Use of saturation-dependent relative mobilities leads to linear flow in wliich the saturations scale as x/t (i.e., position/time). However, experiment and theory liave shown ttiat, in the limit of very large viscosity-ratio, the flow is not linear but fractal. Thus, two questions naturally arise: i) If the actual viscous fingering were fractal, what would be the effect on traditional reservoir simulation? and ii) Is the viscous fingering fratfal 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/t1+ε and fractional flow curves which are functions not solely of saturation S but rather t ε 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 ε : in three dimensions, ε ≈ 1 : in two dimensions, ε ≈ 0.4. Using a standard pore-level model of twodimensional, miscible floods, we have shown that the flow for this model is fractal for a large viscosity-ratio (M=10,000) 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, witfi 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, τ, which increases with viscosity-ratio.as τ =τOM0.17. Once stable, the saturation front advances as x = VOM0.068 t.

Furthermore, using our functional dependency of fractional flow, we showed that the factor M0.068 plays the role of a two-dimensional Koval factor.

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