## 1. Introduction

[2] The general theory of unsaturated flow is partly founded on an understanding of the idealized nonlinear boundary value problem for vertical flow into an initially uniformly dry soil that is instantaneously wetted at the upper surface. The Darcy-Buckingham continuum model leads to the Richards equation, a nonlinear diffusion-convection equation that includes two free functions to represent hydraulic characteristics of a variety of soils. When the dependent variable is chosen to be the volumetric water content, , subject to unidimensional vertical flow,

where *z* and *t* are the depth and time coordinates, is the hydraulic conductivity, and the soil-water diffusivity. The idealized boundary and initial conditions under consideration are,

Here represents the water content at saturation. The initial water content is less than the boundary value. In the following, *K _{s}*,

*K*, , and will be abbreviations for , , , and , respectively. For simplicity, we consider a unique moisture potential, , at which . This ignores the case of positive pressure due to ponding and it also ignores the tension-saturated zone.

_{n}[3] The equivalent depth of water having entered the soil, or cumulative infiltration *i*(*t*), is often expressed as a Philip infiltration series [see *Philip*, 1969],

Within the shifted infiltration , , the cumulative outflow at infinity, has been removed as a contribution to the second term of the infiltration series. The leading coefficient is the sorptivity [*Philip*, 1957a] that depends on the form of the function over the domain , but not on the form of the conductivity . The second and higher infiltration coefficients depend on the form of both functions. Several difficulties remain in predicting the values of the infiltration coefficients. Both *D* and *K* are known to strongly increase with water content. From nonlinear soil-water transport models having this property, explicit calculation of the infiltration coefficients is seldom possible in closed form.

[4] Some insight has been gained from the simple Green–Ampt model that assumes the water content profile to be a step function, separating a tension-saturated zone from a deeper, dryer zone. Driven by a potential gradient between the surface at zero potential and a wetting front at a fixed negative potential, the wetting front progresses deeper into the soil as time increases. It is perceived that the so-called “head at the front” lacks physical meaning except in terms of an average value [*Philip*, 1958]. It may be chosen to match infiltration at early times through sorptivity matching [*Philip*, 1957a]. The Green–Ampt step function water profile results from a limiting case of a soil whose diffusivity is a Dirac delta-function. Unfortunately, this model predicts *S*_{1} to be , which is approximately double that calculated numerically from models with strongly increasing but more realistic continuous functions for and [*Philip*, 1990; *Barry et al.*, 1995; *Ross et al.*, 1996] or inferred from infiltration data [*Talsma*, 1969; *Youngs*, 1995]. In the aftermath of the papers by *Philip* [1992a, 1992b, 1993] three comments debating delta-function soils were published with replies from John Philip [*Barry et al.*, 1994, 1995, 1996; *Philip*, 1994, 1995, 1996]. From *Philip* [1995]: “The topic of ‘delta-function’ soils and their relation to the Green–Ampt model seems to remain in confusion.” More recently, *Barry et al.* [2010] have pointed out that setting a soil's “effective conductivity” to be half the saturated hydraulic conductivity within a Green–Ampt infiltration analysis [*Mailapalli et al.*, 2009; *Rawls et al.*, 1993] has the effect of halving the value of *S*_{1}.

[5] *Barry et al.* [1995] argued that different soil models, each with a limiting delta-function diffusivity and each predicting step function water content profiles, but with different conductivity functions, could have different values for the infiltration coefficients *S _{j}* beyond

*j*= 0. At first this sounds paradoxical, since in the limit of step function water profiles, the cumulative infiltration function

*i*(

*t*) should be determined uniquely by the Green–Ampt parameters that include no information about except its values at the endpoints and . This is a subtle point as its resolution requires knowledge of the infiltration coefficients for one or more one-parameter soil-water models that include a delta-function diffusivity at one limit of the parameter values.

[6] In the absence of realistic soil-water models that allow an explicit closed-form solution of the fundamental boundary value problem (1), (2), an alternative approach has been to use the approximate analytic method devised by Parlange [1971; *Talsma and Parlange*, 1972] and others [*Philip and Knight*, 1974]. In that approach, one leaves the water conservation equation in integral form, within which one may safely make reasonable approximations within integrands without seriously damaging the predicted values of the physical quantities following from water flow. Such derivations are not fully rigorous but they are often useful and they sometimes agree exactly with some fully analytical models. For example, the time-to-ponding obtained in this way by *Parlange and Smith* [1976], assuming constant-flux boundary conditions, agrees with the exactly solvable model calculations of *Broadbridge and White* [1987]. *Parlange et al.* [1982] derive an infiltration relation with a free parameter that approximates soils for which the diffusivity *D* varies rapidly with water content . Mathematically, the situation is approached by considering a delta-function diffusivity. It was claimed that the new parameter introduced by *Parlange et al.* [1982] enables a variation between infiltration according to the well-known Green–Ampt infiltration relation and other infiltration relations that result from assuming a hydraulic conductivity function which behaves differently as the diffusivity approaches a delta-function. A significant result presented by *Parlange et al.* [1982] is an approximate differential equation for the infiltration function *i*(*t*). This infiltration function has some appealing properties, but the question arises as to whether it could result exactly from some soil model with reasonable functions for and .

[7] Some of the above questions on infiltration coefficients may be resolved by referring to a more realistic exactly solvable two-parameter soil-water model for which a series solution was recently constructed by the current authors [*Triadis and Broadbridge*, 2010]. The model includes the nonlinear models of *Broadbridge and White* [1988] and of *Sander et al.* [1988] that were originally used to solve flow problems with constant-flux boundary conditions; the problems were validated numerically by *Broadbridge et al.* [2009]. From the analytically solvable model, infiltration coefficients were constructed unambiguously from a complicated but explicit recurrence relation. Our recent results support the ideas of *Parlange* [1972, 1975] and *Barry et al.* [1995], since along the curves in the two-parameter space that have a limiting delta-function diffusivity, the second infiltration coefficient is not uniquely determined, but it has a limiting value that depends on the form of . Or more directly, it depends on the form of the function that maps the Buckingham soil-water potential to *K*. As usual, and are related through . Furthermore, for the well-accepted model soil with a constant (implying the Gardner soil satisfying with constant), the limiting value of the first infiltration coefficient is , compared to twice that value from the Green–Ampt model. In addition, the exactly solvable model validates the infiltration function obtained by *Parlange et al.* [1982]. This gives us extra confidence to use the approximate analytic approach on a broader class of soil-water models, though we have found it necessary to clarify various features of its earlier formulation.

[8] The analysis also resolves the apparent paradox of how a Green–Ampt-like limit can produce a second infiltration coefficient that is different from that of the Green–Ampt model whose parameters uniquely determine the infiltration parameters. Our calculation shows that the potential at the wet front is not constant but is a function of time, beginning with the traditional Green–Ampt value at *t* = 0 but decreasing in magnitude thereafter. This time-dependent potential at the wet front cannot be calculated without some knowledge of the explicit dependence of *K* on , information that is lacking in the traditional Green–Ampt model.