Diffusion in porous media with hysteresis and bounded speed of propagation

TL;DR

This work analyzes moisture diffusion in porous media with hysteresis captured by a convexifiable Preisach operator, formulating a nonlinear diffusion problem $\theta_t = \mathrm{div}(\kappa(x,\theta)|\nabla u|^{p-2}\nabla u)$ with $\theta = G[u]$. A time-discrete variational scheme is developed, yielding robust a priori bounds and enabling the use of Minty-type arguments to prove existence (and uniqueness when $\kappa$ is θ-independent) of weak solutions with $\nabla u\in L^p$ and $u_t,\theta_t$ in an Orlicz space $L^{\Phi_{log}}$. The paper then establishes finite-speed propagation for $p>3$ by constructing traveling-wave solutions and comparing them to the actual solution, resulting in an explicit front bound $R(t)=R_0+C_p t^{1/p}$. A key finding is that the speed-bound mechanism hinges on the exponent threshold $p>3$, connected to a corresponding doubly nonlinear equation, which clarifies when compactly supported initial data preserve their support. Overall, the results provide rigorous well-posedness and finite-propagation guarantees for hysteretic diffusion in porous media, with clear implications for modeling moisture fronts.

Abstract

It is shown that the problem of moisture propagation in porous media with a nonlinear relation between the mass flux and the pressure gradient as a counterpart of the Darcy law exhibits the property of bounded speed of propagation even in the case of a hysteresis relation between the capillary pressure and the moisture content. The paper specifies conditions for existence and uniqueness of solutions, and provides an upper bound for the moisture propagation speed.

Figures (4)

• Figure 1: Typical experimental hysteresis dependence in porous media between the logarithm soil suction $\psi$ (which can be interpreted as a decreasing function of the pressure) and the volumetric water content $\theta$.
• Figure 2: Primary wetting curve and limit wetting/drying curves in a typical Preisach diagram.
• Figure 3: Dominant traveling wave solution.
• Figure 4: Moving wetting front.

Theorems & Definitions (8)

• Definition 1.1
• Proposition 1.2
• Definition 1.3
• Theorem 1.6
• Theorem 1.8
• Lemma 3.1
• Proposition 3.2
• Proposition 4.1