Elastic potential energy in linear poroelasticity

Biot's theory of linear poroelasticity dates back to the mid-20th century and is still currently the backbone for understanding deformation processes in fluid-saturated porous rocks with applications in geomechanics, hydrogeology, reservoir engineering, and exploration geophysics. Therein, the elastic potential energy is the critical concept to derive the macroscopic stress tensors and constitutive equations. The potential energy is taken as the single function that forms an exact differential in macroscopic strains of the solid and fluid phases. This choice intrinsically implies a reciprocal interaction between the compressive stresses of the two phases. However, it leaves out the possibility that the phases can interact in a nonreciprocal manner, which is sometimes inferred from experimental observations. The limited scope of the Biot theory is overcome by upscaling the pore-scale governing equations using volume averaging aided by the physical argument that the conservation of mass, momentum, energy, and principle of equivalence must hold at all scales. This upscaling reveals that the proper measures of deformations are not only the deformation gradient terms but also a porosity change term that accounts for the emergent pore-interface kinematics at the macroscale. It also reveals that the elastic potential energy of a porocontinuum is the sum of the potential energies of the solid and fluid phases, which are functions of their so-defined respective measure of deformation only. Such potential energy is no longer limited to only the reciprocal interaction of the compressive stresses of the phases, but it enables us to capture nonreciprocal interactions still within the realm of linearity. A nonreciprocal interaction arises whenever the potential energy density is not uniformly distributed at the pore scale but partially localized in the vicinity of pore interfaces or within the bulk part of the phases. A direct consequence is that the coupling coefficients in the linear constitutive equations are expected to be different.