Fractal Analysis
Changes in fingering pattern morphology are closely related to fractal behavior (Nittmann et al. 1985; Kawaguchi et al. 2004). Chen (1989) and May and Maher (1989) noted that repeated tip-splitting is associated with higher fractal dimensions of the fingers. Building on the previous observations, comparative analysis between the fractal dimension (D ) generated based on the statistical theories from Sharma et al. (2012) and Ferer et al. (1993) and fractal dimension (DB ) from the box-counting analysis usingImageJ (Abramoff et al. 2004) was conducted. The main objective of this particular analysis is to characterize the displacement model before and after the finger stability loss in order to better predict the change in displacement pattern caused by the hydrodynamic instability of fingers.
The results demonstrated that while DB falls in the range of [1.47 ~ 1.6], D is observed to be in the range of [1.3 ~ 5.6], which is notably different in the case of CTAB (experiment I) and CTAB+XG (experiment V) as seen in Figure 4 . Two distinct reasons could be responsible for such pronounced discrepancy: (1) considering that Log(l ) was plotted before the fingering instability, the time to reach the normalized finger break time defined as tB/tT for these two experiments is significantly shorter than that of other cases (this point will be further elaborated in the next section), and (2) while tB/tT is relatively short for the experiment I and V, finger development occurred at small-scale for both cases (due to reasons such as hydrodynamic stability and/or delayed chemical interaction), and thereby limiting the application of the method for accuracy. Therefore, the fractal pattern scaling terminology (Equation 18a and 18b) from Ferer et al. (1993) and Sharma et al. (2012) could not be strictly applied for the saturation models of Experiment I and Experiment V.