从郎之万方程出发建立多孔介质微观模型
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摘要
岩土类多孔介质宏观物理力学性质是岩土工程界关注的焦点,其关键是岩土类多孔介质本身复杂的微观结构。综述了当前多孔介质建模的研究现状,简要分析了各种模型建立方法的优势和劣势。针对多孔介质建模的复杂性,从随机动力学相分离过程的朗之万方程出发,将多孔介质的形成过程看作是固相粒子和孔隙相粒子相互作用演化的结果,建立了构造随机多孔介质的微观生长模型。通过不断向空间播撒固相粒子和孔隙相粒子控制多孔介质的演化过程,同时计算了不同参数条件下演化过程中多孔介质孔隙率、欧拉数以及分形维数的变化趋势。由于考虑了固体相和空隙相粒子之间的相互作用力,模型生成的多孔介质与自然界真实的多孔介质更为接近,存在连通和不连通的孔隙结构,孔道迂回曲折。通过与真实样品微观结构的比对,验证了本模型的实用性、可行性。
Macroscopic physical and mechanical properties of porous media like rock and soil are always the focus in geotechnical engineering,the key for understanding these properties is to study the microstructure of such porous media. The current research of porous media modelling,and their advantages and disadvantages were summarized and analysed. A micro-model based on stochastic dynamics of phase separation process was introduced by assuming the formation process of porous medium as an interaction results between solid phase and pore phase. The evolution of porous media was controlled by sowing solid and pore phase particles continuously. Porosity,Euler number and fractal dimension of each evolution step were calculated,the model considered the interaction force between the solid and pore phase particles,so that the generated micro-model was quite similar to the real natural porous media,which contained tortuous porous structures connected or disconnected. The model is validated by comparing with the real samples' SEM photos,which validates the model's feasibility and practicability.
Macroscopic physical and mechanical properties of porous media like rock and soil are always the focus in geotechnical engineering,the key for understanding these properties is to study the microstructure of such porous media. The current research of porous media modelling,and their advantages and disadvantages were summarized and analysed. A micro-model based on stochastic dynamics of phase separation process was introduced by assuming the formation process of porous medium as an interaction results between solid phase and pore phase. The evolution of porous media was controlled by sowing solid and pore phase particles continuously. Porosity,Euler number and fractal dimension of each evolution step were calculated,the model considered the interaction force between the solid and pore phase particles,so that the generated micro-model was quite similar to the real natural porous media,which contained tortuous porous structures connected or disconnected. The model is validated by comparing with the real samples' SEM photos,which validates the model's feasibility and practicability.
引文
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2 Pfeifer P,Avnir D.Chemistry nonintegral dimensions between two and three[J].Journal of Chemical Physics,1984,80(9):4573-4573
3 Eidsath A,Carbonell R G,Whitaker S,et al.Dispersion in pulsed systems-Comparison between theory and experiments for packed beds.Chemical Engineering Science,1983,38(11):1803-1816
4 Nakayama A.PC-aided numerical heat transfer and convective flow[M].New York:CRC Press,1995,319-322
5 Bennion D B,Thomas F B,Formation damage process reducing productivity of low permeability gas reservoirs[C]//Low Permeability Reservoirs Symposium and Exhibition.Denver:Society of Petroleum Engineers,2000:12-15
6 Civan F.Reservoir formation damage:Fundamentals,modeling,assessment,and mitigation[M].2nd ed.Burlington:Gulf Professional Publishing,2007:1091-1114
7李传亮,孔祥言,徐献之,等.多孔介质的双重有效应力[J].自然杂志,1999,21(5):288-291Li Chuanliang,Kong Xiangyan,Xu Xianzhi,et al.Double effective stresses of porous media[J].Chinese Journal of Nature,1999,21(5):288-291
8 Yasuhara H.A mechanistic model for compaction of granular aggregates moderated by pressure solution[J].Journal of Geophysical Research,2003,108(B11)2530:2531
9邓彩华,童亮,陈壁峰,等.多孔介质流动的直接数值模拟[J].武汉理工大学学报,2011,35(6):1257-1260Deng Caihua,Tong Liang,Chen Bifeng,et al.Direct numerical simulation for fluid flow in porous media[J].Journal of Wuhan University of Technology,2011,35(6):1257-1260
10李留仁,赵艳艳,李忠兴,等.多孔介质微观孔隙结构分形特征及分形系数的意义[J].石油大学学报(自然科学版),2004,28(3):105-114Li Liuren,Zhao Yanyan,Li Zhongxing,et al.Fractal characteristics of micropore structure of porous media and the meaning of fractal coefficient[J].Journal of China University of Petroleum(Edition of Natural Science),2004,28(3):105-114
11 Hooman K,Tamayol A,Dahari M,et al.A theoretical model to predict gas permeability for slip flow through a porous medium[J].Applied Thermal Engineering,2014,70(1):71-76
12 Fatt I.The network model of porous media.I.Capillary pressure characteristics[J].Transactions of AIME,1956,207:144-159
13 Joshi M.A class of stoehastic models for porous media[D].Lawrence:University of Kansas,1974
14 Quiblier J A.A new three-dimensional modeling technique for studying porous media[J].Journal of Colloid and Interface Science,1984,98(1):84-102
15周宏伟,谢和平.多孔介质孔隙度与比表面积的分形描述[J].西安矿业学院学报,1997,17(2):97-102Zhou Hongwei,Xie Heping.Fractal description of porosity and specific surface area of porous media[J].Journal of Xi'an Mining Institute,1997,17(2):97-102
16 Hazlett R D.Statistical characterization and stochastic modeling of pore networks in relation to fluid flow[J].Mathematical Geology,1997,29(4):801-822
17 Hidajat I,Rastogi A,Singh M,et al.Transport properties of porous media from thin sections[J].SPE Journal,2002,7(1):40-48
18 Durrett R.Oriented percolation in two dimensions[J].Journal of Applied Probability,1999,12(4):999-1040
19 Perrier E,Bird N,Rieu M.Generalizing the fractal model of soil structure:The pore-solid fractal approach[J].Geoderma,1999,88(3-4):137-164
20 Lehmann P,Stahli M,Papritz A,et al.A fractal approach to model soil structure and to calculate thermal conductivity of soils[J].Transport in Porous Media,2003,52:313-332
21ren P,Bakke S.Process based reconstruction of sandstones and predictions of transport properties[J].Transport in Porous Media,2002,46(2):311-343
22张东辉,金峰,施明恒,等.多孔介质渗流随机模型[J].应用科学学报,2003,21(1):88-92Zhang Donghui,Jin Feng,Shi Mingheng,et al.A random simulation method in porous media[J].Journal of Applied Science,2003,21(1):88-92
23张东辉,金峰,施明恒,等.分形多孔介质中的热传导[J].应用科学学报,2003,21(3):253-257Zhang Donghui,Jin Feng,Shi Mingheng,et al.Heat conduction in fractal porous media[J].Journal of Applied Science,2003,21(3):253-257
24 Wu K,Dijke M I J V,Couples G D,et al.3D stochastic modelling of heterogeneous porous media-applications to reservoir rocks[J].Transport in Porous Media,2006,65:443-467
25袁锦越,杨彬彬,焦阳,等.多孔介质干燥过程分形孔道网络模型与模拟:II.数值模拟与试验验证[J].中国农业大学学报,2007,14(2):55-60Yuan Jinyue,Yang Binbin,Jiao yang,et al.Fractal pore network simulation on the drying of porous media:II.Experimental study and numerical simulation[J].Journal of China Agricultural University,2007,14(2):55-60
26 Okabe H,Blunt M J.Prediction of permeability for porous media reconstructed using multiple-point statistics[J].Physical Review E,2004,70(2):066135
27 Al-Kharusi A S,Blunt M J.Multiphase flow predictions from carbonate pore space images using extracted network models[J].Water Resources Research,2008,44:1-14
28鞠杨,杨永明,宋振铎,等.岩石孔隙结构的统计模型[J].中国科学E辑(技术科学),2008,38(7):1026-1041Ju Yang,Yang Yongming,Song Zhenduo,et al.Statistic model of porous media[J].Science in China(Technological Sciences),2008,38(7):1026-1041
29 Liu X F,Sun J M,Wang H T,et al.Reconstruction of 3-D digital cores using a hybrid method[J].Applied Geophysics,2009,6(2):105-112
30 Biswal B,Manwart C,Hilfer R,et al.Quantitative analysis of experimental and synthetic micro struetures for sedimentary rocks[J].Physica A,1999,273(3):452-475
31ren P,Bakkes.Process based reconstruction of sandstones and predictions of transport properties[J].Transport in Porous Media,2002,46(2):311-343
32 Nsir K,Schfer G.A pore-throat model based on grain-size distribution to quantify gravity-dominated DNAPL instabilities in a water-saturated homogeneous porous medium[J].Comptes RendusGéoscience,2010,342:881-891
33 Hooman K,Tamayol A,Dahari M,et al.A theoretical model to predict gas permeability for slip flow through a porous medium[J].Applied Thermal Engineering,2014,70:71-76
34 Li X C.Heat conduction process in non-uniform porous media[J].Journal of Southeast University,2005,35(5):761-765
35 Kainourgiakis M E,Kikkinides E S,Galani A,et al.Digitally reconstructed porous media:Transport and sorption properties[J].Transport in Porous Media,2005,58(1-2):43-62
36 Carolina V,Diego F,Matteo D,et al.Cellular automata model of phase transition in binary mixtures[J].Industrial&Engineering Chemistry Research,2006,45(8):2892-2896
37 Appert C,Olson J,Rothman D,et al.Phase separation in a threedimensional,two-phase,hydrodynamic lattice gas[J].Journal of Statistical Physics,1995,81(1-2):181-197
38 Yamamoto K,Satake S,Yamashita H,et al.Lattice boltzmann simulation on porous structure and soot accumulation[J].Mathematics&Computers in Simulation,2006,72(2):257-263
39 Ramirez A,Jaramillo D E.Porous media generated by using an immiscible lattice-gas model[J].Computational Materials Science,2012,65:157-164
40 Jude J S,Sarkar S,Sameen A.Reconstruction of 2-D porous media using Karhunen-Lóeve expansion[J].Probabilistic Engineering Mechanics,2013,32:56-65
41 Wang M,Pan N.Numerical analyses of effective dielectric constant of multiphase microporous media[J].Journal of Applied Physics,2007,101(11):1141021-1141028
42 Lin C.Network analysis of filter cake pore structure by high resolution X-ray microtomography[J].The Chemical Engineering Journal,1999,77(1):79-86
43 Vogel H,Tolke J S,Hulz V,et al.Comparison of a lattice-boltzmann model,a full-morphology model,and a pore network model for determining capillary pressure-saturation relationships[J].Vadose Zone Journal,2005(4):380-388
44 Okabe H,Blunt M J.Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics[J].Water Resources Research,2007,43(12):179-183
45 Talabi O,Alsayari S,Iglauer S,et al.Pore-scale simulation of NMR response[J].Journal of Petroleum Science and Engineering,2009,67:168-178
46张挺,卢德唐,李道伦,等.基于二维图像和多点统计方法的多孔介质三维重构研究[J].中国科学技术大学学报,2010,40(3):271-277Zhang Ting,Lu Detang,Li Daolun,et al.A method of reconstruction of porous media using a two-dimensional image and multiplepoint statistics[J].Journal of University of Science and Technology of China,2010,40(3):271-277
47王波,宁正福,姬江.多孔介质模型的三维重构方法[J].西安石油大学学报(自然科学版),2012,27(4):54-61Wang Bo,Ning Zhengfu,Ji Jiang,et al.Study on 3D reconstruction methods of porous medium model[J].Journal of Xi'an Shiyou University(Natural Science Edition),2012,27(4):54-61
2 Pfeifer P,Avnir D.Chemistry nonintegral dimensions between two and three[J].Journal of Chemical Physics,1984,80(9):4573-4573
3 Eidsath A,Carbonell R G,Whitaker S,et al.Dispersion in pulsed systems-Comparison between theory and experiments for packed beds.Chemical Engineering Science,1983,38(11):1803-1816
4 Nakayama A.PC-aided numerical heat transfer and convective flow[M].New York:CRC Press,1995,319-322
5 Bennion D B,Thomas F B,Formation damage process reducing productivity of low permeability gas reservoirs[C]//Low Permeability Reservoirs Symposium and Exhibition.Denver:Society of Petroleum Engineers,2000:12-15
6 Civan F.Reservoir formation damage:Fundamentals,modeling,assessment,and mitigation[M].2nd ed.Burlington:Gulf Professional Publishing,2007:1091-1114
7李传亮,孔祥言,徐献之,等.多孔介质的双重有效应力[J].自然杂志,1999,21(5):288-291Li Chuanliang,Kong Xiangyan,Xu Xianzhi,et al.Double effective stresses of porous media[J].Chinese Journal of Nature,1999,21(5):288-291
8 Yasuhara H.A mechanistic model for compaction of granular aggregates moderated by pressure solution[J].Journal of Geophysical Research,2003,108(B11)2530:2531
9邓彩华,童亮,陈壁峰,等.多孔介质流动的直接数值模拟[J].武汉理工大学学报,2011,35(6):1257-1260Deng Caihua,Tong Liang,Chen Bifeng,et al.Direct numerical simulation for fluid flow in porous media[J].Journal of Wuhan University of Technology,2011,35(6):1257-1260
10李留仁,赵艳艳,李忠兴,等.多孔介质微观孔隙结构分形特征及分形系数的意义[J].石油大学学报(自然科学版),2004,28(3):105-114Li Liuren,Zhao Yanyan,Li Zhongxing,et al.Fractal characteristics of micropore structure of porous media and the meaning of fractal coefficient[J].Journal of China University of Petroleum(Edition of Natural Science),2004,28(3):105-114
11 Hooman K,Tamayol A,Dahari M,et al.A theoretical model to predict gas permeability for slip flow through a porous medium[J].Applied Thermal Engineering,2014,70(1):71-76
12 Fatt I.The network model of porous media.I.Capillary pressure characteristics[J].Transactions of AIME,1956,207:144-159
13 Joshi M.A class of stoehastic models for porous media[D].Lawrence:University of Kansas,1974
14 Quiblier J A.A new three-dimensional modeling technique for studying porous media[J].Journal of Colloid and Interface Science,1984,98(1):84-102
15周宏伟,谢和平.多孔介质孔隙度与比表面积的分形描述[J].西安矿业学院学报,1997,17(2):97-102Zhou Hongwei,Xie Heping.Fractal description of porosity and specific surface area of porous media[J].Journal of Xi'an Mining Institute,1997,17(2):97-102
16 Hazlett R D.Statistical characterization and stochastic modeling of pore networks in relation to fluid flow[J].Mathematical Geology,1997,29(4):801-822
17 Hidajat I,Rastogi A,Singh M,et al.Transport properties of porous media from thin sections[J].SPE Journal,2002,7(1):40-48
18 Durrett R.Oriented percolation in two dimensions[J].Journal of Applied Probability,1999,12(4):999-1040
19 Perrier E,Bird N,Rieu M.Generalizing the fractal model of soil structure:The pore-solid fractal approach[J].Geoderma,1999,88(3-4):137-164
20 Lehmann P,Stahli M,Papritz A,et al.A fractal approach to model soil structure and to calculate thermal conductivity of soils[J].Transport in Porous Media,2003,52:313-332
21ren P,Bakke S.Process based reconstruction of sandstones and predictions of transport properties[J].Transport in Porous Media,2002,46(2):311-343
22张东辉,金峰,施明恒,等.多孔介质渗流随机模型[J].应用科学学报,2003,21(1):88-92Zhang Donghui,Jin Feng,Shi Mingheng,et al.A random simulation method in porous media[J].Journal of Applied Science,2003,21(1):88-92
23张东辉,金峰,施明恒,等.分形多孔介质中的热传导[J].应用科学学报,2003,21(3):253-257Zhang Donghui,Jin Feng,Shi Mingheng,et al.Heat conduction in fractal porous media[J].Journal of Applied Science,2003,21(3):253-257
24 Wu K,Dijke M I J V,Couples G D,et al.3D stochastic modelling of heterogeneous porous media-applications to reservoir rocks[J].Transport in Porous Media,2006,65:443-467
25袁锦越,杨彬彬,焦阳,等.多孔介质干燥过程分形孔道网络模型与模拟:II.数值模拟与试验验证[J].中国农业大学学报,2007,14(2):55-60Yuan Jinyue,Yang Binbin,Jiao yang,et al.Fractal pore network simulation on the drying of porous media:II.Experimental study and numerical simulation[J].Journal of China Agricultural University,2007,14(2):55-60
26 Okabe H,Blunt M J.Prediction of permeability for porous media reconstructed using multiple-point statistics[J].Physical Review E,2004,70(2):066135
27 Al-Kharusi A S,Blunt M J.Multiphase flow predictions from carbonate pore space images using extracted network models[J].Water Resources Research,2008,44:1-14
28鞠杨,杨永明,宋振铎,等.岩石孔隙结构的统计模型[J].中国科学E辑(技术科学),2008,38(7):1026-1041Ju Yang,Yang Yongming,Song Zhenduo,et al.Statistic model of porous media[J].Science in China(Technological Sciences),2008,38(7):1026-1041
29 Liu X F,Sun J M,Wang H T,et al.Reconstruction of 3-D digital cores using a hybrid method[J].Applied Geophysics,2009,6(2):105-112
30 Biswal B,Manwart C,Hilfer R,et al.Quantitative analysis of experimental and synthetic micro struetures for sedimentary rocks[J].Physica A,1999,273(3):452-475
31ren P,Bakkes.Process based reconstruction of sandstones and predictions of transport properties[J].Transport in Porous Media,2002,46(2):311-343
32 Nsir K,Schfer G.A pore-throat model based on grain-size distribution to quantify gravity-dominated DNAPL instabilities in a water-saturated homogeneous porous medium[J].Comptes RendusGéoscience,2010,342:881-891
33 Hooman K,Tamayol A,Dahari M,et al.A theoretical model to predict gas permeability for slip flow through a porous medium[J].Applied Thermal Engineering,2014,70:71-76
34 Li X C.Heat conduction process in non-uniform porous media[J].Journal of Southeast University,2005,35(5):761-765
35 Kainourgiakis M E,Kikkinides E S,Galani A,et al.Digitally reconstructed porous media:Transport and sorption properties[J].Transport in Porous Media,2005,58(1-2):43-62
36 Carolina V,Diego F,Matteo D,et al.Cellular automata model of phase transition in binary mixtures[J].Industrial&Engineering Chemistry Research,2006,45(8):2892-2896
37 Appert C,Olson J,Rothman D,et al.Phase separation in a threedimensional,two-phase,hydrodynamic lattice gas[J].Journal of Statistical Physics,1995,81(1-2):181-197
38 Yamamoto K,Satake S,Yamashita H,et al.Lattice boltzmann simulation on porous structure and soot accumulation[J].Mathematics&Computers in Simulation,2006,72(2):257-263
39 Ramirez A,Jaramillo D E.Porous media generated by using an immiscible lattice-gas model[J].Computational Materials Science,2012,65:157-164
40 Jude J S,Sarkar S,Sameen A.Reconstruction of 2-D porous media using Karhunen-Lóeve expansion[J].Probabilistic Engineering Mechanics,2013,32:56-65
41 Wang M,Pan N.Numerical analyses of effective dielectric constant of multiphase microporous media[J].Journal of Applied Physics,2007,101(11):1141021-1141028
42 Lin C.Network analysis of filter cake pore structure by high resolution X-ray microtomography[J].The Chemical Engineering Journal,1999,77(1):79-86
43 Vogel H,Tolke J S,Hulz V,et al.Comparison of a lattice-boltzmann model,a full-morphology model,and a pore network model for determining capillary pressure-saturation relationships[J].Vadose Zone Journal,2005(4):380-388
44 Okabe H,Blunt M J.Pore space reconstruction of vuggy carbonates using microtomography and multiple-point statistics[J].Water Resources Research,2007,43(12):179-183
45 Talabi O,Alsayari S,Iglauer S,et al.Pore-scale simulation of NMR response[J].Journal of Petroleum Science and Engineering,2009,67:168-178
46张挺,卢德唐,李道伦,等.基于二维图像和多点统计方法的多孔介质三维重构研究[J].中国科学技术大学学报,2010,40(3):271-277Zhang Ting,Lu Detang,Li Daolun,et al.A method of reconstruction of porous media using a two-dimensional image and multiplepoint statistics[J].Journal of University of Science and Technology of China,2010,40(3):271-277
47王波,宁正福,姬江.多孔介质模型的三维重构方法[J].西安石油大学学报(自然科学版),2012,27(4):54-61Wang Bo,Ning Zhengfu,Ji Jiang,et al.Study on 3D reconstruction methods of porous medium model[J].Journal of Xi'an Shiyou University(Natural Science Edition),2012,27(4):54-61
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