Numerical Methods for Estimating the Soil Hydraulic Properties and the Wetting Front in the Soil

M. Hachimi; A. Maslouhi; H. Qanza; K. Tamoh

doi:10.1134/S1064229319110048

Eurasian Soil Science

November 2019, Volume 52, Issue 11, pp 1402–1413 | Cite as

Numerical Methods for Estimating the Soil Hydraulic Properties and the Wetting Front in the Soil

Authors
Authors and affiliations

M. Hachimi
A. Maslouhi
H. Qanza
K. Tamoh

SOIL PHYSICS

First Online: 18 November 2019

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Abstract—

The unsaturated zone in the soil generally plays an important role in the transfer of water and pollutants in the underground environment. To simulate the water flow in this area, the relationships that express the variations of the water content and the hydraulic conductivity of the soil as a function of the pressure must be defined. In this context, determination of the hydrodynamic parameters constitutes an essential step for any study of the water and solutes transfers in the unsaturated zone. The purpose of this article is to identify the soil hydrodynamic parameters by direct method and inverse method, from the infiltration data by using the disc infiltrometer. These two methods of the characterization were applied on two plots: a plot of R’mel, sandy soil, and a M’risa plot, clay-loamy soil in northern Morocco. The results of the analytical and numerical modeling reproduced correctly the experimental measurements. A performance analysis of the models used was implemented to study the reliability of the estimated parameters. The values of RMSE ≈ 0.20 and R² ≈ 0.90 show that the inverse method remains a robust and accurate method for determining the soil hydrodynamic parameters compared to other conventional methods.

Keywords:

unsaturated soil infiltration hydraulic parameters modeling Loukkos Basin

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APPENDIX

Among the analytical solutions for the one-dimensional Richard’s equation, we have chosen that of the Talsma and Parlange (1972). Talsma and Parlange consider unidimensional infiltration with a constant pressure condition h₀ at the infiltration surface and a uniform initial condition of water content given by:

$$\left\{ \begin{gathered} h = {{h}_{0}}\,\,\,\,{\text{for}}\,\,\,\,z = 0\,\,\,{\text{and}}\,\,\,t > 0 \hfill \\ \theta = {{\theta }_{0}}\,\,\,\,{\text{for}}\,\,\,\,z \geqslant 0\,\,\,{\text{and}}\,\,\,t = 0.\, \hfill \\ \end{gathered} \right.$$

(A1)

Talsma and Parlange (1972) propose the following solution:

$${{K}_{s}}t = I - \alpha \left[ {\exp \left( { - \frac{{I - {{K}_{0}}t}}{\alpha }} \right) - 1} \right].$$

(A2)

Where I is the vertical cumulative infiltration [L]; t is time [T]; and K₀ the hydraulic conductivity [L T^–1] corresponding to an initial volumetric water content θ₀; α is the scale parameter.

The equation of Talsma and Parlange is an equation that has been obtained by assuming that the values of the capillary diffusivity of water in the soil D(θ), and the derivative of the hydraulic conductivity dK/dθ, increase rapidly and in a similar way [22, 25].

The infiltration scale parameter α developed by Haverkamp et al. [18], Eq. (A3), permits to take into account a constant pressure head h₀ at the soil surface and a soil texture indicator C_p given by Haverkamp et al. [18].

$$\alpha = \Delta \theta \left( {\frac{{{{K}_{s}}}}{{\Delta K}}{{h}_{0}} - \frac{{{{C}_{p}}}}{2}{{h}_{g}}} \right) = \frac{{{{S}^{2}}\left( {{{\theta }_{0}},\,{{h}_{0}}} \right)}}{{2\Delta K}}.$$

(A3)

Where ${{S}^{2}}\left( {{{\theta }_{0}},\,{{h}_{0}}} \right)$ is sorptivity [L T^–1/2], Eq. (A6), expressed as a function of initial and boundary condition; ΔK = K_s – K₀; Δθ = θ_s – θ₀; h_g = 1/α is a normalization parameter and Cp is a parameter expressed by Eq. (A20).

Replacing the expression of α, Eq. (A3), in Eq. (A2), we find [18]:

$$\begin{gathered} {{K}_{s}}t = I - \Delta \theta \left( {\frac{{{{K}_{s}}}}{{\Delta K}}{{h}_{0}} - \frac{{{{C}_{p}}}}{2}{{h}_{g}}} \right) \\ \times \,\,\left[ {\exp \left( { - \frac{{I - {{K}_{0}}t}}{{\Delta \theta \left( {\frac{{{{K}_{s}}}}{{\Delta K}}{{h}_{0}} - \frac{{{{C}_{p}}}}{2}{{h}_{g}}} \right)}}} \right) - 1} \right], \\ \end{gathered} $$

(A4)

with ΔK = K_s – K₀ and Δθ = θ_s – θ₀.

Therefore, the expression of the cumulative infiltration I(t) becomes:

$$\begin{gathered} I(t) = {{K}_{s}}t - \left( {{{\theta }_{s}} - {{\theta }_{0}}} \right)\left( {\frac{{{{K}_{s}}}}{{{{K}_{s}} - {{K}_{0}}}}{{h}_{0}} - \frac{{{{C}_{p}}}}{2}{{h}_{g}}} \right) \\ \times \,\,\left[ {\exp \left( { - \frac{{I - {{K}_{0}}t}}{{\left( {{{\theta }_{s}} - {{\theta }_{0}}} \right)\left( {\frac{{{{K}_{s}}}}{{{{K}_{s}} - {{K}_{0}}}}{{h}_{0}} - \frac{{{{C}_{p}}}}{2}{{h}_{g}}} \right)}}} \right) - 1} \right]. \\ \end{gathered} $$

(A5)

According to the solution of Parlange [22], the sorptivity is defined by:

$${{S}^{2}}\left( {{{\theta }_{i}}\,,\,{{\theta }_{f}}} \right) = \int\limits_{{{\theta }_{i}}}^{{{\theta }_{f}}} {\left( {{{\theta }_{f}} + \theta - 2{{\theta }_{i}}} \right)} D\left( \theta \right)\,d\left( \theta \right),$$

(A6)

where D(θ) is the capillary diffusivity of water in the soil, defined by:

$$D\left( \theta \right) = K\left( \theta \right)\frac{{dh}}{{d\theta }}.$$

(A7)

To determine the Cp parameter, we will take the van Genuchten (1980) relationship for the water retention curve, θ(h), with the Burdine condition (1953) and the Brooks and Corey (1964) relationship for hydraulic conductivity function K (θ). The function θ(h), Eq. (8), is rewritten as follows:

$$h\left( \theta \right) = {{h}_{g}}{{\left[ {{{{\left( {\frac{{\theta - {{\theta }_{r}}}}{{{{\theta }_{s}} - {{\theta }_{r}}}}} \right)}}^{{ - \tfrac{1}{m}}}} - 1} \right]}^{{\frac{1}{n}}}}.$$

(A8)

Considering θ_r = 0, the derivation of the function h(θ), gives us:

$$\frac{{dh}}{{d\theta }} = - \frac{{{{h}_{g}}}}{{m\,n\,{{\theta }_{s}}}}{{\left[ {{{{\left( {\frac{\theta }{{{{\theta }_{s}}}}} \right)}}^{{ - \tfrac{1}{m}}}} - 1} \right]}^{{\tfrac{1}{n} - 1}}}{{\left( {\frac{\theta }{{{{\theta }_{s}}}}} \right)}^{{ - \tfrac{1}{m} - 1}}}.$$

(A9)

By replacing the equations (9), (A7) and (A9) in equation (A6), we find:

$$\begin{gathered} {{S}^{2}}\left( {0,\,{{\theta }_{s}}} \right) = \int\limits_0^{{{\theta }_{s}}} {\left( {{{\theta }_{s}} + \theta } \right)} \left( { - \frac{{{{K}_{s}}\,{{h}_{g}}}}{{m\,n\,{{\theta }_{s}}}}} \right) \\ \times \,\,{{\left( {\frac{\theta }{{{{\theta }_{s}}}}} \right)}^{{\eta - \tfrac{1}{m} - 1}}}{{\left[ {{{{\left( {\frac{\theta }{{{{\theta }_{s}}}}} \right)}}^{{ - \tfrac{1}{m}}}} - 1} \right]}^{{\tfrac{1}{n} - 1}}}d\theta , \\ \end{gathered} $$

(A10)

Due to the simplicity, we make the following change of variable:

$$\theta {\kern 1pt} * = \frac{\theta }{{{{\theta }_{s}}}}.$$

(A11)

We obtain:

$$\begin{gathered} {{S}^{2}}\left( {0,\,1} \right) \\ = \frac{{{{K}_{s}}\,{{h}_{g}}{{\theta }_{s}}}}{{m\,n\,}}\int\limits_0^1 {\left( {1 + \theta {\kern 1pt} *} \right)} {{\left( {\theta {\kern 1pt} *} \right)}^{{\eta - \tfrac{1}{m} - 1}}}{{\left[ {{{{\left( {\theta {\kern 1pt} *} \right)}}^{{ - \tfrac{1}{m}}}} - 1} \right]}^{{\tfrac{1}{n} - 1}}}d\theta {\kern 1pt} *. \\ \end{gathered} $$

(A12)

Then perform the change of variable $\beta = {{\left( {\theta {\kern 1pt} *} \right)}^{{ - \tfrac{1}{m}}}}$ which give:

$$\begin{gathered} {{S}^{2}}\left( {0,\,1} \right) = - \frac{{{{K}_{s}}\,{{h}_{g}}{{\theta }_{s}}}}{{\,n\,}}{{\int\limits_0^1 {\left( {1 - \beta } \right)} }^{{\tfrac{1}{n} - 1}}}{{\left( \beta \right)}^{{\eta m - \tfrac{1}{n} - 1}}}d\beta \\ + \,\,{{\int\limits_0^1 {\left( {1 - \beta } \right)} }^{{\tfrac{1}{n} - 1}}}{{\left( \beta \right)}^{{\eta m + m - \tfrac{1}{n} - 1}}}d\beta . \\ \end{gathered} $$

(A13)

At this point, we used the Gamma function which is defined by:

$$\Gamma \left( z \right) = \int\limits_0^\infty {{{t}^{{z - 1}}}} {{e}^{{ - t}}}dt,$$

(A14)

with z is a real, as well as the use of the Beta function:

$$\beta \left( {z,\,w} \right) = \int\limits_0^1 {{{t}^{{z - 1}}}} {{\left( {1 - t} \right)}^{{w - 1}}}dt = \frac{{\Gamma \left( z \right)\Gamma \left( w \right)}}{{\Gamma \left( {z + w} \right)}}.$$

(A15)

With these two functions, the expression of the sorptivity becomes:

$$\begin{gathered} {{S}^{2}}\left( {0,\,{{\theta }_{s}}} \right) = - \frac{{{{K}_{s}}\,{{h}_{g}}\,{{\theta }_{s}}}}{n} \\ \times \,\,\left[ {\frac{{\Gamma \left( {m\,\eta - \frac{1}{n}} \right)\Gamma \left( {\frac{1}{n}} \right)}}{{\Gamma \left( {m\,\eta } \right)}} + \frac{{\Gamma \left( {m\,\eta + m - \frac{1}{n}} \right)\Gamma \left( {\frac{1}{n}} \right)}}{{\Gamma \left( {m\,\eta + m} \right)}}} \right]. \\ \end{gathered} $$

(A16)

Among the properties of the Gamma function, we have:

$$\Gamma \left( {z + 1} \right) = z\Gamma \left( z \right).$$

(A17)

By using this property for $z = \frac{1}{n}$, we are getting:

$$\Gamma \left( {\frac{1}{n} + 1} \right) = \frac{1}{n}\Gamma \left( {\frac{1}{n}} \right).$$

(A18)

Hence:

$$\begin{gathered} {{S}^{2}}\left( {0,\,{{\theta }_{s}}} \right) = - {{K}_{s}}\,{{h}_{g}}\,{{\theta }_{s}}\Gamma \left( {\frac{1}{n} + 1} \right) \\ \times \,\,\left[ {\frac{{\Gamma \left( {m\,\eta - \frac{1}{n}} \right)\Gamma \left( {\frac{1}{n}} \right)}}{{\Gamma \left( {m\,\eta } \right)}} + \frac{{\Gamma \left( {m\,\eta + m - \frac{1}{n}} \right)\Gamma \left( {\frac{1}{n}} \right)}}{{\Gamma \left( {m\,\eta + m} \right)}}} \right]. \\ \end{gathered} $$

(A19)

Put:

$$\begin{gathered} {{C}_{p}} = \Gamma \left( {\frac{1}{n} + 1} \right) \\ \times \,\,\left[ {\frac{{\Gamma \left( {m\,\eta - \frac{1}{n}} \right)\Gamma \left( {\frac{1}{n}} \right)}}{{\Gamma \left( {m\,\eta } \right)}} + \frac{{\Gamma \left( {m\,\eta + m - \frac{1}{n}} \right)\Gamma \left( {\frac{1}{n}} \right)}}{{\Gamma \left( {m\,\eta + m} \right)}}} \right]. \\ \end{gathered} $$

(A20)

This relationship defines the parameter Cp.

Finally, the sorptivity becomes:

$${{S}^{2}}\left( {0,\,{{\theta }_{s}}} \right) = - {{K}_{s}}\,{{h}_{g}}\,{{\theta }_{s}}\,\,{{C}_{p}}.$$

(A21)

The term C_p is a parameter that depends only on the shape parameters, i.e. it depends solely on the texture of the soil.

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M. Hachimi
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A. Maslouhi
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H. Qanza
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K. Tamoh
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1.Interdisciplinary Laboratory for Natural Resources and Environment, Department of Physics, Faculty of Sciences, Ibn Tofail UniversityKenitraMorocco

Numerical Methods for Estimating the Soil Hydraulic Properties and the Wetting Front in the Soil

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