Change in effective bulk modulus upon fluid or solid substitution

摘要

Gassmann’s equations, which are frequently used to predict the change in elastic properties of fluid-filled rocks upon substitution of one pore fluid with another, are not applicable for solid-filled rocks, i.e., when the pore material has nonnegligible shear modulus. Examples of such naturally occurring solid materials are heavy oil, cold bitumen, kerogen, clay, etc. Using volume averaging, we derived an exact solid substitution equation for isotropic effective bulk modulus for porous media with solid-filled pores. This exact equation generally requires an additional stiffness, which might not be directly measured or known. However, we established rigorous inequalities between this additional stiffness and other measurable effective stiffnesses. An exact expression had previously been obtained for the effective bulk modulus of solid-filled rocks by invoking a heuristic parameter. We clarified the physical significance of this parameter.We found that the approximate solid substitution equation previously suggested was limited to rocks such as those with identical stiff ellipsoidal pores, and its predictions do not always fall within Hashin-Shtrikman bounds. This is so because this approximation implicitly assumes homogeneous pore pressure and no change in pore-filling shear modulus upon solid substitution. We proposed new solid substitution approximations that provide a better match with numerical simulations and laboratory data, and we provided a step-by-step recipe for practitioners.