## 4 Clogging Models  
Clogging refers to the process in which particles suspended in the fluid (e.g., fine particles, minerals, or sediments) accumulate in the pore spaces, reducing permeability and impeding fluid flow. This process can be driven by size, shape of particle, concentration, and the velocity of the fluid. In this section, we propose a new clogging (pore-filling) model to elucidate the effect of geometry structure on the fluid flow through the pore space of media. In the literature, there are a few works that propose clogging models, e.g., [6; 36; 37; 38]. [PERSON] et al. determined the four classes of pore-filling mechanisms: grain-pore boundary, low-flux, high-flux, and random filling. This pore-filling mechanism is controlled by fluid rates. In [6], the authors introduced the CT image threshold adjustment model and the reactive transport clogging model to estimate the pore structure changing due to carbonate precipitation. The latter model uses chemical reaction theory to evaluate the local volume variation of the precipitate, with the assumption of CO\({}_{2}\) equilibrium conditions; that is, the authors considered the divalent cations' concentration that is derived by solving the reactive transport model. In this work, to elucidate the role of geometric structure, we address the dynamics of particle deposition--the spontaneous attachment of particles to surfaces by means of the probabilistic clogging model. At the initialization step, a total of \(K\) particles are injected into the inlet surface of the rock samples. They invade through the pore spaces in the rock formation, under the pressure imposed between the inlet and the outlet surface, following the random walk rule introduced in Section 3.2. During the particle flow process, we construct the passed array, as described below. The _passed array_ is a one-dimensional array of length equal to the total number of pore and grain voxels in the rock sample, which keeps track of the number of times the particles have passed through each voxel during the random walk process. Because the length of the passed array is the same as the flattened rock sample, it also keeps track of the corresponding voxels' positions in the rock sample. As the particles cannot move to the grain voxels' positions, the elements corresponding to grain voxels on the passed array are always 0 s. By utilizing this, we reduce the memory usage by storing the passed array in a sparse matrix. At some given time instants, the pore voxels will become grain voxels, with the probability given by  
\[f(q,n)=\sum_{i=1}^{n}(1-q)^{i-1}q\equiv 1-(1-q)^{n} \tag{5}\]  
where \(q\in[0,1]\) is the _probability of deposition_, and where \(n\) is the number of times that particles have passed through the corresponding void position. The value of \(n\) is specified by the corresponding element in the passed array. It is easy to verify that the expression in (5) satisfies the requirements of a probability measure as \(f(q,n)\in[0,1]\) for any \(q\in[0,1]\) and that \(n=1,2,3,...\) and \(f(q,n)\) tends to \(1\) as \(n\) approaches infinity. To understand the mathematical meaning of the expression (5), let us consider a sequence of \(i\) Bernoulli experiments \((X_{1},X_{2},\ldots,X_{i})\), which is also called a Bernoulli process. The Bernoulli trials \(X_{j}\), \(j=1,2,\ldots,i\) take either the value \(1\) if the experiment succeeds with probability \(q\) or \(0\) if the experiment fails with probability \(1-q\). Then, the probability of the output \((0,0,0,\ldots,0,1)\), i.e., that the experiment fails on all first \(i-1\) trials and succeeds at the last trial \(i\), is \((1-q)^{i-1}q\). Similarly, each element in the finite series (5) represents the probability of a pore voxel being precipitated at the time instant when the number of times a particle visits that position is exactly \(i\) and failing to be precipitated at all \(i-1\) previously visited times.