A model of two-velocity particles filtration with variable injection

MATEC Web of Conferences, Jan 2018

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.

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A model of two-velocity particles filtration with variable injection

MATEC Web of Conferences A model of two-velocity particles filtration with variable injection Yuri Galagus 0 Galina Safina 0 0 Moscow State University of Civil Engineering , Yaroslavskoe shosse, 26, Moscow, 129337 , Russia 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. 1 Introduction Filtration is one of the most important problems in the design of underground and hydraulic structures [1-3]. 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 [5–8], 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 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. 2 Mathematical model Consider quasi-linear hyperbolic system of equations in the domain   {0  x  1, t  0} (C  S) t C x    0 ; S t  (S)C , C(x,t) x0  1; C(x,t) t0  0 ; S(x,t) t0  0 . 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 x  0 and at the initial moment t  0 : 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 d1 and d2 with velocities 1 and 2 respectively. The filter has pores of two diameters D , D2 and D  d1  D , D  d2  D2 . Thus, both types of 1 1 2 1 particles get stuck in small pores with diameter D1 and pass freely through large pores with diameter D2 .The transport of particles of each type is described by a mass balance equation of type (1) in the dimensionless coordinates linking the concentrations of suspended C1(x, t) , C2 (x, t) and retained particles S1(x, t) , S2 (x, t) . The particles of both types compete for the small pores D1 . The growth rates of the deposits S1 / t and S2 / t are proportional to the concentrations of suspended particles C1 and C2 . The filtration coefficients 1(S) and 2 (S) depend on the amount of the current deposit S(x, t) consisting of both types of retained particles S1 and S2 : S  S1  S2 . In the domain  the concentrations of suspended and retainedparticles satisfy the quasi-linear hyperbolic system (6) (7) (8) (9) (10) (11) (12) with the boundary and initial conditions similar to (3)-(5): (Ci  Si )   t Ci  0 ; i x Si  i (S )Ci , i  1, 2 , t Ci (x, t) x0  fi (t), i  1, 2 ; Ci (x, t) t0  0, i  1, 2 ; Si (x, t) t0  0, i  1, 2 . 0  2   . 1 characteristics t  x / 1 and t  x / 2 : At t  0 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 x  0 (the boundary condition 8). The particles are being transported towards the filter outlet x  1 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. The concentration fronts  1 and  2 are the characteristics defined by the equations t  x / 1 and t  x / 2 respectively. Let The domain  is divided into 3 subdomains by the  0  0  x  1, 0  t   x   ,   0  x  1, 1  1  x  t  1 x   , 2  0  x  1, t  2   x   . 2  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  , 2 , the boundary conditions are set on 1 the characteristics t  , i  1, 2 x i S1 tx  0, 1 S2 tx  0 2 and at the inlet of the porous medium (8). Below the periodic boundary conditions are considered C1 x0  p1 1 k1 sin 1t  , C2 x0  p2 1 k2 sin 2t  . 3 Exact solution for constant filtration coefficients Let i (S)  i  const . 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)–(10)  1x  1x C1  f1(t  x / 1)e 1 , S1  f1(t  x / 1)1e 1 t  x / 1 , (x,t)  1  2; C1  0, S1  0, (x, t)   ; 0 2x C2  f2 (t  x / 1)e 2 , 2x S2  f2 (t  x / 1)2e 2 (t  x / 1), ( x, t)  2; C2  0, S2  0, ( x, t)  0   . 1 The exact solution for the concentrations of suspended and retained particles is given by traveling waves with amplitudes exp(i x / i ); i  1, 2 that depend on the coordinate x . 4 Asymptotic solution The asymptotic solution of the problem (6)-(10) is obtained in the form Ci (x,t)  ci0 (x,t)  ci1(x,t)Smax  O(Sm2ax ), i  1, 2; Si (x,t)  si1(x,t)Smax  O(Sm2ax ), i  1, 2 . Substitution of (13), (14) into (6), (7) yields c 0i  i x ci0  0 , t s1 i  1i 1 s11  s12  ci0 , t c 1i  i x ci1  1i(1 s11  s12 )ci0 . t From (11)-(12) the boundary conditions are determined for equations (15)-(17) si1 tx  0, si2 tx  0 i i ci0 x0  pi 1 ki sin(it), ci1 x0  0, ci2 x0  0 . The main asymptotic terms are determined successively from the system (15), (19)     x    x ci0   pi 1  k1 sin  1  t  i    , t  i ; 0 , t  x ; i  1, 2.  i Summing the two equations of the system (16) we get s1 t 1p1t x1  k11 cos1t x1 12 p2 t x2  k22 cos2 t x2 1 , t  x / 2 ; 1p1t x1  k11 cos1t x1 1 , x / 1  t  x / 2. The solution of (21) is   1  e s1    1  e 5 Numerical calculation The asymptotic solution is compared with the results of the numerical calculation by the finite difference method for 1  0.5; 2  0.25; p1  p2  1; i (S)  i (Smax  S) , i  1, 2 ; 1  1, 2  2 ; Smax  0.1 The asymptotic terms numerically by Euler method, с11, с12 are determined from (17) numerically by a countercurrent scheme [14]. In Figures 1-6 the graphs of the numerical solution (solid line) and the asymptotic solutions Ci (x, t)  ci0 (x, t), i  1, 2 , S(x, t)  s1 (x, t)Smax are presented (dashed line); the asymptotic solutions Ci (x, t)  ci0 (x, t)  ci1(x, t)Smax , Si (x, t)  si1 (x,t)Smax , i  1, 2 are also shown (dotted line). At the filter inlet the suspension with variable concentrations Ci (0, t)  1 ki sin it , i  1, 2 is injected. In Figures 1-3 the parameters k1  1, k2  1 , in Figures 4-6 - k1  1, k2  1 . The 3-D graphs of the solution are given in Figures 3, 6. Fig. 1. a) Suspended particles concentration C1(x,t) x1 b) Suspended particles concentration C2 (x,t) x1 c) Retained particles concentration S(x,t) x1 Fig. 2. a) Suspended particles concentration C1(x,t) t1.5 b) Suspended particles concentration C2 (x,t) t1.5 c) Retained particles concentration S(x,t) t1.5 (21) (22) Fig. 3. a) Suspended particles concentration C1(x,t) b) Suspended particles concentration C2 (x,t) c) Retained particles concentration S(x,t) Fig. 4. a) Suspended particles concentration C1(x,t) x1 b) Suspended particles concentration C2 (x,t) x1 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. 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Yuri Galagus, Galina Safina. A model of two-velocity particles filtration with variable injection, MATEC Web of Conferences, 2018, DOI: 10.1051/matecconf/201819604046