On a generalized diffusion equation arising in petroleum engineering

Suliman Al-Homidan, Ryad Ghanam, Nasser Eddine Tatar

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

In this work we investigate a model which describes diffusion in petroleum engineering. The original model, which already generalizes the standard one (usual diffusion equation) to a non-local model taking into account the memory effects, is here further extended to cope with many other different possible situations. Namely, we consider the Hilfer fractional derivative which by its nature interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative (the one that has been studied previously). At the same time, this kind of derivative provides us with a whole range of other types of fractional derivatives. We treat both the Neumann boundary conditions case and the Dirichlet boundary conditions case and find explicit solutions. In addition to that, we also discuss the case of an infinite reservoir.

Original languageEnglish
Article number349
JournalAdvances in Difference Equations
Volume2013
DOIs
Publication statusPublished - Dec 2013
Externally publishedYes

Fingerprint

Petroleum engineering
Petroleum
Generalized Equation
Diffusion equation
Fractional Derivative
Derivatives
Engineering
Riemann-Liouville Fractional Derivative
Caputo Fractional Derivative
Memory Effect
Neumann Boundary Conditions
Explicit Solution
Dirichlet Boundary Conditions
Boundary conditions
Interpolate
Model
Derivative
Generalise
Range of data
Data storage equipment

ASJC Scopus subject areas

  • Applied Mathematics
  • Algebra and Number Theory
  • Analysis

Cite this

On a generalized diffusion equation arising in petroleum engineering. / Al-Homidan, Suliman; Ghanam, Ryad; Tatar, Nasser Eddine.

In: Advances in Difference Equations, Vol. 2013, 349, 12.2013.

Research output: Contribution to journalArticle

@article{be4336b34d1941c79b41b240caf9592f,
title = "On a generalized diffusion equation arising in petroleum engineering",
abstract = "In this work we investigate a model which describes diffusion in petroleum engineering. The original model, which already generalizes the standard one (usual diffusion equation) to a non-local model taking into account the memory effects, is here further extended to cope with many other different possible situations. Namely, we consider the Hilfer fractional derivative which by its nature interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative (the one that has been studied previously). At the same time, this kind of derivative provides us with a whole range of other types of fractional derivatives. We treat both the Neumann boundary conditions case and the Dirichlet boundary conditions case and find explicit solutions. In addition to that, we also discuss the case of an infinite reservoir.",
author = "Suliman Al-Homidan and Ryad Ghanam and Tatar, {Nasser Eddine}",
year = "2013",
month = "12",
doi = "10.1186/1687-1847-2013-349",
language = "English",
volume = "2013",
journal = "Advances in Difference Equations",
issn = "1687-1839",
publisher = "Springer Publishing Company",

}

TY - JOUR

T1 - On a generalized diffusion equation arising in petroleum engineering

AU - Al-Homidan, Suliman

AU - Ghanam, Ryad

AU - Tatar, Nasser Eddine

PY - 2013/12

Y1 - 2013/12

N2 - In this work we investigate a model which describes diffusion in petroleum engineering. The original model, which already generalizes the standard one (usual diffusion equation) to a non-local model taking into account the memory effects, is here further extended to cope with many other different possible situations. Namely, we consider the Hilfer fractional derivative which by its nature interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative (the one that has been studied previously). At the same time, this kind of derivative provides us with a whole range of other types of fractional derivatives. We treat both the Neumann boundary conditions case and the Dirichlet boundary conditions case and find explicit solutions. In addition to that, we also discuss the case of an infinite reservoir.

AB - In this work we investigate a model which describes diffusion in petroleum engineering. The original model, which already generalizes the standard one (usual diffusion equation) to a non-local model taking into account the memory effects, is here further extended to cope with many other different possible situations. Namely, we consider the Hilfer fractional derivative which by its nature interpolates the Riemann-Liouville fractional derivative and the Caputo fractional derivative (the one that has been studied previously). At the same time, this kind of derivative provides us with a whole range of other types of fractional derivatives. We treat both the Neumann boundary conditions case and the Dirichlet boundary conditions case and find explicit solutions. In addition to that, we also discuss the case of an infinite reservoir.

UR - http://www.scopus.com/inward/record.url?scp=84897848291&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84897848291&partnerID=8YFLogxK

U2 - 10.1186/1687-1847-2013-349

DO - 10.1186/1687-1847-2013-349

M3 - Article

VL - 2013

JO - Advances in Difference Equations

JF - Advances in Difference Equations

SN - 1687-1839

M1 - 349

ER -