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PARAMETRIC MODELS OF RAIN INFILTRATION The boundary-value problem of rain infiltra- tion, as formulated in the previous section, is an idealized mathematical modei in which the fiow equation used is the Richards equation (1931), The analytical or numerical solutions of the Richards equation. linearized or nonlinear, have undoubtedly promoted our knowledge on the mechanics of soil water movement associated with the rainfall.runoff process, but in the past have met with limited application due partly to their time-consuming computations, even with the help of a modem electronic computer and partly to the unrealistic assumptions imposed on the model. Validity of the Richards equation becomes questionable when a natural soil under investigation is nonisothermal, deformable (1.e" swelling or shrinkable such as in clay) and/or produces a counter flow of air upon watering, as mentioned previously. Although the Richards equation was developed for flow through all soils, homogeneous or heterogeneous, isotropic or anisotropic, saturated or unsaturated, and with or without hysteresis, it will be much simpler and more useful in application to describe the infiltration decay' characteristics by means of a small number of parameters combined in certain forms of algebraic equations than by use of the Richards equation, The algebraic infJItration equations, though mostly developed on the basis of empiricism, have increas. ingly gained wide recognition as modeling tools because of their simplicity, There is a problem, however, to evaluate such model parameters which are not physically based, but are essential to the virtual usefulness In the model. This and other related problems concerning the validity of existing algebraic inftltration equations are discussed herein. Many algebraic infiltration equations have been published in the literature. Among them there are the Green.Ampt equation (Green and Ampt, 1911), the Kostiakov equation (Kostiakov, 1932), the Horton equation (Horton, 1940), the Philip equation (Philip, 1957a), the Holtan equation (Holtan, 1961), The major obstacle that has prevented more effective use of the algebraic inf1ltration equations is the difficulty in the evaluation of their parameter values. Several recent studies for validating some of the algebraic infiltration equations in their application to the rain inf1ltration process include the work of Holtan (197 I), Onstad, Olson, and Stone (1972), Smith (1972), Talsma and Parlange (1972), Papadakis and Preul (1973), and Bauer (1974) among many others, In order to have the algebraic infiltration equations widely accepted as predictive models in the sub- surface runoff computation, reappraisal of the methods for determining their parameter values is necessary, All the algebraic infiltration equations will thus be appraised in terms of theoretical concepts behind their developments, physical interpretations, if any. of the parameters involved, and the accuracy in the prediction of the infiltration rate. All the algebraic inflltration equations were developed for computing the infiltration capacity under a particular condition. The term infIltration rate used in this study is defined in a broad sense as the inf1ltration flux or velocity at any instant rather than as the maximum inf1ltration flux, If the inf1ltra- tion capacity is defined as the maximum inf1ltration flux resulting when water at the atmospheric pressure is made freely available at the soil surface, the infJItration rate for water application (rainfall or irrigation) intensities less than the infiltration capacity becomes equal to the application intensity before water ponding on the soil surface, and greater than or equal to the infJItration capacity after water starts ponding, depending upon whether or not there is a non-zero water depth on the soil surface. Unless the time of ponding that separates the above two distinctive inf1ltration stages under rainfall can also be predicted, the algebraic infJItration equations originally formulated for the maximum inf1ltration flux are hardly applicable to the case of rain inf1ltration. Few attempts have been made to over- come this difficulty,however, For example, Mein and Larson (1971 and 1973) have extended the Green. Ampt equation to the rain infiltration rate computa- tion, while Smith (1970) has used the modified Kostiakov equation to evaluate the time of ponding parametrically, Both approaches are nevertheless limited to the case wherein the ponding depth of water on the soil surface is assumed negligibly small. It appears that a more general approach needs to be developed to remove this and other limitations without loss of simplicity which is demanded in principle by any algebraic infiltration equation, The present section is thus specifically directed to in- vestigate the feasibility of developing such a general approach. 31