Analytical Solution of Fractional Order Diffusivity Equation With Wellbore Storage and Skin Effects

Razminia, Kambiz; Razminia, Abolhassan; Machado, J. A. Tenreiro

doi:10.1115/1.4030534

Cited by 14 publications

(6 citation statements)

References 31 publications

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“…Due to the irregular characteristics of formation, flow of fluid is uneven (fluid is uniform only in ideal cases). To solve this problem, Chang and Yortsos (1990) first applied the frac-tal theory (Mandebort 1982) to actual reservoir, they established a fractal reservoir seepage model and obtained its solution, which provided a theoretical basis for further exploration of fractal reservoir. Aprilian et al (1993) found using the fractal theory to study some traditional reservoirs, the results would be much closer to real situation of reservoir.…”

Section: Introductionmentioning

confidence: 99%

The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir

Guo

Zheng

et al. 2020

Comp. Appl. Math.

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By introducing the elastic boundary value condition, the elastic boundary value problem of extended modified Bessel equation is proposed, we can use the following method to solve it. First, two linear independent solutions of extended modified Bessel equation are obtained. Second, the generating function of solution is constructed. Third, the kernel function of solution is constructed using the elastic right value condition. Finally, the solution is obtained by assembling coefficients with the left boundary value condition. As for its application, a fractal homogeneous reservoir seepage model under the elastic outer boundary condition is established, and solution of the model is obtained. Influences of reservoir parameters on characteristic curves corresponding to dimensionless bottom hole pressure and its derivative are analyzed, which provide a new theoretical basis for exploring the flow law of oil. It can be found that seepage model under the elastic outer boundary condition regards three idealized outer boundary conditions (infinite, constant pressure and closed) and homogeneous reservoir seepage model without considering fractal as special cases, so it can reflect real situation of reservoir better and it is helpful to the development of related well test analysis software. Keywords Extended modified Bessel equation • Elasticity • Fractal homogeneous reservoir • The generating function • The kernel function Mathematics Subject Classification 03C98 • 33B99 • 34B60 • 3B340 List of symbols B Oil volume factor (m 3 /m 3) C Well storage coefficient (m 3 /MPa) Communicated by Jorge X. Velasco.

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Section: Introductionmentioning

confidence: 99%

The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir

Guo

Zheng

et al. 2020

Comp. Appl. Math.

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“…In now a days, fractional order derivatives and integrals have found a very pivotal role in different directions of real world problems [4,16,22,25,26,31,32]. Many physical and engineering problems can be narrated with the help of mathematical model that involve fractional PDEs, like that diffusion and wave equations [29], fractional heat-like and wave-like equations having variable coefficients [2,23,24,33], diffusivity equation possessing wellbore storage and skin effects of fractional order [28], fractional Bagley-Torvik equation [11], fractional model of differential-difference equation [14], chemical kinetics system involving a fractional derivative with Mittag-Leffler kernel [30], etc.…”

Section: Introductionmentioning

confidence: 99%

A hybrid analytical algorithm for nonlinear fractional wave-like equations

Kumar

Singh

Purohıt

et al. 2019

Math. Model. Nat. Phenom.

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In this work, we aim to present a hybrid numerical scheme based on the homotopy analysis transform method (HATM) to examine the fractional model of nonlinear wave-like equations having variable coefficients, which narrate the evolution of stochastic systems. The wave-like equation models the erratic motions of small particles that are dipped in fluids and fluctuations of the stochastic behavior of exchange rates. The uniqueness and existence of HATM solution have also been discussed. Some numerical examples are given to establish the accurateness and effectiveness of the suggested scheme. Furthermore, we show that the proposed computational approach can give much better approximation than perturbation and Adomain decomposition method, which are the special cases of HATM. The result exhibits that the HATM is very productive, straight out and computationally very attractive.

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“…Fractional derivatives and integrals have been gaining more and more interest of scientists due to their extensive applications in different directions of science, social science, engineering and finance [1][2][3][4][5][6][7][8][9] when the relaxation process have to accounted for. In this context, Atangana [10] analyzed the fractional non-linear Fisher's reaction-diffusion equation associated with Caputo-Fabrizio (CF) fractional derivative.…”

Section: Introduction Definitions and Preliminariesmentioning

confidence: 99%

Modified Kawahara equation within a fractional derivative with non-singular kernel

2018

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The article addresses a time-fractional modified Kawahara equation through a fractional derivative with exponential kernel. The Kawahara equation describes the generation of non-linear water-waves in the long-wavelength regime. The numerical solution of the fractional model of modified version of Kawahara equation is derived with the help of iterative scheme and the stability of applied technique is established. In order to demonstrate the usability and effectiveness of the new fractional derivative to describe water-waves in the long-wavelength regime, numerical results are presented graphically.

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Analytical Solution of Fractional Order Diffusivity Equation With Wellbore Storage and Skin Effects

Cited by 14 publications

References 31 publications

The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir

The elastic boundary value problem of extended modified Bessel equation and its application in fractal homogeneous reservoir

A hybrid analytical algorithm for nonlinear fractional wave-like equations

Modified Kawahara equation within a fractional derivative with non-singular kernel

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