Shear dispersion in a porous medium. Part 1. An intrusion with a steady shape

Abstract

Many permeable aquifers have vertical variations in the permeability, which are correlated over long lateral distances. When a buoyant high viscosity fluid is injected into a permeable aquifer and displaces a less viscous fluid, the interface that develops has finite extent and travels with constant speed. The effect of the permeability variations leads to fluid in the high permeability regions travelling into the front, where it migrates into the lower permeability part. Subsequently, it lags progressively further behind the advancing front. We explore the influence of this cycling on the dispersion of a tracer or additive in the fluid to determine its distribution within the current as a function of time. At early times tracer is sheared owing to the vertically varying permeability. At later times, cross-aquifer diffusion homogenises the tracer distribution, which spreads longitudinally at a faster rate than by diffusion alone in this shear dispersion regime. Eventually, tracer reaches the front of the current. This acts as a no-flux boundary and the concentration profile transitions to a half-Gaussian, with the maximum concentration at the front. The centre of mass of the tracer spreads backwards relative to the fixed nose at a rate proportional to, where T is time and D is the diffusion coefficient. The initial release of tracer may not be vertically uniform owing to the heterogeneity and we show that this can lead to the centre of mass of tracer initially advancing faster than the mean flow. Although our model is highly idealised, it illustrates how, owing to a vertical variation of permeability, it is possible for a finite pulse of tracer or chemical added after the start of injection to reach the front with important implications for tracer tests and strategies for enhanced recovery.