Abstract
We consider unsteady flow in porous media and focus on the behavior of the coefficients in the unsteady form of Darcy’s equation. It can be obtained by consistent volume-averaging of the Navier–Stokes equations together with a closure for the interaction term. Two different closures can be found in the literature, a steady-state closure and a virtual mass approach taking unsteady effects into account. We contrast these approaches with an unsteady form of Darcy’s equation derived by volume-averaging the equation for the kinetic energy. A series of direct numerical simulations of transient flow in the pore space of porous media with various complexities are used to assess the applicability of the unsteady form of Darcy’s equation with constant coefficients. The results imply that velocity profile shapes change during flow acceleration. Nevertheless, we demonstrate that the new kinetic energy approach shows perfect agreement for transient flow in porous media. The time scale predicted by this approach represents the ratio between the integrated kinetic energy in the pore space and that of the intrinsic velocity. It can be significantly larger than that obtained by volume-averaging the Navier–Stokes equation using the steady-state closure for the flow resistance term.
Highlights
Unsteady flow in porous media can arise from unsteady boundary conditions or unsteady pressure gradients
If the unsteady Darcy equation with constant coefficients (7) was a good model for unsteady flow in porous media, it should be possible to bring both into a comparable form
To investigate unsteady flow in porous media, we focused on the applicability of the unsteady form of Darcy’s equation and its time scale
Summary
Unsteady flow in porous media can arise from unsteady boundary conditions or unsteady pressure gradients. For steady porous media flows, it is generally accepted and confirmed by numerous experimental results that the pressure drop (or hydraulic gradient) on a scale considerably larger than the pore scale can be represented by two terms, a linear and a quadratic one in the space-averaged velocity. For low Reynolds numbers based on pore diameter, the pressure drop increases linearly with the flow velocity, as it is dominated by viscous forces (creeping flow). This regime (Repore < 1) is called the Darcy regime, as it was first discussed by Darcy (1857). The resulting expression can be derived either by a dimensional analysis or by rigorous averaging of the Navier–Stokes equations over a representative elementary volume; see Whitaker (1986, 1996)
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