## 5 Results and Interpretation
### Fluid Flow Invading
\(k=200\) in Estaillades, indicated by the very low porosity at this location in the porosity profile, according to the injection direction shown in Figure 11.  
The particles spread wider in the \(k\) direction in Bentheimer, but over a smaller range compared to Doddington; Among the four rocks, Ketton allowed the particles to spread the widest: there were relatively large numbers of particles that were able to reach the middle way to the outlet surface from time \(t=2000\) and more could reach there when time increased (see the last sub-figure in Figure 5). Figure 3: Evolution of fluid invading four rock samples, each on one row. The injection direction is the increasing direction of \(k\), i.e., from the bottom to the top. Next, we quantitatively investigated the ability of the particles to invade deeply the pore spaces and the corresponding speed. During the long-term random walk simulation processes with two distinct values of velocity, \(v=1\) and \(v=10\), we recorded the number of particles that were able to escape from the outlet surfaces. The results are shown in the left panel of Figure 6. In both cases, no particles were able to escape from the outlet surface of the Estaillades sample within the allotted time \(t=\) 5,000,000. Out of the 100,000 particles injected into the pore space of each rock sample, only a small portion were able to escape from the outlet surfaces of Bentheimer and Doddington. The number of particles that were able to escape from Ketton was much larger than those from Bentheimer and Doddington. Taking only the particles that were able to escape from the outlet surfaces, we defined the _travel time_ of a particle as the number of simulation steps it took for the particles to travel from the inlet surface to the outlet surface in an allotted time. Figure 4: Kernel density estimates for the distribution of particles after the fluid flow simulation finished, i.e., at \(t=5000\). Figure 5: The average flow propagation of the particles (taking from 10 independent trajectories) along the injection direction \(k\) within the pore spaces of the rocks according to time \(t\). Note that, in our simulation, when a particle reached the outlet surface it was not able to come back, because we removed it from the particle moving system for the next simulation steps. The _elapsed time_ for each rock was defined as the average travel time of its escaped particles. The right panel in Figure 6 indicates that the elapsed time of the Ketton rock was remarkably lower than those of Bentheimer and Doddington, despite the fact that the porosity of the Ketton rock was lower than that of the other two (see Table A1). This suggests that the geometric structure of the rock, rather than its porosity, appears to be the factor that affects the transportation speed. The simulation result also reveals that, of the four rocks, Ketton allowed the fastest and highest fluid flow through its pore spaces and had high permeability. Figure 6 also shows the effect of the velocity \(v\) on the number of escaped particles. The simulation result reveals that increasing the value of the velocity by increasing the pressure gradient between the inlet and the outlet surfaces did not necessarily increase the invading process. This may be because as the velocity was too large, the particles lost their flexibility to move in the dispersion direction so as to avoid obstacles in the injection direction. In our simulation, when the velocity increased 10 times, the number of particles escaping from the outlet surface of Ketton decreased by half. These phenomena are similar to the multi-phase fluid displacement mechanisms associated with capillary numbers. A high-velocity condition in our simulation is consistent with a high capillary number condition. Because the injected fluid saturation at a high capillary number is lower than that at a low capillary number [24], our results shown in Figure 6 agree with pore fluid dynamic simulation (i.e., LBM simulation).