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Fig. 1. Schematic of dual-velocity model for colloid transport.
2.The mathematical model and its numerical implementation
An inhomogeneous porous medium is considered, which consists of well permeable (transit) and poorly permeable stagnant zones, the scheme of which is shown in Fig. 1. Parameters in the first zone are indicated with the index 1. There are two sections in the zone, in each of which a colloid particles are with different reversible nonequilibrium kinetics. With the second zone, there is an exchange of particles mass, which we simulate as a sours (sink) term in the fractional time derivative form of concentration in the first zone. Therefore, in contrast to [4], the concentration field in the second zone is not considered. Note, that the fractional approach was previously used in [37,38].
The equations of solute transport in one-dimensional case are written as
where is time, s, x – is distance, m, – is longitudinal dispersion coefficient, , – is pore-water velocity, m/s, – volume particles concentration in liquid, и – concentrations of attached particles, , – porosity, , – medium density, , – retardation factor related to the mass exchange between two zones, , – the order of time derivative, .
The deposition of particles in each of the sections of the first zone is reversible with the difference kinetic equations
(2)
(3)
where , are the coefficients for attachment of colloids from aqueous region onto the solid phase, , are the coefficients for detachment of colloids from the solid into the aqueous phase, .
Let a liquid with a constant concentration of particles be pumped into the medium initially saturated with clean (without particles) liquid from the initial moment of time. Let us consider such time periods where the concentration field does not reach the right boundary of the medium, Under the noted assumptions, the initial and boundary conditions for the problem have the form
(4)
(5)
(6)
The problem (1) - (6) although is linear, obtaining an analytical solution is difficult, because three concentration fields must be found at the same time. Therefore, to solve the problem, we use the finite difference method. In the considered region a uniform grid was introduced
,
where I is a sufficiently large integer chosen so that segment overlaps the area of the calculated change in the fields C1, Sa1 and Sa2, h is the grid step in the х direction.
In the open grid area
equations (1), (2), (3) were approximated as follows
(8)
(9)
where , , – grid values of functions , , at given point .
From the explicit grid equations (8), (9) we determine ,
(10)
(11)
where
,
, .
The grid equations (7) are reduced to the form
(12)
where
,
,
The following order of computing is used. From (10), (11) , are determined, then we solve the system of linear equations (12) by Tomas’ algorithm – Since , schemes (10), (11) are stable, and for (12) the stability conditions of the Tomas’ algorithm are satisfied.
To assess the performance of the proposed model, it is important to compare the results with the corresponding results [4]. To do this, we compare source (stock) member in [4] and in (1). To quantify the proximity of the results based on the curves was calculated
(13)
for a given value of t,where L is the conditional boundary of the region to which the concentration profiles extend,
The proximity of the terms and should guarantee the proximity of the concentration fields determined using the proposed approach and the model [4]. To assess their proximity, we use the standard deviation (13), only for the one determined on the basis of two models, i.e.
where – concentration field for a given t, determined according to [4], and – the same as defined here.
For other moments and different estimates can be obtained for and . In principle, to approximate the two models, it is necessary to pose and solve the corresponding coefficient inverse problems by definition for a given value of or, conversely, definition for a given and
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