Spatiotemporal modelling of decaying evolution of seepage pressure difference in interlayer rock under hydraulic-mechanic coupling
CHENG Jianchao1, 2, LIU Yintong1, 2, WANG Lujun1, 3, HOU Mengdong2, ZHANG Liao2, 4, MAO Tingting2, 4, ZHOU Shenghao2, 4, WANG Jun5, XUE Dongjie1, 2, 4
(1. State Key Laboratory of Water Resource Protection and Utilization in Coal Mining, Shendong Coal Group Co., Ltd., Erdos, Inner Mongolia Autonomous Region 017209, China; 2. School of Mechanics and Civil Engineering, China University of Mining and Technology(Beijing), Beijing 100083, China; 3. National Institute of Low Carbon and Clean Energy, Beijing 102211, China; 4. State Key Laboratory for Tunnel Engineering, China University of Mining and Technology(Beijing), Beijing 100083, China; 5. School of Energy and Mining Engineering, China University of Mining and Technology(Beijing), Beijing 100083, China)
Abstract：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.
[1] 顾大钊. “能源金三角”地区煤炭开采水资源保护与利用工程技术[J]. 煤炭工程，2014，46(10)：34–37.(GU Dazhao. Water resource protection and utilization engineering technology of coal mining in “energy golden triangle” region[J]. Coal Engineering，2014，46(10)：34–37.(in Chinese))
[2] 顾大钊. 煤矿地下水库理论框架和技术体系[J]. 煤炭学报，2015，40(2)：239–246.(GU Dazhao. Theory framework and technological system of coal mine underground reservoir[J]. Journal of China Coal Society，2015，40(2)：239–246.(in Chinese))
[3] 顾大钊. 能源“金三角”煤炭现代开采水资源及地表生态保护技术[J]. 中国工程科学，2013，15(4)：102–107.(GU Dazhao. Water resource and surface ecology protection technology of modern coal mining in China?s energy “Golden Triangle”[J]. Strategic Study of CAE，2013，15(4)：102–107.(in Chinese))
[4] 顾大钊，张 勇，曹志国. 我国煤炭开采水资源保护利用技术研究进展[J]. 煤炭科学技术，2016，44(1)：1–7.(GU Dazhao，ZHANG Yong，CAO Zhiguo. Technical progress of water resource protection and utilization by coal mining in China[J]. Coal Science and Technology，2016，44(1)：1–7.(in Chinese))
[5] 薛东杰，周宏伟，彭瑞东，等. 基于应力降的非连续支承压力强扰动特征研究[J]. 岩石力学与工程学报，2018，37(5)：1 080–1 095. (XUE Dongjie，ZHOU Hongwei，PENG Ruidong，et al. Strong disturbance of discontinuous abutment pressure[J]. Chinese Journal of Rock Mechanics and Engineering，2018，37(5)：1 080–1 095.(in Chinese))
[6] 李 海，韩进忠，李 殿，等. 大倾角煤层多重扰动煤巷围岩应力分布规律[J]. 中国矿业，2021，30(12)：98–105.(LI Hai，HAN Jinzhong，LI Dian，et al. Stress distribution law of complex high-stress roadway in large dipping seam[J]. China Mining Magazine，2021，30(12)：98–105.(in Chinese))
[7] 程建超，贾 震，侯孟冬，等. 砂岩三轴循环加卸载变形特性分析及扩容逾渗模型研究[J]. 岩石力学与工程学报，2024，43(11)：2 687–2 699.(CHENG Jianchao，JIA Zhen，HOU Mengdong，et al. Percolation modeling of dilation deformation evolution of sandstone under tri-axial cyclic loading-unloading[J]. Chinese Journal of Rock Mechanics and Engineering，2024，43(11)：2 687–2 699.(in Chinese))
[8] 程建超，刘殷彤，张 辽，等. 砂岩三轴循环加卸载峰后力学突变行为[J]. 岩石力学与工程学报，2025，44(4)：850–864.(CHENG Jianchao，LIU Yintong，ZHANG Liao，et al. Critical transition of post-peak failure of sandstone under tri-axial cyclic loading and unloading stress[J]. Chinese Journal of Rock Mechanics and Engineering，2025，44(4)：850–864.(in Chinese))
[9] 刘殷彤，程建超，张 辽，等. 压剪大位移下花岗岩粗糙裂隙渗流场演化矢量算子描述[J]. 岩石力学与工程学报，2025，44(3)：651–677.(LIU Yintong，CHENG Jianchao，ZHANG Liao，et al. Vector operator-based description of evolutional velocity field for water flow between rough fractures in granite under large compression-shear displacements[J]. Chinese Journal of Rock Mechanics and Engineering，2025，44(3)：651–677.(in Chinese))
[10] XUE D J，ZHOU J，LIU Y T，et al. A strain-based percolation model and triaxial tests to investigate the evolution of permeability and critical dilatancy behavior of coal[J]. Processes，2018，6(8)：127.
[11] ZHOU H W，XUE D J，JIANG D Y. On fractal dimension of a fracture surface by volume covering method[J]. Surface Review and Letters，2014，21(1)：481–212.
[12] XUE D J，ZHOU H W，LIU J F，et al. Experimental research on the structural behavior of fractured coal under uniaxial compression[J]. Energies，2018，11(10)：2 538.
[13] 李俊平，余志雄，周创兵，等. 水力耦合下岩石的声发射特征试验研究[J]. 岩石力学与工程学报，2006，25(3)：492–498.(LI Junping，YU Zhixiong，ZHOU Chuangbing，et al. Experimental study on acoustic emission characteristics of rock concerning hydromechanical coupling[J]. Chinese Journal of Rock Mechanics and Engineering，2006，25(3)：492–498.(in Chinese))
[14] 来兴平，张 帅，代晶晶，等. 水力耦合作用下煤岩多尺度损伤演化特征[J]. 岩石力学与工程学报，2020，39(增2)：3 217–3 228.(LAI Xingpin，ZHANG Shuai，DAI Jingjing，et al. Multi-scale damage evolution characteristics of coal and rock under hydraulic coupling[J]. Chinese Journal of Rock Mechanics and Engineering，2020，39(Supp.2)：3 217–3 228.(in Chinese))
[15] 朱泽奇，田铠玮，徐启钟，等. 水力耦合作用下玄武岩启裂机制与临界水压研究[J]. 岩土力学，2023，44(增1)：83–90.(ZHU Zeqi，TIAN Kaiwei，XU Qizhong，et al. Research on crack initiation mechanism and critical water pressure of basalt under hydraulic coupling[J]. Rock and Soil Mechanics，2023，44(Supp.1)：83–90.(in Chinese))
[16] WANG L，LIU J，PEI J，et al. Mechanical and permeability characteristics of rock under hydro-mechanical coupling conditions[J]. Environmental Earth Sciences，2015，73：5 987–5 996.
[17] 刘保国，于明圆，孙景来，等. 水–力耦合作用下页岩力学特性及其损伤本构模型研究[J]. 岩石力学与工程学报，2023，42(5)： 1 041–1 054.(LIU Baoguo，YU Mingyuan，SUN Jinglai，et al. Study on mechanical properties and damage constitutive model of shale under hydro-mechanical coupling[J]. Chinese Journal of Rock Mechanics and Engineering，2023，42(5)：1 041–1 054.(in Chinese))
[18] 李均奕，王 伟，曹亚军，等. 考虑水–力耦合和不同应力路径的砂岩卸荷力学特性试验[J]. 河海大学学报：自然科学版，2023，51(3)：135–142.(LI Junyi，WANG Wei，CAO Yajun，et al. Experimental study on unloading mechanical characteristics of sandstone considering water-mechanical coupling and different stress paths[J]. Journal of Hohai University：Natural Sciences，2023，51(3)：135–142.(in Chinese))
[19] BRACE W F，WALSH J B，FRANGOS W T. Permeability of granite under high pressure[J]. Journal of Geophysical Research，1968，73(6)：2 225–2 236.
[20] 王旭升，陈占清. 岩石渗透试验瞬态法的水动力学分析[J]. 岩石力学与工程学报，2006，25(增1)：3 098–3 103.(WANG Xusheng，CHEN Zhanqing. Hydrodynamic analysis of transient method in rock seepage tests[J]. Chinese Journal of Rock Mechanics and Engineering，2006，25(Supp.1)：3 098–3 103.(in Chinese))
[21] 王路军，曹志国，程建超，等. 煤矿地下水库坝基层间岩体破坏及突渗力学模型[J]. 煤炭学报，2023，48(3)：1 192–1 208.(WANG Lujun，CAO Zhiguo，CHENG Jianchao . et al. Failure analysis of rock strata between upper and lower coals under underground reservoir in coal mine and its critical percolation model of jumping permeability[J]. Journal of China Coal Society，2023，48(3)：1 192–1 208.(in Chinese))
[22] CHEN Y F，LIU M M，HU S H，et al. Non-Darcy's law-based analytical models for data interpretation of high-pressure packer tests in fractured rocks[J]. Engineering Geology，2015，199：91–106.
[23] SHAO J F，ZHOU H，CHAU K T. Coupling between anisotropic damage and permeability variation in brittle rocks[J]. International Journal for Numerical and Analytical Methods in Geomechanics，2005，29(12)：1 231–1 247.
[24] ZHAO Y，ZHANG L，WANG W，et al. Transient pulse test and morphological analysis of single rock fractures[J]. International Journal of Rock Mechanics and Mining Sciences，2017，91：139–154.
[25] SI X F，CHEN Z Q，LIAO X X，et al. Pick-up of the permeability parameters of non-Darcy flow in rock by transient permeating method[J]. Journal of Hunan University of Science and Technology：Natural Science Edition，2006，21(3)：16–20.
[26] 杨 帅. 非达西渗流和溶质输运的分数阶导数建模研究[博士学位论文][D]. 北京：中国矿业大学(北京)，2019.(YANG Shuai. Research on fractional derivative modelling of non-Darcian flow and solute transport[Ph. D. Thesis][D]. Beijing：China University of Mining and Technology(Beijing)，2019.(in Chinese))
[27] ZHOU G，ZHANG Q，BAI R，et al. Characterization of coal micro-pore structure and simulation on the seepage rules of low-pressure water based on CT scanning data[J]. Minerals，2016，6(3)：78.
[28] 杨永杰，宋 扬，陈绍杰. 煤岩全应力应变过程渗透性特征试验研究[J]. 岩土力学，2007，28(2)：381–385.(YANG Yongjie，SONG Yang，CHEN Shaojie. Test study on permeability properties of coal specimen in complete stress -strain process[J]. Rock and Soil Mechanics，2007，28(2)：381–385.(in Chinese))
[29] 张 凯，郭俊廷，滕 腾. 弱胶结砂质泥岩注水软化与渗流特性试验研究以—神东矿区为例[J]. 煤炭科学技术，2022，50(2)：195–201.(ZHANG Kai，GUO Junting，TENG Teng. Experimental study on water-softening and seepage characteristics of weakly cemented sandy mudstone：taking Shendong coal mining area as an example[J]. Coal Science and Technology，2022，50(2)：195–201.(in Chinese))
[30] 付宏渊，蒋煌斌，邱 祥，等. 低应力与覆水环境下单裂隙粉砂质泥岩渗流特性[J]. 岩土力学，2020，41(12)：3 840–3 850.(FU Hongyuan，JIANG Huangbin，QIU Xiang，et al. Seepage characteristics of single fracture silty mudstone under low stress and overlying water environment[J]. Rock and Soil Mechanics，2021，41(12)：3 840–3 850.(in Chinese))
[31] SONG Z，ZHANG J. Progressive failure mechanical behaviour and response characteristics of sandstone under stress-seepage coupling[J]. Journal of Geophysics and Engineering，2021，18(2)：200–218.
[32] 中华人民共和国国家标准编写组. GB/T 23561.1—2009 煤和岩石物理力学性质测定方法第1部分：采样一般规定[S]. 北京：中国标准出版社，2009–04–08.(The National Standards Compilation Group of People?s Republic of China. GB/T 23561.1—2009 Methods for determining the physical and mechanical properties of coal and rock-Part 1：General requirements for sampling[S]. Beijing：Standards Press of China，2009–04–08.(in Chinese))
[33] 汪卫华. 非晶态物质的本质和特性[J]. 物理学进展，2013，33(5)：177–351.(WANG Weihua. The nature and properties of amorphous matter[J]. Progress in Physics，2013，33(5)：177–351.(in Chinese))
[34] 刘玉庆，李玉寿，孙明贵. 岩石散体渗透试验新方法[J]. 矿山压力与顶板管理，2002，19(4)：108–110.(LIU Yuqing，LI Yushou，SUN Minggui. A new method of seepage test on the rock bulk[J]. Ground Pressure and Strata Control，2002，19(4)：108–110.(in Chinese))
[35] CONCEZZI M，SPIGLER R. Some analytical and numerical properties of the Mittag-Leffler functions[J]. Fractional Calculus and Applied Analysis，2015，18(1)：64–94.
[36] HILFER R，SEYBOLD H J. Computation of the generalized Mittag-Leffler function and its inverse in the complex plane[J]. Integral Transforms and Special Functions，2006，17(9)：637–652.
[37] INGOC，BARRICK T R，WEBB A G，et al. Accurate Padé global approximations for the Mittag-Leffler function，its inverse，and its partial derivatives to efficiently compute convergent power series[J]. International Journal of Applied and Computational Mathematics，2017，3(2)：347–362.
[38] Research Group on Reasonable Remainder and Abandonment of Fault Pillar under Weishan Lake and Exploitation Technology. A Research Report on Rock Permeability Testing[R]. Xuzhou：China University of Mining and Technology，1992.
[39] 李世平，李玉寿，吴振业. 岩石全应力应变过程对应的渗透率－应变方程[J]. 岩土工程学报，1995，17(3)：13–19.(LI Shiping，LI Yushou，WU Zhenye. The permeability-strain equations relating to complete stress-strain path of the rock[J]. Chinese Journal of Geotechnical Engineering，1995，17(3)：13–19.(in Chinese))
[40] 彭苏萍，屈洪亮，罗立平，等. 沉积岩石全应力应变过程的渗透性试验研究[J]. 煤炭学报，2000，25(2)：113–116.(PENG Suping，QU Hongliang，LUO Liping，et al. An experimental study on the permeability of sedimentary rock during the complete stress-strain path[J]. Journal of China Coal Society，2000，25(2)：113–116.(in Chinese))
[41] 王金安，彭苏萍，孟召平. 岩石三轴全应力应变过程中的渗透规律[J]. 北京科技大学学报，2001，23(6)：489–491.(WANG Jin?an，PENG Suping，MENG Zhaoping. Permeability rule in full strain- stress process of rock under triaxial compression[J]. Journal of University of Science and Technology Beijing，2001，23(6)：489–491.(in Chinese))
[42] MITTAG-LEFFLER G M. Sur la nouvelle function[J]. Comptes rendus del Académiedes Sciences(Paris)，1903，137：554–558.
[43] 李天珍，马 林，张连英. 泥岩非Darcy流渗透特性指标正负号探讨[J]. 采矿与安全工程学报，2007，24(4)：481–485.(LI Tianzhen，MA Lin，ZHANG Lianying. Discussion about positive or negative sign of permeability indexes in non-Darcy flow of mudstone[J]. Journal of Mining and Safety Engineering，2007，24(4)：481–485.(in Chinese))
[44] 孙明贵，黄先伍，李天珍，等. 石灰岩应力–应变全过程的非Darcy流渗透特性[J]. 岩石力学与工程学报，2006，25(3)：484–491.(SUN Minggui，LI Tianzhen，HUANG Xianwu，et al. Seepage properties of non-Darcy flow in complete failure process of limestone[J]. Chinese Journal of Rock Mechanics and Engineering，2006，25(3)：484–491.(in Chinese))
[45] 李天珍，李玉寿，马占国. 破裂岩石非达西渗流的试验研究[J]. 工程力学，2003，20(4)：132–135.(LI Tianzhen，LI Yushou，MA Zhanguo. Testing study of non-Darcian seepage flow in fractured rocks[J]. Engineering Mechanics，2003，20(4)：132–135.(in Chinese))
[46] 孙明贵，李天珍，黄先伍，等. 基于层状岩体渗流失稳条件的煤矿突水机理[J]. 中国矿业大学学报，2005，34(3)：284–288.(SUN Minggui，LI Tianzhen，HUANG Xianwu，et al. Mechanism of water inrush based on the instability of the seepage flow in rock strata[J]. Journal of China University of Mining and Technology，2005，34(3)：284–288.(in Chinese))
[47] LI J S，WANG X D，LIU P C，et al. Review on the development of oil and gas flow in underground porous media[J]. Petroleum Science，2004，1(4)：88–94.
[48] LIU M M，CHEN Y F，HONG J M，et al. A generalized non-Darcian radial flow model for constant rate test[J]. Water Resources Research，2016，52(12)：9 325–9 343.
[49] ZIMMERMAN R W，BODVARSSON G S. Hydraulic conductivity of rock fractures[J]. Transport in Porous Media，1996，23：1–30.
[50] RAGHAVAN R. Fractional derivatives：application to transient flow[J]. Journal of Petroleum Science and Engineering，2011，80(1)：7–13.
[51] ZHOU H W，ZHAO J W，SU T，et al. Using a fractional derivative-based permeability model，characterize gas flow in backfill mining-induced coal seam[J]. International Journal of Rock Mechanics and Mining Sciences，2021，138：104571.
[52] 王 睿，周宏伟，卓 壮，等. 非饱和土空间分数阶渗流模型的有限差分方法研究[J]. 岩土工程学报，2020，42(9)：1 759–1 764. (WANG Rui，ZHOU Hongwei，ZHUO Zhuang，et al. Finite difference method for space-fractional seepage process in unsaturated soil[J]. Chinese Journal of Geotechnical Engineering，2020，42(9)：1 759–1 764.(in Chinese))
[53] MILLER E E，MILLER R D. Physical theory for capillary flow phenomena[J]. Journal of Applied Physics，1956，27(4)：324–332.
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