Abstract
The quasilinear form of Richards’ equation for one-dimensional unsaturated flow in soils can be readily solved for a wide variety of conditions. However, it cannot explain saturated/unsaturated flow and the constant diffusivity assumption, used to linearise the transient quasilinear equation, can introduce significant error. This paper presents a quasi-analytical solution to transient saturated/unsaturated flow based on the quasilinear equation, with saturated flow explained by a transformed Darcy's equation. The procedure presented is based on the modified finite analytic method. With this approach, the problem domain is divided into elements, with the element equations being solutions to a constant coefficient form of the governing partial differential equation. While the element equations are based on a constant diffusivity assumption, transient diffusivity behaviour is incorporated by time stepping. Profile heterogeneity can be incorporated into the procedure by allowing flow properties to vary from element to element. Two procedures are presented for the temporal solution; a Laplace transform procedure and a finite difference scheme. An advantage of the Laplace transform procedure is the ability to incorporate transient boundary condition behaviour directly into the analytical solutions. The scheme is shown to work well for two different flow problems, for three soil types. The technique presented can yield results of high accuracy if the spatial discretisation is sufficient, or alternatively can produce approximate solutions with low computational overheads by using large sized elements. Error was shown to be stable, linearly related to element size.
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References
Chen, C. S.: 1985, Analytical and approximate solutions to radial dispersion from an injection well to a geological unit with simultaneous diffusion into adjacent strata, Water Resour. Res. 21(8), 1069-1076.
Chen, C. J. and Chen, H. C.: 1984, Finite analytic numerical method for unsteady two-dimensional Navier-Stokes equations, J. Comput. Phys. 53, 209-226.
Chen, C. J., Naeri-Neshat, H. and Ho, K. S.: 1981, Finite analytic numerical solution of heat transfer in two-dimensional cavity flow, J. Num. Heat Transfer 4, 179-197.
Connell, L. D., Bailey, M. and Jayatilaka, C. J.: 1998, A quasi-analytical model for groundwater movement in hillslopes, J. Hydrol. 204, 108-123.
Connell, L. D. and Bell, P. R. F.: 1993, Modelling moisture movement in revegetating waste heaps: 1. Development of a finite element model for liquid and vapour transport, Water Resour. Res. 29(5), 1435-1443.
Connell, L. D. and Haverkamp, R.: 1996, A quasi-analytical model for soil solute transport under plant water use, Soil Sci. Soc. Am. J. (to appear).
Fuentes, C., Haverkamp, R. and Parlange, J-Y.: 1992, Parameter constraints on closed-form soil water relationships, J. Hydrol. 134, 117-142.
van Genuchten, M. Th.: 1980, A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. A m. J. 44, 892-898.
Green, W. H. and Ampt, G. A.: 1911, Studies in soil physics: I. The flow of air and water through soils, J. Agric. Sci. 4, 1-24.
Hemker, C. J. and Maas, C.: 1987, Unsteady flow to wells in layered and fissured aquifer systems, J. Hydrol. 90(3/4), 231-249.
Li, S. G., Ruan, F. and McLaughlin, D.: 1992, A space-time accurate method for solving solute transport problems, Water Resour. Res. 28(9), 2297-2306.
de Marsily, G.: 1986, Quantitative Hydrogeology: Groundwater Hydrology for Engineers, Academic Press, San Diego.
Motz, L. H.: 1994, Drawdowns for leaky-aquifer flow with storage in finite width sink, J. Irrig. Drain. Engng. 120(4), 820-827.
Parlange, J.-Y., Haverkamp, R. and Touma, J.: 1985, Infiltration under ponded conditions: 1. Optimal analytical solution and comparison with experimental observations, Soil Sci. 139(4), 305-311.
Parlange, J.-Y., Haverkamp, R. and Fuentes, C.: 1988, A note on linearised infiltration, Soil Sci. 145(4), 310-312.
Philip, J. R.: 1957, The theory of infiltration: 1. The infiltration equation and its solution, Soil Sci. 83, 345-357.
Philip, J. R.: 1969, Theory of infiltration, Adv. Hydrosci. 5, 215-296.
Pullan, A. J.: 1990, The quasilinear approximation for unsaturated porous media flow,Water Resour. Res. 26(6), 1219-1234.
Richards, L. A.: 1931, Capillary conduction of liquids through porous mediums, Physics 1, 318-333.
Ross, P. J.: 1990, Efficient numerical methods for infiltration using Richards' equation,Water Resour. Res. 26(2), 279-290.
Stehfest, H.: 1970, Numerical inversion of Laplace transforms, Commun. ACM 13, 47-49.
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Connell, L.D. A Quasilinear Based Procedure for Saturated/Unsaturated Water Movement in Soils. Transport in Porous Media 36, 1–21 (1999). https://doi.org/10.1023/A:1006504816562
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DOI: https://doi.org/10.1023/A:1006504816562