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Analytic description of the moving moisture front in soils

Bettina Detmann, Chiara Gavioli, Pavel Krejčí, Yanyan Zhang

TL;DR

This work analyzes moisture transport in soils governed by a degenerate Richards equation with bounded suction via the Rossi–Nimmo model. It develops a rigorous analytical framework establishing existence and uniqueness of solutions with compactly supported initial data, and derives explicit upper bounds for front propagation that reveal anisotropy: lateral spreading scales as $\sqrt{t}$ due to diffusion, while downward propagation scales as $t$ due to gravity, with upward capillary rise arrested or reversed depending on soil parameters encoded in $P(u)=f'(u)/g(u)$. The authors construct traveling-wave supersolutions and use geometric envelope arguments to bound the evolving wet region and characterize its shape, confirming an egg-shaped footprint in appropriate regimes and providing detailed asymptotics. Numerical tests across several $P(v)$ profiles corroborate the theory and align with laboratory observations of wetting bulbs in sand, highlighting practical implications for irrigation and contaminant transport modeling.

Abstract

The fact that moisture propagates in soils at a finite speed is confirmed by natural everyday experience as well as by controlled laboratory tests. In this text, we rigorously derive analytical upper bounds for the speed of moisture front propagation under gravity for the solution to the Richards equation with compactly supported initial data. The main result is an explicit criterion describing a competition between gravity and capillarity, where the dominant effect is determined by the characteristics of the soil. If capillarity prevails, the initially wet regions remain wet for all times, while if gravity is dominant, moisture travels downward at a speed that is asymptotically bounded from below and above. As a by-product, we prove the existence and uniqueness of a solution to an initial value problem for the degenerate Richards equation on the whole space. Numerical simulations based on the proposed model confirm the theoretical predictions, with results that closely match experimental observations.

Analytic description of the moving moisture front in soils

TL;DR

This work analyzes moisture transport in soils governed by a degenerate Richards equation with bounded suction via the Rossi–Nimmo model. It develops a rigorous analytical framework establishing existence and uniqueness of solutions with compactly supported initial data, and derives explicit upper bounds for front propagation that reveal anisotropy: lateral spreading scales as due to diffusion, while downward propagation scales as due to gravity, with upward capillary rise arrested or reversed depending on soil parameters encoded in . The authors construct traveling-wave supersolutions and use geometric envelope arguments to bound the evolving wet region and characterize its shape, confirming an egg-shaped footprint in appropriate regimes and providing detailed asymptotics. Numerical tests across several profiles corroborate the theory and align with laboratory observations of wetting bulbs in sand, highlighting practical implications for irrigation and contaminant transport modeling.

Abstract

The fact that moisture propagates in soils at a finite speed is confirmed by natural everyday experience as well as by controlled laboratory tests. In this text, we rigorously derive analytical upper bounds for the speed of moisture front propagation under gravity for the solution to the Richards equation with compactly supported initial data. The main result is an explicit criterion describing a competition between gravity and capillarity, where the dominant effect is determined by the characteristics of the soil. If capillarity prevails, the initially wet regions remain wet for all times, while if gravity is dominant, moisture travels downward at a speed that is asymptotically bounded from below and above. As a by-product, we prove the existence and uniqueness of a solution to an initial value problem for the degenerate Richards equation on the whole space. Numerical simulations based on the proposed model confirm the theoretical predictions, with results that closely match experimental observations.
Paper Structure (10 sections, 8 theorems, 136 equations, 9 figures)

This paper contains 10 sections, 8 theorems, 136 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Existence of solutions
  4. Moisture propagation speed
  5. Traveling wave solutions
  6. Shape of the moving wet region
  7. Lower half-space
  8. Upper half-space
  9. Numerical tests
  10. Conclusions

Key Result

Theorem 2.2

Let Hypothesis hfg1 hold, let $u^*>0$ be a constant, and let $u_0 \in L^\infty(\Omega)$ be given, $0 \le u_0(x) \le u^*$ a. e., $\nabla u_0 \in L^2(\Omega)$. Then there exists a unique solution $u\in L^r(\Omega; C[0,T])$, $1\le r < \infty$, to 1e1 with initial condition e2 and such that $0\le u(x,t)

Figures (9)

  • Figure 1: Comparison of water retention curves ($\theta(h)$). Left: Models with unbounded suction, where $\theta$ only asymptotically approaches zero as $h \to -\infty$. Right: Model with bounded suction (Rossi-Nimmo type), where $\theta$ reaches zero at a finite pressure head $h_*$.
  • Figure 2: Experimental observation of two-dimensional wetting bulb propagation. Left: Time series photographs illustrating the development of the wetting bulb in initially dry sand, shown at eight specific times (1s, 5s, 10s, 30s, 60s, 120s, 300s, 600s). Right: Penetration figures of the wetting front corresponding to the times shown on the left. The water supply is in the origin of the coordinate system.
  • Figure 3: Evolution of the wetting bulb dimensions over time from four experimental tests. The $x$-axis is time in seconds. The wetted horizontal width (left axis, black) and the wetted vertical depth (right axis, gray) are shown in cm. Note the distinct propagation rates (linear vs. square-root growth) that motivate the subsequent analysis.
  • Figure 4: Intersection of the downward wetting front with the plane $E(\boldsymbol{e}_\perp,\boldsymbol{e}_N)$.
  • Figure 5: Moving wetting fronts intersected with the plane $E(\boldsymbol{e}_\perp, \boldsymbol{e}_N)$ in the case $P(0) > 0$ (left) and $P(0) = 0$ (right).
  • ...and 4 more figures

Theorems & Definitions (8)

  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Proposition 4.1