Recent progress with an analytic nonlinear model has provided the exact infiltration coefficients for realistic soil behaviors with nonsingular hydraulic functions, as well as their exact delta-function diffusivity limits. After some correction and reinterpretation of the approximate analytical method, the exactly solvable model validates some previously obtained approximate infiltration functions. The Green–Ampt infiltration function follows from a delta-function diffusivity limit with a hydraulic conductivity that may be, among other possibilities, a linear function of water content. Just as a linear conductivity function is an overestimate for a realistic soil, the second Philip infiltration coefficient S1 in the Green–Ampt infiltration function is too large due to conductivity being overestimated. Better agreement with experiment (halving the value of S1) is obtained from the analytic nonlinear model, with a limiting delta-function diffusivity and a matching Gardner exponential hydraulic conductivity function. In general, infiltration behavior is determined by the limiting forms of the diffusivity and conductivity relative to one another at the saturated water content, or alternatively, the relationship between the conductivity and soil moisture potential. A new infiltration model demonstrates the possible range of S1 for physically valid limiting conductivity functions. We show that in the delta-function diffusivity limit, the solution behaves as if the potential at the wet front were time dependent, decreasing in magnitude from an initial value at the traditional Green–Ampt level.
 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, Ks, Kn, , 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.
 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.
 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 S1 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 : “The topic of ‘delta-function’ soils and their relation to the Green–Ampt model seems to remain in confusion.” More recently, Barry et al.  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 S1.
Barry et al.  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 Sj 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.
 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 , assuming constant-flux boundary conditions, agrees with the exactly solvable model calculations of Broadbridge and White . Parlange et al.  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.  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.  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 .
 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  and of Sander et al.  that were originally used to solve flow problems with constant-flux boundary conditions; the problems were validated numerically by Broadbridge et al. . 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. , 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. . 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.
 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.
2. Infiltration Function From an Analytic Nonlinear Model
 First, define a convenient capillary length scale and gravity timescale,
Note that this generalizes the sorptive length of a Gardner soil for which, , . This length scale is traditionally denoted as , but that symbol will have another use here. Next, define the dimensionless and normalized variables,
and dimensionless versions of the functions of interest,
 If, as the kernel of a distribution, , where is the one-sided Dirac delta-function, then , and we expect that,
where is a one-sided Heaviside step function, so that for and . Triadis and Broadbridge  considered the solution of equation (1) with identical boundary conditions to those described above. For a diffusivity and conductivity of the form,
an analytically derived series solution was used to determine water content profiles from early times, up to times sufficiently large for the solution to be almost identical to the limiting traveling wave. The model parameters C and can be varied to approximate a range of real soil properties with and . The case has constant, implying a Gardner soil with . The case recovers the model studied by Broadbridge and White .
 Note that, , where is the mean diffusivity, averaged over the water content. When comparing with field measurements, it is often more convenient to use the directly measurable sorptivity S0, rather than the mean diffusivity , to rescale the variables. These are related by,
 Infiltration series coefficients were determined uniquely from an analytic series solution method [Triadis and Broadbridge, 2010] for the model (5), not only for the delta-function diffusivity limit but for all values and for all . When the coefficients are expressed as power series in C – 1, the next three infiltration coefficients beyond sorptivity take the form,
 In the nonlinear limit as , the first seven coefficients, all that were calculated, agree exactly with those given implicitly by
The convergence or validity of the small-time asymptotic series expansions at larger times is a separate issue and no statements are made along such lines.
 Within the context of the model (5), the Green–Ampt infiltration function is seen to be inappropriate as it follows from taking , resulting in a linear conductivity function that is independent of C, even as . Of course the wider range of with may still provide excellent approximations to real soils using the leading order approximation ; Barry et al.  inferred a value from Talsma's measurement [Talsma, 1969]. However, this measurement is also consistent with a value provided , which is more realistic than the leading-order approximation . For example, if , Talsma's measurement implies .
 We note with interest Basha , where an infiltration equation with the same functional form as (7) was derived using a traveling wave approximation for a model soil with properties given by equation (5) with and . Agreement with the present work on delta-function soils is observed by taking the limit as in equation (45) of Basha , which correctly reproduces equation (8).
 Although the infiltration series given implicitly in (7) was derived within an approximate analysis based on water conservation equations in integral form, it agrees with an integrable nonlinear model. Therefore, we are encouraged to re-examine the approximate analytical method and to derive from it other useful results for which the exactly solvable model gives validation and increased confidence.
 The analytically solvable model given by (5) obeys the relation
for all . This can be used with the second of (10) to evaluate the integral of equation (11). The result is,
In section 2, analytic solutions already indicated that after taking the limit within the integrable nonlinear model, the exact infiltration function and that obtained from the approximate analytical method are one and the same. Indeed, (13) is readily integrated to give the implicit relation (7) for cumulative infiltration.
 The analytic nonlinear model with parameter and the model of Parlange et al.  with parameter considered in Appendix A, give identical results because in the delta-function diffusivity limit, only the limiting behavior of in the neighborhood of has any effect on infiltration, and the models differ only for .
 We can make progress with the direct integration of equation (11) without considering explicit forms of the diffusivity or conductivity. Differentiating with respect to yields an ordinary differential equation with only and occurring explicitly,
This is readily integrated, revealing an implicit relation for ,
As , the integrand above and approach zero as expected. As from above, we expect singular behavior of the integrand as , which must correspond to . Note that,
so there are no issues with convergence of the integral due to behavior as . In some cases during the derivation of the final form of equation (15), it is convenient to evaluate the integral with respect to at (11) or (14).
 Provided the final forms of (11) and (15) are convenient to work with for particular and functions, they can be used to furnish implicitly using the infiltration rate as a natural parameter with .
4. Infiltration Coefficients by Direct Integration
 We now consider the Philip infiltration series in the form,
Let us consider a series expansion of (11) to highlight the terms of successive order for small . Using a binomial series gives,
The powers of may be treated systematically,
The coefficients are defined according to the rearrangement of the series above. They are dependent on k and the set of coefficients . It is apparent that and we may derive the following relation for :
so that the Wn coefficients can be computed recursively as functions of the qi, making use of those already evaluated.
 Equating terms in (17) quickly yields , consistent with the definition of the sorptivity. For we have the general result,
Equation (18) may be used to evaluate many terms of the infiltration series. Without resorting to computation we can derive,
The first few terms of the infiltration series in the delta-function diffusivity limit are now apparent:
Note that in general, as .
 If we consider the analytic nonlinear model through (12) and (10), it is a simple matter to verify that
As and the Green–Ampt equation is approached, . As , so that , . The first two infiltration coefficients of interest are,
5. Wetting Front Potential
 It will be instructive to consider the Buckingham soil moisture potential,
In the context of the present discussion for soils where infiltration may be approximated by plug flow, we can introduce , the nominal soil moisture potential at the wetting front . In general, the rate of change of the total infiltration is given by,
[Mein and Larson, 1971, 1973]. Eliminating from the equation above using the approximation (A3) and converting to dimensionless variables, we can investigate the wetting front potential using,
where . From the leading order behavior of , it is apparent that at .
 When is very slow to approach a step function relative to the approach of to a delta-function, we may replace in the integrand of (18) by its limiting value 1 at . The integral of (18) then becomes trivial, and the resulting first order differential equation may be solved reproducing (9), the Green–Ampt infiltration equation. Using the simplified form of (18) in (36) we immediately verify that remains constant for .
 This extreme case is equivalent to the approximation of by K1 in equation (10) of Philip . Thus, the analysis of Philip , in fact, only considers a special class of functions, and the Green–Ampt result is not the only result possible for the particular delta-function diffusivity soil considered.
 Let us consider the other extreme, where is very quick to approach a step function relative to the approach of to a delta-function. Here may be replaced by 0 in the integrand of (18) resulting in . The same gravity-free equation is reproduced via the approach of Philip  by assuming that is very close to a step function for , so that should be replaced by 0 rather than K1 in equation (10) of Philip . Using in equation (36), it appears as though . However, this situation effectively corresponds to having identically in the delta-function diffusivity limit. In this gravity-free case, as and the equation used to derive is invalid. When the conductivity does have an effect, as and we expect to be monotonically increasing with,
so that or can be arrived at implicitly using (11) or (15) parametrically with .
 Alternatively, substituting the series form of into equation (23) we arrive at a series representation of the dimensionless wetting front potential,
 For the important case , the integral of (11) can be evaluated and the result combined with equation (8) to obtain ; this can be used to eliminate in (23). Solving the resulting quadratic for before substituting into (8) then yields a single implicit equation for .
 Returning to the moisture potential as a function of , we consider the integration of equation (22):
In the Green–Ampt limit with slow to approach a Heaviside function we immediately have,
At the other extreme, for slow to approach a delta-function, and assuming , we must set in the integrand. We therefore expect for .
where . In agreement with Barry et al. , the Green and Ampt  solution is not the only possibility for a soil with delta-function diffusivity but the infiltration coefficients depend on the form of the single function . As shown below, this gives possibilities much broader than the one-parameter form (7) of the infiltration function.
 When considering the analytic nonlinear model, equation (26) may be integrated even for using the second of (10) as the limiting form of . The result is,
As will be the case in general, the limiting form of is a step function with .
 With regard to , one can use equation (13) to eliminate occurrences of in (7). Subsequently, using (23) to eliminate occurrences of gives an equation which can be solved explicitly for ,
Substituting this expression for into equation (7) then gives a single implicit equation for .
Note that both (30) and (32) give the same value of when . The distinct functional forms of (30) and (32) represent no contradiction when considering (29), as the resulting functions are identical over the integration range from to 0.
6. A Model With Realistic Conductivity
 For the models considered so far, the limit has only applied for . However, Philip [1995, 1996] rightly argues that the conductivity of a delta-function soil must approximate zero for . In this section, we present a new model to demonstrate that a range of infiltration behaviors different from that of equation (8) is possible with a valid limiting conductivity function. The model below is of the general form allowing to approach in a realistic manner while the approach of to a delta-function is determined by the implied smooth relationship.
 The model incorporates a free parameter, . The resulting cumulative infiltration approaches equation (8) as , the Green–Ampt function (9) as , and the gravity-free case discussed in section 5 as .
 We consider the functional form below, where the coefficient is fixed by the satisfaction of (4),
 Despite not being expressible in terms of elementary functions, the equation resulting from (15) nonetheless involves nothing more complex than the dilogarithm function (see Lewin  for definitions, identities, and integrals),
Deriving the above result by hand is somewhat tedious, though it is easily verified numerically. We observe that as and as .
 To derive the soil moisture potential, equation (26) is easily integrated, giving
 To facilitate direct comparison with the nonlinear model of Triadis and Broadbridge , the new model above was designed to produce continuous variation from the gravity-free case to the Green–Ampt result according to the single parameter . Specifying determines the value of the single free parameter or , fixing all with for the model in question. It is easy to imagine multiparameter generalizations of the above models derived through more complicated functional forms.
 One of the conceptual clarifications that results from the present work is the time or depth-dependent nature of , the nominal moisture potential at the wetting front. We find that only remains at the initial value as and the Green–Ampt model is approached. For all other values of , as . Figure 1 illustrates this, and shows expected divergence between the two models at intermediate values of and larger times.
 From equation (26), we see that a particular model of type fixes the relationship between the moisture potential and the hydraulic conductivity, . Figure 2 shows the functions that result for the two models considered, with various values of and . With the -axis scaled exponentially, the diagonal dashed line shows the relationship, , common to both models when . Wetting curve data points for Adelaide Dune Sand from Talsma  are also shown in Figure 2. Parameter values of and provides a reasonable fit using the model of section 6. As and , approaches 0 for the new model introduced. In the Green–Ampt limit as and , for as shown by the horizontal dotted line.
 In agreement with Barry et al. , there is not a single delta-function diffusivity limit for infiltration subject to concentration boundary conditions, but the infiltration function depends on the form of a single additional function, which may be taken to be the hydraulic conductivity as a function of Buckingham soil-water interaction potential (or suction head). In order to demonstrate the consequences of this effect, we have used an analytically solvable model of Triadis and Broadbridge  that is parameterized by two dimensionless parameters C and . The value ensures an exponential form of as in the well-accepted Gardner model. With , the limit results in delta-function diffusivity, step function conductivity as a function of water content, and step function water content profiles. However, the infiltration function is not that of the standard Green–Ampt model. In fact, the limiting cumulative infiltration is given exactly by the implicit equation (8), originally derived by Talsma and Parlange  using an approximate integral analysis. The infiltration function may be formally expressed explicitly as a Lambert W-function but that is defined anyway by an implicit equation [Parlange et al., 2002]. The parametric restriction recovers the model of Broadbridge and White  and this too gives the same infiltration function as the Gardner model, in the limit . By way of contrast, the infiltration function of the original Green and Ampt  model results from taking the limit with . Setting results in a linear conductivity function that is insensitive to the parameter C that governs singular behavior of the diffusivity. This means that the traditional Green–Ampt model, like the linear convection-diffusion model, overestimates the conductivity and therefore overestimates the infiltration coefficient S1 in the Philip infiltration series . In fact, the Green–Ampt value for S1 is exactly twice that predicted by the more realistic Gardner model in the limit of delta-function diffusivity. The latter prediction agrees much better with the measurements [e.g., Talsma, 1969]. The relation (8) should be used as a first approximation for cumulative infiltration in the limit of delta-function diffusivity. The Green–Ampt approach allows us to evaluate the potential at the wet front from the cumulative infiltration function. However, for realistic models, the potential at the wet front depends on time, unlike the traditional Green–Ampt model. For all models with a delta-function diffusivity limit, the potential at the wet front has an initial value, , proportional to the capillary length, with a proportionality constant very close to 1 in most cases. This value increases toward 0 as time increases.
 Validation of the implicit relations (7) and (8), by considering the delta-function diffusivity limit of an exactly solvable realistic model, gives us extra confidence to apply the approximate integral analysis of Parlange et al.  to new model soils. Equation (7) is identical to equation (13) of Parlange et al.  if one equates their parameter to our . A closer inspection reveals that the parameters represent distinct soil models that share a common limiting behavior at the saturated water content. The new model presented demonstrates that a range of values of S1 differing from the limit of Gardner's model can arise from soils with realistic hydraulic conductivity functions and smooth relationships that more closely approximate experimental data.
 For soil properties that are not as well represented by a delta-function diffusivity, we have provided infiltration coefficients that follow uniquely from the complicated explicit recurrence relation of Triadis and Broadbridge  with . For example, the measurement of S1 by Talsma  is consistent with the model of Broadbridge and White  with . These exact infiltration coefficients have been conveniently expressed as corrections to those of (7) that follow from a delta-function diffusivity ( ).
Appendix A:: Re-examining the Approximate Infiltration Relation
 Here we present an alternative derivation of the main results of Parlange et al.  in the hope of making them accessible to a wider audience.
 The Richards equation may be expressed in the form (1) or equivalently as,
Since the initial condition is and the disturbance propagates by a nondegenerate diffusion-convection equation from the boundary at z = 0, we assume , as . We assume that as a delta-function diffusivity is approached through a sequence of smooth physically relevant diffusivity functions, the soil-water content profile, through a sequence of smooth profiles, will approach a sharply defined front located at such that for and for . The sharply defined front in the soil-water content profile is clearly the form of the traveling wave solution expected as . Philip [1957b] has demonstrated the link between a concave conductivity function and a stable traveling wave solution and we will, accordingly, only consider smooth, concave, monotonically increasing functions. For a selection of realistic soils whose diffusivity functions approach a delta-function, we would expect the hydraulic conductivity to take the form of successively steeper functions that approach a Heaviside function with a jump from Kn to Ks located at .
 We proceed by multiplying equation (A1) by and integrating with respect to from to . The left-hand side of equation (A1) can be reduced to a single integral by interchanging the order of integration and using ,
Considering the last term of equation (A5), we note that the integrand is proportional to , so that as we approach a delta-function diffusivity the term (a linear function of the integration variable ) may be replaced by 1. However, we cannot similarly replace the conductivity by in the integrand, as is expected to approach a step function as the diffusivity approaches a delta-function, so that the limit of the combined integrand is ambiguous. Equation (A5) becomes
 The parameter initially appears in equation (10) of Parlange et al. , which is introduced to attempt to interpolate between the Green–Ampt result and the limiting case resulting from when and are proportional.
Assuming a necessary step function jump at so that , a more correct form is,
Note that for singular behavior at the sensible convention is to use one-sided delta-functions and a matching one-sided step function where and . In keeping with this convention and to elicit the required values at , we have used above instead of . Differentiating this limiting form with respect to we arrive at an appropriate replacement for (A8):
 By directly solving equation (11) we can verify that any equation of type (A8) with D proportional to can only lead to equation (8). The fact that no continuously varying free parameters are present is seen immediately upon forming dimensionless variables to obtain .
 On the other hand, equation (A11) may be written in dimensionless form as,
This relationship between and is distinct from equation (12) involving , but we can verify that the form of the resulting infiltration function is identical when (11) is solved using (A12) and (A10) rather than (12) and (10). The identity,