Spatiotemporal modelling of decaying evolution of seepage pressure difference in interlayer rock under hydraulic-mechanic coupling

The safety evaluation of seepage in interlayer rock beneath the underground water reservoir of coal mines is crucial for the secure extraction of lower coal seams. Accurate calculation of the permeability of interlayer rock serves as a fundamental basis for this seepage safety assessment. The transient testing method is frequently employed to measure permeability in low-permeability rocks, relying on a precise description of the decay curve of the seepage pressure difference. However, the spatial distribution of this seepage pressure difference is often overlooked, leading to permeability calculations being influenced by linear Darcy's laws. Using the interlayer rock of the underground reservoir in the Daliuta coal mine as a prototype, we designed complex disturbance stress paths associated with lower coal seam mining in three stages: in-situ stress, mining stress, and cyclic stress. Multiple seepage tests were conducted at characteristic points throughout the entire process on coal, mudstone, and sandstone under varying hydraulic-mechanical coupling conditions using the transient method. These tests revealed the phenomenon of rebound expansion during the mining stress stage of the interlayer rock. Furthermore, we describe the nonlinearity of the relaxation process of the seepage pressure difference decay curve based on the Forchheimer equation and the Mittag-Leffler function, validating the applicability of the Mittag-Leffler function in characterizing the differential pressure decay process of non-Darcy seepage in rocks. Subsequently, a spatiotemporal theoretical model for the decay process of the seepage pressure difference in interlayer rock under hydraulic-mechanical coupling was established, demonstrating that this decay can be represented by the fluid diffusion equation. An accurate description of the spatiotemporal evolution of seepage pressure difference in rock is achieved through fractional-order theory. We also emphasize that the principle underlying the transient method for measuring permeability is to satisfy Darcy’s law spatially, indicating that transients conform to steady-state tests in space, while seepage exhibits non-Darcy characteristics over time. Finally, based on the hysteresis principle, we derive the discrete format of the one-dimensional fractional-order diffusion equation and complete the discretization of spatiotemporal fractional-order equations for the decay of seepage pressure difference in interlayer rock under hydraulic-mechanical coupling. Utilizing the finite-difference method, we solve and visualize the spatiotemporal distribution surfaces of seepage pressure difference in the interlayer rock, providing a foundation for theoretical solutions regarding the spatial distributions of permeability.