Gassmann equations and the constitutive relations for multiple‐porosity and multiple‐permeability poroelasticity with applications to oil and gas shale

Summary Micromechanical characterization of multiple‐porosity and multiple‐permeability fluid‐saturated porous materials from the properties of their single‐porosity constituents is, to date, an open problem in our poromechanics society. This paper offers an in‐depth view to this problem by considering the thermodynamic potential energy density, consistent with Biot's original definition, together with the general thought experiment, which allows for independent control of the sample's confining stress and distinct fluid pore pressures within its individual porosity networks. The complete set of well‐known poroelastic constants, namely, Biot–Willis effective stress, Skempton's pore pressure, and specific storage coefficients, as well as drained, undrained, and Biot moduli for a fluid‐saturated porous material, is herein identified with the reformulated theory. In particular, Gassmann relation for the bulk compressibility of the fluid‐saturated material is accordingly upgraded to the case being addressed in this study. The practical implications of the theory are showcased through a class of analytical solutions to the time‐dependent poroelastic responses of shale to compression, when the hierarchical structure of its porous networks are accounted for at different levels of complexity and inter‐porosity exchange effects. For this purpose, the laboratory setup of a quasi‐2D compression test is considered, in which disk‐shaped fluid‐saturated samples of shale are allowed to drain laterally, while being sealed and confined from the top and bottom. A general closed‐form solution to this problem is derived in the Laplace space, and the inverse numerical results for the cases of single‐porosity, double‐porosity, triple‐porosity, and quadruple‐porosity shale are discussed in the time domain. Copyright © 2015 John Wiley & Sons, Ltd.