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.