Sorptivity is one of the most important parameters for the quantification of water infiltration into soils.

Soil sorptivity represents the capacity of soil to absorb water by capillarity

The squared sorptivity is related to the flux-concentration function,

These two forms, the diffusivity form,

We can thus conclude that the conductivity form,

In this study, we propose a new mixed formulation that overcomes these problems. We compare it to the approaches commonly used to compute sorptivity, i.e., Eqs. (

The paper is organized as follows. The Theory section presents the proposed mixed formulation. Next, the paper analyzes the precision of the mixed formulation by comparing it with the exact analytical formulation for the case of the maximum sorptivity,

To build the mixed formulation,

Concept of the mixed formulation,

Next, we scale sorptivity to separate the respective contributions of scale and shape parameters, as suggested by

Several options exist for the choice of the intermediate water pressure head

In the following section, the mixed formulation

Regarding the computation of sorptivity for very dry initial conditions with

Illustration of the regular strategies for the estimation of the limits for the case of very dry conditions with the estimation of

Note that, for the sake of clarity, the underline in Eq. (

We propose an alternative procedure that employs a specific integrand to compute the same limit (Fig.

The application of the scaling, Eqs. (

A similar approach is often used with the

The integrand specified by

Illustration of the regular strategies for estimating the limits for the case of saturation

As mentioned above, for

The scaled versions of the formulation

In the following sections, we compare these previously used strategies, based on the use of

The validation of the computation of sorptivity with the proposed mixed formulation

The Brooks and Corey (BC) model

The van Genuchten–Burdine (vGB) model combines the

The van Genuchten–Mualem (vGM) model combines the

The Kosugi (KG) model relates the WR function to the soil pore size distribution assuming lognormal distributions

These models involve the following common scale hydraulic parameters: residual water content,

The application of the scaling procedure Eq. (

BC model:

vGB model:

vGM model:

KG model:

The studied hydraulic models exhibit contrasting and challenging features for the computation of sorptivity, including non-null water pressure heads

In this study, we stress the following points: (i) the studied hydraulic models for WR and HC functions exhibit challenging conditions for the computation of sorptivity; (ii) the proposed mixed formulation is an ideal estimator for sorptivity; (iii) the usual methods, based on the use of

The first step, point (i), involves the study of the selected models with regard to the shapes of WR and HC functions. We computed the WR and HC functions for the four selected models, considering the following values of the WR shape index:

For the third goal, point (iii), the estimations provided by the usual strategies were compared to the estimates provided by the proposed mixed formulation,

Finally, regarding point (iv), we computed the dimensional sorptivity for a synthetic loamy soil,

This section presents the four sets of models for describing the water retention and hydraulic conductivity functions, their features, and their dependency on related shape parameters. The features of the WR-HC functions are determinant with regard to the estimation of sorptivity. The BC model has a non-null air-entry water pressure head (Fig.

Water retention (WR) and hydraulic conductivity (HC) curves for different values of the WR shape index

The computations using

The relative error,

Absolute values of relative errors,

The accuracy of the proposed mixed formulation,

In this section, we compare the estimates provided by the strategies commonly considered (i.e.,

In this example, we investigate the accuracy of the limits of the functions

Relative errors of the regular functions

The convergence of the two functions,

For the other hydraulic models, the estimators

The previous results were presented for specific values of the shape parameters in Fig.

Relative errors of estimators

For the BC model, the estimators

The results obtained in this study also revealed particular behaviors for the different formulations used to compute sorptivity, specifically

In the following, we illustrate the need to use accurate sorptivity estimates when modeling water infiltration into soils. We consider the case of one-dimensional (1D) water infiltration into a loamy soil as a function of initial conditions, i.e.,

The studied synthetic loamy soil was defined by

Water retention

Hydraulic parameters, i.e., values of the residual and saturated water contents,

At first, we illustrate the use of the mixed formulation,

Compute the initial and final saturation degrees,

Compute the intermediate water pressure head,

Compute the intermediate saturation degree from the intermediate water pressure head,

Integrate the lower part of the squared sorptivity

Integrate the upper part of the squared sorptivity

Combine the two parts to compute the mixed formulation

Upscale by multiplying the scaled sorptivity,

For the studied case, with

The related scaled water pressure heads are

The intermediate water pressure head takes the value of

The corresponding saturation degree takes the value of

The computation of the lower part of the squared sorptivity gives

The computation of the upper part of the squared sorptivity gives

The combination of the two parts gives the scaled sorptivity:

The upscaling finalizes the computation of sorptivity:

Then, we computed sorptivity with the mixed formulation,

Another interesting point is the difference between the three models. The BC model exhibits much larger sorptivity values, followed by the vGM model and then the KG model. However, the three models are meant to represent the same water retention and hydraulic conductivity functions, i.e., the same soil. That result shows that the selection of the hydraulic model for the water retention and hydraulic conductivity functions substantially impacts the value of sorptivity, even when the models are fitted to the same data and represent the same soil. This point was already raised by

Given the dependence of sorptivity on the choice of the estimator and the hydraulic model, we expected some implications for water infiltration. To demonstrate this, we modeled water infiltration into the studied loamy soil, with a 30 mm water pressure head at the surface (

Cumulative infiltrations as a function of initial

Before investigating the impact of errors, we evaluated the dependence of the cumulative infiltrations as a function of initial and final conditions. For this analysis, we use the mixed formulation for sorptivity with the extended QEI_ext model. The results are depicted in Fig.

As stated above, either a wrong estimation of sorptivity or a wrong choice of the analytical model (QEI instead of QEI_ext model) may lead to erroneous estimates of the cumulative infiltration. The four combinations of the two models and the two estimates of sorptivity were used to model water infiltration for the case of an initial water pressure head of

The proper calculation of sorptivity is crucial to model accurately water infiltration into soils. However, in some cases, e.g., when the initial state is very dry, or the final state corresponds to saturated conditions, the numerical computation of sorptivity using Eq. (

This study demonstrates that, through the use of the new mixed formulation, it is possible to compute sorptivity easily and very accurately. The proposed formulation presents a very practical tool that may be applied to any type of hydraulic model and any value of initial and final water pressure heads and water contents. The proposed approach allows sorptivity to be computed in all cases, thus improving the modeling of water infiltration into soils and the estimation of soil hydraulic properties. In addition, we used the proposed formulation to investigate the sensitivity of sorptivity to initial and final water pressure heads and to the choice of the hydraulic model chosen to quantify the water retention and unsaturated hydraulic curves. This analysis clearly demonstrated that sorptivity increases with the final water pressure head and decreases with the initial water pressure head. We also showed that a proper estimate of sorptivity is crucial with regard to the modeling of water infiltration into soils and that the selection of the model for the hydraulic functions drastically impacts the computation of sorptivity and, consequently, the final amounts of cumulative infiltrations.

All computations were done using Scilab free software. The scripts for the computation of all the results and figures presented in this paper, and in particular the script for the proposed mixed formulation, can be downloaded online:

No data sets were used in this article.

LL established the question, performed the analytical developments, computed the numerical results, and provided the first draft of the manuscript. PEP confirmed the analytical and numerical developments and wrote the first draft with LL. DY, BL, DMF, SDP, and MR verified parts of the numerical computations. JP and JFG helped with the use of the Kosugi model. RDS and MAN helped with the editing, the layout of the manuscript, and the presentation of the results. All the authors contributed to the editing of the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was performed within the INFILTRON project supported by the French National Research Agency (ANR-17-CE04-010).

This research has been supported by the Agence Nationale de la Recherche (grant no. ANR-17-CE04-010).

This paper was edited by Nunzio Romano and reviewed by two anonymous referees.