Numerical research of non-isothermal filtration process in fractal medium with non-locality in time

Abstract

The difference scheme for the numerical solution of boundary problem for a system of equations for non-isothermal filtration with a Caputo derivative of fractional order on time is developed. Stability of the differential scheme is proved. Computational experiment in the analysis of solutions obtained has been done. Physical processes pass slowly in the fractal medium with non-locality in time. It is explained by the fact the occasionally wandering particle is being eliminated from the start place slowly, since not all directions of the movement become available for it. Values of pressure and temperature depending on the coordinate of layer radius and time calculated, and graphs of the dynamics pressure and temperature changes according to the layer radius and in depending on the time are built. Deceleration of the processes with time in the solutions for fractional derivatives which is characteristic for such medium has been established.

Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media

“…In particular, the test carried out within the framework of the studied model showed that the fluid flow process in a fractured porous medium proceeds more slowly than in a continuous medium. This observation is also consistent with the conclusions of the paper [8].…”

“…Some authors [6][7][8] are of the opinion that the fluid motion in a fractured medium cannot be adequately described within the framework of classical theory of fluid flow in porous media, assuming that the nature of the flow depends not only on the current state of the process but also on the previous history of changes in this process. This phenomenon is known as memory in petroleum engineering.…”

“…There are several definitions of the fractional derivative and fractional integral in fractional calculus, an overview of which can be found in [18,19]. In the development of mathematical models of fluid flow in porous media under different physical assumptions, various authors have used fractional time derivatives in the sense of Caputo [7][8][9]20,21], Riemann-Liouville [6,22], Caputo-Fabrizio [23,24], Atangana-Baleanu [25] and others [26] to account for memory effects, whereas the Riemann-Liouville derivative is mainly used to describe nonlocality in the spatial direction [6,27]. In the study of dynamic processes in fractured fractal media, a fractional temporal derivative in the sense of Caputo has an advantage, since it is given by the convolution of the power law kernel and the derivative of the function.…”

Numerical solution of the boundary value problem for the heat equation with fractional Riesz derivative

“…The processes of anomalous diffusion and diffusion of particles in inhomogeneous media were numerically studied by Reviznikov and Slastushenskiy [2] . The works of Liu and Xu [3] , Meerschaert and Tadjeran [4,5] as well as our recent paper [6] are also devoted to numerical methods for solving boundary value problems for partial differential equations of fractional order. The paper of Sweis et al [7] considers a class of differential equations of fractional order with a delay of order ρ and Atangana-Baleanu fractional derivatives in the sense of Caputo fractional derivatives.…”

Transient Heat Conduction in a Semi-Infinite Domain with a Memory Effect: Analytical Solutions with a Robin Boundary Condition

“…Especially in the case of heat conduction and anomalous diffusion, the transient heat conduction via the application of the time-fractional Caputo derivative has been thermodynamically formulated in [7,8] applying the fading memory formalism. Recently, various techniques upon various initial and boundary conditions have been applied to solve time-fractional heat transfer equations [1][2][3][4][5][6][7][8][9][10][11][12][13][14] but a review of these studies is beyond the scope of this work.…”