Rigorous connection between physical properties of porous rocks

Rigorous cross-property bounds that connect the effective thermal conductivity k_* (or the electrical conductivity σ_*) and the effective bulk modulus K_* of any isotropic, two-phase composite were recently derived by the authors. Here we reformulate these bounds and apply them to porous rocks with dry or fluid-filled pores. It is shown that knowledge of the effective conductivity can yield sharp estimates of the effective bulk modulus (and vice versa), even in cases where there is a wide disparity in the phase properties. The bounds yield, in particular, relations between the formation factor and the bulk modulus of the porous medium. By using the same approach we obtain new relations between the bulk moduli of a dry porous material and the bulk modulus of the same material with fluid-filled pores that are more general than the traditional Gassmann equation. The Gassmann formula for the bulk modulus of the fluid-saturated porous medium is shown to correspond to a lower bound on this quantity. Limiting cases that we consider include cracked materials with dry and fluid-saturated pores. Theoretical results are tested against experimental measurements of the effective bulk modulus of dry and water-saturated Westerly granite and sandstone samples. We found good agreement between our cross-property bounds and the experimental data, even when the experimental data depart from the Gassmann formula. Our results add new insight to understanding of the properties of the porous media. They show that the Gassmann approximation works well for rocks with high porosity but needs to be corrected for rocks with high crack-type porosity.