Invasion percolation with viscous forces

Abstract

We study invasion percolation in the presence of viscous forces, as a model of the drainage of a wetting fluid from a porous medium. Using concepts from gradient percolation, we consider two different cases, depending on the magnitude of the mobility ratio M . When M is sufficiently small, the displacement can be modeled by a form of gradient percolation in a stabilizing gradient, involving a particular percolation probability profile. We develop the scaling of the front width and the saturation profile, in terms of the capillary number. In the opposite case, the displacement is described by gradient percolation in a destabilizing gradient and leads to capillary-viscous fingering. This regime is identified in the context of viscous displacements and in general differs from diffusion-limited aggregation, which also describes displacements at large M . Constraints for the validity of the two regimes are developed. Limited experimental and numerical results support the theory of stabilized displacement. The effect of heterogeneity is also discussed. ͓S1063-651X͑97͒08812-0͔PACS number͑s͒: 47.55. Mh, 0.5.40.ϩj, 47.55.Kf

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

“…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). Above this crossover length scale, the flow is destabilized by viscous forces and the displacement takes place in narrow-branched channels (for M < 1).…”

“…This regime is shown to have strong analogies to invasion percolation (Lenormand and Zarcone 1985;Chandler et al 1982;Wilkinson and Willemsen 1983), and the invasion structure is fractal Mandelbrot (1982) ;Feder (1988) with a fractal dimension D c = 1.83 ± 0.01 (Lenormand andZarcone 1985, 1989). If the invasion rate is high, the displacement is either stable or unstable depending on the viscosity contrast M. If a fluid with high viscosity is invading a fluid with low viscosity (M ≥ 1), the resulting pressure field due to the viscous dominated displacement will act against the growth of the invasion front, leading to stabilization of the displacement front at a finite width (Saffman and Taylor 1958;Lenormand et al 1988;Lenormand 1989;Frette et al 1997;Xu et al 1998). On the other hand, if the invading fluid is the less viscous one, the displacement is unstable and falls in the viscous fingering regime (Saffman and Taylor 1958;Måløy et al 1985;Homsy 1987).…”

Fluid Invasion in Porous Media: Viscous Gradient Percolation

“…[4,5,8]. Our network simulations in the high capacitance regime lead to κ ≃ 0.36(2) giving a direct numerical support to Eq.…”

“…which has been widely used in the literature [4,5,6,7,8] and is valid for gradient percolation [3]. However, for viscous gradient percolation, Eq.…”

A Numerical Model of Tracer Transport in a Non-isothermal Two-Phase Flow System for $${\text{ CO}}_2$$ Geological Storage Characterization

“…It should be pointed out that the numerical model is not suitable for regimes with viscous instability as discussed in, e.g., Riaz and Tchelepi (2006) and Riaz et al (2007) or flow with invalidity of volume-averaging (continuum description) as discussed in e.g., Xu et al (1998) and Yortsos et al (2001). Finally, non-equilibrium formulations for capillary pressure-interfacial area-saturation (e.g., Joekar-Niasar et al 2010) have been proposed in the recent literature.…”