## 3 Particle Flow Through Pore Space Governed by Random Walk
### Random Walk Rule Governing Particle Movement  
The random walk method is especially appropriate for describing particle movement in highly disordered and heterogeneous, large-scale media, such as the natural porous formation being studied in this work. The classical random walk particle schemes rely on the equivalence between the Langevin equation and the advection-dispersion equation [30; 31]. This method usually requires high computational costs when there exists low permeability or low-velocity areas in the media when the particles remain trapped in low diffusion zones. The efficiency of the particle-tracking algorithm can be enhanced drastically by moving a particle over a fixed distance imposed by the computational mesh [32; 33]. Variants of this methodology are _continuous time random walk_ (CTRW) and _time domain random walk_ (TDRW). It has been shown that the discretized advection-dispersion equation and the TDRW are equivalent [34; 35]. The TDRW in general moves particles on a lattice with jumps of fixed distance, with the direction and transition time being determined by the local advection and dispersion. In this work, we formulated a TDRW model to describe the particle flow within the porous media under the pressure imposed between the inlet and the outlet surface in the presence of the clogging process. Therefore, the model introduced here works in _time-dependent random environments_, because the geometric structure of the pore space evolves during the deposition process in a stochastic way. The proposed method, though simple, is very powerful, and it has the potential to be applied to a variety of complex large-scale disordered phenomena arising from physics, biology, and engineering; for example, the propagation of heat or diffusion of matter through a medium that is highly irregular, due to defects, impurities, or fluctuation. At a given time instant \(t\), suppose that a particle \(x\) is at the position \((i,j,k)\), with the increasing of the third coordinate \((k)\) representing the injection direction from the inlet to the outlet. In this case, we can identify \(x_{t}\equiv(i,j,k)\). The particle can either stay still or move to one of its pore neighbors defined in Equations (1) and (2). Only when \(\mathcal{N}_{p}(x_{t})=\emptyset\), with \(\emptyset\) representing the empty set, i.e., \(x_{t}\) has degree 0, will the particle \(x\) stay still. Otherwise, it will randomly move to one of its pore neighbors in \(\mathcal{N}_{p}(x_{t})\) in the next time instant \(t+1\), with the following transition probability:  
\[p(x_{t}\to y)=\frac{g(x_{t},y)}{\sum\limits_{z\in\mathcal{N}_{p}(x_{t})}g(x_{t },z)}, \tag{3}\]  
where\[g(x_{t},z)= \begin{cases}v+d&\text{if}\ \ x_{t}\to z\ \ \ \ \text{is injection direction}\\ d&\text{otherwise.}\end{cases}  \tag{4}\]  
Here, \(v\) is called _velocity_, which models the pressure gradient, and \(d\) is the _dispersion_. Both \(v\) and \(d\) are chosen to be non-negative. The positive velocity \(v\) ensures that the particles have more tendency to move in the injection direction than in any other direction. In this sense, the parameter \(v\) can _indirectly_ characterize for the strength of the pressure. The probabilistic jumping rule specified in (3) and (4) models both the advective and diffusive terms. Moreover, the transition probability is spatial-dependent, due to the heterogeneity of the pore structure. From the above construction, the transition probability \(p(x_{t}\to y)\) of the particle is specified as in Table 1:  
There are several key assumptions commonly made in the random walk method: independent particle motion, uniform medium properties, discretization of time and space, neglect of convection or external forces, simplified boundary conditions, and neglect of particle size and shape. Each assumption has its advantages and disadvantages; see Table 2. For example, recall that the classical random walk relies on the Langevin equation, which describes the movement of the non-interacting particle due to advection and uncorrelated random velocity.