Fractional derivatives and their applications in reservoir engineering problems: A review

Obembe, Abiola D.; Al-Yousef, Hasan Y.; Hossain, Mohammed Kamal; Abu-Khamsin, Sidqi A.

doi:10.1016/j.petrol.2017.07.035

Cited by 49 publications

(15 citation statements)

References 128 publications

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“…g(x, t) = Γ(3.5) Γ(1.9) (x 2 − 1)t 0.9 + Γ(3.5) Γ(2.7) (x 2 − 1)t 1.7 + (3x 4 t 4 − 2x 3 t 4 )∕48 − (6x 2 t 6 − 6 6 )∕72 The analytical solutions of this system are u(x, t) = (x − 0.5)t 3 and v(x, t) = (x 2 − 1)t 2.5 . When m = 16, 32 and 64, the numerical and analytical solutions of the system with some m are shown in Figures 1-8.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Xie

Ren

et al. 2019

Numerical Methods Partial

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In the current study, an approximate scheme is established for solving the fractional partial differential equations (FPDEs) with Volterra integral terms via two‐dimensional block‐pulse functions (2D‐BPFs). According to the definitions and properties of 2D‐BPFs, the original problem is transformed into a system of linear algebra equations. By dispersing the unknown variables for these algebraic equations, the numerical solutions can be obtained. Besides, the proof of the convergence of this system is given. Finally, several numerical experiments are presented to test the feasibility and effectiveness of the proposed method.

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Section: Numerical Experimentsmentioning

confidence: 99%

“…Because of the frequent appearance of fractional calculus in physics, science, engineering, mathematics and other applications, fractional calculus has become a hot spot for many researches . At present, most researchers have studied the development of numerical methods for fractional differential and integral equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Xie

Ren

et al. 2019

Numerical Methods Partial

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“…In 1-dimensional case, the space-fractional derivatives of the Riemann-Liouville or the Caputo types are usually used. 35 In multidimensional case, the Riesz potential can be used for this purpose. Thus, we consider the following fractional generalization of Darcy law:…”

Section: Two-dimensional Space-fractional Filtration Equationmentioning

confidence: 99%

Lie group analysis of 2‐dimensional space‐fractional model for flow in porous media

Belevtsov

Lukashchuk

2018

Math Methods in App Sciences

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Using the Lie group analysis approach, we study the symmetry properties of 2‐dimensional space‐fractional filtration equation with the Riesz potential. This equation is derived by a fractional generalization of Darcy law and permits to describe the pressure evolution during 1‐phase fluid flow through naturally fractured porous medium. We construct the prolongation of the point transformation group on the Riesz potential and obtain a generalization of the Leibniz rule for the Riesz potential, which are necessary for investigating symmetry properties using the invariance principle. The Lie group of linearly autonomous point transformations is constructed for the considered equation. In a limiting case of zero‐order potential, the obtained symmetry group coincides with the group of point transformations admitted by the classical integer‐order 2‐dimensional linear heat equation. Also, the asymptotic behavior of a group invariant solution is investigated.

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“…Após a realização do primeiro congresso, a respeito do cálculo fracionário, na Universidade de New Haven em 1974, surgiram muitas aplicações em pesquisas cientificas [12], e muitos autores demonstraram sua enorme capacidade como ferramenta para descrever e solucionar problemas práticos. [13]- [20].…”

Section: Modelagem Reológicaunclassified

Modelagem Reológica Fracionária Do Comportamento De Fluência De Dois Petróleos Brasileiros E Suas Emulsões a/O.

Carmo

Santos

Lenzi

et al. 2018

Anais Do III Simpósio Paranaense De Modelagem, Simulação E Controle De Processos - SIMPROC 2018

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Resumo-O objetivo deste trabalho foi investigar e modelar o comportamento de fluência viscoelástica de dois petróleos de origem brasileira, sendo um petróleo leve (API 24,6) e um petróleo pesado (API 16,8), além de suas emulsões A/O com teores de água variando de 0 a 55%. As emulsões estáveis foram preparadas utilizando um homogeneizador Ultra-Turrax e os dados de fluência viscoelástica foram obtidos a partir de um reômetro de tensão controlada. Para fins de modelagem, testou-se a aplicação de um modelo reológico fracionário simples para descrever o comportamento de fluência no domínio do tempo. O modelo foi capaz de descrever muito bem os dados experimentais obtendo coeficiente de correlação médio (̅̅̅̅) de ,. Observou-se que para as emulsões de ambos os petróleos, a ordem da derivada do modelo é claramente fracionária e dependente do tipo de petróleo. Por fim, os parâmetros do modelo foram confrontados com o teor de fase dispersa nas emulsões, sendo possível estabelecer correlações adequadas entre os parâmetros e o teor de água.

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Fractional derivatives and their applications in reservoir engineering problems: A review

Cited by 49 publications

References 128 publications

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

Lie group analysis of 2‐dimensional space‐fractional model for flow in porous media

Modelagem Reológica Fracionária Do Comportamento De Fluência De Dois Petróleos Brasileiros E Suas Emulsões a/O.

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