Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 57, pp. 1-32.

Title: Modeling of groundwater flow in porous medium layered over inclined impermeable beds

Authors: Petr Girg (Univ. of West Bohemia, Plzen, Czech Republic)
         Lukas Kotrla (Univ. of West Bohemia, Plzen, Czech Republic)

Abstract:
We propose a new mathematical model of groundwater flow in porous medium 
 layered over inclined impermeable beds.
 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.  
 The 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 this article, we analyze qualitative properties of the stationary 
solutions of our model. In particular, we study the existence and regularity 
of weak solutions for the  boundary value problem
$$\displaylines{
-\frac{\rm d}{{\rm d} x}
 \Big[(u(x) + H) |\frac{{\rm d} u}{{\rm d} x}(x) \cos(\varphi) 
 + \sin(\varphi) |^{p - 2}
 \Big(\frac{{\rm d} u}{{\rm d} x}(x) \cos(\varphi) 
 + \sin(\varphi)\Big)\Big]
= f(x),  \quad\ x \in (-1,1)\,,
\cr
 u(-1) = u(1) = 0\,,
}$$
where $p>1$, $H>0$, $\varphi\in (0, \pi/2)$, $f\geq 0$,  
$f\in L^{1}(-1,1)$.
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.


Submitted January 3, 2025. Published May 30, 2025.
Math Subject Classifications: 76S05, 35Q35,34B15, 34B27.
Key Words: Porous medium; filtration; nonlinear Darcy's law; $p$-Laplacian;
   pressure-to-velocity power law.