient) α can be smaller than the porosity ϕ. Further studies are required to understand the applicability of the results to realistic rock geometries.