Filtration of 2-particles suspension in a porous medium

Abstract During the construction of underground storage of hazardous waste, it is necessary to create waterproof walls in the ground. The grout is filtered in the rock, fills the pores and, when hardened, creates a reliable barrier to groundwater. A one-dimensional model of the flow of inhomogeneous particles in a porous medium is considered. The retained particles profiles formed during deep bed filtration are studied. It is shown that when filtering a 2-particle suspension, the deposit is distributed unevenly. The profile of large retained particles is always monotonous, and the profile of small retained particles is nonmonotonic. The monotonicity of the total deposit profile depends on the model parameters. The shape of non-monotonic profiles is time-dependent. At short times, the profile decreases monotonously. At some point, a maximum appears on the profile graph, which shifts from the inlet to the output with increasing time. When the maximum point reaches the outlet, the profile becomes monotonically increasing. With a further increase in time, the retention profiles remain monotonically increasing. Analytical solutions for a filtration model with particles of three or more different types are unknown. Analysis of the retention profiles of the polydisperse suspension requires further study.


Introduction
When constructing tunnels and underground structures, it is necessary to protect buildings from the penetration of groundwater. To create waterproof partitions, a reinforcing solution is pumped into the soil. The liquid grout is pumped in the pores of the soil and forms a waterproof layer during cementation. Mathematical modeling allows to optimize the construction technology [1,2].
The transport of small particles in porous media is accompanied by the formation of deposit. During deep bed filtration the particles are retained in the whole porous medium. Depending on the physical and chemical properties of the particles, the structure and material of the porous medium, various mechanisms of particle capture can prevail: retention in the neck of narrow pores, formation of arch bridges by several particles at the inlet of the pores, attraction to the walls of the pore channels, diffusion into dead-end pores, etc. [3,4]. As a rule, the deposited particles cannot be knocked out from retention sites by moving particles or by the flow of a carrier fluid and remain motionless.
Mass balance of suspended and retained particles and deposit growth equations form a hyperbolic system of deep bed filtration. Unknown functions are volume concentrations of suspended and retained particles. Retention profiles are the dependences of deposit concentration on the spatial coordinate at a fixed time. When filtering a monodisperse suspension, the retention profiles always monotonically decrease [5]. For a polydisperse suspension containing particles of various sizes, the retention profiles become nonmonotonic. This was first shown experimentally [6,7]. Then theoretical  [8,9].
In [10], a bicomponent filtration model with particles of different sizes moving at the same velocities is considered. The non-monotony of the profiles is associated with different particle sizes and depends on the model parameters. The retention profile of large particles always monotonously decreases, and the retention profile of small particles changes the monotony: decreases with a short time and increases with a long time. At intermediate times, a maximum point appears on the profile graph, which shifts from the inlet to the outlet of the porous medium. The monotony of the total retention profile depends on the model parameters. However, the conditions under which the retention profile of the total deposit is nonmonotonic were unknown.
This study contains simple mathematical conditions for the profile nonmonotonicity. It is shown that the total deposit profile monotonically decreases at short times and monotonically increases at a large time. At intermediate time, the profile has a maximum point. Numerical calculations and graphs illustrate the change in the monotony of the profiles.

Mathematical model
(2) Here 00 are positive constants, and 00 12 1 cc  . In equation (2), the deposit growth rate is proportional to the linear filtration and concentration functions. The model uses the blocking filtration function (1 ) b  [12]. For low concentrations of suspended particles, the concentration function is proportional to the first degree of concentration [13].
Boundary and initial conditions determine the unique solution to the system: Conditions (3) specify the injection of a suspension of constant concentration at the inlet of the porous medium 0 x  ; initial conditions (4) mean that at the initial moment 0 t  the porous medium does not contain any suspended and retained particles.
Assume that 12   , i.e. particles of type 1 are larger than particles of type 2.
An analytical solution to problem (1)-(4) is constructed similarly to [14][15][16]. Two types of the suspended particles concentrations are related by equations , cc c c c c cc and the concentrations of suspended and retained particles are related by Riemann invariants -by the relations between solutions on the characteristics of the system [17] The condition of nonmonotonicity of the total deposit profiles follows from equations (7), (8)

Results
Numerical calculations are used in the study of complex models for which the analytical solution is either unknown or has a complex implicit form [18][19][20]. Solutions obtained by numerical methods allow the use of mathematical models to optimize technological processes [21,22].

Discussion
The calculations revealed patterns of change in the monotony of the profiles. Profiles 1 s are always monotonously decreasing, and profiles 2 s are not monotonous. At  (figures 2b, 2c, 3a, 3b, 3c). At large times, the maximum point disappears and the profiles become monotonically increasing (figures 2d, 3d).
With increasing time, the maximum points of the profiles move to the right along the coordinate axis. Before the maximum point at the inlet 0 x  , the profiles monotonously decrease. When the maximum point reaches the outlet 1 x  , the profiles become monotonically increasing. The maximum point on the partial retained concentration profile 2 s always exceeds the maximum point of the total retained concentration profile s, it appears earlier near the inlet and disappears earlier, reaching the outlet of the porous medium.

Summary
The profiles of the retained particles formed during the filtration of a 2-particle suspension in a porous medium are studied. Retained particles profiles of a monodisperse suspension always decrease monotonously. The profiles of large retained particles of a 2-particle suspension also decrease monotonously. Retention profiles of small particles are nonmonotonic.
The monotonicity of the total retention profile depends on the sign of the expression 00 1 1 2 2 Z B c B c . For 0 Z  , the profiles monotonically decrease, for 0 Z  , the profiles are nonmonotonic. The parameters 12 , BB determine the size of the regions occupied by the deposited particles on the porous medium frame.
The shape of non-monotonic profiles is time-dependent. At short times, the profile decreases monotonously. At some point, a maximum appears on the profile graph, which shifts from the inlet to the output with increasing time. When the maximum point reaches the outlet 1 x  , the profile becomes monotonically increasing. With a further increase in time, the retention profiles remain monotonically increasing.
Analytical solutions for a filtration model with particles of three or more different types are unknown. Analysis of the retention profiles of the polydisperse suspension requires further study.