A multi-layer nanocased model to explain the U-shaped evolution of shale gas permeability at constant confining pressure

Understanding the evolution of shale permeability is critical in efficiently recovering gas from shale reservoirs. Two representative profiles of permeability evolution are typically experimentally observed under constant confining pressure and incremented gas pressure - "U-shape" with increasing gas pressure but sometimes absent the second upright limb of the "U" at high gas pressures. Current models fail to address these two different profiles, potentially leading to inappropriate explanation of experiment observations or inaccurate predictions of gas production. In order to determine the mechanistic reason, a multi-layer nanocased model, in which transmissive nanotubes are embedded within a cylindrical sheath of matrix, is proposed. In the model, permeability evolution is defined as a function of the evolving nanotube radius within a total matrix radius. The ensemble structure governs the transition from local deformation of the nanotube wall to the global deformation of the matrix sheath, particularly as sorbing/swelling gas gradually permeates the sheath wall. The finite element method is employed to calculate permeability evolution from an initial equilibrium state to final equilibrium. We develop a series of permeability evolution curves that match various experimental profiles. Contrasting observed permeability responses are attributed to the competition between nanotube strain and matrix global strain. The former term enlarges the nanotube radius while the latter term swells the matrix declining the permeability value. Therefore, when the matrix global strain dominates at late time, permeability decreases. Conversely, a permeability recovery stage is barely observed in the late stage because in most cases global swelling dominates the permeability evolution and an extended observational period is necessary before this swelling appears. This is also the reason why the observed final permeability ratio is rarely greater than unity. A small nanotube radius or a large adsorption strain favors a significant decrease in permeability. When the nanotubes are located within the inorganic matrix, the permeability profile conforms to poro-elastic theory and the decreasing permeability stage is barely apparent or absent. The proposed model thus provides insight into the controls on permeability evolution in shales and the controlling impact of shale matrix properties by considering the inhomogeneities of the shale matrix.