Percolation mechanism of seepage in rough rock fractures
LIU Yintong1, 2, MAO Lingtao2, 3, CHENG Jianchao1, ZHANG Liao1, HOU Mengdong1, MAO Tingting1, #br#
WANG Shuaixiong1, LIU Shenggui1, XUE Dongjie1, 2, 3, 4*
(1. State Key Laboratory for Tunnel Engineering, China University of Mining and Technology(Beijing), Beijing 100083, China;
2. School of Mechanics and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China; 3. State Key Laboratory for Fine Exploration and Intelligent Development of Coal Resources, Beijing 100083, China; 4. State Key Laboratory of Water Resource Protection and Utilization in Coal Mining, China Energy Investment Group Co., Ltd., Beijing 102211, China)
Abstract：In studies of fluid flow through single-fractured rock media, the classical geometric percolation threshold loses its validity, and the threshold pressure gradient (TPG), as an empirical parameter describing the onset condition of actual flow, cannot directly provide a basis for the theoretical determination of the percolation threshold. This is because TPG relies on fluid mechanics mechanisms, whereas the percolation threshold is based on network topology statistics. Therefore, it is essential to define a “transport percolation threshold” that characterizes hydraulic critical behavior. Consequently, this study addresses the determination of the inflection point in Darcy-non-Darcy flow through rough fractures and investigates the transport percolation threshold in seepage flow, focusing on the discontinuous relationship between pressure gradient and flow velocity during transitions from low to high velocities. A comprehensive Bingham fluid seepage model, incorporating fractal dimensions of fracture tortuosity and aperture, is developed. By integrating unit percolation models with diffusion equations, a global percolation model is established, allowing for numerical solutions for full-field percolation in rough fractures. The results indicate that, compared to the assumptions of Darcy flow, the unit percolation model under the Forchheimer flow assumption—characterized by its unique inertial term—more accurately captures the influence of geometric percolation parameters (such as roughness angles and hydraulic aperture) on fluid flow paths during the percolation stage. During phase transitions in percolation, the outlet flow velocity conforms more closely to the nonlinear characteristics described by Forchheimer′s equation rather than Darcy′s linear assumption. This understanding offers a novel perspective for investigating the mechanisms of low-velocity non-Darcian flow.
[1] QIAN J Z，ZHAN H B，ZHAO W D，et al. Experimental study of turbulent unconfined groundwater flow in a single fracture[J]. Journal of Hydrology，2005，311(1/4)：134–142.
[2] BEAR J. Dynamics of fluids in porous media[M]. New York：Dover，1988：131–210.
[3] KUNDU P，KUMAR V，MISHRA I M，et al. Experimental and numerical investigation of fluid flow hydrodynamics in porous media：Characterization of pre-Darcy，Darcy and non-Darcy flow regimes[J]. Powder Technology，2016，303：278–291.
[4] FINENKO M，KONIETZKY H. Numerical simulation of turbulent fluid flow in rough rock fracture：2D Case[J]. Rock Mechanics and Rock Engineering，2024，57：451–479.
[5] CHEN Y F，ZHOU J Q，HU S H，et al. Evaluation of Forchheimer equation coefficients for non-Darcy flow in deformable rough-walled fractures[J]. Journal of Hydrology，2015，529(3)：993–1 006.
[6] EL-ZEHAIRY A A，ABDEL-GAWAD H A A. Pore-network modelling of transverse dispersion in porous media under non-Darcy flow conditions[J]. Advances in Water Resources，2024，185：104626.
[7] FAN J W，GUO W J，LV Y J，et al. A nonlinear mathematical model for fluid flow in low-permeability reservoirs and its effect on well production performance[J]. Geoenergy Science and Engineering，2023，231(A)：212349.
[8] BROADBENT S R，HAMMERSLEY J M. Percolation processes[J]. Mathematical Proceedings of the Cambridge Philosophical Society，1957，53(3)：629–641.
[9] AHARONY A，STAUFFER D. Introduction to percolation theory[M]. Oxford：Taylor and Francis，1992：63–125.
[10] STAUFFER D，AHARONY A. Introduction to Percolation Theory[J]. Physics Today，1993，46(4)：64.
[11] SHI W H，YANG T H，YU S B. Experimental investigation on non-Darcy flow behavior of granular limestone with different porosity[J]. Journal of Hydrologic Engineering，2020，25(8)：06020004.
[12] YAO C，SHAO Y L，YANG J H，et al. Effects of fracture density，roughness，and percolation of fracture network on heat-flow coupling in hot rock masses with embedded three-dimensional fracture network[J]. Geothermics，2020，87：101846.
[13] TIAN W B，LI A F，REN X X，et al. The threshold pressure gradient effect in the tight sandstone gas reservoirs with high water saturation[J]. Fuel，2018，226：221–229.
[14] HOSSEINI E，SARMADIVALEH M，CHEN Z. Developing a new algorithm for numerical modeling of discrete fracture network (DFN) for anisotropic rock and percolation properties[J]. Journal of Petroleum Exploration and Production Technology，2021，11：839–856.
[15] WANG X H，ZHENG J，SUN H Y. Comparative study on interconnectivity between three-dimensional and two-dimensional discrete fracture networks：A perspective based on percolation theory[J]. Journal of Hydrology，2022，609：127731.
[16] HE C，YAO C，JIN Y Z，et al. Effective permeability of fractured porous media with fracture density near the percolation threshold[J]. Applied Mathematical Modelling，2023，117：592–608.
[17] CHENG H，WANG F G，LI S W，et al. From pre-Darcy flow to Darcy flow in porous media：A simple unified model[J]. Water Resources Research，2024，60(7)：e2023WR036902.
[18] CAI J. A fractal approach to low velocity non-Darcy flow in a low permeability porous medium[J]. Chinese Physics B，2014，23(4)：385–389.
[19] DEJAM M，HASSANZADEH H，CHEN Z. Pre-Darcy flow in porous media[J]. Water Resources Research，2017，53(10)：8 187–8 210.
[20] WANG Q，JIANG R，CUI Y，et al. Pre-Darcy flow behavior of CO2 Huff-n-Puff development in Fuyu tight formation：Experiment and numerical evaluation[J]. Journal of Petroleum Science and Engineering，2020，186：106773.
[21] FARMANI Z，FAROKHIAN D，IZADPANAHI A，et al. Pre-Darcy flow and Klinkenberg effect in dense，consolidated carbonate formations[J]. Geotechnical and Geological Engineering，2019，37(4)：3 255–3 270.
[22] DING J C，YANG S L，NIE X R，et al. Dynamic threshold pressure gradient in tight gas reservoir[J]. Journal of Natural Gas Science and Engineering，2014，20：155–160.
[23] SINGH K，CAMULLI H，BRADLEY J. Sediment size effects on non-Darcy flow：Insights from Izbash equation and Forchheimer inertial coefficient analysis[J]. Hydrogeology Journal，2024，32：1 853–1 871.
[24] YOU X C，LI S Y，KANG L，et al. A study of the non-linear seepage problem in porous media via the homotopy analysis method[J]. Energies，2023，16(5)：2 175.
[25] CHAPARIAN E. Yielding to percolation：A universal scale[J]. Journal of Fluid Mechanics，2024，980：A14.
[26] ZAFAR A，SU Y L，LI L，et al. Tight gas production model considering TPG as a function of pore pressure，permeability and water saturation[J]. Petroleum Science，2020，17：1 356–1 369.
[27] ROY S，SINHA S，HANSEN A. Immiscible two-phase flow in porous media：Effective rheology in the continuum limit[J]. Transport in Porous Media，2024，151：1 295–1 311.
[28] OLDENBURG C M，MUKHOPADHYAY S，CIHAN A. On the use of Darcy's law and invasion-percolation approaches for modeling large-scale geologic carbon sequestration[J]. Greenhouse Gases：Science and Technology，2016，6(1)：19–33.
[29] SONI J P，ISLAM N，BASAK P. An experimental evaluation of non-Darcian flow in porous media[J]. Journal of Hydrology，1978，38(3/4)：231–241.
[30] ADLER P M，THOVERT J F，MOURZENKO V V. Fractured porous media[M]. Oxford：Oxford University Press，2012：73–104.
[31] LI L，LI K. Permeability of microcracked solids with random crack networks：role of connectivity and opening aperture[J]. Transport in Porous Media，2015，109(1)：217–237.
[32] ESMAEILPOUR M，GHANBARIAN B，LIANG F，et al. Scale-dependent permeability and formation factor in porous media：Applications of percolation theory[J]. Fuel，2021，301：121090.
[33] KARMAN V T. Progress in the statistical theory of turbulence[J]. Proceedings of the National Academy of Sciences of the United States of America，1948，34(11)：530–539.
[34] PRANDTL L. Essentials of fluid dynamics[M]. New York：Hafner Publishing Company，1952：206–251.
[35] ZHANG Y，YE J H，LI P X. Flow characteristics in a 3D-printed rough fracture[J]. Rock Mechanics and Rock Engineering，2022，55：4 329–4 349.
[36] CHANG A L，SUN H G，ZHANG Y，et al. Spatial fractional Darcy′s law to quantify fluid flow in natural reservoirs[J]. Physica A：Statistical Mechanics and its Applications，2019，519：119–126.
[37] WANG X，SHENG J J. Effect of low-velocity non-Darcy flow on well production performance in shale and tight oil reservoirs[J]. Fuel，2017，190：41–46.
[38] CHEN H，XING J P，JIANG D L，et al. A fractal model of low-velocity non-Darcy flow considering viscosity distribution and boundary layer effect[J]. Fractals，2022，30(1)：2250006.
[39] LI L，HAO Y M，LV Y T，et al. Experimental investigation on low-velocity seepage characteristics and influencing factors in a shale oil reservoir[J]. Journal of Petroleum Science and Engineering，2020，195：107732.
[40] NARASIMHAN A. Essentials of heat and fluid flow in porous media[M]. Cham：Springer Cham，2023：93–150.
[41] ARABJAMALOEI R，RUTH D. Numerical study of inertial effects on permeability of porous media utilizing the Lattice Boltzmann Method[J]. Journal of Natural Gas Science and Engineering，2017，44：22–36.
[42] YAO C S，YANG Y L，HUANG J H，et al. Effects of non-Darcy flow on heat-flow coupling process in complex fractured rock masses[J]. Journal of Natural Gas Science and Engineering，2020，83(1)：103536.
[43] ZHAO C X，ZHANG Z X，WANG S F，et al. Effects of fracture network distribution on excavation-induced coupled responses of pore pressure perturbation and rock mass deformation[J]. Computers and Geotechnics，2022，145：104670.
[44] BLOSHANSKAYA L，IBRAGIMOV A，SIDDIQUI F，et al. Productivity index for Darcy and pre-/post-Darcy flow(Analytical approach)[J]. Journal of Porous Media，2017，20(9)：769–786.
[45] BROWN S R，STOCKMAN H W，REEVES S J. Applicability of the Reynolds Equation for modeling fluid flow between rough surfaces[J]. Geophysical Research Letters，1995，22(18)：2 537–2 540.
[46] FORCHHEIMER P Z. Wasserbewegung durch Boden[J]. Zeitschrift des Vereins Deutscher Ingenieure，1901，45：1 781–1 788.
[47] MOODY L F，PRINCETON N J. Friction factors for pipe flow[J]. Transactions of the American Society of Mechanical Engineers，1944，66(8)：671–681.
[48] YAO Y D，LI G Z，QIN P F. Seepage features of high-velocity non-Darcy flow in highly productive reservoirs[J]. Journal of Natural Gas Science and Engineering，2015，27(3)：1 732–1 738.
[49] AHMED N，SUNADA D K. Nonlinear flow in porous media[J]. Journal of the Hydraulics Division，1969，95(6)：1 847–1 858.
[50] MIAO T J，YU B M，DUAN Y G，et al. A fractal analysis of permeability for fractured rocks[J]. International Journal of Heat and Mass Transfer，2015，81(2)：75–80.
[51] WANG F Y，CHENG H. Effect of gravity on spontaneous imbibition of the wetting phase into gas-saturated tortuous fractured porous media：Analytical solution and diagnostic plot[J]. Advances in Water Resources，2020，142(8)：103657.
[52] SCHWIEBERT M K，LEONG W H. Underfill flow as viscous flow between parallel plates driven by capillary action[C]// IEEE Transactions on Components，Packaging and Manufacturing Technology：Part C. Piscataway：IEEE，1996：133–137.
[53] 薛东杰，侯孟冬，程建超，等. 剪切粗糙裂隙渗流多尺度分形表征[J]. 矿业科学学报，2023，8(5)：663–676.(XUE Dongjie，HOU Mengdong，CHENG Jianchao，et al． Fractal model of shear-induced rough fracture flow by cross-scale description of its geometry[J]. Journal of Mining Science and Technology，2023，8(5)：663–676.(in Chinese))
[54] ACUNA J A，ERSHAGHI I，YORTSOS Y C. Practical application of fractal pressure transient analysis of naturally fractured reservoirs[J]. SPE Formation Evaluation，1995，10(3)：173–179.
[55] HATTON C G，MAIN I G，MEREDITH P G. Non-universal scaling of fracture length and opening displacement[J]. Nature，1994，367(6459)：160–162.
[56] CHANG J，YORTSOS Y C. Pressure-transient analysis of fractal reservoirs[J]. SPE Reservoir Evaluation and Engineering，1990，5(1)：31–38.
[57] CHANG J，YORTSOS Y C. A note on pressure-transient analysis of fractal reservoirs[J]. British Journal of Sports Medicine，1993，1(2)：380–386.
[58] LIU G F，ZHANG Z，CAO Y X，et al. An analogical method on fractal dimension for three-dimensional fracture tortuosity in coal based on CT scanning[J]. Fractals，2023，31(7)：2350072.
[59] XU P，CHEN Z，QIU S X，et al. An analytical model for pore and tortuosity fractal dimensions of porous media[J]. Fractals，2021，29(6)：2150156.
[60] 陈文昭，蒋一凡，刘夕奇. 循环扰动荷载作用下花岗岩损伤及破裂特性研究[J]. 南华大学学报：自然科学版，2022，36(3)：47–53.(CHEN Wenzhao，JIANG Yifan，LIU Xiqi. Damage and fracture characteristics of granite under cyclic disturbance loading[J]. Journal of University of South China：Science and Technology，2022，36(3)：47–53.(in Chinese))
[61] XUE D J，LIU Y T，ZHOU H W，et al. Fractal characterization on anisotropy and fractal reconstruction of rough surface of granite under orthogonal shear[J]. Rock Mechanics and Rock Engineering，2020，53(7)：1 225–1 242.
[62] FANNING J T. A practical treatise on hydraulic and water-supply engineering：Relating to the hydrology，hydrodynamics，and practical construction of water-works，in North America[M]. New York：D. Van Nostrand，1906：582–630.
[63] 李术才，刘人太，张庆松，等. 基于黏度时变性的水泥-玻璃浆液扩散机制研究[J]. 岩石力学与工程学报，2013，32(12)：2 415–2 421. (LI Shucai，LIU Rentai，ZHANG Qingsong，et al. Research on C-S slurry diffusion mechanism with time-dependent behavior of viscosity[J]. Chinese Journal of Rock Mechanics and Engineering，2013，32(12)：2 415–2 421.(in Chinese))
[64] ALKAN H. Percolation model for dilatancy-induced permeability of the excavation damaged zone in rock salt[J]. International Journal of Rock Mechanics and Mining Sciences，2009，46(4)：716–724.
[65] 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)：1–22.
[66] 薛东杰，张泽朋，周宏伟，等. 采动应力分布逾渗模型[J]. 岩石力学与工程学报，2022，41(2)：319–335.(XUE Dongjie，ZHANG Zepeng，ZHOU Hongwei，et al. Percolation model of stress redistribution under excavation[J]. Chinese Journal of Rock Mechanics and Engineering，2022，41(2)：319–335.(in Chinese))
[67] NEUZIL C E. Groundwater flow in low-permeability environments[J]. Water Resources Research，1986，22(8)：1 163–1 195.
[68] PRADA A，CIVAN F. Modification of Darcy's law for the threshold pressure gradient[J]. Journal of Petroleum Science and Engineering，1999，22(04)：237–240.
[69] WEISBACH J，COXE E B. Mechanics of engineering. Theoretical mechanics，with an introduction to the calculus[M]. New York：D. Van Nostrand，1885：896–978.
[70] CHEN Y F，HU S H，HU R，et al. Estimating hydraulic conductivity of fractured rocks from high-pressure packer tests with an Izbash's law-based empirical model[J]. Water Resources Research，2015，51(4)：2 096–2 118.
[71] XING K，QIAN J Z，ZHAO W D，et al. Experimental and numerical study for the inertial dependence of non-Darcy coefficient in rough single fractures[J]. Journal of Hydrology，2021，603：127148.
[72] ZHAO J Y，LIU WT，SHEN J J，et al. Fractal treelike fracture network model for hydraulically and mechanically induced dynamic changes in the non-Darcy coefficient during the process of mine water inrush from collapsed columns[J]. Fractals，2021，29(7)：2150218.
[73] ZHANG S，LIU X H，LIU X L，et al. The friction factor of the fracture-matrix system considering the effects of free flow，seepage flow，and roughness[J]. Journal of Hydrology，2024，628(1)：130602.
[74] FICK A. Ueber Diffusion[J]. Annalen Der Physik，1855，170(1)：59–86.
[75] SØRENSSEN A M. Percolation theory using python[M]. Cham：Springer Cham，2024：195–199.
[76] QIAN J Z，WANG M，ZHANG Y，et al. Experimental study of the transition from non-Darcian to Darcy behavior for flow through a single fracture[J]. Journal of Hydrodynamics，Ser. B，2015，27(5)：679–688.
2026, Vol. 45 
