To the theory of non-local non-isothermal filtration in porous medium

Meilanov, R.R.; Akhmedov, E.N.; Beybalaev, V. D.; Magomedov, R. A.; Ragimkhanov, G. B.; Aliverdiev, A. A.

doi:10.1088/1742-6596/946/1/012076

Cited by 3 publications

(7 citation statements)

References 10 publications

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“…Such dynamics of the pressure and temperature change is inherent to fractal medium. As it known from [4][5][6], in such medium with non-locality in time, as opposed to continuous medium, all physical processes are slower. In such medium not all directions are available for a particle movement, i.e.…”

Section: Convergence Of the Difference Schemementioning

confidence: 99%

“…As is known, the fundamental issues connected with the study of fluid motion in porous medium are actual for today, since there are numerous studies in this field [1][2][3][4][5][6][7]. As it shown in works [4][5][6], peculiarities of multiphase porous structures result in complex nature of heat and mass transfer phenomena. Such structures are characterized by nonlocal memory effects, strong spatial correlations and self-organization.…”

Section: Introductionmentioning

confidence: 99%

“…In this case models build upon the fractional differential equations become relevant [8][9][10][11]. In such models the rate of the fractional derivative allows to take into account non-locality in time (memory effect), and coordinate (spatial correlations), that lead to appearance of self-organization processes [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

“…In papers [5,6] the equations of non-local non-isothermal filtration with the derivatives of fractional order were studied. The present work is devoted to the numerical study of non-isothermal filtration processes in medium with non-locality in time [14,15].…”

Section: Introductionmentioning

confidence: 99%

“…The present work is devoted to the numerical study of non-isothermal filtration processes in medium with non-locality in time [14,15]. In papers [5,6] as the mathematical model of the non-isothermal filtration in such medium is considered the following problem…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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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.

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Section: Convergence Of the Difference Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

et al. 2021

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Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives

Baigereyev

Alimbekova

Oskorbin

2022

JMMCS

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This paper investigates a model of fluid flow in a fractured porous medium under the assumption of a uniform distribution of fractures throughout the volume. This model is based on the use of a fractional differential analogue of Darcy's law, as well as on the assumption that the properties of rock and fluid depend on pressure and its fractional derivative. Unlike previous studies, this study uses a fractional derivative in the Caputo-Fabrizio sense with a non-singular kernel. In this paper, we propose a numerical method for solving this initial boundary value problem and theoretically investigate the order of its convergence. The formulation of a fully discrete scheme is based on application of the finite difference approximation for integer and fractional time derivatives, and the Galerkin method in the spatial variable. A second-order formula is used to approximate both integer derivative and the fractional derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for both semi-discrete and fully discrete schemes, which imply their second-order convergence in time and space variables. A number of computational experiments were carried out on the example of a model problem to validate the accuracy of the scheme. The results of the numerical tests fully confirm the outcome of the theoretical analysis.

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

2021

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The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration.

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To the theory of non-local non-isothermal filtration in porous medium

Cited by 3 publications

References 10 publications

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

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

Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives

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

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