Phenomenological Understanding of Poro-Elasticity via the Micro-Mechanics of a Simple Digital-Rock Model

Poro-elasticity is a material concept that expresses the reversible, macro-scale process interactions that occur in a porous material, such as rocks. These process interactions take place between the pore fluids, and the rock framework (or ‘skeleton’) which contains the pores. The phenomenological basis of poro-elasticity is examined via a micro-mechanics analysis, using a simplified digital-rock model that consists of solid elements in a lattice arrangement, and which hosts a well-connected, lattice-like network of simply-shaped pore elements. The quasi-static poro-mechanical bulk response of this model is defined fully by closed-form equations that provide clear understanding of the process interactions, and which allow key effects to be identified. Several external boundary conditions (non-isotropic strain and stress) are analyzed, with drained and un-drained pore-fluid conditions, along with arbitrary pore pressure states. The calculated responses of the pore-scale model, when translated into continuum-scale equivalent behaviors, indicate significant problems with the existing theories of poro-elasticity that are rooted in an enriched-continuum perspective. Specifically, the results show that the principle of effective stress (and the Biot coefficient alpha) is wrongly attributed to a deficiency in the role of pore pressure. Instead, the micro-mechanics-based phenomenological understanding identifies the change of effective stress, in a characteristically-confined setting, as being the result of changes in the stress components, with a key dependency on the specifics of the far-field constraints. Poro-elasticity is thus not a material characteristic; instead, it is a description of a non-linear system operating at the pore scale. The analysis reveals a discrepancy between the stress states within the model domain and the external stress state. This yet remains to be addressed, in order to translate the micro-scale behavior into an equivalent material law.