Estimation and verication of fractional derivative-based permeability of coals considering mining-induced stresses

To overcome the inaccuracy of the traditional transient pulse test, a new fractional derivative - based permeability estimation formula based on the transient pulse test is proposed to describe the pressure difference decay of a coal body subjected to mining - induced stresses. The permeability of coal specimens under mining disturbance conditions is measured using the MTS815 rock mechanics test system. The experimental results show that the transient pulse test based on the fractional derivative model provides a much better estimation of the coal specimen’s permeability than the conventional exponential decay model. Analyzing the evolution of the coal’s permeability shows that the permeability tends to decrease in the pre - peak compaction stage, following which it gradually increases in the plastic phase, and then increases sharply in the post - peak phase. The significance of the fractional derivative order γ is discussed, and its analysis shows that the solid - liquid interaction inside the specimen becomes complicated when the stress within the coal specimen changes.

masses, is of great theoretical and practical importance for coal mining. In general, permeability measurement methods can be classified into two categories, i.e., steady-state methods and unsteady-state methods (Sander et al. 2017). 0 In the steady-state method, a constant osmotic pressure difference is applied at both ends of the specimen, and the flow rate of the fluid through the specimen per unit time is calculated when the fluid percolation in the specimen reaches a steady state. Thus, the permeability of the specimen can be calculated according to Darcy's law. The steady-state method is mainly suitable for measuring certain materials with high permeability. However, most coal and associated rock bodies have low permeability, and it takes a long time for the flow in the specimens to reach a steady state.
Therefore, the transient pulse method, which is one of the commonly used unsteady-state methods, is often used to measure the permeability of low-permeability rocks (Brace et al. 1968). Although Hsieh et al. (1981) and Neuzil et al. (1981) Brace et al. (1968) is widely used to evaluate the permeability of various geological materials due to its simple formulation (Fedor et al. 2008;Ghabezloo et al. 2009;Metwally et al. 2011;Chen et al. 2011;Mokhtari et al. 2015;.
Many studies have shown that microfracture variations within rocks caused by hydraulic coupling may affect the overall connectivity of the pore structure as well as the permeability of rocks (Zoback et al. 1975;Shao et al. 2005;Mitchell et al. 2008). Fluid flow through a specimen has been found to be characterized by non-Darcy flow when the evolution of rock permeability under triaxial compression is determined by the transient pulse method (Chen et al. 2015;Liu et al. 2016;Wang et al. 2019). In fact, the presence of a large number of microfractures enhances the connectivity inside the rock, resulting in more complex rough contact and seepage paths; the enhanced roughness and flow tortuosity of the fracture network inside the rock also make the fluid-solid interaction stronger, leading to non-Darcy flow (Liu et al. 2016;Zimmerman et al. 1996;Ranjith et al. 2011;Yang et al. 2019  The permeability of the rock specimens is calculated by fitting the pressure difference decay curves of the upper and lower reservoirs. where P0 is the transient pressure pulse (Pa); P(t) is the pressure drop (Pa); K is permeability (m 2 ); Vu and Vd represent the volumes of the upstream and downstream reservoirs (m 3 ), respectively; L is the height of the rock specimens (m) and A is the cross-sectional area (m 2 ); t is the duration of the permeability measurement (s); βω and μ are the compression coefficient (Pa -1 ) and viscosity (Pa·s) of water, respectively. As a result, the widely-used permeability formula can be represented as In Eq. (2), it can be seen that the parameter α is obtained by a best-fit analysis of the test pressure difference data, which is the most important step in calculating the permeability.
Taking the logarithm of both sides of Eq. (1), and defining R = ln(P0/P(t)), Eq. (1) where d is the differential operator, γ is the fractional order.
For the solution of Eq. (6) where 0<γ≤1, I γ is the Riemann-Liouville fractional integral operator given by: where Γ(·) is the Gamma function. Considering the initial condition R(0)=0 and applying the Riemann-Liouville fractional integral in Eq. (8) to Eq. (6), we obtain Therefore, Eq. (9)   The axial and confining pressures increased to a hydrostatic pressure of 10 MPa at a loading rate of 10KN/min, respectively. (2) The axial pressure was loaded in steps of 20% of the peak strength of the specimen to peak stress, while the confining pressure was unloaded gradually by 1 MPa each time; the permeability was measured once after each reduction of the confining pressure, and a total of 4 times before the peak.
(3) when the axial pressure reached the peak stress, the permeability was measured once, and subsequently, the unloading of the axial pressure was prepared, the way of unloading was changed to displacement control, and the unloading rate was set to 0.06 mm/min. (4) The axial pressure was unloaded gradually with a 20% stress gradient of the peak strength and the confining pressure continued to be unloaded at a gradient of 1 MPa; the permeability was measured once after each unloading, and a total of 4 times after the peak was measured.
The loading and unloading stress paths are shown in Fig. 7.   Fig. 7 Test method for measuring coal permeability under mining disturbance

Specimen preparation and test equipment
The coal specimens used for the permeability testing were taken from the No.

Test results and analysis
Due to the development of fractures in the coal specimens selected for the test, it is difficult to measure the permeability under post-peak unloading conditions. The post-peak permeability of the specimens in this test was measured only once before the specimens were destroyed completely and the MTS815 program was stopped. The stress of specimen M1 reached its peak when the fourth permeability measurement was taken, so only three measurements were taken before the peak. The   The curves of the axial strain versus the deviatoric stress and the permeability of the coal specimens are plotted in Fig. 11, where the permeability K and Kγ are calculated using Eq.
(3) and Eq. (10), respectively; the order γ corresponding to the fractional derivative for the determination of permeability is also given, which can be used as an indicator for the determination of Darcy flow. In addition, it can be seen from both Fig. 6 and Fig. 11 that the reference permeability K calculated using Eq. (3) is low compared with the permeability with memory Kγ calculated using Eq. (10)  In addition, although the coal specimens have large discreteness, it can still be seen from Fig. 11 that the permeability of the coal specimens shows a slightly decreasing trend in the pre-peak pressure-density phase; the permeability then gradually increases in the plastic phase and increases sharply in the post-peak phase. It is worth noting that the change in the fractional derivative order γ is complex, as shown in Fig. 11, which indicates that a large number of cracks are generated inside the specimens when the stress in the coal specimens changes, resulting in complex solid-liquid interactions within the specimens.