双重介质中混溶驱动问题的混合元——有限体积元法
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摘要
土壤的结构是相当复杂的,包含于其中的固、液、气三者相互联系,相互制约,构成了一个矛盾的统一体,为生长于其中的植物提供必要的生活条件,是土壤肥力的物质基础。然而,近年来随着工农业生产的发展、工业“三废”大量排放、农药化肥大面积使用、核能利用带来的放射性废物日益增多、人口增长和城市化带来的垃圾不断增加,土壤的化学废物污染已经成为人们关注的又一重点,从而刺激了对于在土壤介质中的溶质运动情况的相关探索。对于溶质迁移过程的研究已经成为当代土壤物理学,特别是在预测杀虫剂、营养矿物质、重金属或其他溶质在土壤中的运动过程中充当着非常重要的工具。
在已有的土壤溶质运移模型中,对流-弥散模型是一类基本的运动方程,该模型具有坚实的物理基础,可以描述质量、热量的运输及反应扩散过程等众多物理现象,且形式简单。但是,由于土壤结构的复杂性,使得该模型不能用于野外大尺度范围的溶质运移问题。在聚合介质中,土壤是由一些列孔隙组成的,溶质通过这些孔隙的速度有快有慢,从而分为裂缝孔隙和岩石孔隙,岩石孔隙就伴随着不动水的存在。在[21]中研究了吸附性多孔介质中化学溶质的横向扩散的相关问题。在多孔介质中,流体分为不动区和动区,流动的水主要存在于大的孔隙中,而溶质在水中的流动也主要存在于这一区域。溶质的在水中迁移主要是基于对流以及纵向的扩散等原因。不动水主要存在于聚合体的内部以及不同聚合体的衔接处。溶质在两种流体区域之间的扩散相容是与它们的浓度差异成比例的,流动液体中的溶质与吸附在土壤颗粒表面上的溶质的动态平衡使得在动水区附近形成了动态的土壤区域,而静止土壤区域主要位于聚合体内的小孔隙附近或者在不动水区边界处。只有在化学溶质弥散渗透穿过不动水区域的流体外层后吸附才会发生。在对流-弥散模型基础上建立起来的非均衡溶质运移两区模型,由于考虑了土壤中不动水体的存在以及不动水体在溶质运移过程中所起的作用和溶质运移过程中的吸附、降解等因素的影响,因此能更准确的描述土壤中溶质的运移过程。
对于较复杂的现实问题,很难用解析的方法求解,而数值模拟是一种有效的求解方法。常用的数值计算方法有有限差分法和有限元法以及有限体积元法,但用标准的有限差分方法求解数学模型时常常失效,根本原因在于对流项的存在。由于对流为主的弥散方程具有双曲性质,中心差分格式虽然关于空间步长具有二阶精度,但会产生数值弥散和非物理力学特性的数值振荡,使数值模拟失真。特征线和经典的有限差分、有限元方法相结合所得到的特征差分、特征有限元方法,可以更好的反映出对流弥散方程的一阶双曲特性,这一类方法考虑沿着特征线方向的离散,利用了物理力学性质,可以有效的克服数值振荡,减少截断误差,大大提高计算精度。崔明老师和张德生老师曾对两区模型应用特征有限元方法进行数值分析。本文就是在以上思想的基础上,在第二章中将混合有限元、有限体积元相结合来求解,给出相应的误差估计并各处相应的一维实例。
In recent years, with the development of agriculture and industry, the discharge of large amount of waste gas, waste water and industrial residue, the widely use of pesti-cides and fertilizer, the increasing growth of radioactive wastes when using the nuclear energy, the increasing garbage caused by population growth and urbanization are caus-ing more and more damages. The intentional or accidental release of chemical wastes on soils has further stimulated current interests in the movement of chemicals. Displace-ment studies have become important tools in soil physics, particularly for predicting the movement of pesticide, nitrates, heavy metals and other solutes through soils.
Convection-Dispersion Equation is one of basic motion equations about solute trans-port in soil. This simple model has a solid physics foundation and could describe the transport of solute, heat, reaction diffusion process, etc. However, according to the complexity of the practical conditions, this model cannot be used in the field of large scale solute transport problems. In aggregated media, soils are composed of slowly and quickly conducting pore sequences, the liquid-filled, dead-end pores or immobile wa-ter exists. One way to account for the transport and immobile water is to partition the liquid phase into mobile and immobile regions, to limit convective-dispersive transport to the mobile liquid region, and to assume that diffusion is responsible for the exchange of solute between the mobile and immobile regions. In [21] the movement of a chemical through a absorbing porous medium with a lateral or intra-aggregated diffusion was con-sidered. Mobile water is located inside the larger pores and Immobile water is located inside aggregates and at the contact points of aggregates and/or particles. A dynamic soil region is located sufficiently close to the mobile water phase for equilibrium be-tween the solute in the mobile liquid and that absorbed by this part of the soil mass. A stagnant soil region, where sorption is diffusion limited, is located mainly around the micro-pores inside the aggregates or along dead-end water pockets. Sorption occurs here only after the chemicals have diffused through the liquid barrier of the immobile liquid phase. The non-equilibrium solute transport two-region model(TRM) is based on the convection-dispersion model. Two-Region Model have taken the immobile water, influ-ence of absorption and degradation into account, which is more accurately to describe the solute transport process in soil.
For more complex problems, it is difficult to present the analytical solutions and nu- merical simulation is a kind of effective method. The basic numerical simulation method is FDM and FEM, but the traditional FDM are not efficient for some models with con-vection term. Due to the hyperbolic properties of convection-dominate dispersion equa-tions, the central difference formula often cause numerical dispersion and oscillation even it has two-order precision in space. The characteristics combing the difference or finite element method can be better to reflect the first-order hyperbolic properties of convection-dispersion equations. The characteristic methods, discrete the equation along the characteristic line according to physical and mechanical properties, can effectively overcome nonphysical oscillation and reduce truncation errors and greatly improve the calculation accuracy. Cui Ming and ZhangDe sheng had impled the characteristic finite element method to two-region model and made an assay of numerical calculation. In this article, In chapter two, based on the same numerical approximation theory, we adopt the mixed finite element method combing the finite volume element method to attain the numerical results and analyze the related error estimates. Meanwhile, we give a one-dimensional simple model.
在已有的土壤溶质运移模型中,对流-弥散模型是一类基本的运动方程,该模型具有坚实的物理基础,可以描述质量、热量的运输及反应扩散过程等众多物理现象,且形式简单。但是,由于土壤结构的复杂性,使得该模型不能用于野外大尺度范围的溶质运移问题。在聚合介质中,土壤是由一些列孔隙组成的,溶质通过这些孔隙的速度有快有慢,从而分为裂缝孔隙和岩石孔隙,岩石孔隙就伴随着不动水的存在。在[21]中研究了吸附性多孔介质中化学溶质的横向扩散的相关问题。在多孔介质中,流体分为不动区和动区,流动的水主要存在于大的孔隙中,而溶质在水中的流动也主要存在于这一区域。溶质的在水中迁移主要是基于对流以及纵向的扩散等原因。不动水主要存在于聚合体的内部以及不同聚合体的衔接处。溶质在两种流体区域之间的扩散相容是与它们的浓度差异成比例的,流动液体中的溶质与吸附在土壤颗粒表面上的溶质的动态平衡使得在动水区附近形成了动态的土壤区域,而静止土壤区域主要位于聚合体内的小孔隙附近或者在不动水区边界处。只有在化学溶质弥散渗透穿过不动水区域的流体外层后吸附才会发生。在对流-弥散模型基础上建立起来的非均衡溶质运移两区模型,由于考虑了土壤中不动水体的存在以及不动水体在溶质运移过程中所起的作用和溶质运移过程中的吸附、降解等因素的影响,因此能更准确的描述土壤中溶质的运移过程。
对于较复杂的现实问题,很难用解析的方法求解,而数值模拟是一种有效的求解方法。常用的数值计算方法有有限差分法和有限元法以及有限体积元法,但用标准的有限差分方法求解数学模型时常常失效,根本原因在于对流项的存在。由于对流为主的弥散方程具有双曲性质,中心差分格式虽然关于空间步长具有二阶精度,但会产生数值弥散和非物理力学特性的数值振荡,使数值模拟失真。特征线和经典的有限差分、有限元方法相结合所得到的特征差分、特征有限元方法,可以更好的反映出对流弥散方程的一阶双曲特性,这一类方法考虑沿着特征线方向的离散,利用了物理力学性质,可以有效的克服数值振荡,减少截断误差,大大提高计算精度。崔明老师和张德生老师曾对两区模型应用特征有限元方法进行数值分析。本文就是在以上思想的基础上,在第二章中将混合有限元、有限体积元相结合来求解,给出相应的误差估计并各处相应的一维实例。
In recent years, with the development of agriculture and industry, the discharge of large amount of waste gas, waste water and industrial residue, the widely use of pesti-cides and fertilizer, the increasing growth of radioactive wastes when using the nuclear energy, the increasing garbage caused by population growth and urbanization are caus-ing more and more damages. The intentional or accidental release of chemical wastes on soils has further stimulated current interests in the movement of chemicals. Displace-ment studies have become important tools in soil physics, particularly for predicting the movement of pesticide, nitrates, heavy metals and other solutes through soils.
Convection-Dispersion Equation is one of basic motion equations about solute trans-port in soil. This simple model has a solid physics foundation and could describe the transport of solute, heat, reaction diffusion process, etc. However, according to the complexity of the practical conditions, this model cannot be used in the field of large scale solute transport problems. In aggregated media, soils are composed of slowly and quickly conducting pore sequences, the liquid-filled, dead-end pores or immobile wa-ter exists. One way to account for the transport and immobile water is to partition the liquid phase into mobile and immobile regions, to limit convective-dispersive transport to the mobile liquid region, and to assume that diffusion is responsible for the exchange of solute between the mobile and immobile regions. In [21] the movement of a chemical through a absorbing porous medium with a lateral or intra-aggregated diffusion was con-sidered. Mobile water is located inside the larger pores and Immobile water is located inside aggregates and at the contact points of aggregates and/or particles. A dynamic soil region is located sufficiently close to the mobile water phase for equilibrium be-tween the solute in the mobile liquid and that absorbed by this part of the soil mass. A stagnant soil region, where sorption is diffusion limited, is located mainly around the micro-pores inside the aggregates or along dead-end water pockets. Sorption occurs here only after the chemicals have diffused through the liquid barrier of the immobile liquid phase. The non-equilibrium solute transport two-region model(TRM) is based on the convection-dispersion model. Two-Region Model have taken the immobile water, influ-ence of absorption and degradation into account, which is more accurately to describe the solute transport process in soil.
For more complex problems, it is difficult to present the analytical solutions and nu- merical simulation is a kind of effective method. The basic numerical simulation method is FDM and FEM, but the traditional FDM are not efficient for some models with con-vection term. Due to the hyperbolic properties of convection-dominate dispersion equa-tions, the central difference formula often cause numerical dispersion and oscillation even it has two-order precision in space. The characteristics combing the difference or finite element method can be better to reflect the first-order hyperbolic properties of convection-dispersion equations. The characteristic methods, discrete the equation along the characteristic line according to physical and mechanical properties, can effectively overcome nonphysical oscillation and reduce truncation errors and greatly improve the calculation accuracy. Cui Ming and ZhangDe sheng had impled the characteristic finite element method to two-region model and made an assay of numerical calculation. In this article, In chapter two, based on the same numerical approximation theory, we adopt the mixed finite element method combing the finite volume element method to attain the numerical results and analyze the related error estimates. Meanwhile, we give a one-dimensional simple model.
引文
[1]Biggar, J.W. and Nielsen, D.R., Miscible displacement:Ⅱ. Behavior of tracers. Soil Sci. Am. Proc.,26(2)(1962),125-128.
[2]Biggar, J.W. and Nielsen, D.R., Miscible displacement:Ⅴ. Exchange processes. Soil Sci. Am. Proc,27(6)(1963),623-627.
[3]Biggar, J.W. and Nielsen, D.R., Spatial variability of the leaching characteristics of a field soil, Water Resour. Res.,12(1976),78-84.
[4]Coats, K.H. and Smith, B.D., Dead-end pore Volume and dispersion in porous media, Soc. Pet. Eng. J.,4(1964),73-84.
[5]D. A. Barry et.al., Velocity scaling of nonequilibrium solute transport in heterogeneous soil:Numerical and experimental results, Water down under 94, Adelaide. Australia, Inst. Eng. Aust.,2(B)(1994),645-648.
[6]J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, Approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO,Anal. Numer.,17 (1983),17-33.
[7]J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO, Anal. Numer.,17 (1983),249-265.
[8]Kutilek, M. and Nielsen, D.R., Soil hydrology Catena Verlag,Geoscience publisher, Cremlingen-Destedt,1994, Germany.
[9]Lapidus, L. and Amundson, N.R., Mathematics of adsorption in beds, Ⅵ. The effects of lon-gitudinal diffusion in ion exchange and chromatographic columns, J. Phys. Chem., 56(1952),984-988.
[10]Ming Cui, Hongsen Chen, R.E. Ewing, Guan Qin, An Alternating direction Galerkin method combined with a modified method of characteristics for miscible displacement influenced by mobile and immobile Water, Int. J. of Numerical Analysis and Modeling, 5(4)(2008),659-672.
[11]Nielsen, D.R. and Biggar, J.W., Miscible displacement in soils:I. Experimental inform-ation, Soil Sci. Am. Proc.,25(1961),1-5.
[12]Nielsen, D.R. and Bigga,J.W., Miscible displacement in soils:Ⅱ. Theoretical considera-tion, Soil Sci. Am. Proc,26(1962),216-221.
[13]Nielsen, D.R. and Biggar,J.W., Miscible displacement:Ⅳ. mixing in galss beads, Soil Sci. Am. Proc.,27(1963),10-13.
[14]Nielsen, D.R., Van Genuchten, M.Th. and Biggar, J.W., Water flow and solute transport processes in the unsaturated zone, Water Resour. Res.,22(9)(1986),89-108.
[15]Nobuo Toride, Leij, F.J., Van Genuchten, M.T., A Comprehensive set of Analytical Solution for Nonequilibrium Solute Transport With First-Order Decay and Zero-Order Production, Water Resour. Res.,29(1993),2167-2182.
[16]Pickens, J.F. and Grisak G.E., Modeling of scale-dependent dispersion in hydrogeo-logic systems, Water Resour. Res.,17(1981),1701-1711.
[17]Scheidegger, A.E., Statistical hydrodynamics in porous media. J. Appl. Phys.,25(1954), 994.
[18]T. F. Russell, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media[J], SLAM J Numerical Analysis, 22(1985),970-1013
[19]U.Hornung, Miscible Displacement in Porous Media Influenced by Mobile and Immobile Water, Nonlinear Partial Differential Equations, Springer, New York,1988.
[20]Van Genuchten M.Th., Davidson, J.M., Wierenga, P.J., An evaluation of kinetic and equi-librium equations for the prediction of pesticide movement through porous media, Soil Sci. Soc. Am. J.,38(1974),29-35.
[21]Van Genuchten M.Th. and Wierenga P.J., Mass transfer studies in sorbing porous media I Analytical solutions, Soil. Sci. Amer. J.,40(1976),473-480.
[22]Van Genuchten, M.Th., Wagenet R.J., Two-site/two-region models for pesticide transport and degradation:Theoretical development and analytical solutions, Soil Sci. Soc. Am. J., 53(1989),1303-1310.
[23]X. Long, Finite volume element method for seawater intrusion problem, Journal Of Shan-dong University, Vol,39, N.3,2004.
[24]Y. Yuan, D. Liang, and H. Rui, Characteristics-finite element method for seawater intrusion numerical simulation and theoretical analysis[J], Acta Mathematical Application Sinaca, 14(1998),11-23.
[25]Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, Computational Science and Engineering Series, Vol.2, SIAM, Philadephia, PA, 2006.
[26]Z. Chen, S. Lyons, and G. Qin, The mechanical behavior of poroelastic media saturated with a Newtonian (?) uid derived via homogenization, International Journal of Numerical Analysis and Modeling,1 (2004),75-98.
[27]Z. Chen and Y. Zhang, Development, analysis and numerical tests of a compositional reser-voir simulator, International Journal of Numerical Analysis and Modeling,5 (2008),86-100.
[28]邓亚珠,孔隙介质中非均衡溶质运移模型的数值解法,2009
[29]袁益让.多孔介质中可压缩可混溶驱动问题的特征-有限元方法[J].计算数学,1992,14(4)：385-400.
[30]袁益让.在多孔介质中完全可压缩、可混溶驱动问题的差分方法[J].计算数学,1993,15(1)：16-28
[2]Biggar, J.W. and Nielsen, D.R., Miscible displacement:Ⅴ. Exchange processes. Soil Sci. Am. Proc,27(6)(1963),623-627.
[3]Biggar, J.W. and Nielsen, D.R., Spatial variability of the leaching characteristics of a field soil, Water Resour. Res.,12(1976),78-84.
[4]Coats, K.H. and Smith, B.D., Dead-end pore Volume and dispersion in porous media, Soc. Pet. Eng. J.,4(1964),73-84.
[5]D. A. Barry et.al., Velocity scaling of nonequilibrium solute transport in heterogeneous soil:Numerical and experimental results, Water down under 94, Adelaide. Australia, Inst. Eng. Aust.,2(B)(1994),645-648.
[6]J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, Approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO,Anal. Numer.,17 (1983),17-33.
[7]J. Douglas, Jr., R. E. Ewing, and M. F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO, Anal. Numer.,17 (1983),249-265.
[8]Kutilek, M. and Nielsen, D.R., Soil hydrology Catena Verlag,Geoscience publisher, Cremlingen-Destedt,1994, Germany.
[9]Lapidus, L. and Amundson, N.R., Mathematics of adsorption in beds, Ⅵ. The effects of lon-gitudinal diffusion in ion exchange and chromatographic columns, J. Phys. Chem., 56(1952),984-988.
[10]Ming Cui, Hongsen Chen, R.E. Ewing, Guan Qin, An Alternating direction Galerkin method combined with a modified method of characteristics for miscible displacement influenced by mobile and immobile Water, Int. J. of Numerical Analysis and Modeling, 5(4)(2008),659-672.
[11]Nielsen, D.R. and Biggar, J.W., Miscible displacement in soils:I. Experimental inform-ation, Soil Sci. Am. Proc.,25(1961),1-5.
[12]Nielsen, D.R. and Bigga,J.W., Miscible displacement in soils:Ⅱ. Theoretical considera-tion, Soil Sci. Am. Proc,26(1962),216-221.
[13]Nielsen, D.R. and Biggar,J.W., Miscible displacement:Ⅳ. mixing in galss beads, Soil Sci. Am. Proc.,27(1963),10-13.
[14]Nielsen, D.R., Van Genuchten, M.Th. and Biggar, J.W., Water flow and solute transport processes in the unsaturated zone, Water Resour. Res.,22(9)(1986),89-108.
[15]Nobuo Toride, Leij, F.J., Van Genuchten, M.T., A Comprehensive set of Analytical Solution for Nonequilibrium Solute Transport With First-Order Decay and Zero-Order Production, Water Resour. Res.,29(1993),2167-2182.
[16]Pickens, J.F. and Grisak G.E., Modeling of scale-dependent dispersion in hydrogeo-logic systems, Water Resour. Res.,17(1981),1701-1711.
[17]Scheidegger, A.E., Statistical hydrodynamics in porous media. J. Appl. Phys.,25(1954), 994.
[18]T. F. Russell, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media[J], SLAM J Numerical Analysis, 22(1985),970-1013
[19]U.Hornung, Miscible Displacement in Porous Media Influenced by Mobile and Immobile Water, Nonlinear Partial Differential Equations, Springer, New York,1988.
[20]Van Genuchten M.Th., Davidson, J.M., Wierenga, P.J., An evaluation of kinetic and equi-librium equations for the prediction of pesticide movement through porous media, Soil Sci. Soc. Am. J.,38(1974),29-35.
[21]Van Genuchten M.Th. and Wierenga P.J., Mass transfer studies in sorbing porous media I Analytical solutions, Soil. Sci. Amer. J.,40(1976),473-480.
[22]Van Genuchten, M.Th., Wagenet R.J., Two-site/two-region models for pesticide transport and degradation:Theoretical development and analytical solutions, Soil Sci. Soc. Am. J., 53(1989),1303-1310.
[23]X. Long, Finite volume element method for seawater intrusion problem, Journal Of Shan-dong University, Vol,39, N.3,2004.
[24]Y. Yuan, D. Liang, and H. Rui, Characteristics-finite element method for seawater intrusion numerical simulation and theoretical analysis[J], Acta Mathematical Application Sinaca, 14(1998),11-23.
[25]Z. Chen, G. Huan, and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, Computational Science and Engineering Series, Vol.2, SIAM, Philadephia, PA, 2006.
[26]Z. Chen, S. Lyons, and G. Qin, The mechanical behavior of poroelastic media saturated with a Newtonian (?) uid derived via homogenization, International Journal of Numerical Analysis and Modeling,1 (2004),75-98.
[27]Z. Chen and Y. Zhang, Development, analysis and numerical tests of a compositional reser-voir simulator, International Journal of Numerical Analysis and Modeling,5 (2008),86-100.
[28]邓亚珠,孔隙介质中非均衡溶质运移模型的数值解法,2009
[29]袁益让.多孔介质中可压缩可混溶驱动问题的特征-有限元方法[J].计算数学,1992,14(4)：385-400.
[30]袁益让.在多孔介质中完全可压缩、可混溶驱动问题的差分方法[J].计算数学,1993,15(1)：16-28