Abstract
Richards’ equation, the governing equation for single phase unsaturated flow in soil, is highly nonlinear and, consequently, is difficult to solve either analytically or numerically. The nonlinearity enters through the soil hydraulic functions, viz., the soil moisture characteristic curve and the unsaturated hydraulic conductivity. We present an exact quasi-analytical solution for the one-dimensional Richards’ equation subject to an arbitrary, time-dependent ponding depth. The solution depends on solving a first-order ordinary differential equation. For certain special cases, such as when the surface ponding depth is constant, fully analytical results can be derived. The theory is used in a number of applications. For example, an existing analytical approximation for cumulative infiltration subject to a ponded, time-dependent surface head is checked. As well, a drainage formula is derived. New results for solute transport coupled with a nonlinear adsorption isotherm are derived by making use of an exact mapping between Richards’ equation and the governing solute transport equation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bajracharya, K., and D. A. Barry. ‘Mixing cell models for nonlinear equilibrium single species adsorption and transport’. J. Contain. Hydrol. 12:227–243.
Barrow, N. J. 1989. ‘Suitability of sorption-desorption models to simulate partitioning and movement of ions in soils’. In B. Bar-Yosef, N. J. Barrow, and J. Goldsmith (eds.)Inorganic Contaminants in the Vadose Zone.Springer-Verlag, Berlin.
Barry, D. A. 1992. ‘Modeling contaminant transport in the subsurface: Theory and computer programs’. In H. Ghadiri, and C. W. Rose (eds.)Modeling Chemical Transport in Soils Natural and Applied ContaminantsLewis Publishers, Boca Raton, Florida.
Barry, D. A., S. J. Barry, and P. J. Hensley. 1993a. ‘Real values of the W function’. Submitted to ACM Trans. Math. Software.
Barry, D. A., J.-Y. Parlange, R. Haverkamp, and J. Tourna. 1992. ‘Comment on “Solution of a nonlinear absorption model of mixed saturated-unsaturated flow” by P. Broadbridge’.Water Resour. Res.28:1737–1738.
Barry, D. A., J.-Y. Parlange, G. C. Sander, and M. Sivapalan. 1993b. ‘A class of exact solutions for Richards’ equation’.J. Hydrol.142:29–46.
Barry, D. A., and G. C. Sander. 1991. ‘Exact solutions for water infiltration with an arbitrary surface flux or nonlinear solute adsorption’.Water Resour. Res.27:2667–2680.
Barry, D. A., G. C. Sander, and I. R. Phillips. 1991. ‘Modelling of solute transport, chemical adsorption and cation exchange’. Int. Hydrology and Water Resources Symp. 2–4 October, Perth, W.A. The Institution of Engineers, Australia, Nat. Conf. Publ. No. 92/19 Preprints of Papers, Vol. 3:924–929.
Broadbridge, P., and I. White. 1988. ‘Constant rate rainfall infiltration: A versatile nonlinear model, 1, Analytic solution’.Water Resour. Res.24:145–154.
Dagan, G. 1989.Flow and Transport in Porous Formations.Springer-Verlag, Berlin.
Dagan, G., and U. Kroszynski. 1973. ‘Drainage of a vertical column’. In A. Hadas, D. Swartzendruber, P. E. Rijtema, M. Fuchs, and B. Yaron (eds.)Physical Aspects of Soil Water and Salts in Ecosystems.Chapman and Hall Limited, London.
Davis, P. J., and I. Polonsky. 1964. ‘Numerical interpolation, differentiation and integration’. In M. Abramowitz and I. A. Stegun (eds.)Handbook of Mathematical Functions With Formulas Graphs and Mathematical Tables.National Bureau of Standards, Appl. Math. Ser. 55, Washington, D.C.
Fritsch, F. N., R. E. Shafer, and R. E. Crowley. 1974. ‘Solution of the transcendental equation we w = x ’ . Commun. Assoc. Comput. Mach.16:123–124.
Gardner, W. R. 1958. ‘Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table’.Soil Sci.85:228–232.
Green, W. H., and G. A. Ampt. 1911. ‘Studies in soil physics: 1. The flow of air and water through soils’. J. Agric. Sci. 4:1–24.
Onstad, C. A., T. C. Olson, and L. R. Stone. 1973. ‘An infiltration model tested with monolith moisture measurements’.Soil Sci.116:13–17.
Parlange, J.-Y. 1975. ‘On solving the flow equation in unsaturated soils by optimization: Horizontal infiltration’.Soil Sci. Soc. Am. Proc.39:415–418.
Parlange, J.-Y., D. A. Barry, M. B. Parlange, and R. Haverkamp. 1992. ‘Note on the sorptivity for mixed saturated-unsaturated flow’.Water Resour. Res.28:2529–2531.
Parlange, J.-Y., D. A. Barry, M. B. Parlange, D. A. Lockington, and R. Haverkamp. 1993. ‘Sorptivity calculation for arbitrary diffusivity’. Transp. Porous Media In Press.
Parlange, J.-Y., R. Haverkamp, and J. Tbuma. 1985. ‘Infiltration under ponded conditions: 1. Optimal analytical solution and comparison with experimental observations’. Soil Sci. 139:305–311.
Parlange, J.-Y., I. Lisle, R. D. Braddock, and R. E. Smith. 1982. ‘The three-parameter infiltration equation’. Soil Sci. 133:337–341.
Philip, J. R. 1992a. ‘Falling head ponded infiltration’.Water Resour. Res.28:2147–2148.
Philip, J. R. 1992b. ‘Falling head ponded infiltration with evaporation’.J. Hydrol.138:591–598.
Rogers, C., M. P. Stallybrass, and D. L. Clements. 1983. ‘On two phase filtration under gravity and with boundary infiltration: Application of a Bäcklund transformation’.Nonlin. Anal. Theory Meth. Appl.7:785–799.
Reichardt, K., D. R. Neilsen, and J. W. Biggar. 1972. ‘Scaling of horizontal infiltration into homogeneous soils’.Soil Sci. Soc. Am. J.36:241–245.
Richards, L. A. 1931. ‘Capillary conduction of liquids through mediums’.Physics1:318–333.
Sander, G. C., J.-Y. Parlange, V. Ktihnel, W. L. Hogarth, D. Lockington, and J. P. J. O’Kane. 1988. ‘Exact nonlinear solution for constant flux infiltration’.J. Hydrol.97:341–346.
Schweich, D., M. Sardin, and J. P. Gaudet. 1983. ‘Measurement of a cation exchange isotherm from elution curves obtained in a soil column: Preliminary results’.Soil Sci. Soc. Am. J.47:32–37.
Spanier, J., and K. B. Oldham. 1987. AnAtlas of Functions.Hemisphere Publishing Corp., New York. Sposito, G. 1989.The Chemistry of Soils.Oxford University Press, New York.
Vachaud, G. 1968.Contribution a L’Etude des Problemes D’Ecoulement en Milieux Poreux NonsatuTees.Ph.D. thesis, Grenoble, France.
Warrick, A. W., D. O. Lomen, and A. Islas. 1990. ‘An analytical solution to Richards’ equation for a draining soil profile’.Water Resour. Res.26:253–258.
Warrick, A. W., A. Islas, and D. O. Lomen. 1991. ‘An analytical solution to Richards’ equation for time-varying infiltration’.Water Resour. Res.27:763–766.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Barry, D.A., Parlange, JY. (1996). Infiltration Subject to Time-Dependent Surface Ponding: Exact Results and Correspondence to Solute Transport with Nonlinear Reaction. In: Singh, V.P., Kumar, B. (eds) Subsurface-Water Hydrology. Water Science and Technology Library, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0391-6_3
Download citation
DOI: https://doi.org/10.1007/978-94-011-0391-6_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4175-1
Online ISBN: 978-94-011-0391-6
eBook Packages: Springer Book Archive