Numerical Modeling of Suspension Filtration in Porous Media with Modified Sedimentation Kinetics

1. Introduction

This paper investigates the mathematical modeling and analysis of suspension filtration in a heterogeneous porous medium, taking into account the deposition of suspended particles on pore walls in a medium saturated with both mobile and immobile liquid phases. Particular attention is given to the effects of sediment accumulation and inter-zone mass transfer on the filtration process.

In recent years, the disposal of various pollutants and suspended impurities through their injection, together with water, into underground reservoirs and permeable geological formations has attracted increasing attention. The practical implementation of this approach requires reliable methods for evaluating and predicting the main process characteristics, which are essential for assessing potential environmental risks and ensuring the safety of underground disposal operations. Mathematical modeling of suspension filtration in porous media is also of considerable importance in the development and optimization of secondary and tertiary oil recovery techniques, underground leaching processes, and related subsurface technologies [1,2,3,4,5].

An adequate mathematical description of suspension filtration is essential for the qualitative and quantitative assessment of transport, deposition, and retention processes in porous media. Such a description should account for the principal physical mechanisms governing the movement of suspended particles and their interaction with the porous matrix. A number of studies have therefore been devoted to the mathematical modeling of suspension filtration and particle transport in porous media [6,7,8,9].

Field observations and operational experience in oil reservoirs with heterogeneous structures indicate that, during waterflooding, fluids in regions characterized by low porosity and permeability may remain essentially immobile or move considerably more slowly than the fluid in the main flow channels, even under substantial pressure gradients. These regions are commonly referred to as stagnant or immobile zones, while the pores associated with them are often described as “dead-end” or “dead” pores. The presence of such zones can significantly affect solute and particle transport, retention, and breakthrough behavior in heterogeneous porous media. Mathematical models describing mass transport in porous media with mobile and immobile regions were developed in a number of studies [10,11,12,13,14,15,16,17,18]. In these models, the exchange of mass between the mobile and immobile zones is generally represented by a first-order kinetic equation.

2. Problem Statement and Mathematical Model

We consider the problem of the filtration of suspended particles, which deposit as sediment within the mobile fluid zone, while also accounting for mass exchange with the second zone. We examine the case where both sedimentation and mass exchange are described by a first-order kinetic equation [19,20,21,22].

Let us consider a porous medium consisting of two zones: 1) ${\Omega }_1$ is a zone with porosity $m_1$, where the pores act as a transit for the liquid - a zone saturated with mobile liquid; 2) ${\Omega }_2$ is a zone with porosity $m_2$ saturated with immobile liquid (Fig. 1).

The porous medium is considered a semi-infinite, one-dimensional domain, with the ${\Omega }_2$ zone evenly distributed within the ${\Omega }_1$ zone.

Figure 1. A two-zone porous medium: transit pores (${\Omega }_1$) and connected, stationary liquid (${\Omega }_2$).

Figure 1. A two-zone porous medium: transit pores (${\Omega }_1$) and connected, stationary liquid (${\Omega }_2$).

Based on the assumptions made, the model for the problem is as follows [22,23,24]:

$$m_1{\displaystyle \frac{\partial c}{\partial t}}+v{\displaystyle \frac{\partial c}{\partial x}}+m_2{\displaystyle \frac{\partial N}{\partial t}}+{\displaystyle \frac{\partial \rho }{\partial t}}=D{\displaystyle \frac{{\partial }^{2}c}{\partial x^2}},$$

(1)

where $c$ is the substance concentration in the ${\Omega }_1$ zone, m³ /m³ , v is the filtration velocity, m/s, $N$ is the substance concentration in the ${\Omega }_2$ zone, and $\rho$ is the precipitate concentration in the ${\Omega }_1$ zone, m³ /m³.

The term ${\displaystyle \frac{\partial N}{\partial t}}$ in (1) describes the intradiffusional mass transfer from the ${\Omega }_1$ zone to the ${\Omega }_2$ zone. To evaluate it, one can use the solution of the diffusion problem for geometric bodies of a certain shape (e.g., cylinders, spheres, etc.) represented as [11]. Here, we use the approach where mass transfer is defined as a first-order kinetic process [19,20]. Thus, the intradiffusional mass exchange is determined by the kinetic equation:

$${\displaystyle \frac{\partial N}{\partial t}}=k\left(c-N\right),$$

(2)

where $k$- const.

The kinetics of precipitate formation is described by a first-order equation

$${\displaystyle \frac{\partial \rho }{\partial t}}={\beta }_{a}c-{\beta }_d\rho ,$$

(3)

${\beta }_a,{\beta }_d$- const .

The system has the following initial and boundary conditions:

$$c\left(0,x\right)=0,N(0,x)=0,\rho \left(0,x\right)=0,c\left(t,0\right)=c_0,c\left(t,\infty \right)=0.$$

(4)

To solve problem (1) - (4), we apply the finite difference method [25,26,27,28,29,30,31,32,33,34].

We introduce a grid in the domain $\Omega =\left\{(t,x),0\le t\le T,0\le x\le \infty \right\}$ where T is the maximum value of time over which the process is studied. To do this, we divide the interval $\left[0,\infty \right)$ with a step of h, and we divide the segment $\left[0,T\right]$ with a step of $\tau$. As a result, we obtain the following grid:

$${\overline{\omega }}_{\tau h}=\left\{(t_j,x_i);t_j=\tau j,x_i=ih,\tau ={\displaystyle \frac{T}{J}},i=\overline{0,I},j=\overline{0,J}\right\},$$

Equations (1), (2), and (3) are approximated as follows:

$$m_1{\displaystyle \frac{c_i^{j+1}-c_i^j}{\tau }}+v{\displaystyle \frac{c_{i+1}^{j+1}-c_{i-1}^{j+1}}{2h}}+m_2{\displaystyle \frac{N_i^{j+1}-N_i^j}{\tau }}+{\displaystyle \frac{{\rho }_i^{j+1}-{\rho }_i^j}{\tau }}=D{\displaystyle \frac{c_{i-1}^{j+1}-2c_i^{j+1}+c_{i+1}^{j+1}}{h^2}},$$

(5)

$${\displaystyle \frac{{\rho }_i^{j+1}-{\rho }_i^j}{\tau }}=k\left(c_i^j-{\rho }_i^{j+1}\right),$$

(6)

$${\displaystyle \frac{S_i^{j+1}-S_i^j}{\tau }}={\beta }_a{c}_i^j-{\beta }_d{S}_i^{j+1},$$

(7)

By transforming the difference schemes (6) and (7), we obtain the following:

$${\rho }_i^{j+1}={\rho }_i^j+\tau k\left(c_i^j-{\rho }_i^{j+1}\right),$$

(8)

$$S_i^{j+1}=S_i^j+\tau \left({\beta }_a{c}_i^j-{\beta }_d{S}_i^{j+1}\right),$$

(9)

3. Results and Discussion

Some results are presented in Figure 2, Figure 3, Figure 4 and Figure 5. An analysis of the graphs shows that due to the entry of substances into the medium, concentration fields c, N, ρ are formed, which move along the reservoir over time. It can be seen from the graphs that over time, all three fields shift towards the interior of the medium, and their values at fixed points simultaneously increase (Figure 2, Figure 3, Figure 4 and Figure 5).

When $k=5\cdot {10}^{-3}$, it can be observed that the distribution of the concentration of suspended particles in the liquid has reached the $0,20m$ part of the medium where $t=900s$, while at $t=1800s$, it has reached $0,25m$ (Figure 2.a).

When $k=5\cdot {10}^{-3}$, the precipitate concentration reaches its maximum value only at the point $x=0$, starting from the time $t=450s$. At larger time values, it is shown that the precipitate concentration also reaches its maximum value at points close to the point $x=0$ (Figure 3. b).

It can be seen that when the value of the coefficient $k$ in equation (2) is decreased $k=2\cdot {10}^{-3}$, the ($c/c_0$) and ($\rho$) profiles spread more widely (Figure 3) and their values at fixed points are greater than when $k=5\cdot {10}^{-3}$. This is explained by the fact that when $k=2\cdot {10}^{-3}$, the substance exchange with the second medium is significantly reduced (Figure 3.b).

Figure 2. Profiles of the change in values of c/c₀ ( a), N (b), ρ (c) at different points in time, where $k=5\cdot {10}^{-3}$c ^-1.

Figure 2. Profiles of the change in values of c/c₀ ( a), N (b), ρ (c) at different points in time, where $k=5\cdot {10}^{-3}$c ^-1.

Figure 3. Profiles of the change in values of c/c₀ ( a), N (b), ρ (c) at different points in time, where $k=2\cdot {10}^{-3}$c ^-1.

Figure 3. Profiles of the change in values of c/c₀ ( a), N (b), ρ (c) at different points in time, where $k=2\cdot {10}^{-3}$c ^-1.

Figure 4. Profiles of the change in values of c/c₀ ( a), N (b), ρ (c) at different points in time, where $k=20\cdot {10}^{-3}$c ^-1.

Figure 4. Profiles of the change in values of c/c₀ ( a), N (b), ρ (c) at different points in time, where $k=20\cdot {10}^{-3}$c ^-1.

Figure 5. Profiles of the change in values of c/c₀ ( a), N (b), ρ (c) at different values of k, where $t=1800$c.

Figure 5. Profiles of the change in values of c/c₀ ( a), N (b), ρ (c) at different values of k, where $t=1800$c.

Conversely, when the value of the coefficient $k$ is increased to $k=20\cdot {10}^{-3}$, it can be seen that the profiles ($c/c_0$) and ($\rho$) spread less (Figure 4), and their values at fixed points are smaller compared to the case where $k=5\cdot {10}^{-3}$. This is explained by a significant increase in mass transfer with the second medium when $k=20\cdot {10}^{-3}$ (Figure 4.b).

These conclusions are more clearly illustrated in Figure 5, which is provided for comparison at different values of the coefficient $k$.

4. Conclusions

A mathematical model for suspension filtration in a heterogeneous porous medium with a two-zone structure and multi-stage deposition kinetics has been developed and numerically investigated. The model accounts for the principal mechanisms governing the filtration process, including suspended particle transport, particle deposition and sediment accumulation, as well as mass exchange between mobile and immobile fluid zones. The resulting system of governing equations was solved using a finite difference scheme, enabling the spatial and temporal evolution of the filtration process to be analyzed.

The numerical simulations demonstrate that the concentrations of suspended particles, particles in the immobile zone, and accumulated sediment exhibit pronounced spatial and temporal variations during filtration. The intensity of mass exchange between the mobile and immobile zones was found to have a significant effect on the evolution and distribution of these quantities. In particular, reducing the mass transfer coefficient leads to broader concentration profiles and higher concentration values at specified locations, whereas increasing the rate of mass exchange produces more localized concentration distributions and lower concentrations. These results confirm the substantial role of inter-zone mass transfer in determining particle transport and deposition in heterogeneous porous media.

The proposed two-zone model with multi-stage deposition kinetics provides a more comprehensive representation of suspension filtration than conventional single-zone models, particularly for porous media containing regions with significantly different flow characteristics. The model can be applied to the analysis of a range of environmental and engineering processes, including contaminant transport in groundwater, subsurface waste disposal, filtration in heterogeneous formations, and enhanced oil recovery. Further research may extend the proposed framework to multidimensional transport, nonlinear and concentration-dependent kinetic processes, and inverse modeling approaches for the estimation and identification of model parameters.