A model of two-velocity particles filtration with variable injection

The filtration problems are actual in the design of underground storage facilities for hazardous waste. When the grout is injected into the porous soil, the fluid penetrates deep into the rock and, when solidified, blocks the pores. A model of suspension flow with 2-size particles moving with different velocities in a porous medium is considered. The proposed model of deep bed filtration generalizes the known equations of mass balance and particle capture kinetics for a fluid flow with various particles. The injection of a suspension with periodically changing concentration is considered. Exact and asymptotic solutions are obtained. The asymptotics rapidly converges to the numerical solution.


Introduction
Filtration is one of the most important problems in the design of underground and hydraulic structures [1][2][3]. Underground storage of toxic and radioactive waste must be reliably isolated from groundwater and reservoir waters. One of the modern ways to construct the waterproof walls in the ground is to inject under pressure a cement-based suspension. The grout is filtered in the porous soil, the concrete grains penetrate into the fine pores of the rock and block them. After the fluid solidifies, the rock becomes firm and water resistant.
The long-term transport of particle suspensions in porous media with particle capture throughout the entire length of the porous rock is called deep bed filtration. There are different mathematical models of deep bed filtration: the classical model for a mono dispersed suspension flow in a porous medium includes the equations of the particle balance and of the capture kinetics [4]; the population balance models for a poly dispersed suspension motion in a porous medium [5][6][7][8], which take into account the size distribution of pores and particles.
All these models assume that the suspended particles are transported with same velocity. However, experimental results have shown that the particle velocities depend on their size [9]. Suspension in the porous medium moves unevenly due to different pore sizes. With increasing the pores diameter the velocity of a fluid with suspended particles becomes greater. Large particles moving only through large pores, so their average velocity is greater than particles of small particles that are transported both over large and small pores.
The fluid flow with two types of solid particles moving with different velocities in a porous medium is considered. A mathematical model which generalizes the known MATEC Web of Conferences 196, 04046 (2018) https://doi.org/10.1051/matecconf/201819604046 XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering equations of mass balance and particle capture kinetics for a flow of fluid with various particles is developed in [10]. When the grout is injected, the concentration of suspended particles in the fluid can vary. The periodic changes of the concentrations at the porous medium inlet are considered in the paper. The filtration problem with variable coefficients does not have an exact solution, so the methods of asymptotic and numerical modeling are used. The asymptotic solution of the problem with a small limit deposit is constructed by methods [11,12]. The asymptotics rapidly converges to the numerical solution.
The mathematical model of 2-size particles transport is presented in Section 2. Section 3 provides the exact solution for constant coefficients. The asymptotic solution for a small limit deposit is obtained in Section 4. Section 5 is devoted to the comparison of the asymptotics with the numerical solution. The conclusion in Section 6 finalizes the paper.

Mathematical model
where ( , ), ( , ) C x t S x t are the suspended and retained particles concentrations;  is the velocity of the suspended particles; ( ) S  is the filtration coefficient. The boundary conditions for the system (1), (2) are set at the inlet of the porous medium 0 x  and at the initial moment 0 t  : The solution of the system (1)-(5) is considered in [13]. Now consider the model of 2-velocity particles transport. Let the suspension flow has particles of two different sizes 1 d and 2 d with velocities 1  and 2  respectively. The filter has pores of two diameters 1 D , 2 D and 1 In the domain  the concentrations of suspended and retainedparticles satisfy the quasi-linear hyperbolic system with the boundary and initial conditions similar to (3)- (5): At 0 t  there are no suspended and retained particles in the porous medium (the initial conditions 9, 10). The suspension with variable concentrations of 2-size particles is injected into the filter inlet 0 x  (the boundary condition 8). The particles are being transported towards the filter outlet 1 x  at different velocities 1  and 2  and gradually fill the entire porous medium. The mobile boundary of the domain which contains particles of a certain size is called a concentration front.
In the domain 0  the porous medium is empty; in the domain 1  only fast particles of type 1 are presence; in the domain 2  there are particles of both types.
To determine the solution in the domains 1  , 2  , the boundary conditions are set on the characteristics , 1, 2 and at the inlet of the porous medium (8).
Below the periodic boundary conditions are considered In this case the system (6), (7) splits into two independent systems of the form (1), (2) for each particle type. The exact solution of the problem (6) ; .

Conclusion
A model of two particles filtration in the porous medium is considered. Particles of two types are injected with variable concentrations into the porous medium and move with different velocities. Particles freely pass through large pores and get stuck at the inlet of small pores due to the size-exclusion mechanism of particles retention. For constant filtration coefficients, an exact solution of the problem is obtained in the form of travelling waves of variable amplitude moving in a porous medium with different velocities.
For linear blocking filter coefficients and periodically varying concentrations at the filter inlet, an asymptotic and numerical solution of the problem are obtained.