Unified Solution for Infiltration and Drainage with Hysteresis: Theory and Field Test

unsaturated hydraulic conductivity, K, expressed as a function of the soil water content, u, or the soil water Hysteresis has been found in both the hydraulic conductivity, K, pressure head, c, and the relation between u and c must vs. pressure head, c, relationship, and the soil water content, u, vs. c relationship. This limits the application of a unified solution for be specified before analytical or numerical models can infiltration and drainage. A Haines’ Jump model of hysteresis is proaccurately predict water flow during infiltration, evapoposed and combined with the Broadbridge and White form of K(u) ration, or drainage. Unfortunately, because of hystereand the diffusivity, D, relationship, D(u). This allows a unified analytisis, these relationships are not simple functions and show cal solution for infiltration and drainage. This solution accounts for a great deal of variation between wetting and drying hysteresis by allowing the inverse macroscopic capillary length scale, cycles. Hysteresis has been found in both c(u) and K(c) a, to be hysteretic. A method of a priori estimating the hysteretic (Haines, 1930; Staple, 1969; Kool and Parker, 1987; nature of a is proposed and tested. The hysteretic change in a can Jaynes, 1992). Studies suggest that there is little hysterebe estimated from other u(c) hysteresis models and then used in sis in K(u) or it is so slight as to be masked by the error combination with the Broadbridge and White hydraulic functions. of the measurements and can be ignored (Gillham et The predicted hysteresis in a was similar to that obtained from inverse procedures. The unified solution was applied to field-measured soil al., 1976; Topp, 1971). water storage during infiltration and drainage. Neglecting hysteresis Considerable effort has been put into the analysis and resulted in poor prediction of water storage during drainage based description of hysteretic soil hydraulic properties. This on hydraulic parameters estimated from infiltration. This was espehas led to numerous models for describing hysteresis in cially true for drainage with high initial water content. Incorporating u(c) (Gillham et al., 1976; Scott et al., 1983; Mualem, the proposed hysteresis model resulted in prediction error less than 1974, 1984; Kool and Parker, 1987; Parlange, 1976; Homeasurement error. In addition, a single unified inverse procedure garth et al., 1988). These models provide a simple means for estimating hydraulic parameters from combined infiltration and for determining scanning curves from a limited amount drainage measurements can now be developed. of data, such as the main wetting and drying hysteresis curves. The models of Parlange (1976) and Mualem (1984) need only one branch of the loop to predict all A description of water infiltration scanning curves. Viaene et al. (1994) compared different under constant-flux boundary conditions in unsatmodels of hysteresis using 10 measured scanning curves urated soils is fundamental to understanding water baland concluded that the best models were the conceptual ance, irrigation, movement of chemicals and, more genmodels needing two branches for calibration. Simulation erally, transport processes occurring in surface soils. studies carried out by Jaynes (1985, 1992) have shown Analytical solutions of Richards’ equation for constantthat none of the models were consistently better than the flux water infiltration into homogeneous soil profiles others. Numerical simulations of flow during transient have been developed using integral procedures (Parinfiltration and redistribution using a variety of hysterelange, 1972; Philip and Knight, 1974; White et al., 1979), sis models did not differ greatly and agreed reasonably Kirchhoff, Hopf-Cole and Storm transforms (Broadwell with experimental water distribution, even when brodge and White, 1988; Warrick et al., 1990, 1991), the scanning curves were not described very accurately and reciprocal Bäcklund transform (Sander et al., 1988, (Kool and Parker, 1987). 1991; Barry and Sander, 1991). Parkin et al. (1992, 1995) Unfortunately, all the models, empirical or theoretipresented analytical solutions for water storage to a cal, do not allow exact unified analytical solutions of fixed depth based on solutions of Broadbridge and infiltration and drainage, even though the hydraulic White (1988) and Warrick et al. (1990). These analytical models of Broadbridge and White (1988) and Sander solutions are useful for assessing the accuracy of numeriet al. (1988) allow exact solution independently for infilcal models and to estimate soil hydraulic properties by tration and drainage. As a result, completely different inverse procedures (Si et al., 1999). Analytical solutions sets of parameters have to be used for infiltration and can also be used to test various inverse techniques for drainage. This greatly inhibits the use of the analytical uniqueness and identifiability of various hydraulic pasolution and our understanding of the role of hysteresis rameters of interest. The model results of Parkin et al. in the application of infiltration and drainage. In this (1992, 1995) can be used directly to interpret time dopaper, we present a model of hysteresis that connects main reflectometry (TDR) measurements. the analytical solution of infiltration with that of drainTo quantitatively predict the movement of water age, thus allowing a unified solution of both drainage through variably saturated soils, detailed knowledge of and infiltration. We apply the model to field-measured the hydraulic properties of the soil are needed. The water storage during infiltration and drainage. To test the approach, the hydraulic parameters estimated from Soil Science Dep., Univ. of Saskatchewan, Saskatoon, SK, Canada S7N 5A8. Received 15 Feb. 1999.* Corresponding author (gary. [email protected]). Abbreviations: RMS, root mean square error; TDR, time domain reflectometry. Published in Soil Sci. Soc. Am. J. 64:30–36 (2000).