Comment on “Simple consistent models for water retention and hydraulic conductivity in the complete moisture range” by A. Peters

Peters [2013] recently presented a new empirical model of soil hydraulic functions over the entire range of soil water potential. His model is based on established approaches for the retention and conductivity function, and assimilates recently gained knowledge about the shape of hydraulic functions in the medium and dry moisture range. Specifically, the retention function reaches a zero soil water content at a water potential corresponding to oven dryness and approaches this point by a linear decrease of water content with the logarithm of suction, which is in agreement with experimental data [Schneider and Goss, 2012] and physical models of water sorption on surfaces [Tokunaga, 2009]. The Peters [2013] retention function does not require more parameters than traditional models, i.e., it is fully parameterized by a saturated water content, a residual water content, and two parameters which describe the location and width of the pore‐size distribution. The unsaturated hydraulic conductivity function represents the flow components capillary flow, film and corner flow, and isothermal vapor flow and requires one additional free parameter as compared to classic models. Peters' model hence offers great potential for modeling soil‐water flow in the full moisture range while simultaneously keeping the number of model parameters at a minimum. The objective of this comment is to reference some shortcomings of the model formulation as published by Peters [2013] and to provide solutions to the following points which we regard as problematic:

1. The function X(h) [‐] defined by equation 3 in Peters [2013] is not differentiable at suction [L], the suction below which X(h) is unity. As a result, the soil water capacity function is not continuous but has a discontinuity at .
2. In cases where the capillary saturation function Γ(h) [‐] does not reach a value of zero at h₀ [L], the suction corresponding to oven dryness, Peters [2013] suggests to compute a corrected function [‐] (see his equation (8)). As X(h) is not differentiable at , the saturation function S_cap is not continuously differentiable if the correction is applied.
3. If the correction discussed under point 2 is applied, closed‐form expressions of the conductivity function K(h) [L T^{– 1}] become unavailable for the models of Kosugi and van Genuchten.
4. It is possible that the correction of Γ(h) is turned on or off during parameter estimation depending on the possible combination of the two parameters describing the pore‐size distribution. Unfortunately, the correction changes a large portion of the retention curve and this may affect the performance of the iterative minimization algorithm by generating spurious secondary minima and discontinuities in the objective function [see Kavetski et al., 2007, for examples from rainfall‐runoff modeling].

In the following, we present solutions to these problems by proposing two modifications of Peters's [2013] retention function and present parameter estimation results for the 10 soils analyzed by him using the new model, which we refer to as Peters‐Durner‐Iden (PDI) model.

The model presented by Peters [2013] expresses soil water content θ [‐] as function of suction h [L] by superimposing an expression for capillary storage and an expression for water storage caused by adsorption of water at solid surfaces. In Peters's [2013] notation, the “residual water content” can be defined as [‐] where [‐] is the saturated water content and w [‐] is a weighting factor. Here, we reformulate Peters's [2013] model of the retention curve as

(1)

where [‐] is the relative saturation of capillary water, and [‐] is the relative saturation of adsorbed water. Thus, the “residual water content” [‐] is the maximum content of adsorbed water. As pointed out by Peters [2013], can be parameterized by classic approaches like those of van Genuchten [1980] and Kosugi [1996] or by models accounting for multimodality of the pore‐size distribution.

Instead of using equation 3 in Peters [2013], we suggest to use a smoothed piecewise linear function to describe the relative saturation of adsorbed water

(2)

where and b [‐] is a smoothing parameter. Note that for h expressed in centimetres, x is the widely used pF value as defined by Schofield [1935]. Equation 2 is a variation of the “smooth plus function” discussed in Chen and Mangasarian [1996]. We refer the reader to Kavetski and Kuczera [2007] for a review of model smoothing techniques in hydrology. Figure 1 (left) shows the function for different values of b. As can be seen, stays at a value of unity for , then shifts smoothly toward a linearly decreasing function for and finally reaches a value of zero at x₀. The smoothness of this shift is controlled by the parameter b: the higher the value of b, the smoother the curve. This is also reflected in the derivative with respect to suction, which is visualized on the right‐hand side of Figure 1 and given by

(3)

Figure 1

(left) Relative saturation of adsorbed water S_ad and (right) its derivative with respect to suction h for different values of the smoothing parameter b.

As opposed to the function X(h) in Peters [2013], the derivative given by equation 3 is continuous and the soil water capacity function can easily be calculated from , i.e., by differentiating equation 1 with respect to h using equation 3. An additional minor advantage of equation 2 is that it does not need the fictitious parameter which was used by Peters [2013].

We have derived a lower and an upper bound for b by sensitivity analysis. The maximum permissible value of b is approximately 0.3 because the derivative given by equation 3 must go toward zero if h approaches zero. A good choice for the minimum value of b is 0.1 because values smaller than 0.1 lead to large maxima of the derivative, equation 3 which can lead to secondary maxima in the capacity function. Sensitivity analyses using the capillary saturation functions of van Genuchten [1980] and Kosugi [1996] enabled us to express b depending on the parameters of the retention function. The criteria for the choice are (i) that no bimodality is introduced to the soil water capacity function in the vicinity of x_a and (ii) that the capacity function is not smoothed too much for porous media with narrow pore‐size distributions. As a practical solution, we propose to scale b in dependence of parameter n of the van Genuchten and parameter σ of the Kosugi function, respectively. A suitable expression for Kosugi's model of capillary saturation is

(4)

where σ is the standard deviation of the lognormal distribution. Given that values for σ range from approximately 0.1 to 3.0, equation 4 results in values of b between 0.1 and approximately 0.3, the range discussed above and also covered by Figure 1. The dependence on guarantees increasing values of b and therefore more smoothing for large values of to avoid secondary maxima in the soil water capacity function around x_a.

By similar reasoning, a suitable expression for b for van Genuchten's model is

(5)

where values of n greater than unity result in values of b between 0.1 and 0.3. As pointed out above, the choice of equations 4 and 5 is empirical and made without physical reasoning, but based on sensitivity analyses using the full possible range of model parameters describing the retention curve. The suitability of this empirical approach is shown in the results below.

Equation 1 states that both S_ad and S_cap must reach zero at h₀ to ensure that soil water content becomes zero at h₀. However, as pointed out by Peters [2013], most models for capillary saturation Γ(h) do not reach zero water content at h₀ in case of a wide pore‐size distribution. Unlike Peters [2013] who solves this problem by multiplying Γ(h) with X(h), we suggest to always rescale Γ(h) by the equation

(6)

which ensures that takes values between zero and unity within the interval . We recommend to always use the rescaled function and to not switch the correction on or off during parameter estimation. Figure 2 illustrates the effect on the relative saturation of capillary water using the van Genuchten model and three parameter sets taken from Carsel and Parrish [1988]. It becomes evident that equation 6 has negligible effect on capillary water retention in sand and silt loam. The rescaling of the silty clay loam curve leads to a moderate change only in the medium to dry range, where adsorbed water constitutes a significant fraction of total water storage and the induced change on is therefore small. There are two advantages resulting from application of equation 6: (i) the derivative of is continuously differentiable and (ii) it is possible to derive closed‐form expressions if Mualem's conductivity model is applied and the models of van Genuchten [1980] and Kosugi [1996] are used to compute Γ(h), because equation 6 rescales Γ(h) linearly.

Figure 2

Relative saturation of water stored in completely filled capillaries S_cap. The dashed lines are the original relative saturation curves calculated with the van Genuchten model. The solid lines are the functions calculated with equation 6. The parameters for the three soil textures are taken from Carsel and Parrish [1988].

The model of unsaturated hydraulic conductivity proposed by Peters [2013] (section 2.1.2) represents the three flow components capillary flow, film and corner flow, and isothermal vapor flow. For the PDI model, the analytical expressions for vapor conductivity and relative film conductivity are identical to those in Peters [2013], i.e., are given by his equations (A1) and (19). As pointed out above, scaling of the retention curve by equation 6 ensures the existence of analytical expressions for . These expressions are, however, not identical to those discussed in the original literature because integration must be done within the limits h₀ to water saturation. For the sake of brevity, we do not present closed‐form expressions in this comment.

We tested the performance of the PDI model by fitting the retention and hydraulic conductivity functions to the 10 data sets analyzed by Peters [2013]. As in Peters [2013], we tested the capillary saturation models of van Genuchten and Kosugi. Model parameters were estimated by nonlinear regression. We used the weighted‐least‐squares objective function given by equation (21) in Peters [2013] and the weights and w_K were set to values identical to those used by Peters [2013]. Since information on column length was not directly available from Peters [2013], we did not use the integral method of Peters and Durner [2006]. We used the trust‐region reflective algorithm by Coleman and Li [1996] which is implemented in Matlab version 7.14 [Matlab, 2012] to minimize .

Figure 3 summarizes the fitting results for Kosugi's capillary saturation function. It becomes obvious that the match between the measured point data and the fitted functions is qualitatively very similar to those visualized in Figures 7–9 in Peters [2013]. This holds for the retention data (shown) and the conductivity data (not shown). The quantitative measures of goodness‐of‐fit which are summarized in Table 1 confirm this when compared to Peters's [2013] results (summarized in his Table 4), with by the PDI model even smaller than Peters's [2013] values in the majority of the 20 cases. The soil water capacity functions of the PDI model shown in Figure 3 are continuous and do not exhibit multimodality or oversmoothing. They appear to be a good basis for numerical simulations. This is also the case if van Genuchten's model of Γ(h) is used (not shown).

Figure 3

Results of fitting the improved model to soils 1–10 from Peters [2013]. Round symbols denote measured water content data, solid black lines the fitted retention functions, and the solid gray lines the soil water capacity functions.

Table 1. Minimum Value of the Objective Function Φ_min and the Root Mean Squared Error of Volumetric Water Content RMSE_θ and Decimal Logarithm of Hydraulic Conductivity RMSE_1gK for the 10 Soils Analyzed

Soil	Kosugi			van Genuchten
	RMSEθ	RMSE1gK	Φmin	RMSEθ	RMSE1gK	Φmin
1	0.0112	0.248	38.2	0.0119	0.285	47.9
2	0.0167	0.220	46.4	0.0172	0.231	50.2
3	0.0115	0.143	25.4	0.0195	0.161	53.5
4	0.0068	0.141	16.9	0.0054	0.136	12.7
5	0.0114	0.337	66.5	0.0108	0.485	108.4
6	0.0082	0.163	10.1	0.0082	0.103	13.3
7	0.0052	0.125	31.1	0.0405	0.059	16.9
8	0.0037	0.045	14.1	0.0036	0.039	13.8
9	0.0042	0.084	21.0	0.0033	0.068	13.2
10	0.0062	0.065	36.2	0.0053	0.059	26.4

Summing up, we have modified Peters's [2013] model of the soil hydraulic properties to guarantee that (i) the soil water retention function is continuously differentiable and (ii) closed‐form equations for capillary conductivity are available for standard capillary saturation functions. The resulting PDI model offers great potential for modeling variably‐saturated water flow while it simultaneously keeps the number of model parameters small and thus honours the principle of parsimony.

Acknowledgments

This study was financially supported within the DFG Research Group FOR 1083 “Multi‐Scale Interfaces in Unsaturated Soil” (MUSIS), grant DU283/10‐1. We thank Andre Peters for providing the soil water retention and conductivity data for soils 1–10. This data are accessible from the sources cited in Peters [2013].