The Green–Ampt limit with reference to infiltration coefficients

Authors


Abstract

[1] 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.

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, math formula, subject to unidimensional vertical flow,

display math

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

display math
display math
display math

Here math formula represents the water content at saturation. The initial water content math formula is less than the boundary value. In the following, Ks, Kn, math formula, and math formula will be abbreviations for math formula, math formula, math formula, and math formula, respectively. For simplicity, we consider a unique moisture potential, math formula, at which math formula. This ignores the case of positive pressure due to ponding and it also ignores the tension-saturated zone.

[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],

display math

Within the shifted infiltration math formula, math formula, 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 math formula over the domain math formula, but not on the form of the conductivity math formula. 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 S1 to be math formula, which is approximately double that calculated numerically from models with strongly increasing but more realistic continuous functions for math formula and math formula [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 S1.

[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 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 math formula except its values at the endpoints math formula and math formula. 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 math formula. 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 math formula 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 math formula and math formula.

[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 math formula. Or more directly, it depends on the form of the function that maps the Buckingham soil-water potential math formula to K. As usual, math formula and math formula are related through math formula. Furthermore, for the well-accepted model soil with a math formula constant (implying the Gardner soil satisfying math formula with math formula constant), the limiting value of the first infiltration coefficient is math formula, 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 math formula, information that is lacking in the traditional Green–Ampt model.

2. Infiltration Function From an Analytic Nonlinear Model

[9] First, define a convenient capillary length scale and gravity timescale,

display math

Note that this generalizes the sorptive length of a Gardner soil for which, math formula, math formula. This length scale is traditionally denoted as math formula, but that symbol will have another use here. Next, define the dimensionless and normalized variables,

display math

and dimensionless versions of the functions of interest,

display math

[10] If, as the kernel of a distribution, math formula, where math formula is the one-sided Dirac delta-function, then math formula, and we expect that,

display math

where math formula is a one-sided Heaviside step function, so that math formula for math formula and math formula. Triadis and Broadbridge [2010] considered the solution of equation (1) with identical boundary conditions to those described above. For a diffusivity and conductivity of the form,

display math

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 math formula can be varied to approximate a range of real soil properties with math formula and math formula. The case math formula has math formula constant, implying a Gardner soil with math formula. The case math formula recovers the model studied by Broadbridge and White [1988].

[11] Note that, math formula, where math formula 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 math formula, to rescale the variables. These are related by,

display math

where h(C) is the function defined implicitly by equation (19) of Broadbridge and White [1988],

display math

[12] The factor math formula, which is the parameter “b” evaluated for more general models by Warrick and Broadbridge [1992], varies between math formula for a constant diffusivity, and math formula for a delta-function diffusivity. The narrow range of this factor improves the robustness of predicted measurable integral quantities as by White and Broadbridge [1988], White and Sully [1987], and Warrick and Broadbridge [1992]. A measurement of C for several well-packed soils gave values between 1.02 and 1.2, from which the factor math formula varied between 0.508 and 0.56 [White and Broadbridge, 1988].

[13] Infiltration series coefficients were determined uniquely from an analytic series solution method [Triadis and Broadbridge, 2010] for the model (5), not only for the math formula delta-function diffusivity limit but for all values math formula and for all math formula. When the coefficients are expressed as power series in C – 1, the next three infiltration coefficients beyond sorptivity take the form,

display math
display math
display math

[14] In the nonlinear limit as math formula, the first seven coefficients, all that were calculated, agree exactly with those given implicitly by

display math

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.

[15] Equation (7) is identical to equation (13) of Parlange et al. [1982] provided one equates math formula and their parameter math formula. As stated by Parlange et al. [1982], it has been verified that when math formula, equation (7) has the simple limiting form,

display math

[Talsma and Parlange, 1972], whereas considering the limit as math formula results in the familiar equation attributed to the Green and Ampt [1911] model,

display math

Considering the limit as math formula in (5),

display math

[16] Within the context of the model (5), the Green–Ampt infiltration function is seen to be inappropriate as it follows from taking math formula, resulting in a linear conductivity function that is independent of C, even as math formula. Of course the wider range of math formula with math formula may still provide excellent approximations to real soils using the leading order approximation math formula; Barry et al. [1995] inferred a value math formula from Talsma's measurement math formula [Talsma, 1969]. However, this measurement is also consistent with a value math formula provided math formula, which is more realistic than the leading-order approximation math formula. For example, if math formula, Talsma's measurement implies math formula.

[17] We note with interest Basha [2011], 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 math formula and math formula. Agreement with the present work on delta-function soils is observed by taking the limit as math formula in equation (45) of Basha [2011], which correctly reproduces equation (8).

[18] 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.

3. Extended Implicit Analysis

[19] Appendix A gives an alternative derivation of equation (9) of Parlange et al. [1982], which we write in the simpler dimensionless form,

display math

[20] The analytically solvable model given by (5) obeys the relation

display math

for all math formula. This can be used with the second of (10) to evaluate the integral of equation (11). The result is,

display math

In section 2, analytic solutions already indicated that after taking the limit math formula 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.

[21] The analytic nonlinear model with parameter math formula and the model of Parlange et al. [1982] with parameter math formula considered in Appendix A, give identical results because in the delta-function diffusivity limit, only the limiting behavior of math formula in the neighborhood of math formula has any effect on infiltration, and the models differ only for math formula.

[22] We can make progress with the direct integration of equation (11) without considering explicit forms of the diffusivity or conductivity. Differentiating with respect to math formula yields an ordinary differential equation with only math formula and math formula occurring explicitly,

display math

This is readily integrated, revealing an implicit relation for math formula,

display math

As math formula, the integrand above and math formula approach zero as expected. As math formula from above, we expect singular behavior of the integrand as math formula, which must correspond to math formula. Note that,

display math

so there are no issues with convergence of the integral due to behavior as math formula. In some cases during the derivation of the final form of equation (15), it is convenient to evaluate the integral with respect to math formula at (11) or (14).

[23] Provided the final forms of (11) and (15) are convenient to work with for particular math formula and math formula functions, they can be used to furnish math formula implicitly using the infiltration rate as a natural parameter with math formula.

4. Infiltration Coefficients by Direct Integration

[24] We now consider the Philip infiltration series in the form,

display math
display math

Let us consider a series expansion of (11) to highlight the terms of successive order for small math formula. Using a binomial series gives,

display math

The powers of math formula may be treated systematically,

display math

The coefficients math formula are defined according to the rearrangement of the series above. They are dependent on k and the set of coefficients math formula. It is apparent that math formula and we may derive the following relation for math formula:

display math

so that the Wn coefficients can be computed recursively as functions of the qi, making use of those already evaluated.

[25] Equating math formula terms in (17) quickly yields math formula, consistent with the definition of the sorptivity. For math formula we have the general result,

display math

Equation (18) may be used to evaluate many terms of the infiltration series. Without resorting to computation we can derive,

display math

The first few terms of the infiltration series in the delta-function diffusivity limit are now apparent:

display math

Note that in general, math formula as math formula.

[26] If we consider the analytic nonlinear model through (12) and (10), it is a simple matter to verify that

display math

As math formula and the Green–Ampt equation is approached, math formula. As math formula, so that math formula, math formula. The first two infiltration coefficients of interest are,

display math

5. Wetting Front Potential

[27] It will be instructive to consider the Buckingham soil moisture potential,

display math

In the context of the present discussion for soils where infiltration may be approximated by plug flow, we can introduce math formula, the nominal soil moisture potential at the wetting front math formula. In general, the rate of change of the total infiltration is given by,

display math

[Mein and Larson, 1971, 1973]. Eliminating math formula from the equation above using the approximation (A3) and converting to dimensionless variables, we can investigate the wetting front potential using,

display math

where math formula. From the leading order behavior of math formula, it is apparent that math formula at math formula.

[28] When math formula is very slow to approach a step function relative to the approach of math formula to a delta-function, we may replace math formula in the integrand of (18) by its limiting value 1 at math formula. 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 math formula remains constant for math formula.

[29] This extreme case is equivalent to the approximation of math formula by K1 in equation (10) of Philip [1995]. Thus, the analysis of Philip [1995], in fact, only considers a special class of math formula functions, and the Green–Ampt result is not the only result possible for the particular delta-function diffusivity soil considered.

[30] Let us consider the other extreme, where math formula is very quick to approach a step function relative to the approach of math formula to a delta-function. Here math formula may be replaced by 0 in the integrand of (18) resulting in math formula. The same gravity-free equation is reproduced via the approach of Philip [1995] by assuming that math formula is very close to a step function for math formula, so that math formula should be replaced by 0 rather than K1 in equation (10) of Philip [1995]. Using math formula in equation (36), it appears as though math formula. However, this situation effectively corresponds to having math formula identically in the delta-function diffusivity limit. In this gravity-free case, math formula as math formula and the equation used to derive math formula is invalid. When the conductivity does have an effect, math formula as math formula and we expect math formula to be monotonically increasing with,

display math

[31] Using equation (11) we can write math formula as a function of math formula,

display math

so that math formula or math formula can be arrived at implicitly using (11) or (15) parametrically with math formula.

[32] Alternatively, substituting the series form of math formula into equation (23) we arrive at a series representation of the dimensionless wetting front potential,

display math

[33] For the important case math formula, the integral of (11) can be evaluated and the result combined with equation (8) to obtain math formula; this can be used to eliminate math formula in (23). Solving the resulting quadratic for math formula before substituting into (8) then yields a single implicit equation for math formula.

[34] Returning to the moisture potential as a function of math formula, we consider the integration of equation (22):

display math

In the Green–Ampt limit with math formula slow to approach a Heaviside function we immediately have,

display math

At the other extreme, for math formula slow to approach a delta-function, and assuming math formula, we must set math formula in the integrand. We therefore expect math formula for math formula.

[35] For the important case where math formula, (26) gives

display math

Note that for the cases math formula briefly considered above, and as must be the case in general, math formula.

[36] Using equation (26), equations (11) and (17) become,

display math

where math formula. In agreement with Barry et al. [1995], the Green and Ampt [1911] 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 math formula. As shown below, this gives possibilities much broader than the one-parameter form (7) of the infiltration function.

[37] When considering the analytic nonlinear model, equation (26) may be integrated even for math formula using the second of (10) as the limiting form of math formula. The result is,

display math

As will be the case in general, the limiting form of math formula is a step function with math formula.

[38] With regard to math formula, one can use equation (13) to eliminate occurrences of math formula in (7). Subsequently, using (23) to eliminate occurrences of math formula gives an equation which can be solved explicitly for math formula,

display math

Substituting this expression for math formula into equation (7) then gives a single implicit equation for math formula.

[39] Equation (26) can also be integrated using (A12) and (A10) representing the model of Parlange et al. [1982], to yield,

display math

Note that both (30) and (32) give the same value of math formula when math formula. The distinct functional forms of (30) and (32) represent no contradiction when considering (29), as the resulting math formula functions are identical over the integration range from math formula to 0.

6. A Model With Realistic Conductivity

[40] For the models considered so far, the limit math formula has only applied for math formula. However, Philip [1995, 1996] rightly argues that the conductivity of a delta-function soil must approximate zero for math formula. 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 math formula allowing math formula to approach math formula in a realistic manner while the approach of math formula to a delta-function is determined by the implied smooth math formula relationship.

[41] The model incorporates a free parameter, math formula. The resulting cumulative infiltration approaches equation (8) as math formula, the Green–Ampt function (9) as math formula, and the gravity-free case discussed in section 5 as math formula.

[42] We consider the functional form below, where the coefficient is fixed by the satisfaction of (4),

display math

The differential equation resulting from (11) is,

display math

The solution of the above equation is not trivial, so we consider a series solution for math formula in the manner of section 4:

display math

The function math formula above is the incomplete beta function [see Abramowitz and Stegun, 1965, p. 6.6.1].

[43] Despite not being expressible in terms of elementary functions, the equation resulting from (15) nonetheless involves nothing more complex than the dilogarithm function math formula (see Lewin [1981] for definitions, identities, and integrals),

display math

Deriving the above result by hand is somewhat tedious, though it is easily verified numerically. We observe that math formula as math formula and math formula as math formula.

[44] To derive the soil moisture potential, equation (26) is easily integrated, giving

display math

7. Discussion

[45] To facilitate direct comparison with the nonlinear model of Triadis and Broadbridge [2010], 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 math formula. Specifying math formula determines the value of the single free parameter math formula or math formula, fixing all math formula with math formula for the model in question. It is easy to imagine multiparameter generalizations of the above models derived through more complicated math formula functional forms.

[46] One of the conceptual clarifications that results from the present work is the time or depth-dependent nature of math formula, the nominal moisture potential at the wetting front. We find that math formula only remains at the initial value math formula as math formula and the Green–Ampt model is approached. For all other values of math formula, math formula as math formula. Figure 1 illustrates this, and shows expected divergence between the two models at intermediate values of math formula and larger times.

Figure 1.

Variation of the nominal moisture potential at the wetting front math formula with time for various values of math formula.

[47] From equation (26), we see that a particular model of type math formula fixes the relationship between the moisture potential math formula and the hydraulic conductivity, math formula. Figure 2 shows the math formula functions that result for the two models considered, with various values of math formula and math formula. With the math formula-axis scaled exponentially, the diagonal dashed line shows the relationship, math formula, common to both models when math formula. Wetting curve data points for Adelaide Dune Sand from Talsma [1970] are also shown in Figure 2. Parameter values of math formula and math formula provides a reasonable fit using the model of section 6. As math formula and math formula, math formula approaches 0 for the new model introduced. In the Green–Ampt limit as math formula and math formula, math formula for math formula as shown by the horizontal dotted line.

Figure 2.

Variation of math formula with math formula for math formula. The dashed line shows the case for math formula, math formula, which appears as a straight line with the math formula axis scaled logarithmically. Data points for Adelaide Dune Sand were taken from Talsma [1970].

8. Conclusion

[48] In agreement with Barry et al. [1995], 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 math formula 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 [2010] that is parameterized by two dimensionless parameters C and math formula. The value math formula ensures an exponential form of math formula as in the well-accepted Gardner model. With math formula, the limit math formula 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 [1972] 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 math formula recovers the model of Broadbridge and White [1988] and this too gives the same infiltration function as the Gardner model, in the limit math formula. By way of contrast, the infiltration function of the original Green and Ampt [1911] model results from taking the limit math formula with math formula. Setting math formula 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 math formula. 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 math formula 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, math formula, proportional to the capillary length, with a proportionality constant very close to 1 in most cases. This value increases toward 0 as time increases.

[49] 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. [1982] to new model soils. Equation (7) is identical to equation (13) of Parlange et al. [1982] if one equates their parameter math formula to our math formula. 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 math formula relationships that more closely approximate experimental data.

[50] 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 [2010] with math formula. For example, the measurement of S1 by Talsma [1969] is consistent with the model of Broadbridge and White [1988] with math formula. These exact infiltration coefficients have been conveniently expressed as math formula corrections to those of (7) that follow from a delta-function diffusivity ( math formula).

Appendix A:: Re-examining the Approximate Infiltration Relation

[51] Here we present an alternative derivation of the main results of Parlange et al. [1982] in the hope of making them accessible to a wider audience.

[52] The Richards equation may be expressed in the form (1) or equivalently as,

display math

through explicit consideration of math formula rather than, math formula (see, for example, Philip [1969]). Integrating this equation from initial water content math formula to math formula then gives equation (4) as written by Parlange et al. [1982],

display math

Since the initial condition is math formula and the disturbance propagates by a nondegenerate diffusion-convection equation from the boundary at z = 0, we assume math formula, as math formula. 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 math formula such that math formula for math formula and math formula for math formula. The sharply defined front in the soil-water content profile is clearly the form of the traveling wave solution expected as math formula. 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 math formula functions. For a selection of realistic soils whose diffusivity functions approach a delta-function, we would expect the hydraulic conductivity math formula to take the form of successively steeper functions that approach a Heaviside function with a jump from Kn to Ks located at math formula.

[53] We proceed by multiplying equation (A1) by math formula and integrating with respect to math formula from math formula to math formula. The left-hand side of equation (A1) can be reduced to a single integral by interchanging the order of integration and using math formula,

display math

returning to (3),

display math

As the diffusivity approaches a delta-function and the soil-water content profile approaches a sharp wetting front at math formula as described above, we expect that

display math

Using the second equation above to simplify (A1) we have,

display math

Using the last of (A3) with the above to simplify equation (A2) we obtain,

display math

Considering the last term of equation (A5), we note that the integrand is proportional to math formula, so that as we approach a delta-function diffusivity the term math formula (a linear function of the integration variable math formula) may be replaced by 1. However, we cannot similarly replace the conductivity math formula by math formula in the integrand, as math formula 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

display math

[54] In agreement with the delta-function diffusivity limit of equation (8) of Parlange et al. [1982], we assume that the sorptivity S0 is given approximately by,

display math

in the limit as math formula becomes proportional to a delta-function (see, for example, Youngs [1968] and Philip and Knight [1974]). Equation (A6) can now be recognized as equation (9) of Parlange et al. [1982].

[55] The parameter math formula initially appears in equation (10) of Parlange et al. [1982], which is introduced to attempt to interpolate between the Green–Ampt result and the limiting case resulting from when math formula and math formula are proportional.

[56] In the delta-function diffusivity limit the general form of the sorptivity is shown in equation (A7). Using this in equation (10) of Parlange et al. [1982] with math formula immediately gives math formula for math formula. Thus, if equation (10) of Parlange et al. [1982] is to remain valid in the delta-function diffusivity limit, considering math formula either contradicts the definition of the sorptivity, (A7), or contradicts the definition of Kn, as math formula.

[57] Second, note the comment below Equation (10) of Parlange et al. [1982] which states that the equation becomes exact if,

display math

If one substitutes this expression for math formula into (A7) defining the limiting value of the sorptivity, the result is, math formula. Alternatively, one can use the above to replace math formula in Equation (8) of Parlange et al. [1982]; here the result contradicts equation (11) of Parlange et al. [1982] for math formula. So, similar to equation (10) of Parlange et al. [1982], the above expression appears to be invalid for math formula.

[58] The form of the original equation (10) of Parlange et al. [1982] in the delta-function diffusivity limit is,

display math

Assuming a necessary step function jump at math formula so that math formula, a more correct form is,

display math

Note that for singular behavior at math formula the sensible convention is to use one-sided delta-functions and a matching one-sided step function where math formula and math formula. In keeping with this convention and to elicit the required values at math formula, we have used math formula above instead of math formula. Differentiating this limiting form with respect to math formula we arrive at an appropriate replacement for (A8):

display math

We can verify that using the above to replace math formula in (A7) causes no contradiction, and substitution into equation (8) of Parlange et al. [1982] correctly reproduces equation (11) of Parlange et al. [1982].

[59] By directly solving equation (11) we can verify that any equation of type (A8) with D proportional to math formula 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 math formula.

[60] On the other hand, equation (A11) may be written in dimensionless form as,

display math

This relationship between math formula and math formula is distinct from equation (12) involving math formula, 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,

display math

is useful when demonstrating this equivalence.

[61] Having gone some way toward validating equation (13) of Parlange et al. [1982], we see that equation (11) of Parlange et al. [1982], which attempts to relate math formula to the integral of math formula, leads to inconsistency. Ambiguity in the definition of math formula has also been discussed in Haverkamp et al. [1990].

Acknowledgment

[62] This work has been supported under project DP1095044 of the Australian Research Council.

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