有限单元法模拟地下水流的水头反常现象——对流量计算的作用
|
摘要
地下水流数值模拟是定量研究地下水水量和水质的重要手段。为揭示有限单元法水头反常现象形成的原因,以三维非均质模型为例,刻画了水头反常现象。储量集中法(Lumped Mass方法)是有限单元法中储释水量计算的替代算法,可以改善水头反常现象。对结点控制区域的均衡性质进行了研究,分析了使用Lumped Mass方法时的水头反常现象,结果表明:结点控制区域的对流量计算受到该结点所在层及其相邻层水力坡度的影响;在抽水初始阶段,抽水井结点水头快速下降,导致相邻层中相邻结点控制区域的侧向流入量为正值,控制区域为储水过程,结点水头上升,表现出水头反常现象。因此,有限单元法中对流量的计算是引起水头反常现象的原因。
Numerical simulations are an important tool for quantitative research on the volume and quality of groundwater. To reveal the cause of an abnormal waterhead phenomenon using the finite element method,a three-dimensional inhomogeneous model is applied as a case study. The Lumped Mass method is an alternative approach for computing storage changes,which can be used to ameliorate abnormal waterheads. In this paper,the Lumped Mass method is applied to an abnormal waterhead phenomenon by analyzing the local conservation. It is observed that the advection fluxes in the control domain of a vertex refer to the hydraulic gradients in that slice and in adjacent slices. As water starts to be pumped,the potential of the well vertex decreases rapidly. This leads to a positive flux flowing into the control domain of an adjacent vertex in an adjacent slice. The storage change of the control domain in this adjacent vertex is positive,and so the potential increases. Hence,the computation of advection fluxes in the finite element method is the cause of the abnormal waterhead phenomenon.
Numerical simulations are an important tool for quantitative research on the volume and quality of groundwater. To reveal the cause of an abnormal waterhead phenomenon using the finite element method,a three-dimensional inhomogeneous model is applied as a case study. The Lumped Mass method is an alternative approach for computing storage changes,which can be used to ameliorate abnormal waterheads. In this paper,the Lumped Mass method is applied to an abnormal waterhead phenomenon by analyzing the local conservation. It is observed that the advection fluxes in the control domain of a vertex refer to the hydraulic gradients in that slice and in adjacent slices. As water starts to be pumped,the potential of the well vertex decreases rapidly. This leads to a positive flux flowing into the control domain of an adjacent vertex in an adjacent slice. The storage change of the control domain in this adjacent vertex is positive,and so the potential increases. Hence,the computation of advection fluxes in the finite element method is the cause of the abnormal waterhead phenomenon.
引文
[1]张文静,周晶晶,刘丹,等.胶体在地下水中的环境行为特征及其研究方法探讨[J].水科学进展,2016,27(4):629-638.(ZHANG W J,ZHOU J J,LIU D,et al.A review:research methods that describe the environmental behavior of colloids in groundwater[J].Advances in Water Science,2016,27(4):629-638.(in Chinese))
[2]张在勇,王文科,陈立,等.非饱和带有限分析数值模拟的误差分析[J].水科学进展,2016,27(1):70-80.(ZHANG Z Y,WANG W K,CHEN L,et al.Analysis of errors in finite analytic numerical simulation of flow in unsaturated zone[J].Advances in Water Science,2016,27(1):70-80.(in Chinese))
[3]DIERSCH H-J G.FEFLOW:finite element modeling of flow,mass and heat transport in porous and fractured media[M].Berlin:Springer,2013.
[4]陈崇希,唐仲华,胡立堂.地下水流数值模拟理论方法及模型设计[M].北京:地质出版社,2014.(CHEN C X,TANG Z H,HU L T.Theory and methods for groundwater numerical simulation and design for models[M].Beijing:Geological Publishing House,2014.(in Chinese))
[5]薛禹群,谢春红,吴吉春.水文地质数值法存在的问题及其对策[J].地球科学进展,1996,11(5):472-474.(XUE Y Q,XIE C H,WU J C.The problems and strategies in hydrogeological numerical methods[J].Advances in Earth Science,1996,11(5):472-474.(in Chinese))
[6]CORDES C,KINZELBACH W.Continuous groundwater velocity fields and path lines in linear,bilinear,and trilinear finite elements[J].Water Resources Research,1992,28(11):2903-2911.
[7]CHIPPADA S,DAWSON C N,MARTINEZ M L,et al.A projection method for constructing a mass conservative velocity field[J].Computer Methods in Applied Mechanics&Engineering,1998,157(1/2):1-10.
[8]LARSON M G,NIKLASSON A J.A conservative flux for the continuous Galerkin method based on discontinuous enrichment[J].Calcolo,2004,41(2):65-76.
[9]CORREA M R,LOULA A F D.Stabilized velocity post-processings for Darcy flow in heterogeneous porous media[J].Communications in Numerical Methods in Engineering,2007,23(6):461-489.
[10]BUSH L,GINTING V.On the application of the continuous Galerkin finite element method for conservation problems[J].Siam Journal on Scientific Computing,2013,35(6):A2953-A2975.
[11]HUGHES T J R,ENGEL G,MAZZEI L,et al.The continuous Galerkin method is locally conservative[J].Journal of Computational Physics,2000,163(2):467-488.
[12]BERGER R C,HOWINGTON S E.Discrete fluxes and mass balance in finite elements[J].Journal of Hydraulic Engineering,2002,128(1):87-92.
[13]TURNER D Z,NAKSHATRALA K B,MARTINEZ M J,et al.Modeling subsurface water resource systems involving heterogeneous porous media using variational multiscale formulation[J].Journal of Hydrology,2012,428(1):1-14.
[14]BUSH L,GINTING V,PRESHO M.Application of a conservative,generalized multiscale finite element method to flow models[J].Journal of Computational and Applied Mathematics,2014,260(1):395-409.
[15]孙讷正.地下水流的数学模型和数值方法[M].北京:地质出版社,1981.(SUN N Z.Numerical models and methods to simulate groundwater[M].Beijing:Geological Publishing House,1981.(in Chinese))
[16]NEUMAN S P,NARASIMHAN T N.Mixed explicit-implicit iterative finite element scheme for diffusion-type problems:I:theory[J].International Journal for Numerical Methods in Engineering,1977,11(2):309-323.
[17]张宏仁.解渗流问题数值方法对比[J].水文地质工程地质,1984(4):27-33.(ZHANG H R.The comparison between numerical methods for seepage flow[J].Hydrogeology and Engineering Geology,1984(4):27-33.(in Chinese))
[18]薛禹群.地下水数值模拟[M].北京:科学出版社,2007.(XUE Y Q.Numerical simulation of groundwater[M].Beijing:Science Press,2007.(in Chinese))
[19]吕康林.关于有限单元法结点水头反常的分析讨论[J].水文地质工程地质,1988(2):42-45.(LYU K L.The analyses and discussions on the abnormal waterhead phenomena[J].Hydrogeology and Engineering Geology,1988(2):42-45.(in Chinese))
[20]WU Q,ZHAO Y,LIN Y F,et al.Locally conservative groundwater flow in the continuous Galerkin method using 3-D prismatic patches[J].Water Resources Research,2016,52(11):9182-9189.
[2]张在勇,王文科,陈立,等.非饱和带有限分析数值模拟的误差分析[J].水科学进展,2016,27(1):70-80.(ZHANG Z Y,WANG W K,CHEN L,et al.Analysis of errors in finite analytic numerical simulation of flow in unsaturated zone[J].Advances in Water Science,2016,27(1):70-80.(in Chinese))
[3]DIERSCH H-J G.FEFLOW:finite element modeling of flow,mass and heat transport in porous and fractured media[M].Berlin:Springer,2013.
[4]陈崇希,唐仲华,胡立堂.地下水流数值模拟理论方法及模型设计[M].北京:地质出版社,2014.(CHEN C X,TANG Z H,HU L T.Theory and methods for groundwater numerical simulation and design for models[M].Beijing:Geological Publishing House,2014.(in Chinese))
[5]薛禹群,谢春红,吴吉春.水文地质数值法存在的问题及其对策[J].地球科学进展,1996,11(5):472-474.(XUE Y Q,XIE C H,WU J C.The problems and strategies in hydrogeological numerical methods[J].Advances in Earth Science,1996,11(5):472-474.(in Chinese))
[6]CORDES C,KINZELBACH W.Continuous groundwater velocity fields and path lines in linear,bilinear,and trilinear finite elements[J].Water Resources Research,1992,28(11):2903-2911.
[7]CHIPPADA S,DAWSON C N,MARTINEZ M L,et al.A projection method for constructing a mass conservative velocity field[J].Computer Methods in Applied Mechanics&Engineering,1998,157(1/2):1-10.
[8]LARSON M G,NIKLASSON A J.A conservative flux for the continuous Galerkin method based on discontinuous enrichment[J].Calcolo,2004,41(2):65-76.
[9]CORREA M R,LOULA A F D.Stabilized velocity post-processings for Darcy flow in heterogeneous porous media[J].Communications in Numerical Methods in Engineering,2007,23(6):461-489.
[10]BUSH L,GINTING V.On the application of the continuous Galerkin finite element method for conservation problems[J].Siam Journal on Scientific Computing,2013,35(6):A2953-A2975.
[11]HUGHES T J R,ENGEL G,MAZZEI L,et al.The continuous Galerkin method is locally conservative[J].Journal of Computational Physics,2000,163(2):467-488.
[12]BERGER R C,HOWINGTON S E.Discrete fluxes and mass balance in finite elements[J].Journal of Hydraulic Engineering,2002,128(1):87-92.
[13]TURNER D Z,NAKSHATRALA K B,MARTINEZ M J,et al.Modeling subsurface water resource systems involving heterogeneous porous media using variational multiscale formulation[J].Journal of Hydrology,2012,428(1):1-14.
[14]BUSH L,GINTING V,PRESHO M.Application of a conservative,generalized multiscale finite element method to flow models[J].Journal of Computational and Applied Mathematics,2014,260(1):395-409.
[15]孙讷正.地下水流的数学模型和数值方法[M].北京:地质出版社,1981.(SUN N Z.Numerical models and methods to simulate groundwater[M].Beijing:Geological Publishing House,1981.(in Chinese))
[16]NEUMAN S P,NARASIMHAN T N.Mixed explicit-implicit iterative finite element scheme for diffusion-type problems:I:theory[J].International Journal for Numerical Methods in Engineering,1977,11(2):309-323.
[17]张宏仁.解渗流问题数值方法对比[J].水文地质工程地质,1984(4):27-33.(ZHANG H R.The comparison between numerical methods for seepage flow[J].Hydrogeology and Engineering Geology,1984(4):27-33.(in Chinese))
[18]薛禹群.地下水数值模拟[M].北京:科学出版社,2007.(XUE Y Q.Numerical simulation of groundwater[M].Beijing:Science Press,2007.(in Chinese))
[19]吕康林.关于有限单元法结点水头反常的分析讨论[J].水文地质工程地质,1988(2):42-45.(LYU K L.The analyses and discussions on the abnormal waterhead phenomena[J].Hydrogeology and Engineering Geology,1988(2):42-45.(in Chinese))
[20]WU Q,ZHAO Y,LIN Y F,et al.Locally conservative groundwater flow in the continuous Galerkin method using 3-D prismatic patches[J].Water Resources Research,2016,52(11):9182-9189.