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

A simple analytic model is proposed which predicts the unsaturated hydraulic conductivity curves by using the moisture content-capillary head curve and the measured value of the hydraulic conductivity at saturation. It is similar to the Childs and Collis-George (1950) model but uses a modified assumption concerning the hydraulic conductivity of the pore sequence in order to take into account the effect of the larger pore section. A computational method is derived for the determination of the residual water content and for the extrapolation of the water content-capillary head curve as measured in a limited range. The proposed model is compared with the existing practical models of Averjanov (1950), Wyllie and Gardner (1958), and Millington and Quirk (1961) on the basis of the measured data of 45 soils. It seems that the new model is in better agreement with observations.

A new model for predicting the hydraulic conductivity of unsaturated porous media

This chapter describes several laboratory methods of determining the hydraulic conductivity and hydraulic diffusivity. Water moves through soil in response to various forces acting upon it. The chemical species water may be transported due to bulk movement of the liquid phase or soil solution, or it may be transported by diffusion relative to the mean motion of the liquid phase. The chapter deals with bulk movement, under isothermal conditions, of the liquid phase in response to mechanical driving forces. However, the transport of water in the gas phase by vapor diffusion will be included in the measured hydraulic conductivity and diffusivity, especially at low water contents. The concept of parameter identification has been applied to the determination of the parameters in the hydraulic conductivity and water retention functions. The method involves the measurement of some aspect of the response of a soil water flow system to a set of applied boundary conditions.

Hydraulic Conductivity and Diffusivity: Laboratory Methods

This report describes the RETC computer code for analyzing the soil water retention and hydraulic conductivity functions of unsaturated soils. These hydraulic properties are key parameters in any quantitative description of water flow into and through the unsaturated zone of soils. The program uses the parametric models of Brooks-Corey and van Genuchten to represent the soil water retention curve, and the theoretical pore-size distribution models of Mualem and Burdine to predict the unsaturated hydraulic conductivity function from observed soil water retention data. The report gives a detailed discussion of the different analytical expressions used for quantifying the soil water retention and hydraulic conductivity functions. A brief review is also given of the nonlinear least-squares parameter optimization method used for estimating the unknown coefficients in the hydraulic models. Several examples are presented to illustrate a variety of program options. The program may be used to predict the hydraulic conductivity from observed soil water retention data assuming that one observed conductivity value (not necessarily at saturation) is available. The program also permits one to fit analytical functions simultaneously to observed water retention and hydraulic conductivity data. The report serves as both a user manual and reference document. Detailed information is given on the computer program along with instructions for data input preparation and sample input and output files. A listing of the source code is also provided.

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The RETC code for quantifying the hydraulic functions of unsaturated soils

. SURFEX is a new externalized land and ocean surface platform that describes the surface fluxes and the evolution of four types of surfaces: nature, town, inland water and ocean. It is mostly based on pre-existing, well-validated scientific models that are continuously improved. The motivation for the building of SURFEX is to use strictly identical scientific models in a high range of applications in order to mutualise the research and development efforts. SURFEX can be run in offline mode (0-D or 2-D runs) or in coupled mode (from mesoscale models to numerical weather prediction and climate models). An assimilation mode is included for numerical weather prediction and monitoring. In addition to momentum, heat and water fluxes, SURFEX is able to simulate fluxes of carbon dioxide, chemical species, continental aerosols, sea salt and snow particles. The main principles of the organisation of the surface are described first. Then, a survey is made of the scientific module (including the coupling strategy). Finally, the main applications of the code are summarised. The validation work undertaken shows that replacing the pre-existing surface models by SURFEX in these applications is usually associated with improved skill, as the numerous scientific developments contained in this community code are used to good advantage.

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The SURFEXv7.2 land and ocean surface platform for coupled or offline simulation of earth surface variables and fluxes

Following the Burdine approach, based on a model developed by Wyllie and Spangler, a theory is presented that develops the functional relationships among saturation, pressure difference, and permeabilities of air and liquid in terms of hydraulic properties of partially saturated porous media. The theory is based only on the capillary pressure-desaturation relationships for porous media. Procedures for determining these hydraulic properties from capillary pressure-desaturation curves are described. Permeabilities to the wetting and nonwetting phases as a function of capillary pressure and saturation are predicted from the experimentally determined hydraulic properties. The results for all media studied are in close agreement with the theory.

R. H. Brooks

Papers

Properties of Porous Media Affecting Fluid Flow