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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...

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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