Abstract
We solve a versatile nonlinear convection-diffusion model for nonhysteretic redistribution of liquid in a finite vertical unsaturated porous column. With zero-flux boundary conditions, the nonlinear boundary-value problem may be transformed to a linear problem which is exactly solvable by the method of Laplace transforms. In principle, this technique applies to arbitrary initial conditions.
The analytic solution for drainage in an initially saturated semi-finite column is compared to previously available approximate analytic solutions, obtained by assuming constant diffusivity, as in the Burgers equation, or by neglecting diffusivity, as in the hyperbolic model. Contrary to popular opinion, the hyperbolic model has more than one shock-free solution in a semi-infinite medium z ⩾ 0, as opposed to an infinite medium z ∈ ℝ. However, both the Burgers equation and an improved hyperbolic model underestimate diffusivity at high liquid contents and consequently overestimate the curvature of the soil liquid content profile.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D.R. Nielsen, J.W. Biggar and K.T. Erh, Spatial variability of field-measured soil-water properties, Hilgardia 42 (1973) 215–259.
M. Mahmoodian-Shooshtari, L.A. Davis, D.M. McWhorter and A. Klute, Unsaturated soil hydraulic properties from redistribution of injected water, Trans. ASAE 27 (1984) 784–790.
A.J. Jones and R.J. Wagenet, In situ estimation of hydraulic conductivity using simplified methods, Water Resour. Res. 20 (1984) 1620–26.
E.C. Childs, The ultimate moisture profile during infiltration in a uniform soil, Soil Sci. 97 (1964) 173–178.
E.G. Youngs and A. Poulovassilis, The different forms of moisture-profile development during the redistribution of soil water after infiltration, Water Resour. Res. 12 (1976) 1007–12.
K.K. Watson and V. Sardana, Numerical study the effect of hysteresis on post-infiltration redistribution, in Infiltration Development and Application, (Ed. Y.-S. Fok), Water resources Research Center, Honolulu (1987).
F.D. Whisler and A. Klute, The numerical analysis of infiltration, considering hysteresis, into a vertical soil column at equilibrium under gravity, Soil Sci. Soc. Am. Proc. 29 (1965) 489–494.
J. Rubin, Numerical method for analysing hysteresis-affected, post-infiltration redistribution of soil moisture, Soil Sci. Soc. Am. Proc. 31 (1967) 13–20.
W.J. Staple, Comparison of computed and measured moisture redistribution following infiltration, Soil Sci. Soc. Am. Proc. 33 (1969) 840–847.
A.J. Peck, Redistribution of soil water after infiltration, Aust. J. Soil Res. 9 (1971) 59–71.
S.J. Perrens and K.K. Watson, Numerical analysis of 2D infiltration and redistribution, Water Resour. Res. 13 (1977) 781–790.
W.R. Gardner, D. Hillel and Y. Benyamini, Post-irrigation movement of soil water; 1. Redistribution, Water Resour. Res. 6 (1970) 851–861.
T. Talsma, The effect of initial moisture content and infiltration quantity on redistribution of soil water, Aust. J. Soil Res. 12 (1974) 15–26.
P.A.C. Raats, Analytic solution of a simplified flow equation, Trans. ASAE 19 (1976) 683–689.
J.R. Philip, Theory of infiltration, Adv. Hyrosci. 5 (1969) 215–296.
A.S. Fokas and Y.C. Yortsos, On the exactly solvable equation occurring in two-phase flow in porous media, SIAM J. Appl. Math. 42 (1982) 318–332.
G. Rosen, Method for the exact solution of a nonlinear diffusion-convection equation, Phys. Rev. Lett. 49 (1982) 1844–46.
C. Rogers, M.P. Stallybrass and D.L. Clements, On two phase filtration under gravity and with boundary infiltration: application of a Bäcklund transformation, J. Nonlinear Anal. 7 (1983) 785–799.
P. Broadbridge and I. White, Constant-rate rainfall infiltration: a versatile nonlinear model; I. Analytic solution, Water Resour. Res. 24 (1988) 145–154.
I. White and P. Broadbridge, Constant-rate rainfall infiltration: a versatile nonlinear model; II. Applications of solutions, Water Resour. Res. 24 (1988) 155–162.
J.R. Philip, The theory of infiltration: 4. Sorptivity and algebraic infiltration equations, Soil Sci. 84 (1957) 257–264.
G. Kirchhoff, Vorlesungen über die Theorie der Wärme, Barth, Leipzig (1894).
M.L. Storm, Heat conduction in simple metals, J. Appl. Phys. 22 (1951) 940–951.
A.R. Forsyth, Theory of Differential Equations, vol. 6, Dover publications, New York (1959). (Previously published in 1906).
E. Hopf, The partial differential equation u t + uu x = µu xx , Commun. Pure Appl. Math. 3 (1950) 201–230.
J.D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951) 225–236.
R.E. Moore, Water conduction from shallow water tables, Hilgardia 12 (1939) 383–426.
J.R. Philip and J.H. Knight, On solving the unsaturated flow equation; 3. New quasi-analytical technique. Soil Sci. 117 (1974) 1–13.
J.H. Knight and J.R. Philip, Exact solutions in nonlinear diffusion, J. Engng. Math. 8 (1974) 219–227.
R.W. Hamming, Numerical Methods for Scientists and Engineers, McGraw-Hill, New York (1962).
J.B. Sisson, A.H. Ferguson and M.Th. van Genuchten, Simple method for predicting drainage from field plots, Soil Sci. Soc. Am. J. 44 (1980) 1147–52.
P.D. Lax, The formation and decay of shock waves, Am. Math. Monthly 79 (1972) 227–241.
B.E. Clothier, J.H. Knight and I. White, Burgers' equation: application to field constant-flux infiltration, Soil Sci. 132 (1981) 255–261.
H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd edition, Clarendon Press, Oxford (1959).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Broadbridge, P., Rogers, C. Exact solutions for vertical drainage and redistribution in soils. J Eng Math 24, 25–43 (1990). https://doi.org/10.1007/BF00128844
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00128844