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

1. INTRODUCTION.

The problems of solute transport and filtration of inhomogeneous fluids are of great practical importance in many branches of engineering and technology. The indicators of many technological processes of oil and gas production are of a random oscillatory nature [1]. Therefore, when studying such processes, the use of deterministic methods of analysis gives only a qualitative picture of the process, and quantitative characteristics are often far from real. To solve such problems, various indicators of variation of time series are often used, such as, variation, variation coefficient and normalized deviation, Theil index, etc. [2]. The above indicators are parametric. Moreover, only they can be used when time series obeys Gaussian distribution. However, very few natural processes can be expressed by normal distribution [2], and therefore, usage of parametric criteria and normal distribution in the analysis of natural processes most of the time is incorrect [4]. So, to formulate such dynamic processes, fractal theory [5] can be used. This theory has wide range of applications in many areas of research [1]. Fractal geometry is used in different processes of oil and gas production, to interpret the results of studies [5], to estimate the filtration properties [6,7,8].

In [9], an analytical solution was obtained for the one-dimensional transport of solutes through homogeneous media. However, the real solute transport tends to depend on the location in the environment. To account for this inhomogeneity, space-dependent dispersion and velocity must be taken into account. Solutions to some problems in the one-dimensional case are given in [10]. For two or three dimensional problems, taking numerical solutions is essential [11, 12]. In [13], a problem was solved for a one-dimensional advection-dispersion equation with variable coefficients using an explicit finite-difference scheme; further, the results were extended to the case of a two-dimensional equation in semi-infinite media [14]. It is known that dispersion generally depends on the flow rate [15]. In [16], it is believed that the dispersion is proportional to the nth power of the velocity with an exponent in the range from 1 to 2. Sometimes the expressions for the velocity and dispersion are written in degenerate form [16]. In the two-dimensional case, the transfer of the solute occurs both in the longitudinal and transverse directions. Significant transport of solute is noted along the transverse direction even at very low transverse velocity and dispersion relative to their longitudinal counterparts. This shows that a two-dimensional model is more suitable than a one-dimensional one.

During filtration and transfer of substances in nonlinear media, as well as during the flow of rheologically complex media, the characteristics often exhibit scale invariance (fractality) in both spatial and temporal parameters. This circumstance makes it possible to develop some general methods for modeling complex environments and in some cases facilitates the description of the processes occurring in them [17]. Therefore, the basic concepts of fractals and examples of the use of fractal characteristics in the analysis of objects with a nonlinear structure are important. Fractals allow one to reveal the unexpected simplicity of constructing complex natural systems and provide methods for their qualitative and quantitative description. For modeling disordered systems, fractal theory plays the same role as random number generators for modeling random processes. In fractal objects, the usual quantitative characteristics (length, area, mass, etc.) turn out to be inapplicable [18,19,20]. In this regard, the dimension of the fractal is used to quantitatively characterize the properties of a fractal.

In [21], a filtration model of flow through a porous medium is proposed. A porous medium is a fractal object, the structure of which is determined by the gap between mating surfaces, consisting of pores and contact areas of wavy and rough mating surfaces. Methods for determining the fractal dimensions of tortuosity and porosity of the medium are presented. The dependences of the leakage on the parameters of the porous and compacted medium, as well as the fractal dimension of the sinuosity and porosity of the compacted medium are obtained.

In [22], an equation to describe fluid filtration which has fractal properties is derived. That the structure of the equation is the same to the known equations’ structure. A real representation of the parameters which are exist in the equation and an experimental determination method are given.

In [23], a mathematical model is presented for two-dimensional solute transport in a semi-infinite inhomogeneous porous medium. Filtration rate and dispersion coefficient are considered as a linear multiple of a spatial and time dependent function. The exponentially decreasing and sinusoidal functions are considered.

This paper considers anomalous solute transport in a two-dimensional porous medium of fractal structure, where the dispersion coefficients and filtration rates are variable in space and time scale.