Numerical solution of filtration in porous rock

Abstract

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 mechanical-geometric 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.

Asymptotics of the Filtration Problem With Almost Constant Coefficients

“…In the domain ͞ Ω 1 , the solution to problem (1)-( 4) coincides with the solution to the Goursat problem (1)-( 3), (5). In characteristic variables τ = t -x/v, y = x, the Goursat problem takes the form numerical and analytical methods are used [11][12][13][14][15][16]. Analytical methods allow to obtain exact and asymptotic solutions and their dependence on parameters.…”

“…Conditions ( 3) and (5) take the form numerical and analytical methods are used [11][12][13][14][15][16]. Analytical methods allow to obtain exact and asymptotic solutions and their dependence on parameters.…”

“…The mathematical model of deep bed filtration includes the equation for the balance of the masses of suspended and retained particles and the kinetic equation of deposit growth, which form a quasilinear hyperbolic system of the first order partial differential equations [10]. To solve filtration problems, both numerical and analytical methods are used [11][12][13][14][15][16]. Analytical methods allow to obtain exact and asymptotic solutions and their dependence on parameters.…”

Numerical methods for solving improper problems of filtration theory

“…Therefore, we restrict ourselves to deriving formulas for calculating the functional derivatives from dJ 2 according to required parameters. In obtaining expressions for the functional derivatives, use the technique given in [18][19][20][21][22][23][24].…”