横观各向同性饱和土的基本方程组
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摘要
两相饱和多孔介质的基本方程组和计算参数的选取在一定程度上是混乱的.本文利用连续介质力学理论,结合空间平均化方法,根据应力-应变关系、运动学关系、连续性方程及广义Darcy定律,建立了横观各向同性液体饱和多孔介质的基本方程组;通过将该基本方程组与Biot理论的基本方程组进行比较,确定了Biot方程组中的弹性常数Bi与弹性常数cij的关系,并得到了确定Biot参数mi和ri的计算公式——Biot参数与渗透率、孔隙率及黏性系数的关系;最后对几种特殊情形进行了讨论,给出了简化方程组.
Although the saturated poroelastic media are given more and more interests,the basic equations and calculating parameters about the media are in confusion to a certain extent.In this work using mechanics of continuum and the method of space average,the basic equations of transversely isotropic fluid-saturated poroelastic media are obtained,in terms of stress-strain's relations,kinetics equations,continuity equations and generalized Darcy law.Comparing those equations with Biot's equations,the relations of elastic constants B-i with elastic constants c-(ij) are determined,and the explicit expressions of Biot's parameters m-i and r-i,i.e.,the relations of Biot's parameters with permeability,viscosity and porosity,are obtained.Finally the particular cases are discussed and the simplified basic equations are given.
Although the saturated poroelastic media are given more and more interests,the basic equations and calculating parameters about the media are in confusion to a certain extent.In this work using mechanics of continuum and the method of space average,the basic equations of transversely isotropic fluid-saturated poroelastic media are obtained,in terms of stress-strain's relations,kinetics equations,continuity equations and generalized Darcy law.Comparing those equations with Biot's equations,the relations of elastic constants B-i with elastic constants c-(ij) are determined,and the explicit expressions of Biot's parameters m-i and r-i,i.e.,the relations of Biot's parameters with permeability,viscosity and porosity,are obtained.Finally the particular cases are discussed and the simplified basic equations are given.
引文
[1]Biot M A.Theory of elastic and consolidation for a porous anisotropicsolid.J.Appl.Phys,1955,26(2):182-185
[2]Biot M A.Theory of elastic waves in a fluid-saturated porous solid.J.Acoust.Soc.Am.,1956,28(1):168-191
[3]Biot M A.Mechanics of deformations and acoustic propagation inporous media.J.Appl.Phys.,1962,33(4):1482-1498
[4]Biot M A.Generalized theory of acoustic propagation in porousdissipative media.J.Acoust.Soc.Am.,1962,34(9):1254-1264
[5]Burrid R,Keller J B.Poroelasticity equations derived from micro-structure.J.Acoust.Soc.Am.,1981,70:1140-1146
[6]Auriault J L,et al.Dynamics of a porous saturated medium,checking of the generalized law of Darcy.J.Acoust.Soc.Am.,1985,77(5):1641-1750
[7]Bowen R M.Compressible porous media models by use of the theoryof mixtures.Int.J.Eng.Sci.,1982,20(6):697-735
[8]Atkin R J,Craine R E.Continuum theories of mixtures:basic theoryand historical development.Q.J.Mech.Appl.Math.,1976,29:209-244
[9]贝尔J著.多孔介质流体动力学.李敬生,陈崇希译.北京:中国建筑工业出版,1983.71-74Bear J.Dynamics of Fluid in Porous Media(in Chinese).Translatedby Li J S,Chen C X.Beijing:Chinese Architecture Press,1983.71-74
[10]樊蔚勋,吴剑国.弹性各向异性体力学及其应用.南京:江苏科学技术出版社,1994.59-62Fan W X,Wu J G.Mechanics of Elastic Anisotropic Body and ItsApplications(in Chinese).Nanjing:Jiangsu Science and TechnologyPress,1994.59-62
[11]黄义,张引科.张量及其在连续介质力学中的应用.北京:冶金工业出版社,2002.169-171Huang Y,Zhang Y K.Tensor and its Applications in ContinuumMechanics(in Chinese).Beijing:Metallurgical Press,2002.169-171
[12]科林斯R E著.流体通过多孔材料的流动.陈钟祥,吴望一译.北京:石油工业出版社,1984.63-64Collins R E.Flow of Liquid through Porous Materials(in Chinese).Translated by Chen Z X,Wu W Y.Beijing:Petroleum IndustryPress,1984.63-64
[13]Johmison D L.Theory of dynamic permeability and tortuosity in fluid-saturated porous media.J.Fluid Mech.,1987,176:379-402
[14]Schmitt P D.Acoustic multipole logging in transversely isotropicporoelastic formations.J.Acoust.Soc.Am.,1989,86(6):2397-2421
[15]刘银斌,李幼铭,吴如山.横向各向同性多孔介质中的地震波传播.地球物理学报,1994,37(4):499-514Liu Y B,Li Y M,Wu R S.Propagation of seismic waves intransversely isotropic poroelastic media.Chinese J.Geophy.(inChinese),1994,37(4):499-514
[16]汪越胜,章梓茂.横观各向同性液体饱和多孔介质中平面波的传播.力学学报,1997,29(3):257-268Wang Y S,Zhang Z M.Propagation of plane waves in transverselyisotropic fluid-saturated poroelastic medium.Acta Mechanica Sinica(in Chinese),1997,29(3):257-268
[17]Kazi-Aoual M N,Bonnet G,Jouanna P.Green’s functions in aninfinite transversely saturated poroelastic medium.J.Acoust.Soc.Am.,1988,84(5):1883-1889
[18]Sharma M D,Gogna M L.Wave propagation in anisotropic liquid-saturated porous solids:Part I.J.Acoust.Soc.Am.,1991,90(2):1068-1073
[19]Paul S.On the displacements produced in a porous elastic half-space by an impulsive line load(non-dissipative case).Pure Appl.Geophys.,1976,114:605-614
[20]Zienkiewicz O C,Chang C T,Bettess P.Drained undrained,consolidating and dynamic behavior assumptions in soil,limits ofvalidity.Geotechnique,1980,30:385-395
[21]Zinkiewicz O C,Shiomi T.Dynamic behavior of saturated porousmedia:generalized Biot formulation and numerical solution.Int.J.Nun.Anal.Meth.Geomech.,1984,8:71-96
[22]黄义,张玉红.饱和土三维非对称Lamb问题.中国科学(E辑),2000,30(4):375-384Huang Y,Zhang Y H.Non-Axisymmetrical Lamb’s problem insaturated soils.Science in China(Series E),2000,30(4):375-384
[23]胡亚元,王立忠,张忠苗等.横观各向同性饱和土体的实用波动方程.振动工程学报,1998,12(2):170-176Hu Y Y,Wang L Z,Zhang Z M,et al.The pragmatic waveequations of transversely isotropic saturated soils.J.VibrationEngineering(in Chinese),1998,12(2):170-176
[24]张引科,黄义.弹性饱和多孔介质在非轴对称荷载下的稳态动力响应.土木工程学报,2002,35(3):41-45Zhang Y K,Huang Y.Steady dynamic response of elastic saturatedporous media under non-axisymmetrical load.Chinese CivilEngineering(in Chinese),2002,35(3):41-45
[2]Biot M A.Theory of elastic waves in a fluid-saturated porous solid.J.Acoust.Soc.Am.,1956,28(1):168-191
[3]Biot M A.Mechanics of deformations and acoustic propagation inporous media.J.Appl.Phys.,1962,33(4):1482-1498
[4]Biot M A.Generalized theory of acoustic propagation in porousdissipative media.J.Acoust.Soc.Am.,1962,34(9):1254-1264
[5]Burrid R,Keller J B.Poroelasticity equations derived from micro-structure.J.Acoust.Soc.Am.,1981,70:1140-1146
[6]Auriault J L,et al.Dynamics of a porous saturated medium,checking of the generalized law of Darcy.J.Acoust.Soc.Am.,1985,77(5):1641-1750
[7]Bowen R M.Compressible porous media models by use of the theoryof mixtures.Int.J.Eng.Sci.,1982,20(6):697-735
[8]Atkin R J,Craine R E.Continuum theories of mixtures:basic theoryand historical development.Q.J.Mech.Appl.Math.,1976,29:209-244
[9]贝尔J著.多孔介质流体动力学.李敬生,陈崇希译.北京:中国建筑工业出版,1983.71-74Bear J.Dynamics of Fluid in Porous Media(in Chinese).Translatedby Li J S,Chen C X.Beijing:Chinese Architecture Press,1983.71-74
[10]樊蔚勋,吴剑国.弹性各向异性体力学及其应用.南京:江苏科学技术出版社,1994.59-62Fan W X,Wu J G.Mechanics of Elastic Anisotropic Body and ItsApplications(in Chinese).Nanjing:Jiangsu Science and TechnologyPress,1994.59-62
[11]黄义,张引科.张量及其在连续介质力学中的应用.北京:冶金工业出版社,2002.169-171Huang Y,Zhang Y K.Tensor and its Applications in ContinuumMechanics(in Chinese).Beijing:Metallurgical Press,2002.169-171
[12]科林斯R E著.流体通过多孔材料的流动.陈钟祥,吴望一译.北京:石油工业出版社,1984.63-64Collins R E.Flow of Liquid through Porous Materials(in Chinese).Translated by Chen Z X,Wu W Y.Beijing:Petroleum IndustryPress,1984.63-64
[13]Johmison D L.Theory of dynamic permeability and tortuosity in fluid-saturated porous media.J.Fluid Mech.,1987,176:379-402
[14]Schmitt P D.Acoustic multipole logging in transversely isotropicporoelastic formations.J.Acoust.Soc.Am.,1989,86(6):2397-2421
[15]刘银斌,李幼铭,吴如山.横向各向同性多孔介质中的地震波传播.地球物理学报,1994,37(4):499-514Liu Y B,Li Y M,Wu R S.Propagation of seismic waves intransversely isotropic poroelastic media.Chinese J.Geophy.(inChinese),1994,37(4):499-514
[16]汪越胜,章梓茂.横观各向同性液体饱和多孔介质中平面波的传播.力学学报,1997,29(3):257-268Wang Y S,Zhang Z M.Propagation of plane waves in transverselyisotropic fluid-saturated poroelastic medium.Acta Mechanica Sinica(in Chinese),1997,29(3):257-268
[17]Kazi-Aoual M N,Bonnet G,Jouanna P.Green’s functions in aninfinite transversely saturated poroelastic medium.J.Acoust.Soc.Am.,1988,84(5):1883-1889
[18]Sharma M D,Gogna M L.Wave propagation in anisotropic liquid-saturated porous solids:Part I.J.Acoust.Soc.Am.,1991,90(2):1068-1073
[19]Paul S.On the displacements produced in a porous elastic half-space by an impulsive line load(non-dissipative case).Pure Appl.Geophys.,1976,114:605-614
[20]Zienkiewicz O C,Chang C T,Bettess P.Drained undrained,consolidating and dynamic behavior assumptions in soil,limits ofvalidity.Geotechnique,1980,30:385-395
[21]Zinkiewicz O C,Shiomi T.Dynamic behavior of saturated porousmedia:generalized Biot formulation and numerical solution.Int.J.Nun.Anal.Meth.Geomech.,1984,8:71-96
[22]黄义,张玉红.饱和土三维非对称Lamb问题.中国科学(E辑),2000,30(4):375-384Huang Y,Zhang Y H.Non-Axisymmetrical Lamb’s problem insaturated soils.Science in China(Series E),2000,30(4):375-384
[23]胡亚元,王立忠,张忠苗等.横观各向同性饱和土体的实用波动方程.振动工程学报,1998,12(2):170-176Hu Y Y,Wang L Z,Zhang Z M,et al.The pragmatic waveequations of transversely isotropic saturated soils.J.VibrationEngineering(in Chinese),1998,12(2):170-176
[24]张引科,黄义.弹性饱和多孔介质在非轴对称荷载下的稳态动力响应.土木工程学报,2002,35(3):41-45Zhang Y K,Huang Y.Steady dynamic response of elastic saturatedporous media under non-axisymmetrical load.Chinese CivilEngineering(in Chinese),2002,35(3):41-45
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