Simple Method for Predicting Drainage from Field Plots 1 J. B. SISSON, A. H. FERGUSON, AND M. TH. VANGENUCHTEN ABSTRACT When the one-dimensional moisture flow equation is simpli- fied by applying the unit gradient approximation, a first-order partial differential equation results. The first-order equation is hyperbolic and easily solved by the method of P. 0. Lax. Three published K(6) relationships were used to generate three analytical solutions for the drainage phase following infiltra- tion. All three solutions produced straight lines or nearly straight lines when log of total water above a depth was plotted versus log of time. Several suggestions for obtaining the required parameters are presented and two example problems are in- cluded to demonstrate the accuracy and applicability of the method. Additional Index Words: Cauchy problem, redistribution, characteristic value problem, hydraulic conductivity, infiltra- tion. Sisson, J. B., A. H. Ferguson, and M. Th. van Genuchten. 1980. Simple method for predicting drainage from field plots. Soil Sci. Soc. Am. J. 44:1147-1152. W ATER DRAINING from soil profiles is an important factor in many contemporary and environmen- tal problems. In the Northern Great Plains this water is responsible for the annual destruction of thousands of hectares of cropland by contributing to the forma- tion and growth of saline seeps (Brown and Ferguson, 1973; Ferguson et al., 1972). In some irrigated areas, subsurface drainage water contributes to river pollu- tion (Wierenga and Patterson, 1972). Estimation of the rate and quantity of drainage water contributing to these problems is essential to finding feasible solutions. But estimating these vari- ables requires predicting the hydrologic behavior of large areas and frequently the characterization of many soils under field conditions. This necessitates the use of simple, yet accurate, models which contain parameters that can be obtained on site as quickly as possible. This paper considers a special class of models based on the assumption of a unit gradient of the total po- tential head. Several studies (cf. Black et al., 1969, and Davidson et al., 1969, among others) have shown that a unit gradient often exists during the redis- tribution and drainage phases when a uniform profile is draining freely in the absence of a shallow water table. Three solutions are presented, each based upon a different conductivity equation. Two example prob- lems are further included to demonstrate the use of the present approach. THEORETICAL CONSIDERATIONS The equation for predicting the one-dimensional flow of water in porous materials is (Taylor and Ash- croft, 1972): 80 3T 9 37 [1] 9 — 0(z,t) is the volumetric moisture content, t = time, z = depth (positive downward), K = K(0) is the hydraulic conductivity, and H - H(6,z) = h(6)-z; i.e., H(hydraulic head) = h (pressure head) — ^(gravitational head). When a unit gradient in the total head H is assumed, 9H/3z = -1, Eq. [1] becomes dz [2] When only the drainage phase is considered, Eq. [2] may be solved subject to the conditions 16 C if z < 0 "<*• " > = W = U if* >o <« where the subscripts c and m denote minimum and maximum obtainable values, respectively. When the profile is saturated initially, O m equals the moisture content at saturation, 0 S . In its general case, however, 6 m may be less than 0 S ; for example, following an irrigation with a flux less than the satu- rated hydraulic conductivity. The initial value problem given by Eq. [2] and [3], also known as a Cauchy or a characteristic value problem, has been the subject of many studies in mathematical and engineering literature (cf. Lax, 1972; Aris and Amundson, 1973, where Aris and Amundson present examples and are an excellent introduction to this subject). The characteristics of Eq. [2] are obtained by solving the following system of ordinary differential equations written in standard form. dz dK/d0 _ Q' [4] The right hand expression implies that d0 must be zero or 9 remains constant for certain values of z and t. Since 0 is constant, dK/dO will be constant and the first two terms in Eq. [4] can be integrated to give z = dK where A = •t = At dK de [5] These results imply that a set of curves (i.e. charac- teristics) propogate from the initial condition Oi- If where 1 Contribution from Rockwell International, Rockwell Han- ford Operations, Richland, Wash. Document RHO-SA-138. Part of a thesis submitted by senior author in partial fulfillment of the M.S. Degree at Montana State University. Received 13 Nov. 1979. Approved 6 Aug. 1980. 2 Senior Soil Physicist, Rockwell International, P.O. Box 800, Richland, WA 99352; Professor, Agronomy Department, Mon- tana State Univ.; and Soil Physicist, U.S. Salinity Lab., River- side, Calif., respectively. 1147