Saffman-Taylor fingering: why it is not a proper upscaled model of viscous fingering in a (even two-dimensional) random porous medium

Abstract:

P.G. Saffman & G. Taylor (1958) studied the stability of the interface between two immiscible fluids of different densities and viscosities when one displaces the other inside a Hele-Shaw (HS) cell. They showed that with a horizontal cell and if the displaced fluid is the more viscous, the interface is unstable and leads to a viscous fingering which they nearly fully modeled [1]. The HS geometry was introduced as a geometry imposing the same flow behavior as the Darcy-scale flow in a two-dimensional (2D) porous medium, and therefore allowing an analogy between the two configurations. This is however not obvious, since capillary forces act at very different scales in the two. Later, researchers performing unstable displacement experiments in HS cells containing random 2D porous media also observed viscous fingering at large viscosity ratios, but with invasion patterns very different from those of Saffman and Taylor (ST) [2-3]. It was however considered that the two processes were both Laplacian growth processes, i.e., processes in which the invasion probability density is proportional to the pressure gradient. Ten years ago, we investigated viscously-unstable drainage in 2D porous media experimentally and measured the growth activity as well as occupation probability maps for the invasion process [4-5]. We concluded that in viscous fingering in 2D porous media, the activity was rather proportional to the square of the pressure gradient magnitude (a so-called DBM model of exponent 2), so that the universality class of the growth/invasion process was different from that of ST viscous fingering. We now strengthen our claim with new results based on the comparison of (i) pressure measurements with the pressure field around a finger such as described by the ST analytical model, and (ii) branching angles in the invasion patterns with those expected for DBMs of various exponents.

[1] Saffman, P. G. and Taylor, G. Proc. Soc. London 1958(Ser A 245), 312-329.

[2] Lenormand, R. Journal of Physics: Condensed Matter 1990(2), SA79.

[3] Måløy, K. J., Feder, J. & Jøssang, T. Viscous fingering fractals in porous media, PRL 1985(55), 2688-2691.

[4] Løvoll, G.; Méheust, Y.; Toussaint, R.; Schmittbuhl, J. & Måløy, K. J. PRE 2004(70), 026301.

[5] Toussaint, R.; Løvoll, G.; Méheust, Y.; Schmittbuhl, J. & Måløy K. J., EPL 2005(71), 583-589.