Modeling Anisotropic Permeability of Coal and Shale with Gas Rarefaction Effects, Matrix–Fracture Interaction, and Adsorption Hysteresis

Permeability of fissured sorbing rocks, such as coal and shale, controls gas transport and is relevant to a variety of scientific problems and industrial processes. Multiple gas transport and rock deformation mechanisms affect permeability evolution, including gas rarefaction effects, gas-sorption-induced anisotropic matrix–fracture interaction, and anisotropic deformation induced by effective stress variation. In this paper, a generic anisotropic permeability model is proposed to address the impacts of the above mechanisms and effects. Specifically, the influence of matrix–fracture interactions on permeability evolution is depicted through the nonuniform matrix swelling caused by the gas diffusion process from fracture walls into the matrix. The following characteristics are also incorporated in this model: (1) anisotropic mechanical and swelling properties, (2) arbitrary box-shaped matrix blocks due to the anisotropic rock structure, (3) adsorbability variation of different matrix blocks because of complex rock compositions, (4) adsorption hysteresis, and (5) dynamic tortuosity. The directional permeability models are derived based on the anisotropic poroelasticity theory and anisotropic swelling equations considering adsorption hysteresis. We use a gas-invaded-volume ratio to describe the nonuniform swelling of matrix blocks. Additionally, swelling of blocks with different adsorption and mechanical properties are characterized by a volume-weighted function. Finally, the anisotropic tortuosity is defined as a power law function of effective porosity. The model is verified against experimental data. Results show that four-stage permeability evolution with time can be observed. Permeability evolution in different directions follows its own ways and depends on anisotropic swelling, mechanical properties, and structures, even when the boundary conditions are identical. Adsorption hysteresis controls the local shrinkage region. Tortuosity variation significantly affects permeability but has the smallest influence on the local swelling region. The existence of multiple matrix types complicates the permeability evolution behavior.