基于等效离散裂隙网络的三维裂隙岩体渗流模型
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摘要
基于等效离散裂隙网络模型,提出一种模拟具有裂隙网络–岩石基质复杂裂隙岩体渗流的三维数值计算方法。该模型将岩石看作由不透水的四面体块体组成的集合,渗流只在相邻块体之间的界面上发生,并遵循Darcy定理。为模拟复杂裂隙岩体的空间结构,开发随机均布三维Delaunay四面体网格生成算法,可以显式模拟三维裂隙网络及其复杂的相交关系。岩石基质中的孔隙被等效界面网络系统代替。为保证该系统拥有与完整岩石相同的宏观渗透性,建立等效离散裂隙网络模型,并提出宏细观渗流参数的转换方法。裂隙也作为界面存在,但具有较高的渗透性,在数值上与基质中界面的处理方式相同,便于程序实现及数值稳定。为验证该方法的有效性,进行几个数值计算,数值结果与解析解吻合良好。最后,通过一系列算例研究裂隙渗透性、大小、方向、相交位置对岩体宏观渗透性的影响,及裂隙密度与逾渗阈值之间的关系。
Based on the equivalent discrete fracture network,a 3D numerical model for fluid flow in fracture network and rock matrix is proposed. The intact porous rock is discretized by an assembly of impermeable tetrahedrons. The flow,following the Darcy law,only happens in interfaces between tetrahedrons. For accurate description of spatial structure of porous rock with complex fracture network,a mesh generation method is developed. Using randomly and uniformly distributed Delaunay tetrahedrons,the geometry of 3D fracture network and complex intersection relations between fractures is directly described. The pore space of the porous matrix is hydraulically replaced by an equivalent interface network system. In order to ensure the system having the same macroscopic permeability with the intact rock,we develop an equivalent discrete fracture network model and propose a method to make connection between the macro and micro hydraulic parameters. Fractures are also treated as interfaces like the matrix,only with a higher hydraulic conductivity,which make the solving system realizable and numerically stable. To verify the efficiency of proposed method,we perform a series of numerical studies,the numerical results agree well with analytical solutions. Finally,the proposed model is applied to study the effects of fracture permeability,size,orientation,intersection position on the macroscopic permeability of rock,and the relationship between fracture density and percolation threshold.
Based on the equivalent discrete fracture network,a 3D numerical model for fluid flow in fracture network and rock matrix is proposed. The intact porous rock is discretized by an assembly of impermeable tetrahedrons. The flow,following the Darcy law,only happens in interfaces between tetrahedrons. For accurate description of spatial structure of porous rock with complex fracture network,a mesh generation method is developed. Using randomly and uniformly distributed Delaunay tetrahedrons,the geometry of 3D fracture network and complex intersection relations between fractures is directly described. The pore space of the porous matrix is hydraulically replaced by an equivalent interface network system. In order to ensure the system having the same macroscopic permeability with the intact rock,we develop an equivalent discrete fracture network model and propose a method to make connection between the macro and micro hydraulic parameters. Fractures are also treated as interfaces like the matrix,only with a higher hydraulic conductivity,which make the solving system realizable and numerically stable. To verify the efficiency of proposed method,we perform a series of numerical studies,the numerical results agree well with analytical solutions. Finally,the proposed model is applied to study the effects of fracture permeability,size,orientation,intersection position on the macroscopic permeability of rock,and the relationship between fracture density and percolation threshold.
引文
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[4]张奇华,邬爱清.三维任意裂隙网络渗流模型及其解法[J].岩石力学与工程学报,2010,29(4):720-730.(ZHANG Qihua,WU Aiqing.Three-dimensional arbitrary fracture network seepage model and its solution[J].Chinese Journal of Rock Mechanics and Engineering,2010,29(4):720-730.(in Chinese))
[5]LONG J C S,REMER J S,WILSON C R,et al.Porous media equivalents for networks of discontinuous fractures[J].Water Resources Research,1982,18(3):645-658.
[6]KOUDINA N,GARCIA R G,THOVERT J F,et al.Permeability of three-dimensional fracture networks[J].Physical Review E,1998,57(4):4 466-4 479.
[7]JIANG Q H,YE Z Y,YAO C,et al.A new variational inequality formulation for unconfined seepage flow through fracture networks[J].Science China Technological Sciences,2012,55(11):1-12.
[8]JIANG Q,YAO C,YE Z,et al.Seepage flow with free surface in fracture networks[J].Water resources research,2013,49(1):176-186.
[9]JIANG Q,YE Z,ZHOU C.A numerical procedure for transient free surface seepage through fracture networks[J].Journal of hydrology,2014,519(PA):881-891.
[10]姚池,姜清辉,叶祖洋,等.裂隙网络无压渗流分析的初流量法[J].岩土力学,2012,33(6):1 896-1 903.(YAO Chi,JIANG Qinghui,YE Zuyang,et al.Initial flow method for unconfined seepage problems of fracture networks[J].Rock and Soil Mechanics,2012,33(6):1 896-1 903.(in Chinese))
[11]YAO C,JIANG Q H,WEI W,et al.The variational inequality formulation for unconfined seepage through three-dimensional dense fracture networks[J].Science China Technological Sciences,2013,56(5):1 241-1 247.
[12]叶祖洋,姜清辉,姚池,等.岩体裂隙网络非稳定渗流分析与数值模拟[J].岩土力学,2013,34(4):1 171-1 178.(YE Zuyang,JIANG Qinghui,YAO Chi,et al.Formulation and simulation of non-steady seepage flow through fracture network in rock masses[J].Rock and Soil Mechanics,2013,34(4):1 171-1 178.(in Chinese))
[13]叶祖洋,姜清辉,姚池,等.三维裂隙网络非稳定渗流分析的变分不等式方法[J].力学学报,2013,45(6):878-887.(YE Zuyang,JIANG Qinghui,YAO Chi,et al.A variational iequality approach for non-steady seepage flow through three-dimensional fracture network[J].Chinese Journal of Theoretical and Applied Mechanics,2013,45(6):878-887.(in Chinese))
[14]张有天,陈平,王镭.有自由面渗流分析的初流量法[J].水利学报,1988,8(1):18-26.(ZHANG Youtian,CHEN Ping,WANGLei.Initial flow method for seepage analysis with free surface[J].Journal of Hydraulic Engineering,1988,8(1):18-26.(in Chinese))
[15]毛昶熙.渗流计算分析与控制[M].北京:中国水利水电出版社,1988:1-62.(MAO Changxi.Seepage calculation and control[M].Beijing:China Water Power Presss,1988:1-62.(in Chinese))
[16]BARENBLATT G I,ZHELTOV I P,KOCHINA I N.Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks strata[J].Journal of Applied Mathematics and Mechanics,1960,24(5):1 286-1 303.
[17]WARREN J E,ROOT P J.The behavior of naturally fractured reservoirs[J].Society of Petroleum Engineers Journal,1963,3(3):245-255.
[18]GERKE H H,GENUCHTEN M T V.A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media[J].Water Resources Research,1993,29(2):305-319.
[19]黎水泉,徐秉业.双重孔隙介质流固耦合理论模型[J].水动力学研究与进展,2001,16(4):460-466.(LI Shuiquan,XU Bingye.Theory model of fluid flow through deformable dual porosity media[J].Journal of Hydrodynamics,2001,16(4):460-466.(in Chinese))
[20]张玉军.遍有节理岩体的双重孔隙/裂隙介质热-水-应力耦合模型及有限元分析[J].岩石力学与工程学报,2009,28(5):947-955.(ZHANG Yujun.Coupled thermo-hydro-mechanical model and finite element analyses of dual-porosity fractured medium for ubiquitous-joint rock mass[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(5):947-955.(in Chinese))
[21]BOGDANOV I I,MOURZENKO V V,THOVERT J F,et al.Effective permeability of fractured porous media in steady state flow[J].Water Resources Research,2003,39(1):257-263.
[22]YAO C,JIANG Q H,SHAO J F.A numerical analysis of permeability evolution in rocks with multiple fractures[J].Transport in Porous Media,2015,108(2):289-311.
[23]杨钦,徐永安,陈其明,等.三维约束Delaunay三角化的研究[J].计算机辅助设计与图形学学报,2000,(8):590-594.(YANG Qin,XU Yongan,CHEN Qiming,et al.Research on 3D constrained Delaunay triangulation[J].Journal of Computer Aided Design and Computer Graphics,2000,(8):590-594.(in Chinese))
[24]杨忱瑛.限定Delaunay四面体网格划分算法的研究与实现[硕士学位论文][D].南京:南京航空航天大学,2009.(YANG Chenying.The research and implement of constrained delaunay tetrahedralization algorithm[M.S.Thesis][D].Nanjing:Nanjing University of Aeronautics Astronautics,2009.(in Chinese))
[25]刘振平,刘建,杜根明,等.基于Hovland改进模型与约束Delaunay TIN的三维边坡稳定性分析方法[J].岩石力学与工程学报,2017,36(增1):3 214-3 223.(LIU Zhenping,LIU Jian,DUGenming,et al.Three-dimensional slope stability analysis method based on Hovland improved model and constrained Delaunay TIN[J].Chinese Journal of Rock Mechanics and Engineering,2017,36(Supp.1):3 214-3 223.(in Chinese))
[26]CHI Y,JIANG Q H,WEI W,et al.The variational inequality formulation for unconfined seepage through three-dimensional dense fracture networks[J].Science China Technological Science,2013,56(5):1 241-1 247.
[27]SI H.TetGen.http://tetgen.berlios.de.2007.
[28]LOUGH M F,LEE S H,KAMATH J.An efficient boundary integral formulation for flow through fractured porous media[J].Journal of Computational Physics,1998,143(2):462-483.
[29]李新强,陈祖煜,杨昭冬.利用三维裂隙网络确定岩体渗透张量的方法[J].工程地质学报,2007,(增1):638-643.(LI Xinqiag,CHENZuyu,YANG Zhaodong.3D equivalent conductivity method of fracture network and porous media[J].Journal of Engineering Geology,2007,(Supp.1):638-643.(in Chinese))
[30]BALBERG I,ANDERSON C H,ALEXANDER S,et al.Excluded volume and its relation to the onset of percolation[J].Physical Review B,1984,30(7):3933.
[31]BERKOWITZ B.Analysis of fracture network connectivity using percolation theory[J].Mathematical Geology,1995,27(4):467-483.
[32]THOVERT J F,MOURZENKO V V,ADLER P M.Percolation in three-dimensional fracture networks for arbitrary size and shape distributions[J].Physical Review E,2017,95(4):042112.
[33]CHARLAIX E,GUYON E,RIVIER N.A criterion for percolation threshold in a random array of plates[J].Solid state communications,1984,50(11):999-1 002.
[34]POLLARD D D.On the form and stability of open hydraulic fractures in the Earth?s crust[J].Geophysical Research Letters,2013,3(9):513-516.
[35]BAECHER G B,LANNEY N A.Trace length biases in joint surveys[J].1978.
[36]HUSEBY O,J-F T,ADLER P M.Geometry and topology of fracture systems[J].Journal of Physics A General Physics,1997,30(30):1 415.
[2]XU T,APPS JA,PRUESS K.Numerical simulation of CO2 disposal by mineral trapping in deep aquifers[J].Applied Geochemistry,2004,19:917-936.
[3]TSANG C F,BERNIER F,DAVIES C.Geohydromechanical processes in the excavation damaged zone in crystalline rock,rock salt,and indurated and plastic clays-in the context of radioactive waste disposal[J].International Journal of Rock Mechanics and Mining Sciences,2005,42(1):109-125.
[4]张奇华,邬爱清.三维任意裂隙网络渗流模型及其解法[J].岩石力学与工程学报,2010,29(4):720-730.(ZHANG Qihua,WU Aiqing.Three-dimensional arbitrary fracture network seepage model and its solution[J].Chinese Journal of Rock Mechanics and Engineering,2010,29(4):720-730.(in Chinese))
[5]LONG J C S,REMER J S,WILSON C R,et al.Porous media equivalents for networks of discontinuous fractures[J].Water Resources Research,1982,18(3):645-658.
[6]KOUDINA N,GARCIA R G,THOVERT J F,et al.Permeability of three-dimensional fracture networks[J].Physical Review E,1998,57(4):4 466-4 479.
[7]JIANG Q H,YE Z Y,YAO C,et al.A new variational inequality formulation for unconfined seepage flow through fracture networks[J].Science China Technological Sciences,2012,55(11):1-12.
[8]JIANG Q,YAO C,YE Z,et al.Seepage flow with free surface in fracture networks[J].Water resources research,2013,49(1):176-186.
[9]JIANG Q,YE Z,ZHOU C.A numerical procedure for transient free surface seepage through fracture networks[J].Journal of hydrology,2014,519(PA):881-891.
[10]姚池,姜清辉,叶祖洋,等.裂隙网络无压渗流分析的初流量法[J].岩土力学,2012,33(6):1 896-1 903.(YAO Chi,JIANG Qinghui,YE Zuyang,et al.Initial flow method for unconfined seepage problems of fracture networks[J].Rock and Soil Mechanics,2012,33(6):1 896-1 903.(in Chinese))
[11]YAO C,JIANG Q H,WEI W,et al.The variational inequality formulation for unconfined seepage through three-dimensional dense fracture networks[J].Science China Technological Sciences,2013,56(5):1 241-1 247.
[12]叶祖洋,姜清辉,姚池,等.岩体裂隙网络非稳定渗流分析与数值模拟[J].岩土力学,2013,34(4):1 171-1 178.(YE Zuyang,JIANG Qinghui,YAO Chi,et al.Formulation and simulation of non-steady seepage flow through fracture network in rock masses[J].Rock and Soil Mechanics,2013,34(4):1 171-1 178.(in Chinese))
[13]叶祖洋,姜清辉,姚池,等.三维裂隙网络非稳定渗流分析的变分不等式方法[J].力学学报,2013,45(6):878-887.(YE Zuyang,JIANG Qinghui,YAO Chi,et al.A variational iequality approach for non-steady seepage flow through three-dimensional fracture network[J].Chinese Journal of Theoretical and Applied Mechanics,2013,45(6):878-887.(in Chinese))
[14]张有天,陈平,王镭.有自由面渗流分析的初流量法[J].水利学报,1988,8(1):18-26.(ZHANG Youtian,CHEN Ping,WANGLei.Initial flow method for seepage analysis with free surface[J].Journal of Hydraulic Engineering,1988,8(1):18-26.(in Chinese))
[15]毛昶熙.渗流计算分析与控制[M].北京:中国水利水电出版社,1988:1-62.(MAO Changxi.Seepage calculation and control[M].Beijing:China Water Power Presss,1988:1-62.(in Chinese))
[16]BARENBLATT G I,ZHELTOV I P,KOCHINA I N.Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks strata[J].Journal of Applied Mathematics and Mechanics,1960,24(5):1 286-1 303.
[17]WARREN J E,ROOT P J.The behavior of naturally fractured reservoirs[J].Society of Petroleum Engineers Journal,1963,3(3):245-255.
[18]GERKE H H,GENUCHTEN M T V.A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media[J].Water Resources Research,1993,29(2):305-319.
[19]黎水泉,徐秉业.双重孔隙介质流固耦合理论模型[J].水动力学研究与进展,2001,16(4):460-466.(LI Shuiquan,XU Bingye.Theory model of fluid flow through deformable dual porosity media[J].Journal of Hydrodynamics,2001,16(4):460-466.(in Chinese))
[20]张玉军.遍有节理岩体的双重孔隙/裂隙介质热-水-应力耦合模型及有限元分析[J].岩石力学与工程学报,2009,28(5):947-955.(ZHANG Yujun.Coupled thermo-hydro-mechanical model and finite element analyses of dual-porosity fractured medium for ubiquitous-joint rock mass[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(5):947-955.(in Chinese))
[21]BOGDANOV I I,MOURZENKO V V,THOVERT J F,et al.Effective permeability of fractured porous media in steady state flow[J].Water Resources Research,2003,39(1):257-263.
[22]YAO C,JIANG Q H,SHAO J F.A numerical analysis of permeability evolution in rocks with multiple fractures[J].Transport in Porous Media,2015,108(2):289-311.
[23]杨钦,徐永安,陈其明,等.三维约束Delaunay三角化的研究[J].计算机辅助设计与图形学学报,2000,(8):590-594.(YANG Qin,XU Yongan,CHEN Qiming,et al.Research on 3D constrained Delaunay triangulation[J].Journal of Computer Aided Design and Computer Graphics,2000,(8):590-594.(in Chinese))
[24]杨忱瑛.限定Delaunay四面体网格划分算法的研究与实现[硕士学位论文][D].南京:南京航空航天大学,2009.(YANG Chenying.The research and implement of constrained delaunay tetrahedralization algorithm[M.S.Thesis][D].Nanjing:Nanjing University of Aeronautics Astronautics,2009.(in Chinese))
[25]刘振平,刘建,杜根明,等.基于Hovland改进模型与约束Delaunay TIN的三维边坡稳定性分析方法[J].岩石力学与工程学报,2017,36(增1):3 214-3 223.(LIU Zhenping,LIU Jian,DUGenming,et al.Three-dimensional slope stability analysis method based on Hovland improved model and constrained Delaunay TIN[J].Chinese Journal of Rock Mechanics and Engineering,2017,36(Supp.1):3 214-3 223.(in Chinese))
[26]CHI Y,JIANG Q H,WEI W,et al.The variational inequality formulation for unconfined seepage through three-dimensional dense fracture networks[J].Science China Technological Science,2013,56(5):1 241-1 247.
[27]SI H.TetGen.http://tetgen.berlios.de.2007.
[28]LOUGH M F,LEE S H,KAMATH J.An efficient boundary integral formulation for flow through fractured porous media[J].Journal of Computational Physics,1998,143(2):462-483.
[29]李新强,陈祖煜,杨昭冬.利用三维裂隙网络确定岩体渗透张量的方法[J].工程地质学报,2007,(增1):638-643.(LI Xinqiag,CHENZuyu,YANG Zhaodong.3D equivalent conductivity method of fracture network and porous media[J].Journal of Engineering Geology,2007,(Supp.1):638-643.(in Chinese))
[30]BALBERG I,ANDERSON C H,ALEXANDER S,et al.Excluded volume and its relation to the onset of percolation[J].Physical Review B,1984,30(7):3933.
[31]BERKOWITZ B.Analysis of fracture network connectivity using percolation theory[J].Mathematical Geology,1995,27(4):467-483.
[32]THOVERT J F,MOURZENKO V V,ADLER P M.Percolation in three-dimensional fracture networks for arbitrary size and shape distributions[J].Physical Review E,2017,95(4):042112.
[33]CHARLAIX E,GUYON E,RIVIER N.A criterion for percolation threshold in a random array of plates[J].Solid state communications,1984,50(11):999-1 002.
[34]POLLARD D D.On the form and stability of open hydraulic fractures in the Earth?s crust[J].Geophysical Research Letters,2013,3(9):513-516.
[35]BAECHER G B,LANNEY N A.Trace length biases in joint surveys[J].1978.
[36]HUSEBY O,J-F T,ADLER P M.Geometry and topology of fracture systems[J].Journal of Physics A General Physics,1997,30(30):1 415.
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