## 5 Results and Interpretation
### Random Walk-Based Clogging
#### 5.2.2 Multi-Phase Clogging  
For the multi-phase clogging, we chose the simulation parameter setting as follows: \(K=100,000\), \(v=1\), \(d=0.01\), \(N=5000\), \(M=500\). Three different values of probability of deposition, \(q=[0.0001,0.001,0.01]\), were performed, to investigate its effect on the clogging process. To visualize the clogging process through pore rock, we will show the voxel-to-grain process on a slice in the middle of the rock that was perpendicular to the injection direction, \(k=200\) of the Bentheimer rock with \(q=0.0001\) in Figure 8. In these sub-figures, the grain voxels are in gray, the pore voxels are in white, and the precipitated voxels are in red. After the 500 th phase, the porosity in this two-dimensional slice reduced from 0.2488 to 0.1981; that is, the porosity reduced approximately 20% after the 500 th phase clogging process. Next, let us consider the effect of the probability of deposition \(q\) on the clogging process of the four types of rocks. In Figure 9, we depict the time evolution of porosities during the multi-phase process on four rock samples with three different values of probability of deposition, \(q=0.0001,0.001,0.01\). It is interesting that the larger the value of \(q\), the faster the porosities decreased at the beginning of the process, while with the smaller the value of \(q\), the more effective the deposition was in the long term. This behavior was observed in all the rock samples. In the single-phase case, we see that the porosity  
Figure 7: Average porosity decrease taken from 10 independent simulations for each parameter setting (\(q\), rock) with respect to deposition probability \(q\). decrease can be described as a monotonically non-increasing function of \(q\), which leads to the conclusion that the larger the probability of deposition the more pore voxels were precipitated. However, this was not so in the multi-phase case. The reason is that, during the multi-phase process the geometric structure of the pore space evolved according to time; as a result, the changes in permeability due to deposition affected the fluid behavior within the geologic formations. When we set the value for \(q\) as large, then the deposition happened rapidly at the beginning of the trajectories (see the slopes of the porosity curves corresponding to \(q=0.01\) in Figure 9, from phase 0 to phase 25). However, at these time intervals, most of the particles had not yet reached near enough to the outlet surface, meaning that most of the precipitated voxels were near the inlet surfaces. This narrowed down or obutrated the capillaries and, therefore, prevented the particles from moving toward the injection direction. Figure 8: Evolution of the multiple-phase clogging process according to time in the middle slice along the injection direction, i.e., \(k=200\), of Bentheimer: grain voxels (gray), pore voxels (white), mineralized pore voxels (red) with \(q=0.0001\). Figure 9: Effect of deposition probability \(q\) on the multi-phase clogging processes of four types of rocks. Let us observe the distribution of the precipitated voxels along the injection direction at the end of the multi-phase clogging process. Figure 10 shows a histogram of the precipitated voxels on equal-length bins along the \(k\) direction. More precisely, we divided the rock sample derived at the end of the multi-phase clogging into 50 disjoint, three-dimensional blocks along the injection direction. Each block consisted of 20 consecutive layers, e.g., the first block consisted of 20 layers, from \(k=1\) to \(k=20\), and the 50-th block consisted of 20 layers, from \(k=981\) to \(k=1000\). We counted the number of precipitated voxels on each block and used the bar plot to visualize the distribution. According to Figure 10, when the chosen probability of deposition \(q\) was quite large--say, \(q=0.01\)--it blocked the capillaries, which prevented the particles from approaching the outlet surfaces, as seen in Bentheimer, Doddington, and Estaillades.