Dispersion in heterogeneous porous media: 1. Local volume averaging and large‐scale averaging

Abstract

The problem of diffusion and dispersion in porous media is considered in the absence of either adsorption or chemical reaction. For this situation, local volume averaging has been used to obtain both a convection‐dispersion equation and a method of closure that provides for a direct theoretical prediction of the dispersion tensor. This approach is associated with three length scales, lβ, the pore diameter, ro, the radius of the averaging volume, and L the macroscopic length scale, and makes use of a representative unit cell to determine the dispersion tensor. Previous work suggests that extremely complex unit cells must be used to capture the observed dispersion phenomena, and complex unit cells naturally introduce another length scale, i.e., the length scale associated with the local heterogeneities. To deal with this problem, we propose the use of large‐scale averaging to account for the influence of the local heterogeneities. This leads to a large‐scale convection‐dispersion equation that contains additional terms involving time derivatives which arise as a result of the presence of heterogeneities and which do not appear in the conventional local volume average convection‐dispersion equation. The first new term contains the second derivative with respect to time and is likely to be unimportant for time scales appropriate for groundwater transport processes; however, it may be important for small‐scale laboratory experiments. The second new term contains a mixed space‐time derivative which may be important for many cases of practical interest. A closure scheme is presented that allows for the theoretical determination of all the coefficients that appear in the large‐scale convective‐convection‐dispersion equation.

Transport in porous packings: Statistical characterization of transport, role of fluctuation and data analysis

“…time Of is therefore given by the average value of the Figures 8 and 9 show a comparison between the theoretical prediction of the stochastic model developed above and experimental data on dispersion in packed beds (Plumb and Whitaker 1988a;1988b;Carberry and Bretton 1958;Ebach and White 1958;Edwards and Richardson 1968) and in chromatographic columns (Ostergren and 'I!raghd 1989). The particle Peclet number Pep = vdp/D, where dp is the size of the granular material constituting the packing, can be assumed proportional to the Auctuational Peclet number Pef.…”

“…The present analysis opens up interesting perspectives for a more sophisticated analysis of dispersion in porous structures. This can be achieved by considering the porous medium as a periodic structure characterized by a complex unit pore cell (Plumb and Whitaker 1988a). The dispersion features can then be obtained by solving (18) numerically on the considered unit pore cell.…”

Extent of reaction in open systems with multiple heterogeneous reactions

“…Models must include a spectrum of reactions, ranging from equilibrium-controlled to kinetically-controlled, with the possibility that the nature of any given reaction may change in the spatial domain of interest. It is well accepted (Plumb and Whitaker, 1988) that groundwater transport can be described in many situations by the convection-dispersion equations. Dispersion through porous media is largely a tortuous flow phenomenon and is usually independent of diffusion, making the partial-differential operator like that considered by Shapiro (1962).…”

Investigation of the effect of transverse thermal dispersion on forced convection in porous media

“…In recent years, there has been considerable interest in modeling the thermal dispersion effect in porous media (Plumb [1], Hong and Tien [2], Plumb and Whitaker [3], [4], Hunt and Tien [5], Hsu and Cheng [6], Nakayama [7], Hsieh and Lu [8], Nield and Bejan [9]). This is because many modern applications of porous media are characterized by significant filtration velocities, when in addition to the molecular diffusion of heat it is also necessary to account for an additional heat transfer which appears due to hydrodynamic mixing of the interstitial fluid at the pore scale.…”