Abstract
Experiments in which calcium chloride displaced sodium chloride from four cores showed the extent of asymmetry in the resulting effluent concentration profiles. These results provided a check on how validly the mixing process is modeled by a differential (i.e., not finite-stage) capacitance mathematical model. The effluent concentration profile from two consolidated cores exhibited considerable asymmetry, while two unconsolidated cores yielded nearly symmetrical profiles. All runs resulted in breakthrough of the 30 per cent concentration significantly before one pore volume was injected. In addition, velocity appreciably affected the effluent concentration profile from a Torpedo sandstone core. The differential capacitance model matched the data significantly better than a simple diffusion model. The capacitance model allows determination of the amount of dead-end pore space in a porous matrix and the effect of velocity on the rate of diffusion into this space. An experimental program yielding insight into the physical validity of the capacitance effect is described.
Introduction
Axial dispersion - the mixing accompanying the flow of miscible fluids through porous media- has been the subject of many relatively recent studies and a comprehensive review of the topic has been given by Perkins and Johnston. This dispersion is of practical interest in studies of the miscible displacement process, fixed-bed chemical reactors, and the adsorption of solutes from a flowing stream onto the surface of a porous medium. In the latter case, the effect of dispersion must be considered when adsorption parameters are determined from the nature of concentration profiles.In general, early studies of dispersion assumed applicability of a simple diffusion equation and were concerned with correlation of the experimentally determined "effective" diffusion coefficient with system properties over a large range of the latter. Recent investigators have been concerned with the deviations between the asymmetrical effluent concentration profiles observed and the symmetrical ones predicted by the diffusion model.In the present study, effluent concentration profiles were obtained from consolidated and unconsolidated cores. These profiles were compared with those predicted by a differential (i.e., not finite-stage) capacitance model. Solutions to the simple diffusion model, for three sets of boundary conditions, were compared with one another and with the experimental profiles.
SUMMARY OF PREVIOUS WORK
The reader is referred to Perkins and Johnston for an extensive review of studies of dispersion in porous media. Many investigators have employed the simple diffusion model characterized by Eq. 1 below:
(1)
The dispersion coefficient D for unconsolidated systems is correlated by
....................(2)
for 2 less than less than 50, where v is interstitial velocity and dp is particle diameter. Since heterogeneity of the sand pack affects the mixing, this equation is also expressed as
(3)
where a is proportional to the degree of heterogeneity and is about 3.5 for random packs of unconsolidated sand. Eq. 3 holds for. For a homogeneous (regular) type of packing, a should be 1 or less. Aris and Amundson and Carberry and Bretton consider a to be the number of particle lengths per mixing cell in the finite-stage model. Data from consolidated cores indicate a dp to be about 0.36 cm for outcrop rocks, Torpedo sandstone having a reported value of 0.17 cm.
SPEJ
P. 73^
Anomalous diffusion in porous media