Figure 11—\({\mathbf{t}_{\mathbf{D}}}^{\mathbf{e}}\mathbf{s(x,t)}\) vs. ( \({\mathbf{x/}\mathbf{t}_{\mathbf{D}}}^{\mathbf{1+e}}\)) using DB.
The saturation profiles are expected not to scale linearly with position and time when using traditional linear-flow models. With the collapse of curves when\({t_{D}}^{e}s(x,t)\) was plotted with (x/\({t_{D}}^{1+e}\)), however, self-similarity is expected to be observed (in accordance with the results from the previous literature). We plotted time vs. saturation for time steps before the finger break and the time steps after the finger break (e.g., t>600s). Time steps after the finger break are indicated in a red rectangle in the legend box of Figures 5-11 and their corresponding saturation was recorded. As can be seen, complete collapsed saturation profiles could not be obtained regardless of the adequacy of the range of fractal numbers (for both D andDB) . However, saturation profiles allowed us to observe that there exists a significant difference between the fingering behavior prior to the finger break (which often results in droplet generation—this will be elaborated further in the next section) and after the finger break. And they tend to follow a similar trend of each section; in the earlier stages of experiments, fingers did not reach significant portions of the Hele-Shaw sample resulting in a lack of data points and accordingly smooth curves for most samples. After the finger break, however, significant fluctuations in the graphs can be observed due to the expansion of saturated profiles (in the form of fingers or droplets). Therefore, it can be inferred that hydrodynamic instability (in the form of droplet generation) is the determining factor in generating different types of “self-similar” fingering behaviors.
Figure 12 displays the relationship between fractal dimension (DB ) and classic capillary number (Equation 1). Since Dvalues for Experiment I and Experiment V were observed to be inordinate, (while the rest were consistent with DB) ,DB were selected for further analysis. As seen, with the capillary number increase, fractal dimensions increase until reaching a peak followed by a relatively slow decrease. Such behavior can be explained by the interplay between surface tension and the viscosity effect. As the surface tension decreases, the fractal number increases (due to the high tip-splitting), however, after the capillary number reaches a certain point (10-3~10-3), the viscosity becomes the dominant force in the establishment of finger morphology (smoothing the frontal mobility line) which leads to a reduction in fractal number.