Modeling of groundwater flow in porous medium layered over inclined impermeable bed

TL;DR

This work develops a nonlinear groundwater-flow model for a porous medium layered on an inclined bed using a generalized Dupuit-Forchheimer framework with a power-law constitutive relation, producing a parabolic PDE with a $p$-Laplacian-type operator. The authors establish well-posedness and qualitative properties of steady-state solutions, proving existence of weak solutions, regularity ($C^{1,\beta}$ under small forcing), and a priori bounds, and they derive first-order reformulations of the weak problem. A central technique is linearization around the trivial solution, which yields a spatially varying diffusion coefficient $D(x)$ and allows the use of Green's function to prove Weak and Strong Maximum Principles. The results provide rigorous insight into nonlinear groundwater flow over inclined beds, including explicit derivative bounds and a framework for further analysis and numerical simulation, with several open directions highlighted for broader parameter ranges and bed geometries.

Abstract

We propose a new mathematical model of groundwater flow in porous medium layered over inclined impermeable bed. In its full generality, this is a free-surface problem. To obtain analytically tractable model, we use generalized Dupuit-Forchheimer assumption for inclined impermeable bed. In this way, we arrive at parabolic partial differential equation which is a generalization of the classical Boussinesq equation. Novelty of our approach consists in considering nonlinear constitutive law of the power type. Thus introducing $p$-Laplacian-like differential operator into the Boussinesq equation. Unlike in the classical case of the Boussinesq equation, the convective term cannot be set aside from the main part of the diffusive term and remains incorporated within it. In the sequel of the paper, we analyze qualitative properties of the stationary solutions of our model. In particular, we study existence and regularity of weak solutions for the following boundary value problem \begin{equation*} \begin{aligned} & - \frac{\rm d}{{\rm d} x} \left[ (u(x) + H) \left|\frac{{\rm d} u}{{\rm d} x}(x) \cos(\varphi) + \sin(\varphi) \right|^{p - 2} \left(\frac{{\rm d} u}{{\rm d} x}(x) \cos(\varphi) + \sin(\varphi)\right) \right] & \begin{aligned} & = f(x)\,, & \qquad\qquad x \in (-1,1)\,, & u(-1) = u(1) = 0\,,& \end{aligned} \end{aligned} \end{equation*} where $p>1$, $H>0$, $\varphi\in (0, π/2)$, $f\geq 0$, $f\in L^{1}(-1,1)$. In the case of $p>2$, we study validity of Weak and Strong Maximum Principles as well. We use methods based on the linearization of the $p$-Laplacian-type problems in the vicinity of known solution, error estimates, and analysis of Green's function of the linearized problem.

Figures (3)

• Figure 1: Classical Dupuit-Forchheimer assumption (D1). The flow is horizontal with the velocity distribution profile and the piezometric head being constant along vertical line segment $\mathrm{A}\mathrm{B}$.
• Figure 2: Geometric relations between $h$, $\widehat{h}$ and $\varphi$. Due to (eD1) Piezometric head $h$ is assumed to be constant along the line segment $AB$. The fictive piezometer is measuring piezometric head at point $C$.
• Figure 3: State and flow variables in the case of the flow in porous medium layered over inclined impermeable layer under generalized Dupuit-Forchheimer assumption (eD1). The averaged velocity is assumed to be constant on the line segment AB and parallel to the inclined bedrock. Here, $h(x,t)= \widehat{h}(x,t)\cos\varphi + x\sin\varphi$, $v=v(x,t)=|\partial h/\partial x|^{m-1} \partial h/\partial x$, and the groundwater flow per unit width is $Q_{\hbox{\tiny p.u.w.}}(x,t)=\widehat{h}(x,t)v(x,t)$.

Theorems & Definitions (27)

• Definition 2.1: Weak solution
• Remark 2.2
• Remark 3.1
• Theorem 3.2
• Corollary 3.3
• Lemma 3.4
• Theorem 3.5
• Theorem 3.6: $C^1$-regularity of a weak solution
• Corollary 3.7
• Theorem 3.8