Smoothed Particle Hydrodynamics Modeling of Gravity Currents on a Dry Porous Medium

Gravity currents flowing over porous media occur in many environmental processes and industrial applications, such as irrigation, benthic boundary layers, and oil spills. The coupling of the flow over the porous surface and the infiltration of the fluid in the porous media is complex and difficult to model. Of particular interest is the prediction of the position of the runoff front and the depth of the infiltration front. We present here a model for the flow of a finite volume of a highly viscous Newtonian fluid over a dry, homogenous porous medium. The Navier-Stokes equations describing the runoff flow are coupled to the Volume Averaged Navier-Stokes equations for the infiltration flow. The numerical solution of these equations is challenging because of the presence of two free surfaces (runoff and infiltration waves), the lack of fixed boundary conditions at the runoff front, and the difficulties in defining appropriate conditions at the surface of the porous medium. The first two challenges were addressed by using Smoothed Particle Hydrodynamics, which is a Lagrangian, mesh-free particle method particularly suitable for modelling free surface flows. Two different approaches were used to model the flow conditions at the surface of the porous medium. The Two Domain Approach (TDA) assumes that runoff and infiltration flows occur in two separate homogenous domains; here, we assume the continuity of velocity and stresses at the interface of the two domains. The One Domain Approach (ODA) models runoff and infiltration flows as occurring through a medium whose hydraulic properties vary continuously in space. The transition from the hydraulic properties of the atmosphere and the porous medium occur in a layer near the surface of the porous medium. Expressions listed in literature were used to compute the thickness of this transition layer and the spatial variation of porosity and permeability within it. Our results showed that ODA led to slower velocities of the runoff front and enhanced infiltration when compared to the implemented formulation of TDA. In the ODA, depending on the description of the transition layer, the maximum distances travelled by the runoff front and the maximum depth of infiltration varied over a range of ±15% and ±50% when compared to their respective averaged values.