Gassmann-Consistency of Inclusion Models

Abstract:

Mathematical inclusion theories predict the effective elastic properties of a porous medium with idealized-shape inclusions as a function of the elastic moduli of the host matrix and those of the inclusions. These effective elastic properties depend on the volumetric concentration of the inclusions (the porosity of the host frame) and the aspect ratio of an inclusion (the ratio between the thickness and length). Seemingly, these models can solve the problem of fluid substitution and solid substitution: any numbers can be used for the bulk and shear moduli of the inclusions, including zero for empty inclusions (dry rock). In contrast, the most commonly used fluid substitution method is Gassmann’s (1951) theory. We explore whether inclusion based fluid substitution is consistent with Gassmann’s fluid substitution. We compute the effective bulk and shear moduli of a matrix with dry inclusions and then conduct Gassmann’s fluid substitution, comparing these results to those from directly computing the bulk and shear moduli of the same matrix but with the inclusions having the bulk modulus of the fluid. A number of examples employing the differential effective medium (DEM) model and self-consistent (SC) approximation indicate that the wet-rock bulk moduli as predicted by DEM and SC are approximately Gassmann-consistent at high aspect ratio and small porosity. However, at small aspect ratios and high porosity, these inclusion models are not Gassmann-consistent. For all cases, the shear moduli are not Gassmann-consistent at all, meaning that the wet-rock shear modulus as given by DEM or SC is very different from the dry-rock moduli as predicted by the same theories. We quantify the difference between the two methods for a range of porosity and aspect ratio combinations.