Homogenization of a multiscale model for water transport in vegetated soil

Abstract

In this paper we consider the multiscale modelling of water transport in vegetated soil. In the microscopic model we distinguish between subdomains of soil and plant tissue, and use the Richards equation to model the water transport through each. Water uptake is incorporated by means of a boundary condition on the surface between root tissue and soil. Assuming a simplified root system architecture, which gives a cylindrical microstructure to the domain, the two-scale convergence and periodic unfolding methods are applied to rigorously derive a macroscopic model for water transport in vegetated soil. The degeneracy of the Richards equation and the dependence of root tissue permeability on the small parameter introduce considerable challenges in deriving macroscopic equations, especially in proving strong convergence. The variable-doubling method is used to prove the uniqueness of solutions to the model, and also to show strong two-scale convergence in the non-linear terms of the equation for water transport through root tissue.

Figures (1)

• Figure 1: An illustration of the domain $\Omega$ comprised of bulk soil $B^\varepsilon$, rhizosphere soil $R^\varepsilon$. and root tissue $P^\varepsilon$; (a) from above; (b) cross-section of the $x_2-x_3$ plane at $x_1 = {L_1}/{2}$.

Theorems & Definitions (26)

• Remark 3.2
• Definition 3.3
• Remark 3.4
• Theorem 3.5
• proof
• Remark 3.6
• Theorem 3.7
• Lemma 3.8
• proof
• proof : Proof of Theorem \ref{['theorem_microscopic_models_uniqueness']}