The hydraulic conductivity function, which is required to solve the Richards equation, is difficult to measure. Therefore prediction methods are frequently used where the shape of the conductivity function is estimated from the more easily measured water retention characteristic. Errors in conductivity estimations can arise either from an invalidity of the prediction model for a given soil, or from an incorrect description of the retention data. This second error source is particularly important for soils with heterogeneous pore systems that cannot be adequately described by the usually used retention functions. To describe the retention characteristics of such soils, a flexible θ(ψ) function was formed by superimposing unimodal retention curves of the van Genuchten (1980) type. By combining this retention model with the conductivity prediction model of Mualem (1976), conductivity estimations for soils with heterogeneous pore systems are obtained. Estimated conductivities by this model and the classical van Genuchten‐Mualem method can differ by orders of magnitude. Thus reported disagreements between measured and estimated conductivities may in some cases be due to an inadequate description of the retention data rather than due to a failure of the prediction model.
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: 2013] is not differentiable at suction h5h a [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 h a . (8)). As X(h) is not differentiable at h a , the saturation function S cap is not continuously differentiable if the correction is applied. In cases where the capillary saturation function C(h)[ If the correction discussed under point 2 is applied, closed-form expressions of the conductivity functionbecome unavailable for the models of Kosugi and van Genuchten.4. It is possible that the correction of C(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 S cap ðhÞ5XðhÞCðhÞ 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.
[1] The commonly used models for characterizing hydraulic conductivity of porous media rely on pore bundle concepts that account for capillary flow only and neglect film flow. Experimental evidence suggests that water flow at medium to low water contents in unsaturated porous media can be significantly underestimated by these capillary bundle models. We present a new model that combines a simple film flow function with the capillary flow model of Mualem. This new model can easily be coupled to any water retention function. Moreover, due to its mathematical simplicity, it can easily and efficiently be implemented in existing codes for the numerical solution of unsaturated flow problems. We investigated a set of soil water retention and conductivity data from the literature that all reached dry conditions and were poorly described by existing capillary bundle models. These data were well described with the new model if the model was coupled with an appropriate retention function. Investigation of conductivity data from the UNsaturated SOil hydraulic DAtabase (UNSODA) database showed that, in 75% of all data sets, the new model achieved the best performance using a modified version of Akaike's information criterion. The numeric simulation of an evaporation scenario using Richards's equation showed that by neglecting film flow, the evaporation rate may be underestimated by more than an order of magnitude.Citation: Peters, A., and W. Durner (2008), A simple model for describing hydraulic conductivity in unsaturated porous media accounting for film and capillary flow, Water Resour. Res., 44, W11417,
In this note we derive a closed-form expression representing the hydraulic conductivity function for soils with a multi-modal pore size distribution. By combining the multi-modal representation of the retention function of Durner with the conductivity representation model of Mualem and following van Genuchten, we derive a simple analytical expression for the conductivity of soils with heterogeneous pore systems. Examples for the representation of bi-modal and trimodal retention and corresponding conductivity curves demonstrate the applicability of the simple analytical expressions. It is concluded that the usefulness of the multi-modal representation of retention functions is increased by providing an analytical expression to directly calculate unsaturated hydraulic conductivities.
Knowledge of hydraulic functions is required for various hydrological and plant‐physiological studies. The evaporation method is frequently used for the simultaneous determination of hydraulic functions of unsaturated soil samples, i.e., the water‐retention curve and hydraulic‐conductivity function. All methodic variants of the evaporation method suffer from the limitation that the hydraulic functions can only be determined to a mean tension of ≈ 60 kPa. This is caused by the limited measurement range of the tensiometers of typically 80 kPa on the dry end. We present a new, cost‐ and time‐saving approach which overcomes this restriction. Using the air‐entry pressure of the tensiometer's porous ceramic cup as additional defined tension value allows the quantification of hydraulic functions up to close to the wilting point. The procedure is described, uncertainties are discussed, and measured as well as simulated test results are presented for soil samples of various origins, different textures (sand, loam, silt, clay, and peat) and variable dry bulk density. The experimental setup followed the system HYPROP which is a commercial device with vertically aligned tensiometers that is optimized to perform evaporation measurements. During the experiment leaked water from the tensiometer interior wets the surrounding soil of the tensiometer cup and can lead to a tension retardation as shown by simulation results. This effect is negligible when the tensiometers are embedded vertically. For coarsely textured soils and horizontal tensiometer alignment, however, the retardation must be considered for data evaluation.
The evaporation method is frequently used for simultaneous determination of soil water retention and hydraulic conductivity relationships. Tension is measured at two depths within a short soil column as water evaporates from its surface. Water content and flux are determined by weighing the column. Tensions, water contents, and fluxes are used to derive the water retention curve and the unsaturated hydraulic conductivity function. The measurement range of the conventional procedure is limited on the wet end by the inability of pressure transducers to accurately register very small tension differences. Hence, the resulting calculated hydraulic gradient in the vertical direction is associated with large uncertainties. On the dry end, water cavitation in the tensiometer, which typically occurs around 70 to 90 kPa, is the limitation. We present here a new design based on improved tensiometers that resist cavitation to much higher tensions, some reaching values as high as 435 kPa. On the wet end, data from a simple steady‐state method were used to supplement the evaporation method. On the dry end, applying the new tensiometers enabled the quantification of hydraulic functions up to 293 kPa average tension. Experimental results and soil water simulation affirmed the validity of the linearization assumption, even on the dry end when nonlinear tension–depth profiles emerge. The application of evaporation functions as a supplement for frequent weighing reduces costs for the equipment and increases the effectiveness of the method. Their validity for deriving fluxes was confirmed for the extended range, too. Results are presented for soil samples of different textures (sand, loam, silt, clay, and peat), various origins, and various dry bulk densities.
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