The Theory and Application of Upwind Finite Difference Fractional Steps Procedure for Seawater Intrusion

ABSTRACT

Numerical simulation and theoretical analysis of seawater intrusion is the mathematical basis for modern environmental science. Its mathematical model is the nonlinear coupled system of partial differential equations with initial-boundary problems. For a generic case of a three-dimensional bounded region, two kinds of finite difference fractional steps pro- cedures are put forward. Optimal order estimates in norm are derived for the error in the approximation solution. The present method has been successfully used in predicting the consequences of seawater intrusion and protection projects.

Numerical simulation and theoretical analysis of seawater intrusion is the mathematical basis for modern environmental science. Its mathematical model is the nonlinear coupled system of partial differential equations with initial-boundary problems. For a generic case of a three-dimensional bounded region, two kinds of finite difference fractional steps pro- cedures are put forward. Optimal order estimates in norm are derived for the error in the approximation solution. The present method has been successfully used in predicting the consequences of seawater intrusion and protection projects.

Cite this paper

Y. Yuan, H. Rui, D. Liang and C. Li, "The Theory and Application of Upwind Finite Difference Fractional Steps Procedure for Seawater Intrusion,"*International Journal of Geosciences*, Vol. 3 No. 5, 2012, pp. 972-991. doi: 10.4236/ijg.2012.325098.

Y. Yuan, H. Rui, D. Liang and C. Li, "The Theory and Application of Upwind Finite Difference Fractional Steps Procedure for Seawater Intrusion,"

References

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[2] Y. Yuan, D. Liang and H. Rui, “The Mathematics Model of the Consequence of Predictive Seawater Intrusion and Protection Projects,” In: E. Jiang, Ed., Proceeding of the 2nd Higher Mathematics Research of Shandong Province, Qingdao Ocean University Press, Qingdao, 1994, pp. 1-5.

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[4] G. Segol, G. F. Pinder and W.G. Gray, “A Galerkin Finite Element Technique for Calculating the Transient Position of the Saltwater Front,” Water Resources Research, Vol. 11, No. 2, 1975, pp. 343-347. doi:10.1029/WR011i002p00343

[5] P. S. Huyakorn, P. S. Anderson and J. W. Mercer, “Saltwater Intrusion in Aquifers: Development and Testing of a Three Dimensional Finite Element Model,” Water Resources Research, Vol. 23, No. 2, 1987, pp. 293-312.doi:10.1029/WR023i002p00293

[6] A. D. Gupta and D. D. Yapa, “Saltwater Encroachment in an Aquifer: A Case Study,” Water Resources Research, Vol. 18, No. 3, 1982, pp. 546-556.doi:10.1029/WR018i003p00546

[7] J. Douglas Jr. and T. F. Russell, “Numerical Methods for Convection-Dominated Diffusion Problems Based on Combining the Method of Characteristics with Finite Element or Finite Difference Procedures,” SIAM Journal of Numerical Analysis, Vol. 19, No. 5, 1982, pp. 781-895.

[8] J. Douglas Jr., “Simulation of Miscible Displacement in Porous Media by a Modified Method of Characteristic Procedure,” Lecture Notes in Mathematics 912, Numerical Analysis, Dundee, 1981.

[9] J. Douglas Jr., “Finite Difference Methods for Two-Phase Incompressible Flow in Porous Media,” SIAM Journal of Numerical Analysis, Vol. 20, No. 4, 1983, pp. 681-696.doi:10.1137/0720046

[10] R. E. Ewing, T. F. Russell and M. F. Wheeler, “Convergence Analysis of an Approximation of Miscible Displacement in Porous Media by Mixed Finite Elements and a Modified Method of Characteristics,” Computer Methods in Applied Mechanics and Engineering, Vol. 47, No. 1-2, 1984, pp. 73-92. doi:10.1016/0045-7825(84)90048-3

[11] A. Bermudez, M. R. Nogueriras and C. Vazuez, “Numerical Analysis of Convection-Diffusion-Reaction Problems with Higher Order Characteristics/Finite Elements. Part I: Time Discretization,” SIAM Journal of Numerical Analysis, Vol. 44, No. 5, 2006, pp. 1829-1853.doi:10.1137/040612014

[12] A. Bermudez, M. R. Nogueriras and C. Vazuez, “Numerical Analysis of Convection-Diffusion-Reaction Problems with Higher Order Characteristics/Finite Elements. Part II: Fully Discretized Scheme and Quadrature Formulas,” SIAM Journal of Numerical Analysis, Vol. 44, No. 5, 2006, pp. 1854-1876. doi:10.1137/040615109

[13] J. Douglas Jr. and J. E. Roberts, “Numerical Method for a Model for Compressible Miscible Displacement in Porous Media,” Mathematics of Computation, Vol. 4, No. 164, 1983, pp. 441-459.

[14] Y. Yuan, “Time Stepping along Characteristics for the Finite Element Approximation of Compressible Miscible Displacement in Porous Media,” Mathematica Numerica Sinica, Vol. 14, No. 4, 1992, pp. 385-400.

[15] Y. Yuan, “Finite Difference Methods for a Compressible Miscible Displacement Problem in Porous Media,” Mathematica Numerica Sinica, Vol. 15, No. 1, 1993, pp. 16-28.

[16] R. E. Ewing, “The Mathematics of Reservoir Simulation,” Society for Industrial and Applied Mathematics, Philadelphia, 1983. doi:10.1137/1.9781611971071

[17] J. Douglas Jr. and Y. Yuan, “Numerical Simulation of Immiscible Flow in Porous Media Based on Combining the Method of Characteristics with Mixed Finite Element Procedure,” In: M. F. Wheeler, Ed., Numerical Simulation in Oil Recovery, Spring-Verlag, Minnesota, 1986, pp. 119-131.

[18] Y. Yuan, “Characteristic Finite Difference Methods for Moving Boundary Value Problem of Numerical Simulation of Oil Deposit,” Science in China, Series A, Vol. 37, No. 12, 1994, pp. 1412-1453.

[19] Y. Yuan, “The Characteristic Mixed Finite Element Method and Analysis for Three-Dimensional Moving Boundary Value Problem,” Science in China, Series A, Vol. 39, No. 3, 1996, pp. 276-288.

[20] Y. Yuan, “Finite Difference Method and Analysis for Three-Dimensional Semiconductor Device of Heat Conduction,” Science in China, Series A, Vol. 39, No. 11, 1999, pp. 1140-1151.

[21] O. Axelsson and I. A. Gustafasson, “Modified Upwind Scheme for Convective Transport Equations and the Use of a Conjugate Gradient Method for the Solution of Non-Symmetric Systems of Equations,” IMA Journal of Applied Mathematics, Vol. 23, No. 3, 1979, pp. 321-337.

[22] R. E. Ewing, R. D Lazarov and A. T. Vassilevski, “Finite Difference Scheme for Parabolic Problems on Composite Grids with Refinement in Time and Space,” SIAM Journal on Numerical Analysis, Vol. 31, No. 6, 1994, pp. 1605-1622. doi:10.1137/0731083

[23] R. D. Lazarov, I. D. Mishev and P. S. Vassilevski, “Finite Volume Method for Convection-Diffusion Problems,” SIAM Journal on Numerical Analysis, Vol. 33, No. 1, 1996, pp. 31-55. doi:10.1137/0733003

[24] D. W. Peaceman, “Fundamentals of Numerical Reservoir Simulation,” Elsevier, Amsterdam, 1980.

[25] G. I. Marchuk, “Splitting and Alternating Direction Me- thod,” In: P. G. Ciarlet and J. L. Lions, Eds., Handbook of Numerical Analysis, Elsevior Science Publishers BV, Paris, 1990, pp. 197-460. doi:10.1016/S1570-8659(05)80035-3

[26] J. Douglas Jr. and J. E. Gunn, “Two Order Correct Difference Analogues for the Equation of Multidimensional Heat Flow,” Mathematics of Computation, Vol. 17, No. 81, 1963, pp. 71-80. doi:10.1090/S0025-5718-1963-0149676-2

[27] J. Douglas Jr. and J. E. Gunn, “A General Formulation of Alternation Methods. Part 1. Parabolic and Hyperbolic Problems,” Numerische Mathematik, Vol. 9, No. 5, 1964, pp. 428-453.

[28] R. E. Ewing, “Mathematical Modeling and Simulation for Multiphase Flow in Porous Media. In Numerical Treatment of Multiphase Flow in Porous Media,” Lecture Notes in Physics, Vol. 552, 2000, pp. 43-57.

[29] A. A. Samarski and B. B. Andreev, “Finite Difference Methods for Elliptic Equation,” Science Press, Beijing, 1984.

[30] Y. Xue and C. Xie, “Study on Joint-Surface of Saltwater and Freshwater in Seawater Intrusion Problem,” Nanjing University Press, Nanjing, 1991, pp. 43-94.

[1] J. Bear, “Dynamics of Fluids in Porous Media,” American Elsevier Publishing Company, Inc., Amsterdam, 1972.

[2] Y. Yuan, D. Liang and H. Rui, “The Mathematics Model of the Consequence of Predictive Seawater Intrusion and Protection Projects,” In: E. Jiang, Ed., Proceeding of the 2nd Higher Mathematics Research of Shandong Province, Qingdao Ocean University Press, Qingdao, 1994, pp. 1-5.

[3] H. R. Henry, “Effects of Dispersion on Salt Water Encroachment in Coastal Aquifers,” US Geological Surrey, Water Supply Paper, 1964, pp. 1613-1624.

[4] G. Segol, G. F. Pinder and W.G. Gray, “A Galerkin Finite Element Technique for Calculating the Transient Position of the Saltwater Front,” Water Resources Research, Vol. 11, No. 2, 1975, pp. 343-347. doi:10.1029/WR011i002p00343

[5] P. S. Huyakorn, P. S. Anderson and J. W. Mercer, “Saltwater Intrusion in Aquifers: Development and Testing of a Three Dimensional Finite Element Model,” Water Resources Research, Vol. 23, No. 2, 1987, pp. 293-312.doi:10.1029/WR023i002p00293

[6] A. D. Gupta and D. D. Yapa, “Saltwater Encroachment in an Aquifer: A Case Study,” Water Resources Research, Vol. 18, No. 3, 1982, pp. 546-556.doi:10.1029/WR018i003p00546

[7] J. Douglas Jr. and T. F. Russell, “Numerical Methods for Convection-Dominated Diffusion Problems Based on Combining the Method of Characteristics with Finite Element or Finite Difference Procedures,” SIAM Journal of Numerical Analysis, Vol. 19, No. 5, 1982, pp. 781-895.

[8] J. Douglas Jr., “Simulation of Miscible Displacement in Porous Media by a Modified Method of Characteristic Procedure,” Lecture Notes in Mathematics 912, Numerical Analysis, Dundee, 1981.

[9] J. Douglas Jr., “Finite Difference Methods for Two-Phase Incompressible Flow in Porous Media,” SIAM Journal of Numerical Analysis, Vol. 20, No. 4, 1983, pp. 681-696.doi:10.1137/0720046

[10] R. E. Ewing, T. F. Russell and M. F. Wheeler, “Convergence Analysis of an Approximation of Miscible Displacement in Porous Media by Mixed Finite Elements and a Modified Method of Characteristics,” Computer Methods in Applied Mechanics and Engineering, Vol. 47, No. 1-2, 1984, pp. 73-92. doi:10.1016/0045-7825(84)90048-3

[11] A. Bermudez, M. R. Nogueriras and C. Vazuez, “Numerical Analysis of Convection-Diffusion-Reaction Problems with Higher Order Characteristics/Finite Elements. Part I: Time Discretization,” SIAM Journal of Numerical Analysis, Vol. 44, No. 5, 2006, pp. 1829-1853.doi:10.1137/040612014

[12] A. Bermudez, M. R. Nogueriras and C. Vazuez, “Numerical Analysis of Convection-Diffusion-Reaction Problems with Higher Order Characteristics/Finite Elements. Part II: Fully Discretized Scheme and Quadrature Formulas,” SIAM Journal of Numerical Analysis, Vol. 44, No. 5, 2006, pp. 1854-1876. doi:10.1137/040615109

[13] J. Douglas Jr. and J. E. Roberts, “Numerical Method for a Model for Compressible Miscible Displacement in Porous Media,” Mathematics of Computation, Vol. 4, No. 164, 1983, pp. 441-459.

[14] Y. Yuan, “Time Stepping along Characteristics for the Finite Element Approximation of Compressible Miscible Displacement in Porous Media,” Mathematica Numerica Sinica, Vol. 14, No. 4, 1992, pp. 385-400.

[15] Y. Yuan, “Finite Difference Methods for a Compressible Miscible Displacement Problem in Porous Media,” Mathematica Numerica Sinica, Vol. 15, No. 1, 1993, pp. 16-28.

[16] R. E. Ewing, “The Mathematics of Reservoir Simulation,” Society for Industrial and Applied Mathematics, Philadelphia, 1983. doi:10.1137/1.9781611971071

[17] J. Douglas Jr. and Y. Yuan, “Numerical Simulation of Immiscible Flow in Porous Media Based on Combining the Method of Characteristics with Mixed Finite Element Procedure,” In: M. F. Wheeler, Ed., Numerical Simulation in Oil Recovery, Spring-Verlag, Minnesota, 1986, pp. 119-131.

[18] Y. Yuan, “Characteristic Finite Difference Methods for Moving Boundary Value Problem of Numerical Simulation of Oil Deposit,” Science in China, Series A, Vol. 37, No. 12, 1994, pp. 1412-1453.

[19] Y. Yuan, “The Characteristic Mixed Finite Element Method and Analysis for Three-Dimensional Moving Boundary Value Problem,” Science in China, Series A, Vol. 39, No. 3, 1996, pp. 276-288.

[20] Y. Yuan, “Finite Difference Method and Analysis for Three-Dimensional Semiconductor Device of Heat Conduction,” Science in China, Series A, Vol. 39, No. 11, 1999, pp. 1140-1151.

[21] O. Axelsson and I. A. Gustafasson, “Modified Upwind Scheme for Convective Transport Equations and the Use of a Conjugate Gradient Method for the Solution of Non-Symmetric Systems of Equations,” IMA Journal of Applied Mathematics, Vol. 23, No. 3, 1979, pp. 321-337.

[22] R. E. Ewing, R. D Lazarov and A. T. Vassilevski, “Finite Difference Scheme for Parabolic Problems on Composite Grids with Refinement in Time and Space,” SIAM Journal on Numerical Analysis, Vol. 31, No. 6, 1994, pp. 1605-1622. doi:10.1137/0731083

[23] R. D. Lazarov, I. D. Mishev and P. S. Vassilevski, “Finite Volume Method for Convection-Diffusion Problems,” SIAM Journal on Numerical Analysis, Vol. 33, No. 1, 1996, pp. 31-55. doi:10.1137/0733003

[24] D. W. Peaceman, “Fundamentals of Numerical Reservoir Simulation,” Elsevier, Amsterdam, 1980.

[25] G. I. Marchuk, “Splitting and Alternating Direction Me- thod,” In: P. G. Ciarlet and J. L. Lions, Eds., Handbook of Numerical Analysis, Elsevior Science Publishers BV, Paris, 1990, pp. 197-460. doi:10.1016/S1570-8659(05)80035-3

[26] J. Douglas Jr. and J. E. Gunn, “Two Order Correct Difference Analogues for the Equation of Multidimensional Heat Flow,” Mathematics of Computation, Vol. 17, No. 81, 1963, pp. 71-80. doi:10.1090/S0025-5718-1963-0149676-2

[27] J. Douglas Jr. and J. E. Gunn, “A General Formulation of Alternation Methods. Part 1. Parabolic and Hyperbolic Problems,” Numerische Mathematik, Vol. 9, No. 5, 1964, pp. 428-453.

[28] R. E. Ewing, “Mathematical Modeling and Simulation for Multiphase Flow in Porous Media. In Numerical Treatment of Multiphase Flow in Porous Media,” Lecture Notes in Physics, Vol. 552, 2000, pp. 43-57.

[29] A. A. Samarski and B. B. Andreev, “Finite Difference Methods for Elliptic Equation,” Science Press, Beijing, 1984.

[30] Y. Xue and C. Xie, “Study on Joint-Surface of Saltwater and Freshwater in Seawater Intrusion Problem,” Nanjing University Press, Nanjing, 1991, pp. 43-94.