A Strain-Driven Model for Anisotropic Permeability Evolution of Shale and Coal Incorporating Creep Deformation, Anisotropic Internal Swelling/Shrinkage, and Gas Rarefaction Effects

Abstract Permeability of shale and coal is a main controlling factor for gas migration and is sensitive to effective stress, sorption/desorption-induced internal swelling/shrinkage (swelling/shrinkage at fracture/pore surfaces), and gas rarefaction effects. The dependence of gas permeability on effective stress and rarefaction effects has been extensively studied. However, the impacts of anisotropic strains and their time-dependent evolution (creep deformation) on permeability variation were still not fully understood, which makes it difficult to accurately predict permeability evolution and simulate gas transport, especially for deep coal. To fill this knowledge gap, a modified sugar-cube conceptual model that captures the structural anisotropy of coal and shale is used to develop a generic fully anisotropic strain-driven permeability model incorporating anisotropic creep deformation, directional internal matrix swelling/shrinkage, and gas rarefaction effects. The time-dependent creep deformation is described by the Nishihara quasi-static rheological model with elastic, viscoelastic, and visco-plastic strain elements. Unlike previous studies where anisotropic internal swelling/shrinkage is ignored or simulated by simply using three sets of independent Langmuir pressure and swelling strain constants, a mechanical-property-based swelling model is used to truly couple directional internal swelling/shrinkage strain with mechanical anisotropy according to the energy balance theory. The Beskok-Karniadakis model is employed to accurately characterize full-Knudsen-number-ranged gas rarefaction effects. The proposed permeability model is verified against coal permeability measurement data. Analyses results indicate that the permeability evolution in each direction shows unique features depending on the anisotropic structure, directional internal swelling, and mechanical properties. The permeability reduction contributed by three-stage creep deformation can be larger than 90%. Internal swelling strain variation in all directions also exhibits a noticeable impact on the magnitude of permeability, which is more obvious at the third stage. The overall influence of the gas rarefaction phenomenon turns heavier as time increase due to the continuous narrowing of flow channel. Due to its analytical feature, the proposed model is suitable for different permeability measurement conditions, including constant effective stress, constant confining pressure, and constant average pore pressure conditions. It can be easily incorporated into a more complex and realistic Multiphysics framework for field-scale simulation and well production prediction.