## 5 Results and Interpretation  
### Fluid Flow Invading  
A total of \(K=100,000\) particles is injected into the inlet surface of each rock sample, following the scheme described in Section 3.1. Under the pressure gradient imposed between the inlet and the outlet, these particles will move deeper into the pore spaces. The velocity and dispersion parameters were chosen as \(v=1\), \(d=0.01\) in this study. The simulations are performed for a total of 5000 time steps. With this setting, as \(v>~{}>d\), the particles have a much stronger tendency to move along the injection direction compared to the others. In the free lattice--that is, without the grain voxels--the expected position of a particle after 5000 simulation steps along the injection direction is  
\[\left[(-1)\times\frac{d}{v+6d}+1\times\frac{v+d}{v+6d}\right]\times 5000= \frac{5000v}{v+6d}\approx 4717.\]  
When considering the movement along the injection direction, only the position transfer forward and backward along the \(k\)-coordinate needs to be counted, because movement along the \(i,j\)-directions does not contribute. However, as the porous rock samples we used in this study contained a large number of grain voxels, i.e., a small proportion of pore voxels (see Table 11) the speed of fluid flow along the injection direction was much lower. This is illustrated in Table 3, which represents the measurements of the average propagation depths of the particles at the 5000 th step of the random walk (RW) simulation. As shown in the table, the average propagation depth in Ketton is twice that of Bentheimer and Doddington, and three times of that of Estaillades. Figure 3 shows samples of the evolution of fluid invading the pore spaces of the rock. During the random walk movement, the particles that reached the outlet surfaces were removed from the simulation for the next random walk step, as they were considered to have escaped. At the end of the simulations, there were no particles escaping from the outlet surface of Bentheimer and Estaillades, while in Doddington and Ketton the numbers of escaped particles were 5 and 581, respectively. Comparing the invading evolutions, we see that among the four rocks, the propagation of the particles along the injection direction increased from Estaillades to Bentheimer to Doddington and, finally, to Ketton. We further investigated the distribution of particles along the injection direction at the end of the simulations, i.e., \(t=5000\). Figure 4 shows the kernel density estimates for the distributions. The log scale was used to better distinguish the differences, especially in the long tails of the distributions. To quantify this propagation, we counted the number of particles present at several slices along the injection direction (\(k=50;100;150;\dots;1000\)) according to the simulation time (\(t=200;400;600;\dots;5000\)). As the random walk was stochastic, we used the aggregate (average) results on multiple (10) trials to access the robustness of the propagation on each rock sample. Figure 5 illustrates the flow propagation of the particles along the injection direction in the the pore spaces of the rocks according to time. The values on the surfaces were the average values taken on the 10 trials. Looking at the change along the space axis (i.e., \(k\)), we observe that in Estaillades, the particles were almost concentrated near the inlet surface (small \(k\)). This may have been caused by the existence of a \"bottleneck\" near \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Bentheimer** & **Doddington** & **Estaillades** & **Ketton** \\ \hline Average propagation depth & 108 & 114 & 72 & 212 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average propagation depth after the 5000 th step of the RW simulation. It was calculated by taking the average of the z-coordinate of the particles at the 5000 th time step. The corresponding coordinates of the escaped particles were assigned a value of 1000.