This paper presents the theory and formulation of the compressibility, porosity, and permeability of shale reservoirs by considering the effects of stress shock causing slope discontinuity and loading/unloading hysteresis. The slope discontinuity happens because the relative contributions of the matrix and fracture change at a critical effective stress at which the fractures close or open depending on whether the process is loading or unloading. The hysteresis phenomenon occurs as a result of partially reversible and irreversible deformations of the various shale rock constituents by various processes during loading and unloading. Two semi-analytical modeling approaches are developed for describing the stress-dependency of the petrophysical properties of porous rock formations. The first approach implements a kinetic model and the second approach applies an elastic cylindrical pore-shell model. Both approaches yield high-quality correlations of the various petrophysical properties of porous rocks with effective stress by honoring the slope discontinuity observed in the compressibility, porosity, and permeability of rocks at critical effective stress.
Shale rock formations include matters of inorganic (quartz, clay, etc.) and organic (kerogen) in the rock matrix, gas at various states (dissolved, adsorbed, and free gases), and brine (water and dissolved salts). The pore system in naturally fractured rocks contains both the matrix and fracture porosity. Some induced fractures are generated in brittle rocks during stress deformation and hydraulic fracturing. The contribution of the interconnectivity of pores and the porosity in the inorganic and organic matters, and the fracture system to the overall effective porosity and permeability of shale rocks depends on the effective stress.
The effective stress σ acting upon porous rocks is determined by an amended Biot's (1941) law as the difference between the total confining stress σc and some degree of participation of the pore fluid pressure p (Biot and Willis, 1957; Kümpel, 1991, Kwon et al., 2001, Walls and Nur, 1979, Zimmermann, 1991, Zoback and Byerlee, 1975a, b):