Inertial Effects on Fluid Flow through Disordered Porous Media

Abstract

We investigate the origin of the deviations from the classical Darcy law by numerical simulation of the Navier-Stokes equations in two-dimensional disordered porous media. We apply the Forchheimer equation as a phenomenological model to correlate the variations of the friction factor for different porosities and flow conditions. At sufficiently high Reynolds numbers, when inertia becomes relevant, we observe a transition from linear to nonlinear behavior which is typical of experiments. We find that such a transition can be understood and statistically characterized in terms of the spatial distribution of kinetic energy in the system. [S0031-9007(99)09541-1] PACS numbers: 47.55.Mh, 47.11. + j A standard approach in the investigation of singlephase fluid flow in microscopically disordered and macroscopically homogeneous porous media is to characterize the system in terms of Darcy's law [1][2][3], which assumes that a global index, the permeability k, relates the average fluid velocity V through the pores with the pressure drop DP measured across the system,Here L is the length of the sample in the flow direction and m is the viscosity of the fluid. However, in order to understand the interplay between porous structure and fluid flow, it is necessary to examine local aspects of the pore space morphology and relate them to the relevant mechanisms of momentum transfer (viscous and inertial forces). This has been accomplished in previous studies [4][5][6][7][8][9][10] where computational simulations based on a detailed description of the pore space have been quite successful in predicting permeability coefficients and validating well-known relations on real porous materials. In spite of its great applicability, the concept of permeability as a global index for flow should be restricted to viscous flow conditions or, more precisely, to small values of the Reynolds number. Unlike the sudden transition from laminar to turbulent flow in pipes and channels where there is a critical Reynolds number value separating the two regimes, experimental studies on flow through porous media have shown that the passage from linear (Darcy's law) to nonlinear behavior is more likely to be gradual (see Dullien [1], and references therein). It has then been argued [1] and confirmed by numerical simulations [11,12] that the contribution of inertia to the flow in the pore space should also be examined in the framework of the laminar flow regime before assuming that fully developed turbulence effects are present and relevant to momentum transport. Here we show by direct simulation of the Navier-Stokes equations that the departure from Darcy's law in flow through high porosity percolation structures (e . e c , when e c is the critical percolation porosity) and at sufficiently high Reynolds numbers can also be explained in terms of the inertial contribution to the laminar fluid flow through the void space. The calculations we perform do not apply for unstable or turbulent Reynolds conditions. We then demonstrate that it is possible to statisti...

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