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Advances in modeling of water in the unsaturated zone

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Abstract

This paper reviews recent advances in analytical and numerical solution of problems of water flow through rigid soils in the unsaturated zone. The Richards model remains the most widely accepted and fertile framework for water flow analyses. More general formulations are reserved for the analysis of problems involving macroporosity, thermal effects, and air pressure effects. Many exact and approximate solutions have been derived for particular boundary value problems of homogeneous soils using methods such as quasi-linear analysis, Green-Ampt analysis, perturbation, and the kinematic wave approximation. Numerical simulators have become bigger and more accurate due to improvements in the areas of nonlinear solution procedures, mass conservation, computational efficiency, and computer hardware. Problems of natural heterogeneity have been addressed primarily through application of various stochastic methods to the Richards model. The stochastic formulations generally refute the concept of simple ‘equivalent’ homogeneous properties, but do themselves offer a certain limited potential for a predictive capability.

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References

  • Abriola, L. M., 1986, Finite element solution of the unsaturated flow equations using hierarchic basic functions, in A. sa da Costa et al. (eds.), Proc. Sixth Intl. Conf. Finite Elements in Water Resources, Springer-Verlag, New York.

    Google Scholar 

  • Ahuja, L. R., 1983, Modeling infiltration into crusted soils by the Green-Ampt approach, Soil Sci. Soc. Am. J. 47, 412–418.

    Google Scholar 

  • Allen, M. B. and Murphy, C. L., 1986, A finite-element collocation method for variably saturated flow in two space dimensions, Water Resour. Res. 22, 1537–1542.

    Google Scholar 

  • Andersson, J. and Shapiro, A. M., 1983, Stochastic analysis of one-dimensional steady state unsaturated flow: A comparison of Monte Carlo and perturbation methods, Water Resour. Res. 19, 121–133.

    Google Scholar 

  • Babu, D. K., 1979, Steady state moisture flows in unsaturated soil containing a plane, isolated, impermeable barrier, Adv. Water Resour. 2, 35–46.

    Google Scholar 

  • Batu, V., 1982, Time-dependent, linearized two-dimensional infiltration and evaporation from nonuniform and nonperiodic strip sources, Water Resour. Res. 18, 1725–1733.

    Google Scholar 

  • Batu, V., 1983, Time-dependent, linearized two-dimensional infiltration and evaporation from nonuniform and period strip sources, Water Resour. Res. 19, 1523–1529.

    Google Scholar 

  • Baveye, P. and Sposito, G., 1984, The operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers, Water Resour. Res. 20, 521–530.

    Google Scholar 

  • Beese, F. and van der Ploeg, R. R., 1976, Influence of hysteresis on moisture flow in an undisturbed soil monolith, Soil Sci. Soc. Am. J. 40, 480–484.

    Google Scholar 

  • Beven, K. J. and Clarke, R. T., 1986, On the variation of infiltration into a homogeneous soil matrix containing a population of macropores, Water Resour. Res. 22, 383–388.

    Google Scholar 

  • Beven, K. J. and Germann, P., 1982, Macropores and water flow in soils, Water Resour. Res. 18, 1311–1325.

    Google Scholar 

  • Boulier, J. F., Tourna, J., and Vauclin, M., 1984, Flux-concentration relation-based solution of constant-flux infiltration equation, 1, Infiltration into nonuniform initial moisture profiles, Soil Sci.Soc. Am. J. 48, 245–251.

    Google Scholar 

  • Bresler, E. and Dagan, G., 1983a, Unsaturated flow in spatially variable fields, 2, Application of water flow models to various fields, Water Resour. Res. 19, 421–428.

    Google Scholar 

  • Bresler, E. and Dagan, G., 1983b, Unsaturated flow in spatially variable fields, 3, Solution transport models and their application to two fields, Water Resour. Res. 19, 429–435.

    Google Scholar 

  • Broadbridge, P., 1987, Integrable flow equations that incorporate spatial heterogeneity, Transport in Porous Media 2, 129–144.

    Google Scholar 

  • Brutsaert, W., 1982, Some exact solutions for nonlinear desorptive diffusion, J. Appl. Math. Phys. 33, 540–546.

    Google Scholar 

  • Bybordi, M., 1968, Moisture profiles in layered porous materials during steady-state infiltration, Soil Sci. 105, 379–383.

    Google Scholar 

  • Camillo, P. J., Gurney, R. J., and Schmugge, T. J., 1983, A soil and atmospheric boundary layer model for evapotranspiration and soil moisture studies, Water Resour. Res. 19, 371–380.

    Google Scholar 

  • Celia, M. A. and Pinder, G. F., 1986, An alternating-direction collocation solution for the unsaturated flow equation, in A. sa da Costa et al. (eds.), Proc. Sixth Intl. Conf. Finite Elements in Water Resources, Springer-Verlag, New York.

    Google Scholar 

  • Childs, E. C. and Bybordi, M., 1969, The vertical movement of water in stratified porous material, 1, Infiltration, Water Resour. Res. 5, 446–459.

    Google Scholar 

  • Clapp, R. B., 1982, A wetting front model of soil water dynamics, PhD dissertation, Dept. of Environ. Sci., Univ. of Va., Charlottesville.

    Google Scholar 

  • Clapp, R. B., Hornberger, G. M., and Cosby, B. J., 1983, Estimating spatial variability in soil moisture with a simplified dynamic model, Water Resour. Res. 19, 739–745.

    Google Scholar 

  • Cooley, R. L., 1983, Some new procedures for numerical solution of variably saturated flow problems, Water Resour. Res. 19, 1271–1285.

    Google Scholar 

  • Curtis, A. A. and Watson, K. K., 1984, Hysteresis affected water movement in scale heterogeneous profiles, Water Resour. Res. 20, 719–726.

    Google Scholar 

  • Cushman, J. H., 1984, On unifying the concepts of scale, instrumentation, and stochastics in the development of multiphase transport theory, Water Resour. Res. 20, 1668–1676.

    Google Scholar 

  • Cushman, J. H., 1986, On measurement, scale, and scaling, Water Resour. Res. 22, 129–134.

    Google Scholar 

  • Dagan, G. and Bresler, E., 1983, Unsaturated flow in spatially variable fields, 1, Derivation of models of infiltration and redistribution, Water Resour. Res. 19, 413–420.

    Google Scholar 

  • Dane, J. H. and Mathis, F. H., 1981, An adaptive finite difference scheme for the one-dimensional water flow equation, Soil Sci. Soc. Am. J. 45, 1048–1054.

    Google Scholar 

  • Davidson, M. R., 1984, A Green-Ampt model of infiltration in a cracked soil, Water Resour. Res. 20, 1685–1690.

    Google Scholar 

  • Davidson, M. R., 1985, Asymptotic behavior of infiltration in soils containing crack or holes, Water Resour. Res. 21, 1345–1353.

    Google Scholar 

  • de Vries,D. A., 1958, Simultaneous transfer of heat and moisture in porous media, EOS Trans. AGU 39, 909–916.

    Google Scholar 

  • Diment, G. A. and Watson, K. K., 1983, Stability analysis of water movement in unsaturated porous materials, 2, Numerical studies, Water Resour. Res. 19, 1002–1010.

    Google Scholar 

  • Diment, G. A. and Watson, K. K., 1985, Stability analysis of water movement in porous materials, 3, Experimental studies, Water Resour. Res. 21, 979–984.

    Google Scholar 

  • Diment, G. A., Watson, K. K., and Blennerhassett, P. J., 1982, Stability analysis of water movement in unsaturated porous materials, 1, Theoretical considerations, Water Resour. Res. 18, 1248–1254.

    Google Scholar 

  • Edwards, W. M., van der Ploeg, R. R., and Ehlers, W., 1979, A numerical study of the effects of noncapillary-sized pores upon infiltration, Soil Sci. Soc. Am. J. 43, 851–856.

    Google Scholar 

  • Freeze, R. A., 1971, Three-dimensional transient saturated-unsaturated flow in a groundwater basin, Water Resour. Res. 7, 347–366.

    Google Scholar 

  • Freeze, R. A., 1980, A stochastic-conceptual analysis of rainfall-runoff processes on a hillslope. Water Resour. Res. 16, 391–408.

    Google Scholar 

  • 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.

    Google Scholar 

  • Germann, P. F., 1985, Kinematic wave approach to infiltration and drainage into and from soil macropores, Trans. Amer. Soc. Agric. Eng. 28, 745–749.

    Google Scholar 

  • Germann, P. F. and Beven, K., 1985, Kinematic wave approximation to infiltration into soils with sorbing macropores, Water Resour. Res. 21, 990–996.

    Google Scholar 

  • Germann, P. F. and Beven, K., 1986, A distribution function approach to water flow in soil macropores based on kinematic wave theory, J. Hydrol. 83, 173–183.

    Google Scholar 

  • Green, W. G. and Ampt, G., 1911, Studies of soil physics, I, The flow of air and water through soils, J. Agric. Sci. 4, 1–24.

    Google Scholar 

  • Hassanizadeh, M. and Gray, W. G., 1979a, General conservation equations for multiphase systems, 1, Averaging procedure, Adv. Water Resour. 2, 131–144.

    Google Scholar 

  • Hassanizadeh, M. and Gray, W. G., 1979b, General conservation equations for multiphase systems, 2, Mass, momenta, energy, and entropy equations, Adv. Water Resour. 2, 191–203.

    Google Scholar 

  • Hasanizadeh, M. and Gray, W. G., 1980, General conservation equations for multiphase systems, 3, Constitutive theory for porous media flow, Adv. Water Resour. 3, 25–40.

    Google Scholar 

  • Herkelrath, W. N., 1981, Comment on ‘Analysis of water and heat flow in unsaturated-saturated porous media’ by Marios Sophocleous, Water Resour. Res. 17, 255.

    Google Scholar 

  • Hromadka, T. V., Guymon, G. L., and Pardoen, G. C., 1981, Nodal domain integration model of unsaturated two-dimensional soil-water flow: development, Water Resour. Res. 17, 1425–1430.

    Google Scholar 

  • Hurley, D. G. and Pantelis, G. 1985, Unsaturated and saturated flow through a thin porous layer on a hillslope, Water Resour. Res. 21, 821–824.

    Google Scholar 

  • Huyakorn, P. S., Springer, E. P., Guvanasen, V., and Wadsworth, T. D., 1986, A three-dimensional finite-element model for simulating water flow in variably saturated porous media, Water Resour. Res. 22, 1790–1808.

    Google Scholar 

  • 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, 1099–1115.

    Google Scholar 

  • Jury, W. A., Sposito, G., and White, R. E., 1986, A transfer function model of solute transport through soil, 1, fundamental concepts, Water Resour. Res. 22, 243–247.

    Google Scholar 

  • Lees, S. J. and Watson, K. K., 1975, The use of a dependent domain model of hysteresis in numerical soil water studies, Water Resour. Res. 11, 943–948.

    Google Scholar 

  • Lighthill, M. J. and Whitham, G. B., 1955, On kinematic waves I. Flood movement in long rivers,Proc. Royal Soc. of London, Ser. A 229, 281–316.

    Google Scholar 

  • Lomen, D. O. and Warrick, A. W., 1974, Time-dependent linearized infiltration, II, Line sources, Soil Sci. Soc. Am. J. 38, 568–572.

    Google Scholar 

  • Luxmoore, R. J. and Sharma, M. L., 1980, Runoff responses to soil heterogeneity: Experimental and simulation comparisons for two contrasting watersheds, Water Resour. Res. 16, 675–684.

    Google Scholar 

  • Mahrer, Y., Naot, O., Rawitz, E., and Katan, J., 1984, Temperature and moisture regimes in soils mulched with transparent polyethylene, Soil Sci. Soc. Am. J. 48, 362–367.

    Google Scholar 

  • Mantoglou, A. and Gelhar, L. W., 1987a, Stochastic modeling of large-scale transient unsaturated flow systems, Water Resour. Res. 23, 37–46.

    Google Scholar 

  • Mantoglou, A. and Gelhar, L. W., 1987b, Capillary tension head variance, mean soil moisture content, and effective specific soil moisture capacity of transient unsaturated flow in stratified soils,Water Resour. Res. 23, 47–56.

    Google Scholar 

  • Mantoglou, A. and Gelhar, L. W., 1987c, Effective hydraulic conductivities of transient unsaturated flow in stratified soils, Water Resour. Res. 23, 57–67.

    Google Scholar 

  • Miller, E. E. and Miller, R. D., 1956, Physical theory for capillary flow phenomena, J. Appl. Phys. 27, 324–332.

    Google Scholar 

  • Milly, P. C. D., 1982, Moisture and heat transport in hysteretic, inhomogeneous porous media: A matric head-based formulation and a numerical model, Water Resour. Res. 18, 489–498.

    Google Scholar 

  • Milly, P. C. D., 1984a, A linear analysis of thermal effects on evaporation from soil, Water Resour. Res. 20, 1075–1085.

    Google Scholar 

  • Milly, P. C. D., 1984b, A simulation analysis of thermal effects on evaporation from soil, Water Resour. Res. 20, 1087–1098.

    Google Scholar 

  • Milly, P. C. D., 1985a, Stability of the Green-Ampt profile in a delta function soil, Water Resour.Res. 21, 399–402.

    Google Scholar 

  • Milly, P. C. D., 1985b, A mass-conservative procedure for time-stepping in models of unsaturated flow, Adv. Water Resour. 8, 32–36.

    Google Scholar 

  • Milly, P. C. D., 1986, An event-based simulation model of moisture and energy fluxes at a bare soil surface, Water Resour. Res. 22, 1680–1692.

    Google Scholar 

  • Milly, P. C. D. and Eagleson, P. S., 1980, The coupled transport of water and heat in a vertical soil column under atmospheric excitation, Tech. Rep. 258, Dep. of Civ. Eng., Mass. Inst. of Technol.,Cambridge.

    Google Scholar 

  • Milly, P. C. D. and Eagleson, P. S., 1982, Infiltration and evaporation at inhomogeneous land surfaces, Tech. Rep. 278, Dep. of Civ. Eng., Mass. Inst. of Technol., Cambridge.

    Google Scholar 

  • Milly, P. C. D. and Eagleson, P. S., 1987, Effects of spatial variability on average water balance, Water Resour. Res. 23, 2135–2143.

    Google Scholar 

  • Morel-Seytoux, H. J., 1973, Two-phase flows in porous media, Adv. Hydrosci. 9, 119–202.

    Google Scholar 

  • Morel-Seytoux, H. J., 1984, From excess infiltration to aquifer recharge: A derivation based on the theory of flow of water in unsaturated soils, Water Resour. Res. 20, 1230–1240.

    Google Scholar 

  • Morel-Seytoux, H. J. and Billica, J. A., 1985a, A two-phase numerical model for prediction of infiltration: Application to a semi-infinite soil column, Water Resour. Res. 21, 607–615.

    Google Scholar 

  • Morel-Seytoux, H. J. and Billica, J. A., 1985b, A two-phase numerical model for prediction of infiltration: Case of an impervious bottom, Water Resour. Res. 21, 1389–1396.

    Google Scholar 

  • Nakano, Y., 1980a, Particular solutions to the problem of horizontal flow of water and air through porous media near a water table, Adv. Water Resour. 3, 19–24.

    Google Scholar 

  • Nakano, Y., 1980b, Particular solutions to the problem of horizontal flow of water and air through porous media near a wetting front, Adv. Water Resour. 3, 81–85.

    Google Scholar 

  • Nakano, Y., 1980c, Particular solutions to the problem of vertical flow of water and air through porous media near a water table, Adv. Water Resour. 3, 125–133.

    Google Scholar 

  • Narasimhan, T. N., Witherspoon, P. A., and Edwards, A. L., 1978, Numerical model for saturated-unsaturated flow in deformable porous media, 2, The algorithm, Water Resour. Res. 14, 255–261.

    Google Scholar 

  • Neuman, S. P., 1973, Saturated-unsaturated seepage by finite elements, Proc. ASCE, J. Hydraul, Div. 99(HY12), 2233–2250.

    Google Scholar 

  • Nielsen, D. R., Jackson, R. D., Cary, J. W., and Evans, D. D. (eds.), 1972, Soil Water, Amer. Soc.Agron., Madison, Wisconsin.

    Google Scholar 

  • Nielsen, D. R., Van Genuchten, M. Th., and Biggar, J. W., 1986, Water flow and solute transport processes in the unsaturated zone, Water Resour. Res. 22, 89S-108S.

    Google Scholar 

  • Parlange, J.-Y., Lisle, I. G., Braddock, R. D., and Smith, R. E., 1982, The three-parameter infiltration equation, Soil Sci. 133, 337–341.

    Google Scholar 

  • Parlange, J.-Y., Lisle, I. G., Prasad, S. N., and Romkens, M. J. M., 1984, Wetting front analysis of the nonlinear diffusion equation, Water Resour. Res. 20, 636–638.

    Google Scholar 

  • Peck, A. J., Luxmoore, R. J., and Stolzy, J. L., 1977, Effects of spatial variability of soil hydraulic properties in water budget modeling, Water Resour. Res. 13, 348–354.

    Google Scholar 

  • Perroux, K. M., Smiles, D. E., and White, I., 1981, Water movement in uniform soils during constant-flux infiltration, Soil Sci. Soc. Am. J. 45, 237–240.

    Google Scholar 

  • Philip, J. R., 1957a, The theory of infiltration: 1. The infiltration equation and its solution, Soil Sci. 83, 345–357.

    Google Scholar 

  • Philip, J. R., 1957b, The theory of infiltration, 4, Sorptivity and algebraic infiltration equations, Soil Sci. 84, 257–264.

    Google Scholar 

  • Philip, J. R., 1957c, Evaporation, and moisture and heat fields in the soil, J. Atmos. Sci. 14, 354–366.

    Google Scholar 

  • Philip, J. R., 1967, Sorption and infiltration in heterogeneous media, Aust. J. Soil Res. 5, 1–10.

    Google Scholar 

  • Philip, J. R., 1968a, Absorption and infiltration in two- and three-dimensional systems, and Discussion, in P. E. Rijtema and H. Wassink (eds.), Water in the Unsaturated Zone, Vol. I, Proc. Wageningen Symp., IASH.

  • Philip, J. R., 1968b, Steady infiltration from buried point sources and spherical cavities, Water Resour. Res. 4, 1039–1047.

    Google Scholar 

  • Philip, J. R., 1968c, The theory of absorption in aggregated media, Aust. J. Soil Res. 6, 1–19.

    Google Scholar 

  • Philip, J. R., 1972, Steady infiltration from buried, surface, and perched point and line sources in heterogeneous soils, 1, Analysis, Soil Sci. Soc. Am. Proc. 36, 268–273.

    Google Scholar 

  • Philip, J. R., 1973, On solving the unsaturated flow equation, 1, The flux-concentration relation, Soil Sci. 116, 328–335.

    Google Scholar 

  • Philip, J. R., 1975, Stability analysis of infiltration, Soil Sci. Soc. Am. Proc. 39, 1042–1049.

    Google Scholar 

  • Philip, J. R., 1980, Field heterogeneity: some basic issues, Water Resour. Res. 16, 443–448.

    Google Scholar 

  • Philip, J. R., 1984a, Steady infiltration from circular cylindrical cavities, Soil Sci. Soc. Am. J. 48,270–278.

    Google Scholar 

  • Philip, J. R., 1984b, Steady infiltration from spherical cavities, Soil Sci. Soc. Am. J. 48(4), 724–729.

    Google Scholar 

  • Philip, J. R., 1985a, Scattering functions and infiltration, Water Resour. Res. 21, 1889–1894.

    Google Scholar 

  • Philip, J. R., 1985b, Approximate analysis of the borehole permeater in unsaturated soil. Water Resour. Res. 21, 1025–1033.

    Google Scholar 

  • Philip, J. R., 1986a, Steady infiltration from buried discs and other sources, Water Resour. Res. 22,1058–1066.

    Google Scholar 

  • Philip, J. R., 1986b, Steady infiltration from spheroidal cavities in isotropic and anisotropic soils, Water Resour. Res. 22, 1874–1880.

    Google Scholar 

  • Philip, J. R., 1987, An analog for infiltration and unsaturated seepage, EOS Trans. AGU 68, 153–154.

    Google Scholar 

  • Philip, J. R. and de Vries,D. A., 1957, Moisture movement in porous materials under temperature gradients, EOS Trans. AGU 38, 222–232.

    Google Scholar 

  • Pollock, D. W., 1986, Simulation of fluid flow and energy transport processes associated with high-level radioactive waste disposal in unsaturated alluvium, Water Resour. Res. 22, 765–775.

    Google Scholar 

  • Prasad, S. N. and Romkens, M. J. M. 1982, An approximate integral solution of vertical infiltration under changing boundary conditions, Water Resour. Res. 18, 1022–1028.

    Google Scholar 

  • Raats, P. A. C., 1970, Steady infiltration from line sources and furrows, Soil Sci. Soc. Am. Proc. 34,709–714.

    Google Scholar 

  • Richards, L. A., 1931, Capillary conduction of liquids through porous mediums, Physics 1, 318–333.

    Google Scholar 

  • Rubin, J., 1967, Numerical method for analyzing hysteresis-affected, post-infiltration redistribution of soil moisture, Soil Sci. Soc. Am. Proc. 31, 13–20.

    Google Scholar 

  • Russo, D. and Bresler, E., 1982, A univariate versus a multivariate parameter distribution in a stochastic-conceptual analysis of unsaturated flow, Water Resour. Res. 18, 483–488.

    Google Scholar 

  • Schieldge, J. P., Kahle, A. B., and Alley, R. E., 1982, A numerical simulation of soil temperature and moisture variations for a bare field, Soil Sci. 134, 197–207.

    Google Scholar 

  • Schmitz, G., Vauclin, M., and Seus, G. J., 1984, A time variant computational mesh technique to simulate a large scale ponding test, in J. P. Laible et al. (eds.), Proc. Fifth Intl. Conf. Finite Elements in Water Resources, Springer-Verlag, New York.

    Google Scholar 

  • Sharma, M. L. and Luxmoore, R. J., 1979, Soil spatial variability and its consequence on simulated water balance, Water Resour. Res. 15, 1567–1573.

    Google Scholar 

  • Sisson, J. B., Ferguson, A. H., and Van Genuchten, M. Th., 1980, Simple methods for predicting drainage from field plots, Soil Sci. Soc. Am. J. 44, 1147–1152.

    Google Scholar 

  • Smettem, K. R. J., 1986, Analysis of water flow from cylindrical macropores, Soil Sci. Soc. Am. J. 50, 1139–1142.

    Google Scholar 

  • Smith, R. E., 1983a, Unsaturated soil water flow, Rev. Geophysics and Space Physics 21, 755–760.

    Google Scholar 

  • Smith, R. E., 1983b, Approximate soil water movement by kinematic characteristics, Soil Sci. Soc. Am. J. 47, 3–8.

    Google Scholar 

  • Smith, R. E. and Hebbert, R. H. B., 1979, A Monte Carlo analysis of the hydrologic effects of spatial variability of infiltration, Water Resour. Res. 15, 419–429.

    Google Scholar 

  • Sophocleous, M., 1979, Analysis of water and heat flow in unsaturated-saturated porous media, Water Resour. Res. 15, 1195–1206.

    Google Scholar 

  • Sposito, G., 1978a, The statistical mechanical theory of water transport through unsaturated soil, 1, The conservation laws, Water Resour. Res. 13, 474–478.

    Google Scholar 

  • Sposito, G., 1978b, The statistical mechanical theory of water transport through unsaturated soil, 2, Derivation of the Buckingham-Darcy flux law, Water Resour. Res. 14, 479–484.

    Google Scholar 

  • Sposito, G., Gupta, V. K., and Bhattacharya, R. N., 1979, Foundational theories of solute transport in porous media: A critical review, Adv. Water Resour. 2, 59–68.

    Google Scholar 

  • Sposito, G., Jury, W. A., and Gupta, V. K., 1986, Fundamental problems in the stochastic convection-dispersion model of solute transport in aquifers and field soils, Water Resour. Res. 22, 77–88.

    Google Scholar 

  • Stagnitti, F., Parlange, M. B., Steenhuis, T. S., and Parlange, J.-Y., 1986, Drainage from a uniform soil layer on a hillslope, Water Resour. Res. 22, 631–634.

    Google Scholar 

  • Touma, J. and Vauclin, M., 1986, Experimental and numerical analysis of two-phase infiltration in a partially saturated soil, Transport in Porous Media, 1, 27–55.

    Google Scholar 

  • Van der Heijde, P., Bachmat, Y., Bredehoeft, J., Andrews, B., Holtz, D. and Sebastian, S., 1985, Groundwater Management: The use of Numerical Models, 2nd edn, Water Resources Monograph 5,American Geophysical Union, Washington.

    Google Scholar 

  • Van Genuchten, M. Th., 1983, An Hermitian finite element solution of the two-dimensional saturated-unsaturated flow equation, Adv. Water Resour. 6, 106–111.

    Google Scholar 

  • Van Genuchten, M. Th. and Jury, W. A., 1987, Progress in unsaturated flow and transport modeling,Rev. Geophys. 25, 135–140.

    Google Scholar 

  • Waechter, R. T. and Philip, J. R., 1985, Steady two-and three-dimensional flows in unsaturated soil: The scattering analog, Water Resour. Res. 21, 1875–1887.

    Google Scholar 

  • Wang, J. S. Y. and Narasimhan, T. N., 1985, Hydrologic mechanisms governing fluid flow in a partially saturated, fractured, porous medium, Water Resour. Res. 21, 1861–1874.

    Google Scholar 

  • Warrick, A. W., 1974, Time-dependent linearized infiltration, I, Point sources, Soil Sci. Soc. Am. J. 38, 383–386.

    Google Scholar 

  • Warrick, A. W. and Lomen, D. O., 1976, Time-dependent linearized infiltration, III, Strip and disc sources, Soil Sci. Soc. Am. J. 40, 639–643.

    Google Scholar 

  • Warrick, A. W. and Lomen, D. O., 1983, Linearized moisture flow with root extraction over two-dimensional zones, Soil Sci. Soc. Am. J. 47, 869–872.

    Google Scholar 

  • Weir, G. J., 1987, Steady infiltration from small shallow circular ponds, Water Resour. Res. 23, 733–736.

    Google Scholar 

  • Willis, W. O., 1960, Evaporation from layered soils in the presence of a water table, Soil Sci. Soc. Am. Proc. 24, 239–242.

    Google Scholar 

  • Wooding, R. A., 1968, Steady infiltration from a shallow circular pond, Water Resour. Res. 4,1259–1273.

    Google Scholar 

  • Yeh, G.-T., 1981, On the computation of Darcian velocity and mass balance in the finite element modeling of groundwater flow, Water Resour. Res. 17, 1529–1534.

    Google Scholar 

  • Yeh, T.-C. J., Gelhar, L. W., and Gutjahr, A. L., 1985a, Stochastic analysis of unsaturated flow in heterogeneous soils, 1, Statistically isotropic media, Water Resour. Res. 21, 447–456.

    Google Scholar 

  • Yeh, T.-C. J., Gelhar, L. W., and Gutjahr, A. L., 1985b, Stochastic analysis of unsaturated flow in heterogeneous soils, 2, Statistically anisotropic media with variable α, Water Resour. Res. 21, 457–464.

    Google Scholar 

  • Yeh, T.-C. J., Gelhar, L. W., and Gutjahr, A. L., 1985c, Stochastic analysis of unsaturated flow in heterogeneous soils, 3, Observations and applications, Water Resour. Res. 21, 465–471.

    Google Scholar 

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    P. C. D. Milly

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Milly, P.C.D. Advances in modeling of water in the unsaturated zone. Transp Porous Med 3, 491–514 (1988). https://doi.org/10.1007/BF00138613

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