Numerical solution of filtration in porous rock

The filtration problem is one of the most relevant in the design of retaining hydraulic structures, water supply channels, drainage systems, in the drainage of the soil foundation, etc. Construction of transport tunnels and underground structures requires careful study of the soil properties and special work to prevent dangerous geological processes. The model of particle transport in the porous rock, which is based on the mechanicalgeometric interaction of particles with a porous medium, is considered in the paper. The suspension particles pass freely through large pores and get stuck in small pores. The deposit concentration increases, the porosity and the permissible flow of particles through large pores changes. The model of one-dimensional filtration of a monodisperse suspension in a porous medium with variable porosity and fractional flow through accessible pores is determined by the quasi-linear equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth. This complex system of differential equations has no explicit analytical solution. An equivalent differential equation is used in the paper. The solution of this equation by the characteristics method yields a system of integral equations. Integration of the resulting equations leads to a cumbersome system of transcendental equations, which has no explicit solution. The system is solved numerically at the nodes of a rectangular grid. All calculations are performed for non-linear filtration coefficients obtained experimentally. It is shown that the solution of the transcendental system of equations and the numerical solution of the original hyperbolic system of partial differential equations by the finite difference method are very close. The obtained solution can be used to analyze the results of laboratory research and to optimize the grout composition pumped into the porous soil.


Introduction
The study of water filtration in soils and various porous materials is of great practical interest in solving a variety of engineering problems in the field of hydraulic engineering, construction of water supply and wastewater facilities. Calculations of filtration problems are used in the design of retaining hydraulic structures (earthen dams, lintels, dams), water supply channels, drainage systems, when draining the soil foundation, etc. In the construction of underground structures, cementation of soils is the most common method of strengthening porous rock [1]. Therefore, research and development of improved solutions for the soil strengthening is one of the most pressing problems in the field of construction.
Solid particles are carried by the fluid flow through the sinuous channels of a porous medium, some particles are locked in the pores and are retained on the porous frame, which leads to a change in its structure, reducing its porosity and accessible fractional flow. [2]. There are various mechanisms of particle capture by porous rocks: size exclusion in thin pores, bridging, gravity segregation, and diffusion [3], [4]. A mathematical model of particle motion in a filter based on size-exclusion mexanism of particle retention in a porous medium is considered in [5], [6] . The suspension particles freely pass through large pores and lock pores that are smaller than the particle size. The considered model takes into account the change in porosity and accessible particle flow through large pores with increasing deposit concentration. At the initial time the porous medium does not contain suspended and retained particles. The suspension of constant concentration is injected at the filter inlet. It gradually fills the porous medium. The concentration front of suspended and retained particles moves with a constant velocity from the inlet to the outlet. Before the front, the porous medium is empty and the concentrations of suspended and retained particles are zero. Behind the front, particles are filtered, the concentrations of suspended and retained particles are positive.
There are various filtration models of monodisperse and polydisperse suspensions [7], [8]. In a number of important special cases, it is possible to obtain an exact or asymptotic solution of the problem [9], [10]. In the general case the model have no explicit analitic solution and numerical methods have to be used. The filtration problem is reduced to a system of transcendental equations. In this paper, the system solution was found using the Matlab application package. The solution of the transcendental system is compared with the numerical solutions of the original hyperbolic system and the equivalent partial differential equation, obtained using difference schemes with second-order approximation.

A mathematical model
Quasilinear hyperbolic system of first order equations ( ( ) ) ( ( ) ) 0 ( at the initial time 0  t the concentrations of suspended and retained particles of the porous medium are zero The concentration front of suspended and retained particles, determined by a linear In the domain 0  the solutions ( , ) C x t and ( , ) S x t are zero, in the domain 1  both solutions are positive. At the concentration front the solution ( , ) C x t has a discontinuity of the first kind, the solution ( , ) S x t is continuous and has discontinuous derivatives. In [11] it is shown that from the system of equations (1)-(4) it is possible to obtain the equation where the initial function is determined implicitly by the relation Let us rewrite equation (5) as To and condition (6) is written in the form: The solution of the problem (10), (11) has the form To obtaine the solution to problem (5), (6), it is necessary to express characteristic variables 0 ,  t in formula (12) in terms of Cartesian variables x, t. Substituting expression (12) into equations (9) and integration over the variable τ with respect to initial conditions yields:  (17) For known solution ( , ) S x t the solution ( , ) C x t can be determined from the relation between the solutions obtained in [12].

Numerical methods
The system of equations (16), (17)  ()     g S g g S g S g S ; S , moreover, the filtration coefficient is zero at a certain positive value S , otherwise the concentration of retained particles increases indefinitely and is infinite in the limit.
Then the cubic polynomial ()  S can be represented as Then, at each node of the domain 1  it is possible to solve the system of equations (18), (19), substituting the current values j t and i x into the right-hand sides of the equations. Values of ( , ) C x t are calculated in the same nodes. To test the obtained solution the problem (1)-(4) was numerically solved by the method of finite differences, similarly to [14]- [16]. The integrals from the left side of equation (1) over a rectangular grid cell were approximated by the formulas of trapezoids and rectangles, which have a second order of accuracy. The replacement of the partial derivative in equation (2)

Discussion
The one-dimensional filtration problem of monodisperse suspension in a porous medium with variable porosity and accessible fractional flow is considered in the paper. A mathematical model based on size-exclusion particles capture is a system of partial differential equations of the first order. Continuing the works [17], [18], this article presents a numerical solution of the problem.
The standard approach for the numerical solution of the filtration problem is to solve the original nonlinear system (1)-(4) using the finite difference method [19], [20]. In contrast to the generally accepted method, in this paper the filtration problem is reduced to a system of transcendental equations that are solved numerically at the nodes of a rectangular grid. This method increases the accuracy and reduces the time of calculations. The results of the calculations are close to the standard numerical solution of the original problem.
The obtained solution of filtration problem allows to analyze data from laboratory experiments and optimize the composition of grout for the porous rock.

Conclusions
The problem (1)-(4) of one-dimensional filtration of a monodisperse suspension in a porous medium with variable porosity and accessible fractional flow does not have an explicit analytical solution. The paper considers the transition from system (1)-(2) to the equivalent equation (5) and the solution of this equation by the characteristics method. The obtained transcendental system of equations (18)- (19) is solved numerically at each node of the rectangular grid in the domain 1  . The solution of the filtration problem (18)-(19) is compared with the numerical solutions of problems (1)-(2) and (9), obtained by the finite difference method with the second order approximation by integrating over a rectangular grid cell