Growth activity during fingering in a porous Hele-Shaw cell

Abstract

We present in this paper an experimental study of the invasion activity during unstable drainage in a two-dimensional random porous medium, when the (wetting) displaced fluid has a high viscosity with respect to that of the (nonwetting) displacing fluid, and for a range of almost two decades in capillary numbers corresponding to the transition between capillary and viscous fingering. We show that the invasion process takes place in an active zone within a characteristic screening length lambda from the tip of the most advanced finger. The invasion probability density is found to only depend on the distance z to the latter tip and to be independent of the value for the capillary number Ca. The mass density along the flow direction is related analytically to the invasion probability density, and the scaling with respect to the capillary number is consistent with a power law. Other quantities characteristic of the displacement process, such as the speed of the most advanced finger tip or the characteristic finger width, are also consistent with power laws of the capillary number. The link between the growth probability and the pressure field is studied analytically and an expression for the pressure in the defending fluid along the cluster is derived. The measured pressure is then compared with the corresponding simulated pressure field using this expression for the boundary condition on the cluster.

Influence of Viscous Fingering on Dynamic Saturation–Pressure Curves in Porous Media

“…Since the observed structure is highly dependent on the imposed flow rate, the latter method has the advantage that invasion happens at a constant well-defined capillary number. This ensures that the length scales characteristic of the present flow regimes stay constant during the experiments (Løvoll et al 2004;Toussaint et al 2005).…”

“…Where displacement will take place in narrow channels and branched loop-less structures, as shown in Fig. 5 (Løvoll et al 2004;Toussaint et al 2005). At scales below this, capillary fluctuations dominate the dynamics, and the resulting structure is characteristic of the capillary fingering regime, with droplets of wetting fluid remaining trapped in loops of the invading non wetting fluid.…”

“…In the first case, the flow is driven by slowly changing the pressure difference over the model. In the second case, a preset constant flow rate is imposed by means of a syringe pump or a tailored gravity pump (Méheust et al 2002;Løvoll et al 2004) (see Fig. 1).…”

“…Viscous fingering in disordered porous media is different from standard Saffman-Taylor fingering (Saffman and Taylor 1958), obtained in empty straight channels (Hele-Shaw cells), where the fingers are compact and occupy 1/2 of the system width W . In contrast, in disordered porous media these structures are branched, the invasion structure is fractal (Måløy et al 1985) and it has been demonstrated that the invasion structure occupies a smaller fraction of the system (0.4 W ) (Løvoll et al 2004;Toussaint et al 2005).…”

“…Where the proportionality factor β 1 (Løvoll et al 2004;Toussaint et al 2005), in the following we will use: l c = a/Ca. Note that this experimental result is different from the theoretical result for invasion percolation in a destabilizing gradient (Yortsos et al 1997;Xu et al 1998).…”