24 Mar 2021
24 Mar 2021
Scaling procedure for straightforward computation of sorptivity
 ^{1}Univ Lyon, Université Claude Bernard Lyon 1, CNRS, ENTPE, UMR5023 LEHNA, F69518, VaulxenVelin, France
 ^{2}GERSLEE, Univ Gustave Eiffel, IFSTTAR, F44344 Bouguenais, France
 ^{3}Civil Engineering Department, Engineering Faculty, Munzur University, Tunceli, Turkey
 ^{4}Manaaki Whenua  Landcare Research, 7640 Lincoln, New Zealand
 ^{5}Department of Regional Geographic Analysis and Physical Geography, University of Granada, Granada, 18071, Spain
 ^{6}Departamento de Suelo y Agua, Estación Experimental de Aula Dei, Consejo Superior de Investigaciones Cientíﬁcas (CSIC), PO Box 13034, 50080 Zaragoza, Spain
 ^{7}Agricultural Department, University of Sassari, Viale Italia, 39, 07100 Sassari, Italy
 ^{8}Department of Soil Science and Engineering, Faculty of Agriculture, University of Maragheh, Maragheh, Iran
 ^{9}Forschungszentrum Jülich GmbH, Institute of Bio and Geosciences: Agrosphere (IBG3), Jülich, Germany
 ^{10}School of Plant and Environmental Sciences, Virginia Polytechnic Institute and State University, Blacksburg, VA, United States
 ^{11}Department of Land, Air and Water Resources, University of California, Davis, CA 95616, United States
 ^{12}University of Montpellier,UMR LISAH, IRD, Montpellier, France
 ^{1}Univ Lyon, Université Claude Bernard Lyon 1, CNRS, ENTPE, UMR5023 LEHNA, F69518, VaulxenVelin, France
 ^{2}GERSLEE, Univ Gustave Eiffel, IFSTTAR, F44344 Bouguenais, France
 ^{3}Civil Engineering Department, Engineering Faculty, Munzur University, Tunceli, Turkey
 ^{4}Manaaki Whenua  Landcare Research, 7640 Lincoln, New Zealand
 ^{5}Department of Regional Geographic Analysis and Physical Geography, University of Granada, Granada, 18071, Spain
 ^{6}Departamento de Suelo y Agua, Estación Experimental de Aula Dei, Consejo Superior de Investigaciones Cientíﬁcas (CSIC), PO Box 13034, 50080 Zaragoza, Spain
 ^{7}Agricultural Department, University of Sassari, Viale Italia, 39, 07100 Sassari, Italy
 ^{8}Department of Soil Science and Engineering, Faculty of Agriculture, University of Maragheh, Maragheh, Iran
 ^{9}Forschungszentrum Jülich GmbH, Institute of Bio and Geosciences: Agrosphere (IBG3), Jülich, Germany
 ^{10}School of Plant and Environmental Sciences, Virginia Polytechnic Institute and State University, Blacksburg, VA, United States
 ^{11}Department of Land, Air and Water Resources, University of California, Davis, CA 95616, United States
 ^{12}University of Montpellier,UMR LISAH, IRD, Montpellier, France
Abstract. Sorptivity is a parameter of primary importance in the study of unsaturated flow in soils. This integral parameter is often considered for modeling the computation of water infiltration into vertical soil profiles (1D or 3D axisymmetric geometry). Sorptivity can be directly estimated from the knowledge of the soil hydraulic functions (water retention of hydraulic conductivity), using the integral formulation of Parlange (Parlange, 1975). However, it requires the prior determination of the soil hydraulic diffusivity and its numerical integration between the initial and the final saturation degrees, which may be tricky for some instances (e.g., coarse soil with diffusivity functions quasiinfinite close to saturation). In this paper, we present a specific scaling procedure for the computation of sorptivity considering slightly positive water pressure heads at the soil surface and initial dry conditions (corresponding to most water infiltration on the field). The square sorptivity is related to the square dimensionless sorptivity (referred to as c_{p} parameter) corresponding to a unit soil (i.e., unit values of all the scaled parameters and zero residual water content) utterly dry at the initial state and saturated at the final state. The c_{p} parameter was computed numerically and analytically for five current models: delta functions (Green and Ampt model), Brooks and Corey, van GenuchtenMualem, van GenuchtenBurdine, and Kosugi models as a function of the shape parameters. The values are tabulated and can be easily used to determine any dimensional sorptivity value for any case. We propose brandnew analytical expressions for some of the models and validate previous formulations for the other models. Our numerical results also showed that the relation between the c_{p} parameters and shape parameters strongly depends on the chosen model, with either increasing or decreasing trends when moving from coarse to fine soils. These results highlight the need for carefully selecting the proper model for the description of the water retention and hydraulic conductivity functions for the rigorous estimation of sorptivity. Present results show the need to understand better the hydraulic model's mathematical properties, including the links between their parameters, and, secondly, to better relate these properties to the physical processes of water infiltration into soils.
Laurent Lassabatere et al.
Status: final response (author comments only)

RC1: 'Comment on hess2021150', Anonymous Referee #1, 05 May 2021
It is a great work where a rescaling procedure is developed to calculate the sorptivity in which, through the calculation of the parameter $ c_p $, different models are compared which start from different hypotheses, making these results remarkable.
Authors are recommended to add and / or replace the work of Brutsaert (1976) with the work of Parlange (1975).
Finally, a general revision is recommended to correct typos.

AC1: 'Reply on RC1', Laurent Lassabatere, 10 May 2021
Dear Reviewer,
The authors warmly thank you for the time dedicated to reviewing our paper and his positive feedback. The authors will check for typos carefully. They will add a few sentences in the introduction about the alternative flux concentration functions available for the computation of sorptivity. In this paper, the authors considered Parlange’s equation for the flux concentration function. However, other flux concentration functions may be used in the integral that defines the sorptivity, like those proposed by Crank (1979), Philip and Knight (1974), Parlange (1975), Brutsaert (1976).
The authors intend to deal with the sensitivity of the computation of sorptivity with regard to the choice of the flux concentration function. The authors will also discuss the proper selection of these functions with regard to the type of soil. All those topics will be the subject of another paper.
Best regards,
Laurent Lassabatere on behalf of the authors.
Brutsaert, W. 1976. The concise formulation of diffusive sorption of water in a dry soil, Water Resources Research, 12(6), 1118–1124.
Crank, J. 1979. The mathematics of diffusion, Oxford university press.
Philip, J. R. a and Knight, J. H. 1974: On solving the unsaturated flow equation: 3. new quasianalytical technique, Soil Science, 117(1), 1–13, 1974.
Parlange, J.Y. 1975. On Solving the Flow Equation in Unsaturated Soils by Optimization: Horizontal Infiltration, Soil Science Society Of America Journal, 39(3), 415–418, 1975.

AC1: 'Reply on RC1', Laurent Lassabatere, 10 May 2021

RC2: 'Comment on hess2021150', Anonymous Referee #2, 20 May 2021
This is an excellent paper that provides scaling equations to estimate sorptivity for a wide range of hydraulic functions as well as initial and final soil moisture status. The mathematical derivation is thorough and accurate to the best of what I was able to follow. I have two main comments and a few minor corrections.
Comment 1. Eq. (22) gives rise to contrasting values of sorptivity for the different hydraulic conductivity functions. The authors attribute this difference to the dependence of the parameter c_{p} on the hydraulic functions (see section 4.4). However, sorptivity as defined in Eq 22 also varies with h_{g} and 2 h_{a}*. Indeed, the authors defined a variable cp' = c_{p}2 h_{a}*. Therefore, consider deriving shape indices for c_{p}'.
Comment 2. What is the value of h_{a}? I suspect it is equal to h_{g} for the Delta and BC models and zero for the others. If that is the case, h_{a}* = 1 for the former two and 0 for the others (see the top of Page 5). Thus, c_{p}' = c_{p } 1 or c_{p}' = c_{p . }
If you plot c_{p}', the curves for c_{p,d}' and c_{p, BC}' in Figure 3 would be lowered by 1 and the in (a) and (c). This would reduce the dissimilarity between the various hydraulic functions a bit.
Other Small Comments
 In the first line of the introduction, verify if sorptivity is actually used for desorption.
 Eq (4), Eq (5), and elsewhere there is no need to show the detailed stepbystep derivation of straightforward algebraic manipulations.
 In the last paragraph of Page 3, rewrite the sentence that starts with "Secondly, ...".
 In the same paragraph as above, define "BEST."
 In the same paragraph as above, introduce hydraulic functions starting with the delta function to be consistent with how the equations are presented.
 Rewrite equation (6) using the Heaviside function since H is defined underneath and later references use H as well.
 Postpone the introduction of the scaling parameters section 2.1, where they are used.
 Consider moving Eq (23) (definition of c_{p}) to just after Eq (15), where c_{p} is initially introduced. Also, provide more information of what assumptions were used by Haverkamp et al. in deriving c_{p}.
 Edit the incomplete first sentence of section 2.2.2.

AC2: 'Reply on RC2', Laurent Lassabatere, 25 May 2021
Dear Reviewer,
The authors thank you for your very positive review and the time dedicated to reviewing the paper.
Detailed answers to comments.
Regarding the two main comments, we will revise the paper to make it clear that sorptivity has two parts, the unsaturated sorptivity that corresponds to cp’ and the saturated part that corresponds to 2ha*. The saturated part is related to the airentry water pressure head ha* that equals unity only for the Delta and the BC functions. In that case, the saturated part equals 2*ha*=2. Note that by convention, for the Delta and BC functions, the scale water pressure head, hg, is taken equal to the air entry pressure head, so that ha*=1. For the other hydraulic functions, ha* = 0, and the saturated part of sorptivity is null. We will implement the reviewer's suggestions on discussing both parts of the sorptivity as a function of the shape indexes and revise Figure 3.
Comment 1 by Reviewer RC2. Eq. (22) gives rise to contrasting values of sorptivity for the different hydraulic conductivity functions. The authors attribute this difference to the dependence of the parameter cp on the hydraulic functions (see section 4.4). However, sorptivity as defined in Eq 22 also varies with hg and 2 ha*. Indeed, the authors defined a variable cp' = cp2 ha*. Therefore, consider deriving shape indices for cp'.
Authors’ answer: In the result section (section 3.2), we plotted the scaled dimensionless sorptivity, cp, as a function of shape indexes. For the Delta and BC functions, the scaled sorptivity lumps the saturated parts, equal to 2 ha* = 2 plus the unsaturated part, cp’. For the other functions, the scaled sorptivity corresponds directly to the unsaturated part, cp’, since ha* and the saturated part of sorptivity are null. We will insist on this point and discuss the evolution of the unsaturated part cp’ as a function of the shape index, as suggested by the reviewer. Note that the computation of dimensional sorptivity with equation 22 requires both the saturated and unsaturated parts of sorptivity.
Comment 2 by Reviewer RC2. What is the value of ha? I suspect it is equal to hg for the Delta and BC models and zero for the others. If that is the case, ha* = 1 for the former two and 0 for the others (see the top of Page 5). Thus, cp' = cp  1 or cp' = cp. If you plot cp', the curves for cp,d' and cp, BC' in Figure 3 would be lowered by 1 and the in (a) and (c). This would reduce the dissimilarity between the various hydraulic functions a bit.
Authors’ answer: The reviewer is correct to state that ha* = 1 by convention for the Delta and BC functions and zero for the other hydraulic functions. Thus, the unsaturated part of the sorptivity equals cp' = cp – 2 for the Delta and BC functions and cp' = cp for the others. We will add cp' for the BC hydraulic functions in Figure 3a and discuss this point in the revised version of the paper.
Detailed answers to the reviewer’s suggestions.
We will carefully search for typos and revise the manuscripts. We will rewrite the inappropriate sentences. More importantly, the reviewer suggested changing the structure of section 2.1 with the presentation of first all the dimensional equations before the scaling procedure. The authors will investigate his proposal and attempt to revise this section.
 Reviewer: In the first line of the introduction, verify if sorptivity is actually used for desorption. Authors: Regarding sorptivity and its definition (line 22 of the manuscript), it corresponds to the following description of sorptivity proposed by Minasmy and Cook (2011): “Sorptivity is a measure of the capacity of the medium to absorb or desorb liquid by capillarity.” The concept of sorptivity can be considered regarding the two sides of the same coin, i.e., water adsorption and desorption, with hysteresis effects.
 Reviewer: Eq (4), Eq (5), and elsewhere there is no need to show the detailed stepbystep derivation of straightforward algebraic manipulations. Authors: We will carefully look at equations and simplify them when necessary. Our goal is to help the reader to retrieve all equations and derivations. When the derivations were relatively simple, all the steps were kept. When derivations were more complicated, we tried to keep only the main steps.
 Reviewer: (i) In the last paragraph of Page 3, rewrite the sentence that starts with "Secondly, ..." // (ii) In the same paragraph as above, define "BEST." // (iii) In the same paragraph as above, introduce hydraulic functions starting with the delta function to be consistent with how the equations are presented. Authors: we will rewrite this paragraph and address these points.
 Reviewer: Rewrite equation (6) using the Heaviside function since H is defined underneath and later references use H as well. Authors: we will rewrite the equation accordingly.
 Reviewer: (i) Postpone the introduction of the scaling parameters section 2.1, where they are used // (ii) Consider moving Eq (23) (definition of cp) to just after Eq (15), where cp is initially introduced. Also, provide more information of what assumptions were used by Haverkamp et al. in deriving cp. Authors: we will rewrite section 2.1 accordingly.
 Reviewer: Edit the incomplete first sentence of section 2.2.2. Authors: we will rewrite the sentence.
Again, the authors thank the reviewer for his suggestions and contributions.
Best regards,
Laurent Lassabatere on behalf of the authors.
Laurent Lassabatere et al.
Model code and software
Scilab script for sorptivity Laurent Lassabatere https://zenodo.org/record/4587160
Laurent Lassabatere et al.
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