Semianalytical Solution to Richards' Equation for Layered Porous Media
Publication: Journal of Irrigation and Drainage Engineering
Volume 124, Issue 6
Abstract
Traditional finite-difference and finite-element solutions to Richards' equation can exhibit stability problems and mass balance errors when sharp saturation fronts migrate across material interfaces. Several semianalytical solutions to Richards' equation have been developed for layered media, but none allow for arbitrary constitutive relationships between capillary pressure, water saturation, and relative water permeability. This paper develops a more general semianalytical solution to Richards' equation that can simulate unsaturated flow in layered media and utilize arbitrary constitutive relationships. Based on a combination of the Runge-Kutta and shooting methods, initial-boundary-value problems are solved for layered systems without experiencing the types of stability problems commonly associated with numerical models. The proposed numerical scheme is computationally efficient and numerically stable, and it compares favorably with other analytical solutions. Results of two example simulations demonstrate the versatility of the method for solving both horizontal and vertical unsaturated flow problems. The primary disadvantages of the numerical approach are that it is limited to one-dimensional flow and that it requires the development of specific iteration schemes for different categories of boundary-value problems.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Acton, F. S. (1990). Numerical methods that work; 1990 corrected addition. Math. Assn. of Am., Washington, D.C.
2.
Bear, J. (1972). Dynamics of fluids in porous media. Dover Publications, Inc., Mineola, N.Y., 764.
3.
Bear, J. (1979). Hydraulics of groundwater. McGraw-Hill Inc., New York, 569.
4.
Brooks, R. H., and Corey, A. T. (1964). “Hydraulic properties of porous media.”Hydro. Paper No. 3, Colorado State Univ., Fort Collins, Colo.
5.
Brooks, R. H., and Corey, A. T.(1966). “Properties of porous media affecting fluid flow.”J. Irrig. and Drain. Div., ASCE, 92(2), 61–87.
6.
Celia, M. A., Bouloutas, E. T., and Zarba, R. L.(1990). “A general mass-conservative numerical solution for the unsaturated flow equation.”Water Resour. Res., 26(7), 1483–1496.
7.
Corey, A. T. (1994). Mechanics of immiscible fluids in porous media. Water Resources Publications, Highlands Ranch, Colo.
8.
Dullien, F. A. L. (1992). Porous media: fluid transport and pore structure. Academic Press, Inc., San Diego, Calif., 574.
9.
Gardner, W. R. (1958). Some steady-state solutions for the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci., 85, 228–232.
10.
Hillel, D. (1971). Soil and water, physical principals and processes. Academic Press, Inc., New York, 288.
11.
Jury, W. A., Gardner, W. R., and Gardner, W. H. (1991). Soil physics, 5th Ed., John Wiley & Sons, Inc., New York, 328.
12.
Keller, H. B. (1968). Numerical methods for two-point boundary-value problems. Blaisdell, Waltham, Mass.
13.
Lenhard, R. J.(1992). “Measurement and modeling of three-phase saturation-pressure hysteresis.”J. Contaminant Hydro., 9, 243–269.
14.
Lopez-Bakovic, I. L., and Nieber, J. L. (1989). “Analytic steady-state solution to one-dimensional unsaturated water flow in layered soils.”Unsaturated flow in hydrologic modeling, H. J. Morel-Seytoux, ed., Kluwer Academic Publishers, Boston, Mass.
15.
Marinelli, F. (1996). “Semi-analytical solutions for one-dimensional flow of immiscible fluids in layered porous media,” PhD dissertation, Colorado State Univ., Fort Collins, Colo.
16.
McWhorter, D. B. (1971). “Infiltration affected by flow of air.”Hydro. Paper No. 49, Colorado State Univ., Fort Collins, Colo., 43.
17.
McWhorter, D. B., and Marinelli, F. (1998). “Theory of soil-water flow.”Drainage design, J. vanSchilfgaarde, ed., Am. Soc. of Agronomy, Madison, Wis. (in press).
18.
McWhorter, D. B., and Sunada, D. K.(1990). “Exact integral solutions for two-phase flow.”Water Resour. Res., 26(3), 399–413.
19.
Milly, P. C. D.(1985). “A mass conservative procedure for time-stepping in models of unsaturated flow.”Advances in Water Resour., 8, 32–36.
20.
Moldrup, P., Rolston, D. E., and Hansen, J. A.(1989). “Rapid and numerically stable simulation of one-dimensional, transient water flow in unsaturated, layered soils.”Soil Sci., 148(3), 219–226.
21.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992). Numerical recipes in Fortran, the art of scientific computing, 2nd Ed., Cambridge University Press, New York.
22.
Richards, L. A.(1931). “Capillary conduction of liquids through porous mediums.”Physics, 1, 318–333.
23.
Ross, P. J.(1990). “Efficient numerical methods for infiltration using Richards' equation.”Water Resour. Res., 26(2), 279–290.
24.
Ross, P. J., and Bristow, K. L.(1990). “Simulating water movement in layered and gradational soils using the Kirchoff transform.”Soil Sci. Soc. Am. J., 54(6), 1519–1524.
25.
Srivastava, R. and, and Yeh, J.(1991). “Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils.”Water Resour. Res., 27(5), 753–762.
26.
van Genuchten, M. T.(1980). “A closed-form equation for predicting the hydraulic conductivity of saturated soils.”Soil Sci. Soc. Am. J., Proc., 44, 892–898.
Information & Authors
Information
Published In
Journal of Irrigation and Drainage Engineering
Volume 124 • Issue 6 • November 1998
Pages: 290 - 299
Copyright
Copyright © 1998 American Society of Civil Engineers.
History
Published online: Nov 1, 1998
Published in print: Nov 1998
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.