From Darcy flow to convective flow: pore-scale study of density-driven currents in porous media

We conducted a series of pore-scale numerical simulations on convective flow in porous media, with a fixed Schmidt number of 400 and a wide range of Rayleigh numbers. The porous media are modeled using regularly arranged square obstacles in a Rayleigh-B\'enard (RB) system. As the Rayleigh number increases, the flow transitions from a Darcy-type regime to an RB-type regime, with the corresponding $Sh-Ra_D$ relationship shifting from sublinear scaling to the classical 0.3 scaling of RB convection. Here, $Sh$ and $Ra_D$ represent the Sherwood number and Rayleigh-Darcy number, respectively. For different porosities, the transition begins at approximately $Ra_D = 4000$, at which point the characteristic horizontal scale of the flow field is comparable to the size of a single obstacle unit. When the thickness of the concentration boundary layer is less than about one-sixth of the pore spacing, the flow finally enters the RB regime. In the Darcy regime, the scaling exponent of $Sh$ and $Ra_D$ decreases as porosity increases. Based on the Grossman-Lohse theory (J. Fluid Mech., vol. 407, 2000; Phys. Rev. Lett., vol. 86, 2001), we provide an explanation for the scaling laws in each regime and highlight the significant impact of mechanical dispersion effects within the boundary layer. Our findings provide some new insights into the validity range of the Darcy model.