A modeling approach to stress-dependent porosity and permeability decays of rocks

The rock porosity or permeability highly depends on stress, a crucial property in resource exploitation and geological storage engineering. However, due to factors such as rock types, loading paths, and loading ranges, the stress-porosity/permeability relationships are pretty different, exhibiting linear, nonlinear, or even heavy-tailed characteristics. The exponential and power-law models, two mainstream empirical relationships for describing the rock permeability and porosity decays, are used to fit the data with “heavy tail” characteristics but yield poor fitting or outrageous predictions for specific stress ranges. Based on the physical interpretation of the compaction-induced microstructural evolution inside the rock, this paper proposes fractional-order relaxation equations, which consider the memory effect of permeability/porosity variations with stress, leading to accurate descriptions of effective stress-porosity/permeability relationships by the Mittag-Leffler (ML) law. The fitting on low-permeability shales and relatively high-permeability sandstones shows that the ML law agrees better with the experimental data, especially with “heavy tail” characteristics than the two classical laws. Moreover, the numerical solutions for the proposed ML models are presented via the predictor-corrector algorithm. The relationship between the ML, exponential, and power laws is also discussed.