Asymptotic model of suspension filtration in a porous media is considered. Asymptotic solution of the fil-tration problem is constructed for large values of time. To determine the asymptotics an integral form of solution is used. The asymptotics is compared with numerical solution.
The filtration problem in a porous medium is an important part of underground hydromechanics. Filtration of suspensions and colloids determines the processes of strengthening the soil and creating waterproof walls in the ground while building the foundations of buildings and underground structures. It is assumed that the formation of a deposit is dominated by the size-exclusion mechanism of pore blocking: solid particles pass freely through large pores and get stuck at the inlet of pores smaller than the diameter of the particles. A one-dimensional mathematical model for the filtration of a monodisperse suspension includes the equation for the mass balance of suspended and retained particles and the kinetic equation for the growth of the deposit. For the blocking filtration coefficient with a double root, the exact solution is given implicitly. The asymptotics of the filtration problem is constructed for large time. The numerical calculation of the problem is carried out by the finite differences method. It is shown that asymptotic approximations rapidly converge to a solution with the increase of the expansion order.
A classic pursuit problem is considered in which the pursuer is always moving towards the target. The shape of the mechanical trajectory is established, and the time of motion is calculated. An integral for the length of the pursuit curve is constructed, its asymptotics is calculated and compared with the result of the numerical computation.
Filtration problem of suspension with identical particles in a porous medium is considered. Physical and mathematical models for size-exclusion capture mechanism of suspended particles at the inlet of filter pores are presented. An exact solution is constructed for the forward and back flow of suspension in a porous medium in case of linear filtration coefficient.
Filtration of the suspension in a porous medium is important when strengthening the soil and creating watertight partitions for the construction of tunnels and underground structures. A model of deep bed filtration with variable porosity and fractional flow, and a size-exclusion mechanism of particle retention are considered. A global asymptotic solution is constructed in the entire domain in which the filtering process takes place. The obtained asymptotics is close to the numerical solution.
The mathematical model of filtering the suspension in a porous medium and its asymptotic solution behind the concentration front depend on several parameters. These parameters are determined by the given values of concentrations of suspended and retained particles at the filter inlet and outlet. An estimation of the method`s error is given.
The study of filtration as one of the problems of underground hydromechanics is necessary for the design and construction of tunnels, underground and hydraulic structures. Deep bed filtration of suspension in a porous medium with variable porosity and permeability and with an initial deposit is considered. An asymptotic solution to a model with small limit deposit is constructed; the asymptotics is compared with numerical calculation
A size-exclusion model of solid particle capture for a flow of suspension in a porous media is considered. For a quasi-linear system of equations for the concentration of suspended and retainrd particles the asymptotic solution is constructed near the filter inlet. For linear filtration coefficient the numerical comparison of the asymptotics and the exact solution is performed.
Filtration of the suspension in a porous medium with a geometric particle capture mechanism is considered. The porous medium has an initial deposit unevenly distributed across the filter. The nonlinear model of deep bed filtration suggests that the porosity and permeability of the porous medium depend on the deposit. The asymptotics of the movable boundary of the two phases is determined. The asymptotic solution of the problem is constructed and calculated near the filter inlet.