Numerical Solution of the Problem of Anomalous Solute Transport in a Two-Dimensional Nonhomogeneous Porous Medium

2. STATEMENT OF PROBLEM.

Consider a two-dimensional object, the diagram of which is shown in Fig. 1. Let a solution with a certain concentration be supplied at some point in the medium. From such a point source , the solution spreads into the medium in mutually perpendicular directions and . The components of the flow velocity along and directions at a given point of the region will be denoted by and , respectively. Both of these components satisfy Darcy's law. Through and will denote the longitudinal and transverse components of the hydrodynamic dispersion, respectively, in the directions and [14, 16]. If the medium has a fractal structure, the transfer process has an anomalous character, which can be modeled by differential equations of fractional order.

FIGURE 1. Scheme of a two-dimensional environment, and are chosen so that it was , respectively.

When the anomalous diffusion with respect to spatial coordinates is taken into account, the transport equation can be written in the form

(1)

where is the concentration of the solution transported through the medium at a point of during , .

In order to solve the two-dimensional advection-dispersion equation (1), it is necessary to set the initial and boundary conditions.

Initially, let the medium be filled with a clean (no substance) liquid. Starting from the initial moment from the point (0, 0), a solution with a certain concentration is supplied for a certain time . At infinity in the directions and the conditions of the absence of substance consumption are accepted. Then the initial and boundary conditions can be written in the form

, (2)

(3)

, (4)

where is the concentration of the substance supplied to the medium.

Since the medium is inhomogeneous, two components of the velocity, i.e. and , are considered linear functions of the corresponding coordinates and . In addition, the velocities are considered to be dependent on , i.e. some functional dependence of the velocity component on is taken into account . Thus, the components of the fluid velocity are taken in the form

(5)

where and are the parameters of inhomogeneity in the directions of and , is a known function, is a parameter, , . Different values and express different characteristics of heterogeneity.

It is known that the diffusion (hydrodynamic dispersion) coefficients depend on the velocity of fluid motion [14]. Here the following dependence is accepted

. (6)

where is a given function, , .

In (5), (6), the coefficients , , , , can be interpreted as homogeneous coefficients of the velocity of movement of the diffusion coefficients, respectively, along the and directions.

System (1) is written as

(7)

where , .

Fractional derivatives in (7) are understood here by the definition of Caputo [24]

where is the gamma function.