Oscillatory Darcy Flow in Porous Media

We investigate the flow in a porous medium subjected to an oscillatory (sinusoidal) pressure gradient. Direct numerical simulation (DNS) has been performed to benchmark the analytical solutions of the unsteady Darcy equation with two different expressions of the time scale: one given by a consistent volume averaging of the Navier-Stokes equation with a steady-state closure for the flow-resistance term, another given by volume averaging of the kinetic energy equation with a closure for the dissipation rate. For small and medium frequencies, the analytical solutions with the time scale obtained by the energy approach compare well with the DNS results in terms of amplitude and phase lag. For large dimensionless frequencies (omega tau greater than or similar to 10), we observe a slightly smaller damping of the amplitude than predicted by the unsteady Darcy equation with the low-frequency time scale. This can be explained by a change in the velocity fields towards a potential flow solution. Note that at those high frequencies, the flow amplitudes remain below 1 % of those of the steady-state flows. Our DNSs, however, indicate that the time scale predicted by the steady-state closure for the flow-resistance term is too small. In general, this study supports the use of the unsteady form of Darcy's equation with constant coefficients to solve time-periodic Darcy flow, provided the proper time scale has been found.