Dear Julio,

The linear diffusion equation (which is alike heat equation, plus maybe a first order term - the advection-diffusion equation) is not a good mathematical model for modelling moisture in soils. For this you have to consider non-linear models. A commonly accepted one is the Richards equation, in which the diffusion coefficient (or tensor) is replaced by a nonlinear function depending on the moisture saturation/water content.

Depending on the specific situation you look at, you may also include more complex effects such as hysteresis, or dynamic capillarity,...

One particular feature of such models is that, since the diffusion depends now on the water content, it may become 0 at locations which are completely dry, or infinite at locations where the medium is fully saturated (i.e. all pores are filled by water, no air is present in the medium at that location). This makes the problem become degenerate parabolic and requires proper numerical schemes (e.g. regularisation-based). Also, since the model is non-linear, if you use implicit schemes you may have to find good linearisation schemes to solve the time discrete or fully discrete equation obtained at each time-step.

You may find a lot of such papers on this subject in the literature (including some authored by the undersigned :))

Good luck,

Sorin

Maybe some percolation models can do the trick. See the following:

Article Percolation Theory and its Application for Interpretation of...

Article Finite element/percolation theory modelling of the micromech...

Article Percolation Theory and Network Modeling Applications in Soil Physics

In my opinion, the soils should be described by PDEs and therefore simulated using finite elements methods. Otherwise, you can miss the spatial properties of the soil.

It seems to me that the model will depend on the very problem. For instance, if you investigate just up-down penetration, you may use finite elements models. If dissipation and hydration bonds factors ought to be involved, the only stochastic and probability models seems to be relevant.

It depends on what you want to model. You can use PDEs to implement diffusion equation or cellular automata for simple percolation models. Cellular automata give you much greater freedom in describing the fine structure of the earth. On top of it, CAs are more flexible in the definition of local rules of water propagation through the earth.

In CAs, the diffusion equation of agent V looks in 1D as this

V(i, t+1) = Const [V(i-1, t) -2 V(i, t) + V(i+1, t)]

so the central cell is mixing values V from neighbors and cell itself.

You can add easily there other terms representing reactions of water and its levels of retention in various types and layers of the earth.

Finally, you have agent-based models and lattice-Boltzmann models, which can model water penetration in the finest details.

For links and details about those approaches look at my latest review paper covering all major complex systems approaches in biology and medicine.

https://www.researchgate.net/publication/330546521_Complex_Systems_and_Their_Use_in_Medicine_Concepts_Methods_and_Bio-Medical_Applications

It is evident that there is a whole spectrum of possible approaches. For example, code of text-based models goes to the page of NetLogo where are provided inline examples of various codes solving a whole variety of problems including yours.

Respected Julio Trujillo,

To begin with, I suggest you

Application of the Finite Element Method (FEM) for a detailed analysis of the soil structure in the observed domain.

1. Sahimi, M. and Imdakm, A.O.: 1988, ‘The effect of morphological disorder on hydrodynamic dispersion in flow through porous media’, Journal of Physics A: Math. Gen. 21, 3833–3870.

2. Adler, P.M.: 1992, Porous Media: Geometry and Transports, Butterworth-Heinemann, Boston.

3. Adler, P.M.: 1994, ‘The method of reconstructed porous media’, Current Topics in the Physics of Fluids 1, 277–306.

4. PERCOLATION THEORY AND NETWORK MODELING APPLICATIONS IN SOIL PHYSICS BRIAN BERKOWITZ1 and ROBERT P. EWING2 USA (Now at: Department of Agronomy, Iowa State University, Ames, Iowa 50011, USA

With respect

Dear Julio,

The linear diffusion equation (which is alike heat equation, plus maybe a first order term - the advection-diffusion equation) is not a good mathematical model for modelling moisture in soils. For this you have to consider non-linear models. A commonly accepted one is the Richards equation, in which the diffusion coefficient (or tensor) is replaced by a nonlinear function depending on the moisture saturation/water content.

Depending on the specific situation you look at, you may also include more complex effects such as hysteresis, or dynamic capillarity,...

One particular feature of such models is that, since the diffusion depends now on the water content, it may become 0 at locations which are completely dry, or infinite at locations where the medium is fully saturated (i.e. all pores are filled by water, no air is present in the medium at that location). This makes the problem become degenerate parabolic and requires proper numerical schemes (e.g. regularisation-based). Also, since the model is non-linear, if you use implicit schemes you may have to find good linearisation schemes to solve the time discrete or fully discrete equation obtained at each time-step.

You may find a lot of such papers on this subject in the literature (including some authored by the undersigned :))

Good luck,

Sorin