Abstract
Groundwater flow with steep gradients in a vertical plane of infinite horizontal extension due to arbitrary non-symmetric strip sources and/or sinks can be described by the 2D Laplace equation. Notwithstanding the strongly nonlinear character of the free surface boundary condition, the exact analytical solution to this problem is developed in a closed form by employing neither the Dupuit assumption nor any other form of linearization. The first section of the development, still including the unsteady case, leads via conformal mapping and transformation procedures to a singular integro-differential-equation for the transient groundwater table. From this point onwards we restrict ourselves to the steady case for which the exact solution of the 2D Laplace equation for the pressure head and the location of the groundwater table was achieved. The solution is expressed exclusively in algebraic terms without the need for iterative procedures. It can not only be applied to real world phenomena, including a simple solution of the inverse problem, but also provide a new transparency regarding the solution characteristics and may serve as a standard for investigating numerical solutions and the domain of validity of simplified approaches. The computer program can be downloaded from www.tu-dresden.de/fghhihm/hydrologie.html
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References
Bear, J., Hydraulics of Groundwater, McGraw-Hill, 1979.
Childs, E. C.: 1959, A Treatment of the capillary fringe in drainage theory, J. Soil Sci. 10, 83-100.
Dagan, G.: 1966, The solution of the linearized equations of free-surface flow in porous media, J. Mec. 5(2), 207-215.
Edenhofer, J. Haucke, J. and Schmitz, G.: 1993, Comprehensive analytical modelling of a transient phreatic surface in arid regions, In: H. Lieth and A. Masoom (eds), Towards the Rational Use of High Salinity Tolerant Plants, vol. 1, pp. 511-518.
Engelund, F.: 1951, Mathematical discussion of drainage problems, Trans. Danish Acad. Tech. Sci., Nr. 3.
Guo Weixing: 1997, Transient groundwater flow between reservoirs and water-table aquifer, J. Hydrol. 195, 370-384.
Gustafsson, Y.: 1946, Untersuchungen über die Strömungsverhältnisse in gedräntem Boden, Acta Agr. Suecana 2, 1-157.
Hooghoudt, S. B.: 1940, Bijdragen tot de Kennis van enige Natuurkundige Grootheden van de Grond, 7, Algemeene Beschouwing van het Probleem van de Detail Ontwatering en de Infiltratie door middel van Parallel Loopende Drains, Greppels, Slooten en Kanalen, Versl. Landbouwk. Ond. 46, 515-707.
Kirkham, D.: 1966, Steady state theories for drainage, J. Irr. Drain. Div. ASCE 92(1), 19-39.
Manglik A., Rai S. N. and Singh R. N.: 1997, Response of an unconfined aquifer induced by time varying recharge from a rectangular basin, Water Res. Management 11, 185-196.
Muskat, M.: 1937, The Flow of Homogeneous Fluids through Porous Media, McGraw-Hill, New York.
Polubarinova-Kochina, P. Y.: 1962, Theory of Ground-Water Movement, Princeton University Press, Princeton, NJ.
Powell, N. L. and Kirkham, D.: 1976, Tile drainage in bedded soil or a draw, Soil Sci. Soc. Am. J. 40, 625-630.
Rai, S. N. and Singh, R. N.: 1995, Two-dimensional modelling of water table fluctuation in response to localised transient recharge, J. Hydrol. 167, 167-174.
Read, W. W. and Volker, R. E.: 1993, Series solutions for steady seepage through hillsides with arbitrary flow boundaries, Water Resour. Res. 29(8), 2811-2880.
Schmitz, G. and Edenhofer, J.: 1988, Semi analytical solutions for the groundwater mound problem, Adv. Water Resour. 11, 21-24.
Schmitz, G. and Edenhofer, J.: 2000, Exact closed-form solution of the two-dimensional Laplace equation for steady groundwater flow with nonlinearized free-surface boundary condition, Water Resour. Res. 36(7), 1975-1980.
Serrano S. E.: 1995, Analytical solutions of the nonlinear groundwater flow equation in unconfined aquifers and the effect of heterogeneity, Water Resour. Res. 31(11), 2733-2742.
Serrano S. E. and Workman, S. R.: 1998, Modelling transient stream/aquifer interaction with the nonlinear Boussinesq equation and its analytical solution, J. Hydrol. 206, 245-255.
Tóth, J.: 1963, A Theoretical analysis of groundwater flow in small drainage basins, J. Geophys. Res. 68, 4795-4812.
van Deemter, J. J.: 1950, Bijdragen tot de kennis van enige natuurkundige grootheden van de grond, 11, Theoretische en numerieke behandeling van ontwatering-en infiltratie-stromingsproblemen, Versl. Landbouwk. Ond. 56, Nr. 7.
van de Giesen, N. C., Parlange, J. Y and Steenhuis, T. S.: 1994, Transient flow to open drains: comparison of linearized solutions with and without the Dupuit assumption, Water Resour. Res. 30(11), 3033-3039.
van Schilfgaarde, J., Kirkham, D. and Frevert, R. K.: 1956, Physical and Mathematical Theories of Tile and Ditch Drainage and Their Usefulness in Design. Bulletin No. 436, Iowa Agric. Experiment Sta., Ames, Iowa, pp. 681-683.
Workman S. R., Serrano, S. E. and Liberty, K.: 1997, Development and application of an analytical model of stream/aquifer interaction, J. Hydrol. 200, 149-163.
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Edenhofer, J., Schmitz, G.H. Pressure Distribution in a Semi-infinite Horizontal Aquifer with Steep Gradients Due to Steady Recharge and/or Drainage: The Exact Explicit Solution. Transport in Porous Media 45, 345–364 (2001). https://doi.org/10.1023/A:1012061618468
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DOI: https://doi.org/10.1023/A:1012061618468