The inclusion of molecular diffusion effects in the network modelling of hydrodynamic dispersion in porous media

Hydrodynamic dispersion of tracers in flow through porous media has attracted a considerable amount of experimental and theoretical study over the past thirty years. A number of mathematical approaches have been used to analyse dispersion, which take account of its stochastic origins. These include methods based on ensemble averaging of particle motions in conjectural networks of tubes or idealised particle beds or, alternatively, Monte Carlo simulations of the motions of large samples of particles through a particular network structure. All of these methods attempt to evaluate appropriate statistics from distance travelled or arrival time distributions of the marker particles, from which the dispersion coefficient may be calculated. In this paper, we derive a network model of hydrodynamic dispersion in porous media which takes into account the effects of molecular diffusion of the particles propagated through the network. Our approach is based on a description of the particle jump statistics in the network elements (capillaries) which takes proper account of the molecular diffusion of the particles over all Peclet numbers, Pe, and aspect ratio ranges of the tube. We demonstrate that the convection-diffusion equation in the "Taylor limit" is inappropriate to use over very wide flow regimes for calculating these single-element jump statistics; other (Monte Carlo) methods are used to obtain these quantities. When the statistical parameters derived in this way are incorporated into a full network model, the calculated dispersion behaviour predicted over a very wide range of network Peclet numbers, Pe_N, agrees very well with experimental results presented in the literature. By plotting the calculated quantities in an appropriate way, the diffusion-dominated, mixed and convection-dominated flow regimes are clearly reproduced by our model. In particular, we note that in the flow regime which is mainly convection-dominated, but where molecular diffusion still plays a role (the "mixed" regime), our model predicts that D increases faster than a linear variation with fluid flow velocity, U;D varies as U^? where ? is in the range 1.19-1.25. Most of the data on dispersion presented in the literature have been for low molecular weight species in both consolidated and unconsolidated packs. Recently, some data have been published on both polymer and tracer dispersion in the same flow experiment. It appears that polymer dispersion is larger than that of tracer by a factor of between 2 and 4 in certain flow regimes. In this paper we analyse this effect in terms of the widely different diffusion constants of the polymer and tracer molecules. © 1991.