Fractal nature of viscous fingering in two‐dimensional pore level models

Abstract

Use of saturation-dependent relatiue mobilities leads to linear flow; howeuer, experiment and theory show that, in the limit of very lurge uiscosity ratio, the flow is not linear but fractal. Generally, fractional flows and relatiue mobilities depend M O Oh8 acts as a 2-D Koual factor IntroductionComposition-dependent relative mobilities are used in all traditional modeling of flow in porous media, such as Buckley-Levcrett or Koval (Collins, 1961; Rhec et al., 1986). Their use predicts flow in which the saturation front advances linearly with time. However, it has been shown that the limit of infinite viscosity ratio ( M -+x, where the injected fluid has zero viscosity) is accurately described by diffusion-limited aggregation (DLA), a process that is known to form fractal objects with nonuniform densities (Vicsek, 1989; Witten and Sander, 1981;Meakin, 1983a). For fractal flow, the saturation front advances faster than linearly with time. Two questions naturally arise: ( I ) If the actual viscous fingering were fractal, what would be the effect on traditional simulation? and (2) Is the viscous fingering fractal for realistic, unstable viscosity ratios? As we will see, our modeling indicates that large-scale flows are not fractal, but that small-scale flows are fractal. Furthermore, this crossover from small-scale fractal flows to large-scale linear flows leads to definite predictions regarding the dependence of flow velocity upon viscosity ratio.First, what is a fractal? The classic signature of a fractal object is a non-Euclidean relationship between mass and size. If one has an ordinary solid disk, the mass is proportional to R2, but for a circular fractal object, the mass is proportional to R"{, with a noninteger fractal dimension, D,; for example, for DLA, Df = 1.7. Therefore, a fractal object is less dense than an ordinary object due to material voids inside the object. However, it should be emphasized that neither the mass density nor the compensating voids are uniformly distributed; indeed, the mass density decreases with R while the void density must increase with R. Formation of these fractals is an unstable, nonequilibrium process. A number of excellent reviews discuss a wide variety of these fractal growth phenomena, including material deposition, dielectric breakdown, and two-phase flow in porous media, the topic of this article (Vicsek, 1989;Mandelbrot, 1982; Feder, 1988).Second, what is viscous fingering? If the flow is "unstable"(viscosity ratio M > 11, the injected fluid fingers into the displaced fluid (Saffman and Taylor, 1958). This effect has been widely studied in "Hele-Shaw" cells, where a high viscosity fluid occupies the space between two flat glass plates, and a low-viscosity fluid is injected at the center. If the viscosity ratio is large enough, the viscous fingering patterns satistji a fractal relationship with fractal dimension D, = 1.70&0.05 (Daccord et al., 1986). If the space between the Hele-Shaw plates is filled with a bead pack, mimicking a porous medium, analysis of the fingeri...

Two-Phase Flow in Porous Media: Predicting Its Dependence on Capillary Number and Viscosity Ratio

“…This assumed behavior is predicted using saturation-dependent relative permeabilities in an averaged Darcy's law, as in the treatments of Buckley-Leverett or Koval (Dullien 1979). For the past quarter century, it has been appreciated that flow in porous media is fractal (i.e., non-Euclidean) in certain well-defined limits (Nittmann et al 1985;Lenormand et al 1988;Blunt and King 1990;Sahimi 1994;Ferer and Smith 1994;Ferer et al 1995Ferer et al , 1996Frette et al 1997). For fractal flow in two dimensions (2D) with injection along a side of width W , the fractal dimension, D f , relates a typical length scale like the center of mass (i.e., average position), x , of the injected fluid to the total amount of injected fluid, V , which is directly proportional to time, t, given a constant injection rate, so that…”

University/NETL Student Partnership Program

“…A crossover from fractal to compact flow has been observed with this model for miscible flow in horizontal porous media, where gravitational effects are negligible [20]. The present work centers on the interplay between buoyancy and viscous forces and their effect on interfacial stability using this pore-level network model.…”

“…Now, using these two Jacobi, (20) and (21), the Kalman gain matrix K k can be obtained. These Jacobi are updated for every time step.…”

“…The microporous structure comprises interconnected pores of different shapes and sizes. 20 The size of these micropores is generally considered to be less than 2 nm in diameter. 12,16 The aperture of the passageways that interconnect the micropores may be less than 10 Å.…”

Gas-Oil Relative Permeability and Residual Oil Saturation as Related to Displacement Instability and Dimensionless Numbers

“…The relative permeability functions of porous medium are strongly correlated with the flow regimes and are strongly dependent on the capillary number and viscosity ratio (Avraam and Payatakes, 1995a;Singh et al, 2001;Tsakiroglou et al, 2003). Moreover, invasion percolation and gradient percolation models have also been employed to interpret quantitatively the growth of stabilized and destabilized fronts under the interactions of gravity, capillarity, and viscous forces (Wilkinson, 1986;Gouyet et al, 1988;Cieplak and Robbins, 1990;Ferer et al, 1995;Zhang and Liu, 1998;Xu et al, 1998;Meheust et al, 2002;Ferer et al, 2003;Lovell et al, 2005). For gas-oil gravity drainage with high viscosity contrast, where the interplay between capillary, gravity and viscous forces controls the invasion front, interaction of transport functions and prevailing forces becomes important.…”