Axisymmetric gravity currents in porous media under the Darcy–Forchheimer regime

Abstract

We investigate the spreading of axisymmetric Newtonian gravity currents (GCs) propagating within an infinite, saturated porous medium bounded below by a horizontal impermeable surface. The flow is driven by buoyancy, and the intruding fluid advances due to its higher density relative to the lighter, stationary saturating fluid. Different injection scenarios are considered, depending on the exponent gamma γ $\gamma$ : gamma equals 0 γ = 0 $\gamma =0$ and gamma equals 1 γ = 1 $\gamma =1$ correspond to instantaneous and constant injection, respectively, while gamma less than 1 γ < 1 $\gamma \lt 1$ and gamma greater than 1 γ > 1 $\gamma \gt 1$ describe waning and waxing inflow conditions, respectively. The temporal evolution of the current volume therefore follows upper V tilde upper T Superscript gamma V ∼ T γ $V\sim T^{\gamma }$ . The flow is assumed to obey the Darcy–Forchheimer law, which extends Darcy’s model by introducing a nonlinear inertial correction that scales quadratically with the seepage velocity. The resulting nonlinear problem is solved numerically to determine the temporal evolution of the GC profiles. A special regime, referred to as the high-Forchheimer regime, also admits a similarity solution, valid when the local Forchheimer number is much greater than unity. This condition depends on a dimensionless group upper N N $N$ , which controls the relative importance of nonlinear resistance and may vary widely around unity in practical applications. Theoretical predictions are validated through an extensive laboratory investigation, involving 18 experiments performed with porous media of different particle size ranges and under different injection conditions. The experimental results show good agreement with the theoretical predictions, both in terms of front propagation and current profiles. Finally, a simplified subsurface application is presented to illustrate the range of conditions under which inertial effects may become non-negligible.