Soil Water Retention and Hydraulic Conductivity fitting

This web app implements key water retention models along with our recently proposed Inverse Laplace Transform (ILT) model (Cardinali et al., 2024; Cardinali et al., 2025) for fitting your data. The ILT model has proven effective in capturing complex, multi-modal soil porosity structures, providing a continuous spectrum of pore size distributions. In addition, we have extended our approach to explore both Multi-Exponential and Multi-Gaussian models:

\( \small \begin{array}{c} \text{Exponential:} \quad \theta(h) = \sum\limits_{i=1}^n \omega_i \exp\left( -\frac{h}{h_{0,i}} \right) \\[1em] \text{Gaussian:} \quad \theta(h) = \sum\limits_{i=1}^n \omega_i \exp\left[ -\frac{1}{2} \left( \frac{h}{h_{0,i}} \right)^2 \right] \end{array} \)

where \(\omega\) is the amplitude and \(h_0\) its respective characteristic matric potential, which are physically interpreted as the amount of pores that empty at that suction head. Based on the ILT-Gaussian approach and using cappilary bundle model, the closed-form equation to unsaturated hydraulic conductivity for ILT-gauss is

\( \small K(h) = K_s \,\Theta(h)^\ell \, \left( \frac{ \sum\limits_{i=1}^n \, \frac{\omega_i}{h_{0,i}} \left\{ 1 - \text{erf} \left[ \sqrt{ \text{ln} \left( \frac{1}{\theta_s \, \Theta_{i}(h)} \right) } \right] \right\} }{ \sum\limits_{i=1}^n \, \frac{\omega_i}{h_{0,i}} } \right )^q \)

where \(K_s\) is the saturated hydraulic conductivity, and \(\ell\), \(\beta\), and \(q\) are fitting parameters of the \(K(h)\) function. For all models, the hydraulic conductivity was obtained using general capillary bundle model equation:

\( \small K(\Theta) = K_s \, \Theta^\ell \left[ \frac{\int_0^\Theta \frac{1}{h^\beta (\Theta')} \, d\Theta'}{ \int_0^1 \frac{1}{h^\beta (\Theta')} \, d\Theta'} \right]^q \)

where \(K_s\), \(\ell\), \(\beta\), and \(q\) are fitting parameters, with \(\ell = 0.5\), \(\beta = 1\), and \(q = 2\) when fixed (Mualem).

More details are available in our papers (see citation). Please remember to cite our work if you use this webapp.

Questions? Watch the tutorial video: https://www.youtube.com/watch?v=UKdAfjYKiS4

Supporters

M. C. B. Cardinali, G. V. Von Atzingen, T. B. Moraes

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