Differences in miscible viscous fingering of finite width slices with positive or negative log-mobility ratio

When a sample of fluid of finite size is displaced in a porous medium by another miscible fluid, viscous fingering may occur when the two fluids have different viscosities. Depending whether the sample is more or less viscous than the carrier fluid, the log-mobility ratio R [defined as R=ln(mu_{2}mu_{1}) where mu_{2} and mu_{1} are the viscosities of the sample and of the carrier] is respectively positive or negative. In the case of a linear displacement of a finite slice of fluid, R>0 leads to fingering of the rear interface of the sample where the less viscous carrier invades the more viscous sample. If R<0 , it is on the contrary the frontal interface of the sample that develops fingers where the less viscous sample displaces the more viscous bulk solution. We investigate here numerically the differences in fingering dynamics between the positive and negative log-mobility ratio cases leading to the growth of fingers against or along the direction of the flow, respectively. To do so, we integrate Darcy's law coupled to a convection-diffusion equation for the concentration of a solute ruling the viscosity of the finite-size sample. The statistical moments of the solute's concentration distribution are analyzed as a function of dimensionless parameters of the problem such as the length of the slice l , the log-mobility ratio R , and the ratio between transverse and axial dispersion coefficients . We find that, on average, the mixing zones and the width of the sample broadening due to fingering are larger for negative R than for positive R . This is due to the fact that fingers travel quicker in the flow direction than against the flow. Relevance of our results are discussed for interpretation of experimental results obtained in chromatographic separation and for understanding conditions of enhanced spreading of contaminants in aquifers.