A new and relatively simple equation for the soil-water content-pressure head curve, 8(h), is described in this paper. The particular form of the equation enables one to derive closedform analytical expressions for the relative hydraulic conductivity, Kr, when substituted in the predictive conductivity models of N.T. Burdine or Y. Mualem. The resulting expressions for Kr(h) contain three independent parameters which may be obtained by fitting the proposed soil-water retention model to experimental data. Results obtained with the closed-form analytical expressions based on the Mualem theory are compared with observed hydraulic conductivity data for five soils with a wide range of hydraulic properties. The unsaturated hydraulic conductivity is predicted well in four out of five cases. It is found that a reasonable description of the soil-water retention curve at low water contents is important for an accurate prediction of the unsaturated hydraulic conductivity. Additional Index Words: soil-water diffusivity, soil-water retention curve. van Genuchten, M. Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892-898. T USE OF NUMERICAL MODELS for simulating fluid flow and mass transport in the unsaturated zone has become increasingly popular the last few years. Recent literature indeed demonstrates that much effort is put into the development of such models (Reeves and Duguid, 1975; Segol, 1976; Vauclin et al., 1979). Unfortunately, it appears that the ability to fully characterize the simulated system has not kept pace with the numerical and modeling expertise. Probably the single most important factor limiting the successful application of unsaturated flow theory to actual field problems is the lack of information regarding the parameters entering the governing transfer equations. Reliable estimates of the unsaturated hydraulic conductivity are especially difficult to obtain, partly because of its extensive variability in the field, and partly because measuring this parameter is time-consuming and expensive. Several investigators have, for these reasons, used models for calculating the unsaturated conductivity from the more easily measured soil-water retention curve. Very popular among these models has been the Millington-Quirk method (Millington and Quirk, 1961), various forms of which have been applied with some success in a number of studies (cf. Jackson et al., 1965; Jackson, 1972; Green and Corey, 1971; Bruce, 1972). Unfortunately, this method has the disadvantage of producing tabular results which, for example when applied to nonhomogeneous soils in multidimensional unsaturated flow models, are quite tedious to use. Closed-form analytical expressions for predicting 1 Contribution from the U. S. Salinity Laboratory, AR-SEA, USDA, Riverside, CA 92501. Received 29 June 1979. Approved 19 May I960. 'Soil Scientist, Dep. of Soil and Environmental Sciences, University of California, Riverside, CA 92521. The author is located at the U. S. Salinity Lab., 4500 Glenwood Dr., Riverside, CA 92502. the unsaturated hydraulic conductivity have also been developed. For example, Brooks and Corey (1964) and Jeppson (1974) each used an analytical expression for the conductivity based on the Burdine theory (Burdine, 1953). Brooks and Corey (1964, 1966) obtained fairly accurate predictions with their equations, even though a discontinuity is present in the slope of both the soil-water retention curve and the unsaturated hydraulic conductivity curve at some negative value of the pressure head (this point is often referred to as the bubbling pressure). Such a discontinuity sometimes prevents rapid convergence in numerical saturated-unsaturated flow problems. It also appears that predictions based on the Brooks and Corey equations are somewhat less accurate than those obtained with various forms of the (modified) Millington-Quirk method. Recently Mualem (1976a) derived a new model for predicting the hydraulic conductivity from knowledge of the soil-water retention curve and the conductivity at saturation. Mualem's derivation leads to a simple integral formula for the unsaturated hydraulic conductivity which enables one to derive closed-form analytical expressions, provided suitable equations for the soil-water retention curves are available. It is the purpose of this paper to derive such expressions using an equation for the soil-water retention curve which is both continuous and has a continuous slope. The resulting conductivity models generally contain three independent parameters which may be obtained by matching the proposed soil-water retention curve to experimental data. Results obtained with the closedform equations based on the Mualem theory will be compared with observed data for a few soils having widely varying hydraulic properties. THEORETICAL Equations Based on Mualem's Model The following equation was derived by Mualem (1976a) for predicting the relative hydraulic conductivity (Kr) from knowledge of the soil-water retention curve

A closed-form equation for predicting the hydraulic conductivity of unsaturated soils

This paper reviews the importance of large continuous openings (macropores) on water flow in soils. The presence of macropores may lead to spatial concentrations of water flow through unsaturated soil that will not be described well by a Darcy approach to flow through porous media. This has important implications for the rapid movement of solutes and pollutants through soils. Difficulties in defining what constitutes a macropore and the limitations of current nomenclature are reviewed. The influence of macropores on infiltration and subsurface storm flow is discussed on the basis of both experimental evidence and theoretical studies. The limitations of models that treat macropores and matrix porosity as separate flow domains is stressed. Little-understood areas are discussed as promising lines for future research. In particular, there is a need for a coherent theory of flow through structured soils that would make the macropore domain concept redundant.

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Macropores and water flow in soils

Water Retention: Laboratory Methods

Numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems. Numerical solution based on the standard h-based form of Richards equation generally yields poor results, characterized by large mass balance errors and erroneous estimates of infiltration depth. Conversely, numerical solutions based on the mixed form of Richards equation can be shown to possess the conservative property, so that mass is perfectly conserved. This leads to significant improvement in numerical solution performance, while requiring no additional computational effort. However, use of the mass-conservative method does not guarantee good solutions. Accurate solution of the unsaturated flow equation also requires use of a diagonal time (or mass) matrix. Only when diagonal time matrices are used can the solution be shown to obey a maximum principle, which guarantees smooth, nonoscillatory infiltration profiles. This highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow.

A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation

DOI 10.1007/s10040-0010176-2 Abstract Various techniques are available to quantify recharge; however, choosing appropriate techniques is often difficult. Important considerations in choosing a technique include space/time scales, range, and reliabili- ty of recharge estimates based on different techniques; other factors may limit the application of particular tech- niques. The goal of the recharge study is important be- cause it may dictate the required space/time scales of the recharge estimates. Typical study goals include water-re- source evaluation, which requires information on re- charge over large spatial scales and on decadal time scales; and evaluation of aquifer vulnerability to contam- ination, which requires detailed information on spatial variability and preferential flow. The range of recharge rates that can be estimated using different approaches should be matched to expected recharge rates at a site. The reliability of recharge estimates using different tech- niques is variable. Techniques based on surface-water and unsaturated-zone data provide estimates of potential recharge, whereas those based on groundwater data gen- erally provide estimates of actual recharge. Uncertainties in each approach to estimating recharge underscore the need for application of multiple techniques to increase

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Choosing appropriate techniques for quantifying groundwater recharge

A Comparison of Numerical Simulation Models For One-Dimensional Infiltration1

Transient two-dimensional water flow is studied in relation to the recharge of a water table aquifer. The approach is based on the physics of water transfer in the complete domain defined by both the saturated and unsaturated zones of soil. Experimental data were first obtained in a slab of soil (3 m in length, 2 m in height, and 5 cm in thickness) in which the changes of water content and water pressure occurring in the flow domain were measured throughout an artificial recharge event. These were compared with the predictions of a numerical model based on the continuity of transfer between the unsaturated and saturated zones. The validity of the model is proved by the excellent agreement between simulated and experimental results. Furthermore, comparison with the results obtained by a viscous flow analog reveals the errors associated with the traditional free surface approach, which neglects transfer in the unsaturated zone.

Experimental and numerical study of a transient, two‐dimensional unsaturated‐saturated water table recharge problem

Experimental results dealing with flow processes involving hysteresis effects in the suction-water content relationship were obtained on a vertical column of soil submitted to the following successive controlled flow conditions: (1) redistribution of water following a constant head infiltration in an initially air-dried column. (2) constant head infiltration followed by redistribution, and (3) constant flux infiltration followed by controlled evaporation and redistribution.

Water contents were measured by using gamma ray attenuation, and water suctions were obtained simultaneously from 10 pressure transducers distributed along the soil column, each one being connected to a tensiometer. A data acquisition system permitted fully automated measuring sequences. All the data were punched on tape and treated with a computer.

Examination of the changes of water content and water suction at a given depth permitted us to obtain precise information on the hysteretic behavior of Ψ (θ). This analysis showed that the scanning curves for a single reversal can be defined uniquely with reference to the transition water content θ*, that the independent domain theory appears to be inadequate for describing soil water hysteresis completely, and that k(θ) can be considered practically unique.

Georges Vachaud

Papers

A Comparison of Numerical Simulation Models For One-Dimensional Infiltration1

Experimental and numerical study of a transient, two‐dimensional unsaturated‐saturated water table recharge problem

Hysteresis During Infiltration and Redistribution in a Soil Column at Different Initial Water Contents

A Test of the Uniqueness of the Soil Moisture Characteristic During Transient, Nonhysteretic Flow of Water in a Rigid Soil1

Modeling water quality in an urban river using hydrological factors – Data driven approaches