Flux-Based Alternative Formulation for Variably Saturated Subsurface Flow
Publication: Journal of Hydrologic Engineering
Volume 12, Issue 5
Abstract
Finite-element and finite-difference solutions to different forms of the Richards equation can exhibit stability problems as well as mass balance errors. These problems are more pronounced for sharp wetting fronts in soils with very dry initial conditions and at material interfaces for layered soil profiles. The pressure head form can suffer from large mass balance errors while the moisture content form conserves mass perfectly but has difficulty handling material boundaries in layered soil profiles and near saturation conditions. This paper presents a conservative mixed formulation for the solution of the Richards equation of unsaturated flow by the finite-element method in which the discharge velocity (volumetric flux) and pressure head are the primary field variables. Solution techniques for various boundary conditions including prescribed constant pressure head and constant flux are also presented. The ability of the formulation to handle variably saturated domain and material heterogeneity is also established. Example solutions for infiltration into homogeneous and heterogeneous unsaturated and variably saturated soil profiles are compared with finite-element and finite-difference solutions of pressure head and mixed form of Richards equation. Results indicate that the alternative formulation is mass conservative, stable, and can better handle heterogeneous boundaries and variably saturated flow domains.
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Acknowledgments
This research was partially supported with funds from NSERC (Natural Sciences and Engineering Research Council) Discovery Grants awarded to D. F. Stolle and J. E. Smith, and a CRESTech Grant to J. E. Smith. The writers will also like to acknowledge the funding provided by the Center for Effective Design of Structures at McMaster University.
References
Baca, R. G., Chung, J. N., and Mulla, D. J. (1997). “Mixed transform finite element method for solving the non-linear equation for flow in variably saturated porous media.” Int. J. Numer. Methods Fluids, 24(5), 441–455.
Baker, D. L. (1995). “Darcian weighted interblock conductivity means for vertical unsaturated flow.” Ground Water, 33(3), 385–390.
Bouloutas, E. T. (1989). “Improved numerical methods for modeling flow and transport processes in partially saturated porous media.” Ph.D. thesis, Massachusetts Inst. of Technology, Dept. of Civil Engineering, Cambridge, Mass.
Broadbridge, P., and White, I. (1988). “Constant rate rainfall infiltration: A versatile nonlinear model. 1: Analytical solution.” Water Resour. Res., 24(1), 145–154.
Brutsaert, W. F. (1971). “A functional iteration technique for solving the Richards equation applied to two dimensional infiltration problems.” Water Resour. Res., 7(6), 1583–1596.
Celia, M. A., Boulouta, 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.
Chen, J. M., Tan, Y.-C., and Chen, C.-H. (2001). “Multidimensional infiltration with arbitrary surface fluxes.” J. Irrig. Drain. Eng., 127(6), 370–377.
Cooley, R. L. (1983). “Some new procedures for numerical solution of variably saturated flow problems.” Water Resour. Res., 19(5), 1271–1285.
Downer, C. W., and Ogden, F. L. (2004). “Appropriate vertical discretization of Richards equation for two-dimensional watershed-scale modeling.” Hydrolog. Process., 18(1), 1–22.
Dussaillant, A. R., Wu, C. H., and Potter, K. W. (2004). “Richards equation model of a rain garden.” J. Hydrol. Eng., 9(3), 219–225.
El-Kadi, A. I., and Ling, G. (1993). “The Courant and Péclet number criteria for the numerical solution of the Richards equation.” Water Resour. Res., 29(10), 3485–3494.
Forsyth, P. A. (1988). “Comparison of the single-phase and two-phase numerical model formulation for saturated-unsaturated groundwater flow.” Comput. Methods Appl. Mech. Eng., 69(2), 243–259.
Forsyth, P. A., Wu, Y. S., and Pruess, K. (1995). “Robust numerical techniques for saturated-unsaturated flow with dry initial conditions in heterogeneous media.” Adv. Water Resour., 18(1) 25–38.
Freeze, R. A. (1971). “Three-dimensional transient saturated-unsaturated flow in a groundwater basin.” Water Resour. Res., 7(2), 347–366.
Gottardi, G., and Venutelli, M. (1993). “Richards: Computer program for the numerical simulation of one-dimensional infiltration into unsaturated soil.” Comput. Geosci., 19(9), 1239–1266.
Gottardi, G., and Venutelli, M. (2001). “PF: Two-dimensional finite-element groundwater flow model for saturated-unsaturated soils.” Comput. Geosci., 27(2), 179–189.
Guarracino, L., and Quintana, F. (2004). “A third-order accurate time scheme for variably saturated groundwater flow modeling.” Commun. Numer. Methods Eng., 20(5), 379–389.
Gui, S., Zhang, R., Turner, J. P., and Xue, X. (2000). “Probabilistic slope stability analysis with stochastic soil hydraulic conductivity.” J. Geotech. Geoenviron. Eng., 126(1), 1–9.
Hao, X., Zhangab, R., and Kravchenkoc, A. (2005). “A mass-conservative switching method for simulating saturated-unsaturated flow.” J. Hydrol., 311(1–4), 254–265.
Haverkamp, R., Vauclin, M., Touma, J., Wierenga, P. J., and Vachaud, G. (1977). “A comparison of numerical simulation models for one dimensional infiltration.” Soil Sci. Soc. Am. J., 41(2), 285–294.
Hills, R. G., Porro, I., Hudson, D. B., and Wierenga, P. J. (1989). “Modeling one-dimensional infiltration into very dry soils. Part 1: Water content versus pressure head based algorithms.” Water Resour. Res., 25(6), 1259–1269.
Hsu, S. M., Ni, C. F., and Huang, P. F. (2002). “Assessment of three infiltration formulas based on model fitting on Richards equation.” J. Hydrol. Eng., 7(5), 373–379.
Huang, K., Mohanty, B. P., and van Genuchten, M. T. (1996). “A new convergence criterion for the modified Picard iteration method to solve the variably saturated flow equation.” J. Hydrol., 178, 69–91.
Huyakorn, P. S., Mercer, J. W., and Ward, D. S. (1985). “Finite-element matrix and mass balance computational schemes for transport in variably-saturated porous media.” Water Resour. Res., 21(3), 346–358.
Huyakorn, P. S., Thomas, S. D., and Thompson, B. M. (1984). “Techniques for making finite elements competitive in modeling flow in variably saturated media.” Water Resour. Res., 20(8), 1099–1115.
Ju, S.-H., and Kung, K.-J. S. (1997). “Mass types, element orders, and solution schemes for Richards equation.” Comput. Geosci., 23(2), 175–187.
Khire, M. V., Benson, C. H., and Bosscher, P. J. (2000). “Capillary barriers: Design variables and water balance.” J. Geotech. Geoenviron. Eng., 126(8), 695–708.
Kim, S., Kavvas, M. L., and Yoon, J. (2005). “Upscaling of vertical unsaturated flow model under infiltration condition.” J. Hydrol. Eng., 10(2), 151–159.
Kirkland, M. R., Hills, R. G., and Wierenga, P. J. (1992). “Algorithms for solving Richards equation for variably saturated soils.” Water Resour. Res., 28(8), 2049–2058.
Lehmann, F., and Ackerer, P. (1998). “Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media.” Transp. Porous Media, 31(3), 275–292.
Li, C. W. (1993). “A simplified Newton iteration method with linear finite elements for transient unsaturated flow.” Water Resour. Res., 29(4), 965–971.
Milly, P. C. D. (1985). “A mass-conservative procedure for time-stepping in models of unsaturated flow.” Adv. Water Resour., 8(1), 32–36.
Neuman, S. P. (1973). “Saturated-unsaturated seepage by finite elements.” J. Hydr. Div., 99(12), 2233–2250.
Pan, L., Warrick, A. W., and Wierenga, P. J. (1996). “Finite element methods for modeling water flow in variably saturated porous media: Numerical oscillation and mass-distribution schemes.” Water Resour. Res., 32(6), 1883–1889.
Parlange, J.-Y. (1972). “Theory of water movement in soils. VIII: One-dimensional infiltration with constant flux at the surface.” Soil Sci., 114(1), 1–4.
Philip, J. R. (1969). “Theory of infiltration.” Adv. Hydrosci., Vol. 5, V. T. Chow, ed., Academic, New York, 215–296.
Rathfelder, K., and Abriola, L. M. (1994). “Mass conservative numerical solutions of the head based Richards equation.” Water Resour. Res., 30(9), 2579–2586.
Richards, L. A. (1928). “The usefulness of capillary potential to soil moisture and plant investigators.” J. Agric. Res., 37(12), 719–742.
Richards, L. A. (1931). “Capillary conduction of liquids through porous mediums.” Physics (N.Y.), 1(5), 318–333.
Romano, N., Brunone, B., and Santini, A. (1998). “Numerical analysis of one-dimensional unsaturated flow in layered soils.” Adv. Water Resour., 21(4), 315–324.
Segerlind, L. J. (1984). Applied finite element analysis, 2nd Ed., Wiley, New York.
Serrano, S. E. (2004). “Modeling infiltration with approximate solutions to Richards equation.” J. Hydrol. Eng., 9(5), 421–432.
Sharma, R. S., White, D. J., and Schaefer, V. R. (2004). “Implications of changes in suction and moisture regime in highway foundations and embankments.” Proc., Geo-Trans 2004, Geotechnical Special Publication No. 126(2), ASCE/Geo Institute Reston, Va., 2115–2122.
Šimúnek, J., Šejna, M., and van Genuchten, M. T. (1998). The HYDRUS-1D software package for simulating the movement of water, heat, and multiple solutes in variably saturated media, version 2.0, U.S. Salinity Laboratory, USDA-ARS, Riverside, Calif.
Šimúnek, J., Šejna, M., and van Genuchten, M. T. (1999). The HYDRUS-2D software package for simulating two-dimensional movement of water, heat, and multiple solutes in variably saturated media, version 2.0, IGWMC-TPS-53, International Ground Water Modeling Center, Colorado School of Mines, Golden, Colo.
Šimúnek, J., Vogel, T., and van Genuchten, M. T. (1994). “The SWMS 2D code for simulating water flow and solute transport in two dimensional variably saturated media.” Res. Rep. No. 132, U.S. Salinity Lab., Agric. Res. Service, USDA, Riverside, Calif.
Skaggs, T. H., Trout, T., Šimúnek, J., and Shouse, P. J. (2004). “Comparison of HYDRUS-2D simulations of drip irrigation with experimental observations.” J. Irrig. Drain. Eng., 130(4), 304–310.
Stormont, J. C., and Zhou, S. (2005). “Impact of unsaturated flow on pavement edgedrain performance.” J. Transp. Eng., 131(1), 46–53.
Tan, T.-S., Phoon, K.-K., and Chong, P.-C. (2004). “Numerical study of finite element method based solutions for propagation of wetting fronts in unsaturated soil.” J. Geotech. Geoenviron. Eng., 130(3), 254–263.
Tang, Y. K., and Skaggs, R. W. (1980). “Drain depth and subirrigation in layered soils.” J. Irrig. and Drain. Div., 106(2), 113–122.
Tracy, F. T. (1995). “1-D, 2-D, and 3-D analytical solutions of unsaturated flow in groundwater.” J. Hydrol., 170(1–4), 199–214.
Vanderborght, J., et al. (2005). “A set of analytical benchmarks to test numerical models of flow and transport in soils.” Vadose Zone J., 4(1), 206–221.
van Genuchten, M. T. (1980). “A closed-form equation for predicting the hydraulic conductivity of unsaturated soils.” Soil Sci. Soc. Am. J., 44(5), 892–898.
Warrick, A. W. (1991). “Numerical approximations of Darcian flow through unsaturated soil.” Water Resour. Res., 27(6), 1215–1222.
Warrick, A. W., Lomen, D. O., and Yates, S. R. (1985). “Generalized solution to infiltration.” Soil Sci. Soc. Am. J., 49(1), 34–38.
Wheeler, J. A., Wheeler, M. F., and Yotov, I. (2002). “Enhanced velocity mixed finite element methods for flow in multiblock domains.” Comput. Geosci., 6(3–4), 315–322.
Yeh, G.-T. (1999). Computational subsurface hydrology: Fluid flows, Kluwer Academic, Boston.
Zaidel, J., and Russo, D. (1992). “Estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils.” Water Resour. Res., 28(9), 2285–2295.
Zhang, X., Bengough, A. G., Crawford, J. W., and Young, I. M. (2002). “Efficient methods for solving water flow under prescribed flux infiltration.” J. Hydrol., 260(1–4), 75–87.
Zornberg, J. G., LaFountain, L., and Caldwell, J. A. (2003). “Analysis and design of evapotranspirative cover for hazardous waste landfill.” J. Geotech. Geoenviron. Eng., 129(5), 427–438.
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Published In
Journal of Hydrologic Engineering
Volume 12 • Issue 5 • September 2007
Pages: 501 - 512
Copyright
© 2007 ASCE.
History
Received: Jul 27, 2005
Accepted: Nov 17, 2006
Published online: Sep 1, 2007
Published in print: Sep 2007
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