On Some Remarkable Observations of Laboratory Dispersion Using Computed Tomography (CT)
Authors: Walsh M.P. , Withjack E.M. 
Published: March 2010

Abstract
Fickian dispersion theory is traditionally and routinely used to analyze laboratory dispersive flows. We present evidence, however, that shows laboratory dispersion in Berea sandstone is not Fickian. The Berea mixing zone growth does not grow proportional to the square root of time as required by Fickian theory, rather it grows more nearly proportional to time. Mathematical modeling efforts confirm that this non-Fickian flow behavior is attributed to very small yet significant spatial permeability variations. It is likely that these conclusions apply to other laboratory-scale consolidated rock samples, especially media more heterogeneous than Berea sandstone. This work leads to a deeper understanding of dispersion in porous media.

Introduction
Dispersion is a well known phenomenon which affects fluid flow in porous media(1). Dispersion is particularly important in oil recovery processes involving fluid injection because it directly affects the efficiency with which the injected fluid displaces and recovers oil(2). Dispersion is also important in slug processes such as miscible and chemical flooding because it directly affects slug dissipation and controls injected fluid requirements(3,4). This work is restricted to a discussion of dispersion in equal-viscosity, equal-density fluid displacements.
To study and predict the effects of dispersion, most investigators employ a Fickian dispersion model. By Fickian, we mean a model based on some variation of Fick's first law of diffusion(5). Accordingly, the dispersive flux of a chemical species will depend strongly on its concentration gradient. To describe longitudinal dispersion in uni-directional laboratory displacements, most investigators employ a longitudinal dispersion coefficient which is an active function of molecular diffusion and convective dispersion(6). This type of model dictates that at sufficiently low flow rates, molecular diffusion dominates; conversely at sufficiently high flow rates, molecular diffusion is negligible and convective dispersion dominates. This type of (Fickian) dispersion model is strongly supported by experimental observations in uniform sand or bead packs and has been shown to accurately model the elution history performance(7,8) and mixing zone growth behavior(9,10). This type of model has also been successful at modeling the elution history performance from consolidated cores(11).
Despite the successes of Fickian dispersion theory, a complete study validating its applicability to consolidated porous media has not been presented to date. For example, Fickian dispersion theory has not been widely shown to model mixing zone growth in consolidated cores. Fickian dispersion theory has also yielded some persistent inconsistencies when comparing the results of synthetic and consolidated porous media. For example, Fickian dispersion theory yields essentially length-independent dispersivities in synthetic media whereas it predicts strongly length-dependent dispersivities in consolidated media. Fickian dispersion theory also requires the use of empirical constants to account for the apparent greater dispersivities in consolidated media than in synthetic media(6). These observations are a possible sign that Fickian dispersion theory may not be applicable to consolidated media.
One purpose of this work is to re-examine the applicability of Fickian dispersion theory to consolidated porous media by analyzing the recent experimental results of Withjack(12). Aided by computed tomography (CT), Withjack performed laser tests in a 91 cm long Berea core and measured tracer fluid