二维卤水动力学问题数值解法研究
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摘要
卤水是一种重要的自然资源,随着工农业生产的迅速发展,对卤水的开采规模越来越大,这就必然要提出精确预测、合理开采、正确设计与综合利用的问题。本文以察尔汗盐湖别勒滩区段为背景,根据青海省察尔汗盐湖别勒滩区段的水文地质和水化学的实际,概化得到该区卤水动力学地质概念模型,并在此基础上建立了卤水动力学数学模型,拟合该区卤水开采过程中卤水的动态变化规律。实际的水文地质条件往往是比较复杂的,用解析法求解很困难,数值法为研究这类问题开辟了新的途径。它以渗流理论为基础,从整个计算区域的水量平衡或质量守恒出发,建立反映整个计算区域卤水运动状况的数学模型。解水动力问题的数值方法有多种,但最通用的还是有限差分法(FDM)和有限元法(FEM,也称有限单元法,有限元素法)。本文分别用有限差分法和有限元法求解了所建立的数学模型,并给出了有限元法的程序设计图。最后分别对有限差分法和有限元法的时间复杂度进行了分析,同时对两种数值解法作了详细的评述。
Brine is an important natural resource. With the rapid development of industrial and agricultural production, brine is explored more and more. So there are bound to make accurate forecasts , proper design and utilization issues . This paper is based on the Bieletan section, Qarhan Salt Lake in Qinghai Province . According to Hydrogeology and water chemistry of the Bieletan section, Qarhan Salt Lake in Qinghai Province, Brine dynamic geological conceptual model is established. And on this basis, a brine dynamic mathematical is established.The actual hydrogeological conditions are often complicated. So it is difficult to use analytic method to solve the problem. But numerical method opens up new avenues to study these problems . It is based on percolation theory. According the water balance or mass balance of the entire computational domain, the calculation to reflect the situation of the regional groundwater model is established. There are several ways to solve the prolem of brine dynamic. But the most common methods are finite difference method (FDM) and finite element method (FEM, also known as finite element method, finite element method). The finite difference method and the finite element method are used for solving the mathematical model in this paper. Then the program design of the finite element method is gived.Finally, finite difference method and finite element method of time complexity is analyzed respectively. At the same time ,the two numerical methods are commented in detail.
Brine is an important natural resource. With the rapid development of industrial and agricultural production, brine is explored more and more. So there are bound to make accurate forecasts , proper design and utilization issues . This paper is based on the Bieletan section, Qarhan Salt Lake in Qinghai Province . According to Hydrogeology and water chemistry of the Bieletan section, Qarhan Salt Lake in Qinghai Province, Brine dynamic geological conceptual model is established. And on this basis, a brine dynamic mathematical is established.The actual hydrogeological conditions are often complicated. So it is difficult to use analytic method to solve the problem. But numerical method opens up new avenues to study these problems . It is based on percolation theory. According the water balance or mass balance of the entire computational domain, the calculation to reflect the situation of the regional groundwater model is established. There are several ways to solve the prolem of brine dynamic. But the most common methods are finite difference method (FDM) and finite element method (FEM, also known as finite element method, finite element method). The finite difference method and the finite element method are used for solving the mathematical model in this paper. Then the program design of the finite element method is gived.Finally, finite difference method and finite element method of time complexity is analyzed respectively. At the same time ,the two numerical methods are commented in detail.
引文
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[2]祁长青,吴青柏,施斌,等.青藏铁路冻土路基温度场随机有限元分析[J].工程地质学报,2005,13(3):330-335.
[3]刘宁,刘光延.大体积混凝土结构温度场的随机有限元算法[J] .清华大学学报(自然科学版),1996,36(1):41-47。
[4]复旦大学数学系主编,1979,数学物理方程,高等教育出版社。
[5]边际,1984,边界单元法在地下水非稳定流计算中的初步应用,水利学报,第4期。
[6]于升松等.察尔汗盐湖首采区钾卤水动态及其预测,科学出版社,2001年1月
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[8]杨永征等.察尔汗盐湖钾矿资源开采中的环境问题分析与对策[J] .青海地质,2001,10V.15N.Dec.1999
[9]倪志耀.矿物溶解-结晶的反应动力学.地质地球化学.Vol.26,No.2,1998
[10]岑况.地质流体-岩石反应的地球化学动力学方法.现代地质,Vol.12,No.3,sep,1998
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[13]李庆和等编,1981,数值分析,华中工学院出版社.
[14]陈崇希,1983,地下水不稳定井流计算方法,地质出版社.
[15]陈崇希、陈明佑、陈爱光、区永和,1985,矿坑涌水量计算方法研究,武汉地质学院出版社.
[16]张蔚榛,1983,地下水非稳定流计算和地下水资源评价,科学出版社.
[17]陈金甫、关治,1987,偏微分方程数值解法,清华大学出版社.
[18]杨天行、傅泽周、刘金山、林学钰,1980,地下水流向井的非稳定运动的原理及计算方法,地质出版社.
[19]薛禹群、谢春红,1980,水文地质学的数值方法,煤炭工业出版社.
[20] G.E.福雪斯,W.R.华沙著,胡祖炽、吴文达、陈永和译,1979,偏微分方程的有限差分方法,上海科学技术出版社。
[21]王大纯,张人权等。水文地质学基础。北京:地质出版社,1998
[22]薛禹群,朱学愚等.地下水动力学。北京:地质出版社,2000
[23]首都地区地卜水资源和环境调查评价项目组。首都地区地下水资源和环境调查评价报告,2003
[24]曲兴辉.平原区地h水系统模拟模型研究.东北水利水电2005,22(5): 18-20
[25]石教英,蔡文立.科学计算可视化算法与系统[M] .北京:科学出版社,1996.1-18
[26]唐泽圣,孙延奎,邓俊辉.科学计算可视化理论与应用研究进展.清华大学学报[J] ,2001,41(45):200一202
[27]戴会超,田斌.科学计算可视化仿真及其在水利行业中的应用[J] .水力发电学报,2005,24(6):88一90
[28]韩程辉,刘文生.地下水模拟系统(GMS)与矿井防治水[J] .矿业安全与环保,2005,1(32):25一26
[29]贺国平,张彤,赵月芬,周东.GMS数值建模方法研究综述[J] .地下水,2007,29(3):32-35梁煦枫,王文科,曾永刚.GMS在水文地质结构可视化方面的应用[J] .东北水利水电,2006,9(24):64-66
[30] J.Bear著,李竞生、陈崇希译,孙纳正校,1983,多孔介质流体动力学,中国建筑工业出版社.
[31] B.Carnahan,H.A.Luther and J.O.Wilkes,1969,Applied Numerical Methods,John Wiley﹠ Sons,Inc.
[32] D.R.Croft and D.G.Lilley,1977,Heat Transfer Calcu lation Using Finite Difference Equations,Applied Science Publishers LTD. London.
[33] D.W.Peaceman,1977,Fundamantals of Numerical Reservoir Simulation,Elsevier Scientific Publishing Company.
[34] H.F.Wangand M.P.Anderson,1982,Introduction to Groundwater Modeling,W.H.Freman and Comapany.
[35] J.A.Liggett and P.L.Liu, 1979,Unsteady flow in confined aquifers:A comparison of two boundary integral methods,Water Resources Research,Vol.15,NO.4.
[36] P.M.Morse and H.Feshbach,1953,Methods of Theoretical Physics,Part.I.
[37] T.N.Narasimhan,1982,Recent Trends in Hydrogeology,The Geological Society of America,Inc.
[38] M.Klenke H. Thiem. Acoupled flow and transport model for simulation of groundwater reservoirs. Mathematics and Computer Simulation,2002(1):45-49
[39] J.A.P.H. Vermulst,W.J.De Lange. An analytic-based approach for coupling models for unsaturated and saturated groundwater flow at different scales. Journal of Hydrology,2000(7): 35-37
[40] Richard C. Peralta. Optimal perennial yield planning for complex nonlinear aquifers: Methods and examples. Advances in Water Resources1999(11): 55-60
[41] Charles S. Sawver,Madhavi Kamakoti. Optimal flow rates and well locations for soil vapor extraction design. Journal of Contaminant Hvdrology,1998(11):21-24
[42] Mark S. Nemeth,Helena M. Solo-Gabriele. Evaluation of the use of reach transmissivitv to quanta exchange between groundwater and surface water. Journal of Hydrology, 2003(2):18-22
[43] John Marler Shemin Ge. The Permeability of the Elkhorn Fault Zone, South Park,Colorado. Ground Water, 2003,41(3): 223-233
[44] Wu Qiang,Yin Zhongmin,Wu Xun. Groundwater Study in the Weerselo Area,Overijssel,the Netherlands. Beijing:Petroleum Industry Press, 2001,47-74
[45] Moench A F. Importance of the Vadose Zone in Analyses of Unconfined Aquifer Tests. GroundWater, 2004, 42(2): 223-233
[46] Lee D T. and Schachter B J. Two Algorithms for Constructing a Delaunay Triangulation,Int. Computer and Information Sciences,1980,9(3)
[47] Watson D F. Computing the n2dimension Delaunay Tesselation with Application to Voronoi Polytopes. Computer Journal,1981(24):167一172
[48] McCaullaghM T. and RossC G. Delaunay Triangulation of a Random Data Set For Iirarithmic Mapping. The Cartographic Journa1,1980(17):93一99
[49]FrancoisRisacher,HugoAlonso.TheoriginofbrineandsaltsinChileeansalars:ahydrochemical review.Earth-Science Review .2003,63:249一293.
[50] M .Grobe,H.G .Machel.Origin and evolution of saline groundwater in the Munsterland Cretaceous Basin, Germany:oxygen,hydrogen,and strontium isotope evidence.Journal of Geochemical Exploration.2000,69-70:5一9.
[51] H.T.Iampen,B.J.Rostron. Hydrogeochemistry of pre-Mississippian brines, Williston Basin, Canada-USA. Journal of Geochemical Exploration.2000, 69-70:29一35.
[52] David Banks,Howard Markland,Pau1 V.Smith,Carlos Mendez., Distribution, salinity and pH dependence of elements in surface waters of the catchment areas of the Salars of Coipasa and Uyuni,Bolivian Altiplano, Journal of Geochemical Exploration.2004,84: 141一166.
[53] V.Carmona,J.J.Pueyo.Solute inputs in the Salar de Atacama.Journal of Geochemical Exploation.2000,69-70 :449-452.
[54] Francois Risacher,Hugo Alonso,Carlos Salazar. Hydrochemistry of two adjacent acid saline lakes in the Andes of northern Chile.Chemical Geology.2002,187:39一57.
[55] Vesna Zic,Marko Branica.The distribution of iodate and iodide in Rogoznica Lake(Easr Adriatic Coast).Estuarine,Coastal and Shelf Science.2006,66:55一66.
[56] R. J. Spencer, H. P. Eugster and B. F. Jones.Geochemistry of Great Salt Lake, Utah I Hydrochemistry since 1850. Geochimica et Cosmochimica Acta,1985,49(3):727-737.
[2]祁长青,吴青柏,施斌,等.青藏铁路冻土路基温度场随机有限元分析[J].工程地质学报,2005,13(3):330-335.
[3]刘宁,刘光延.大体积混凝土结构温度场的随机有限元算法[J] .清华大学学报(自然科学版),1996,36(1):41-47。
[4]复旦大学数学系主编,1979,数学物理方程,高等教育出版社。
[5]边际,1984,边界单元法在地下水非稳定流计算中的初步应用,水利学报,第4期。
[6]于升松等.察尔汗盐湖首采区钾卤水动态及其预测,科学出版社,2001年1月
[7]唐渊等.中国盐湖自然资源及其开发利用[M] .科学出版社,1999.
[8]杨永征等.察尔汗盐湖钾矿资源开采中的环境问题分析与对策[J] .青海地质,2001,10V.15N.Dec.1999
[9]倪志耀.矿物溶解-结晶的反应动力学.地质地球化学.Vol.26,No.2,1998
[10]岑况.地质流体-岩石反应的地球化学动力学方法.现代地质,Vol.12,No.3,sep,1998
[11]孙纳正,1981,地下水流的数学模型和数值方法,地质出版社.
[12]朱学愚、钱孝星、刘新仁,1987,地下水资源评价,南京大学出版社.
[13]李庆和等编,1981,数值分析,华中工学院出版社.
[14]陈崇希,1983,地下水不稳定井流计算方法,地质出版社.
[15]陈崇希、陈明佑、陈爱光、区永和,1985,矿坑涌水量计算方法研究,武汉地质学院出版社.
[16]张蔚榛,1983,地下水非稳定流计算和地下水资源评价,科学出版社.
[17]陈金甫、关治,1987,偏微分方程数值解法,清华大学出版社.
[18]杨天行、傅泽周、刘金山、林学钰,1980,地下水流向井的非稳定运动的原理及计算方法,地质出版社.
[19]薛禹群、谢春红,1980,水文地质学的数值方法,煤炭工业出版社.
[20] G.E.福雪斯,W.R.华沙著,胡祖炽、吴文达、陈永和译,1979,偏微分方程的有限差分方法,上海科学技术出版社。
[21]王大纯,张人权等。水文地质学基础。北京:地质出版社,1998
[22]薛禹群,朱学愚等.地下水动力学。北京:地质出版社,2000
[23]首都地区地卜水资源和环境调查评价项目组。首都地区地下水资源和环境调查评价报告,2003
[24]曲兴辉.平原区地h水系统模拟模型研究.东北水利水电2005,22(5): 18-20
[25]石教英,蔡文立.科学计算可视化算法与系统[M] .北京:科学出版社,1996.1-18
[26]唐泽圣,孙延奎,邓俊辉.科学计算可视化理论与应用研究进展.清华大学学报[J] ,2001,41(45):200一202
[27]戴会超,田斌.科学计算可视化仿真及其在水利行业中的应用[J] .水力发电学报,2005,24(6):88一90
[28]韩程辉,刘文生.地下水模拟系统(GMS)与矿井防治水[J] .矿业安全与环保,2005,1(32):25一26
[29]贺国平,张彤,赵月芬,周东.GMS数值建模方法研究综述[J] .地下水,2007,29(3):32-35梁煦枫,王文科,曾永刚.GMS在水文地质结构可视化方面的应用[J] .东北水利水电,2006,9(24):64-66
[30] J.Bear著,李竞生、陈崇希译,孙纳正校,1983,多孔介质流体动力学,中国建筑工业出版社.
[31] B.Carnahan,H.A.Luther and J.O.Wilkes,1969,Applied Numerical Methods,John Wiley﹠ Sons,Inc.
[32] D.R.Croft and D.G.Lilley,1977,Heat Transfer Calcu lation Using Finite Difference Equations,Applied Science Publishers LTD. London.
[33] D.W.Peaceman,1977,Fundamantals of Numerical Reservoir Simulation,Elsevier Scientific Publishing Company.
[34] H.F.Wangand M.P.Anderson,1982,Introduction to Groundwater Modeling,W.H.Freman and Comapany.
[35] J.A.Liggett and P.L.Liu, 1979,Unsteady flow in confined aquifers:A comparison of two boundary integral methods,Water Resources Research,Vol.15,NO.4.
[36] P.M.Morse and H.Feshbach,1953,Methods of Theoretical Physics,Part.I.
[37] T.N.Narasimhan,1982,Recent Trends in Hydrogeology,The Geological Society of America,Inc.
[38] M.Klenke H. Thiem. Acoupled flow and transport model for simulation of groundwater reservoirs. Mathematics and Computer Simulation,2002(1):45-49
[39] J.A.P.H. Vermulst,W.J.De Lange. An analytic-based approach for coupling models for unsaturated and saturated groundwater flow at different scales. Journal of Hydrology,2000(7): 35-37
[40] Richard C. Peralta. Optimal perennial yield planning for complex nonlinear aquifers: Methods and examples. Advances in Water Resources1999(11): 55-60
[41] Charles S. Sawver,Madhavi Kamakoti. Optimal flow rates and well locations for soil vapor extraction design. Journal of Contaminant Hvdrology,1998(11):21-24
[42] Mark S. Nemeth,Helena M. Solo-Gabriele. Evaluation of the use of reach transmissivitv to quanta exchange between groundwater and surface water. Journal of Hydrology, 2003(2):18-22
[43] John Marler Shemin Ge. The Permeability of the Elkhorn Fault Zone, South Park,Colorado. Ground Water, 2003,41(3): 223-233
[44] Wu Qiang,Yin Zhongmin,Wu Xun. Groundwater Study in the Weerselo Area,Overijssel,the Netherlands. Beijing:Petroleum Industry Press, 2001,47-74
[45] Moench A F. Importance of the Vadose Zone in Analyses of Unconfined Aquifer Tests. GroundWater, 2004, 42(2): 223-233
[46] Lee D T. and Schachter B J. Two Algorithms for Constructing a Delaunay Triangulation,Int. Computer and Information Sciences,1980,9(3)
[47] Watson D F. Computing the n2dimension Delaunay Tesselation with Application to Voronoi Polytopes. Computer Journal,1981(24):167一172
[48] McCaullaghM T. and RossC G. Delaunay Triangulation of a Random Data Set For Iirarithmic Mapping. The Cartographic Journa1,1980(17):93一99
[49]FrancoisRisacher,HugoAlonso.TheoriginofbrineandsaltsinChileeansalars:ahydrochemical review.Earth-Science Review .2003,63:249一293.
[50] M .Grobe,H.G .Machel.Origin and evolution of saline groundwater in the Munsterland Cretaceous Basin, Germany:oxygen,hydrogen,and strontium isotope evidence.Journal of Geochemical Exploration.2000,69-70:5一9.
[51] H.T.Iampen,B.J.Rostron. Hydrogeochemistry of pre-Mississippian brines, Williston Basin, Canada-USA. Journal of Geochemical Exploration.2000, 69-70:29一35.
[52] David Banks,Howard Markland,Pau1 V.Smith,Carlos Mendez., Distribution, salinity and pH dependence of elements in surface waters of the catchment areas of the Salars of Coipasa and Uyuni,Bolivian Altiplano, Journal of Geochemical Exploration.2004,84: 141一166.
[53] V.Carmona,J.J.Pueyo.Solute inputs in the Salar de Atacama.Journal of Geochemical Exploation.2000,69-70 :449-452.
[54] Francois Risacher,Hugo Alonso,Carlos Salazar. Hydrochemistry of two adjacent acid saline lakes in the Andes of northern Chile.Chemical Geology.2002,187:39一57.
[55] Vesna Zic,Marko Branica.The distribution of iodate and iodide in Rogoznica Lake(Easr Adriatic Coast).Estuarine,Coastal and Shelf Science.2006,66:55一66.
[56] R. J. Spencer, H. P. Eugster and B. F. Jones.Geochemistry of Great Salt Lake, Utah I Hydrochemistry since 1850. Geochimica et Cosmochimica Acta,1985,49(3):727-737.