The Fractal Nature of Viscous Fingering: Saturation Profiles and Fractional Flows From Modeling of Miscible, Two-Component Flows in Two-Dimensional Pore Level Models

Abstract

Use of saturation-dependent relative mobilities leads to linear flow in which the saturations scale as x/t (i.e., position/time). However, experiment and theory have shown that, in the limit of very large viscosity-ratio, the flow is not linear but fractal. Thus, two questions naturally arise:If the actual viscous fingering were fractal, what would be the effect on traditional reservoir simulation? andIs the viscous fingering fractal for viscosity-ratios of real reservoirs? To study the first question, we used general arguments to show that fractal flow produces saturations which scale not as x/t but rather as x/t and fractional flow curves which are functions not solely of saturation S but rather t S. Thus, when the flow is fractal, fractional flows and the relative mobilities depend on time t and saturation S not solely on saturation. Earlier studies have determined the values of : in three dimensions, ; in two dimensions, 0.4. Using a standard pore-level model of two-dimensional, miscible floods, we have shown that the flow for this model is fractal for a large viscosity ratio (M=l0,000) and that the functional dependencies of the saturation and fractional flow agree with the results of our general arguments but not with the results from the traditional models, such as Buckley-Leverett or Koval. We addressed the second question using our model, with viscosity-ratios, M = 3 300. Our modeling of the finite viscosity-ratio showed that, initially, the flows are fractal; but that they become compact or stable on a time scale,, which increases with viscosity-ratio, as. Once stable, the saturation front advances as. Furthermore, using our functional dependency of fractional flow, we showed that the factor M plays the role of a two-dimensional Koval factor. Introduction Composition dependent relative mobilities are used in all traditional modeling of flow in porous media, such as Buckley-Leverett, Koval, etc. Use of composition dependent relative mobilities leads to simulations in which the saturation front advances linearly with time. However, it has been shown that the limit of infinite viscosity-ratio (M) is accurately described by diffusion limited aggregration (DLA), a process which is known to form fractal objects with non-uniform densities. For fractal flow the saturation front advances faster than linearly with time. Two questions naturally arise:if the actual viscous fingering were fractal, what would be the effect on traditional reservoir simulation? andis the viscous fingering fractal for viscosity-ratios in the reservoir range? First, what is a fractal? The classic signature of a fractal object is a non-Euclidean relationship between mass and size. That is, if one has an ordinary solid disk the mass is proportional to R2, but for a circular fractal object the mass is proportional to RDf, where the fractal dimension, Df, is not an integer, e.g. 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, usually non-equilibrium, process. A number of excellent reviews discuss a wide variety of these fractal growth phenomena including material deposition, dielectric breakdown, as well as two-phase flow in porous media. P. 435

Experimental and computational advances on the study of Viscous Fingering: An umbrella review

“…Regarding the modeling of the MVF fractal nature, Ferer conducted one of the very first numerical studies on this regard in 2 [57]. The purpose was to compare the conventional reservoir simulators, which assumed that the saturation front was lineal, against simulations where the saturation front was fractal.…”

Monitoring and Characterizing the Finger Patterns Developed by Miscible Displacement in Fractured Heavy Oil Systems

“…Later, Peters and Cavalero illustrated that injection rate and mobility ratios slightly influenced the fractal nature of induced fingers in first contact miscible displacements. Ferer et al noted that if fractal flows dominate viscous fingering, saturation will not be a function of fractional flow. Zhang and Liu interpreted that the effective fractal dimension of fingers decreases by an increase of viscosity ratio, and also that low fractal dimension corresponds to severely ramified viscous fingering fractal structure and short breakthrough time.…”