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

(First edition: 1998, this reprint: 2004). This publication presents an updated procedure for calculating reference and crop evapotranspiration from meteorological data and crop coefficients. The procedure, first presented in FAO Irrigation and Drainage Paper No. 24, Crop water requirements, in 1977, allows estimation of the amount of water used by a crop, taking into account the effect of the climate and the crop characteristics. The publication incorporates advances in research and more accurate procedures for determining crop water use as recommended by a panel of high-level experts organised by FAO in May 1990. The first part of the guidelines includes procedures for determining reference crop evapotranspiration according to the FAO Penman-Monteith method. These are followed by updated procedures for estimating the evapotranspiration of different crops for different growth stages and ecological conditions.

Crop evapotranspiration : guidelines for computing crop water requirements

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

A remote sensing surface energy balance algorithm for land (SEBAL)-1. Formulation

http://cursosihlla.bdh.org.ar/curso%20posgrado-tdi1/3_Bibliografia/1997%20-%20RSE_Pv%20Carlson.pdf

On the relation between NDVI, fractional vegetation cover, and leaf area index

Canopy temperatures, obtained by infrared thermometry, along with wet- and dry-bulb air temperatures and an estimate of net radiation were used in equations derived from energy balance considerations to calculate a crop water stress index (CWSI). Theoretical limits were developed for the canopy air temperature difference as related to the air vapor pressure deficit. The CWSI was shown to be equal to 1 - E/Ep, the ratio of actual to potential evapotranspiration obtained from the Penman-Monteith equation. Four experimental plots, planted to wheat, received postemergence irrigations at different times to create different degrees of water stress. Pertinent variables were measured between 1340 and 1400 each day (except some weekends). The CWSI, plotted as a function of time, closely paralleled a plot of the extractable soil water in the 0- to 1.1-m zone. The usefulness and limitations of the index are discussed.

Canopy temperature as a crop water stress indicator

Canopy temperatures were measured on durum wheat grown in six differentially irrigated plots. Soil water content was measured by using a neutron-scattering technique at two locations within each plot. Water contents, in 20-cm increments to 160 cm, were determined two to five times per week. Using a sliding cubic smoothing technique, we calculated daily water contents and thus water depletion rates for the entire growing season. Canopy temperatures were measured daily between 1330 and 1400 hours. Air temperatures measured at 150 cm above the soil surface were subtracted from the canopy temperatures to form the difference Tc – Ta. The summation of Tc – Ta over time yielded a factor termed the ‘stress degree day’ (SDD). The SDD concept shows promise as an indicator for determining the times and amounts of irrigations. An expression relating evapotranspiration (ET) to net radiation and Tc – Ta was simplified and tested by using ET measurements with a lysimeter. The expression was used to predict water use by wheat in the six plots. Predicted ET and measured water used agreed reasonably well. The expression may be useful in determining amounts of irrigation water to apply.

Wheat canopy temperature: A practical tool for evaluating water requirements

Relations between evaporation coefficients and vegetation indices studied by model simulations

Our research efforts with durum wheat have led to the development of the SDD concept. Its application makes possible crop yield estimates from remotely acquired canopy temperatures and auxiliary air temperature measurements obtained during the period from head emergence to the cessation of head growth. Canopy albedo measurements appear adequate to delineate this critical period, making the technique potentially adaptable to predictions of crop yields by remote-sensing. The trifactor nomograms produced from combinations of the linear regression equations also suggest that the SDD concept may be used for scheduling irrigations by remote-sensing.

https://science.sciencemag.org/content/sci/196/4285/19.full.pdf

Remote-sensing of crop yields.

Analysis of an empirical model for soil heat flux under a growing wheat crop for estimating evaporation by an infrared-temperature based energy balance equation

Robert J. Reginato

Papers

Canopy temperature as a crop water stress indicator

Wheat canopy temperature: A practical tool for evaluating water requirements

Relations between evaporation coefficients and vegetation indices studied by model simulations

Remote-sensing of crop yields.

Analysis of an empirical model for soil heat flux under a growing wheat crop for estimating evaporation by an infrared-temperature based energy balance equation