Author(s): Alessandro Lenci; Sepideh Majdabadi Farahani; Luca Chiapponi; Sandro Longo; Vittorio Di Federico
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Keywords: Gravity currents; Porous media; Pure Forchheimer flow; Self-similar solution
Abstract: Consider a plane gravity current of density ρ injected at the origin and advancing through a homogeneous porous medium with a flat, impermeable, horizontal bottom. At equilibrium, the current is governed by the balance between viscous drag and buoyancy, the latter arising from the density difference Δρ between the current and the ambient fluid. We assume a sharp interface described by the function z = h(x, t), and neglect both surface tension effects and the motion in the ambient fluid, under the assumption that h ≪ h₀, the ambient thickness. We assume that lubrication theory and hydrostatic approximation are valid. In the Darcy-Forchheimer regime, the flow law, the mass balance equation, and the global and local boundary and initial conditions are given respectively by ∂h/∂x = −u/K − bu|u|, φ∂h/∂t + ∂(hu)/∂x = 0, φ∫₀^{x_N} h(x, t)dx = qt^α, h(x_N, t) = 0, h(x, 0) = 0, where x and t are the spatial and temporal coordinates, x_N is the unknown nose position, u the seepage velocity, φ the porosity, K the hydraulic conductivity, b the Forchheimer coefficient, α and q are non-negative constants. Here, we consider a pure Forchheimer flow dropping the first term on the r.h.s. of equation (1). This approximation is justified when N = (Δρ/ρ)gbk²/ν² ≫ 1, with k the intrinsic permeability and ν the fluid kinematic viscosity.
Year: 2025