### Abstract

In reservoir engineering, an oil reservoir is commonly modeled using Darcys diffusion equation for a porous medium. In this work we propose a fractional diffusion equation to model the pressure distribution, p(x, t), of fluid in a horizontal one-dimensional homogeneous porous reservoir of finite length, L, and uniform thickness. A chief concern in this work is to examine the sensitivity of the pressure distribution, p(x, t), to different forms of pseudo-diffusivity, K, including cases when it depends upon the order of the fractional derivative (α), 0 ≥ α < 1 (e.g., K ∝ (1-α)), which may be more realistic for some types of rock formations. In all cases the systems show a near-linear increase in the pressure difference P(x, t) = (p(x, t)-p in the reservoir for large times, where p = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with α, but there is a crossover at x/L = 0.4 and this trend reverses for x/L > 0.4. When K = 10k (k is the conventional permeability when α = 0), the solutions are almost independent of α, and when K = 0.1k the rate of increase in P depends upon α. This effect is enhanced when K = (1-α)k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as α → 1. These results show that the pressure distribution in a porous reservoir is very sensitive to the dependence of the pseudo-diffusivity on the order of the fractional derivative, α. ;copy 2015 Published by NRC Research Press.

Original language | English |
---|---|

Pages (from-to) | 18-36 |

Number of pages | 19 |

Journal | Canadian Journal of Physics |

Volume | 93 |

Issue number | 1 |

DOIs | |

Publication status | Published - 22 Dec 2014 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Canadian Journal of Physics*,

*93*(1), 18-36. https://doi.org/10.1139/cjp-2013-0387

**Sensitivity of the pressure distribution to the fractional order α in the fractional diffusion equation.** / Malik, Nadeem A.; Ghanam, Ryad; Al-Homidan, S.

Research output: Contribution to journal › Article

*Canadian Journal of Physics*, vol. 93, no. 1, pp. 18-36. https://doi.org/10.1139/cjp-2013-0387

}

TY - JOUR

T1 - Sensitivity of the pressure distribution to the fractional order α in the fractional diffusion equation

AU - Malik, Nadeem A.

AU - Ghanam, Ryad

AU - Al-Homidan, S.

PY - 2014/12/22

Y1 - 2014/12/22

N2 - In reservoir engineering, an oil reservoir is commonly modeled using Darcys diffusion equation for a porous medium. In this work we propose a fractional diffusion equation to model the pressure distribution, p(x, t), of fluid in a horizontal one-dimensional homogeneous porous reservoir of finite length, L, and uniform thickness. A chief concern in this work is to examine the sensitivity of the pressure distribution, p(x, t), to different forms of pseudo-diffusivity, K, including cases when it depends upon the order of the fractional derivative (α), 0 ≥ α < 1 (e.g., K ∝ (1-α)), which may be more realistic for some types of rock formations. In all cases the systems show a near-linear increase in the pressure difference P(x, t) = (p(x, t)-p in the reservoir for large times, where p = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with α, but there is a crossover at x/L = 0.4 and this trend reverses for x/L > 0.4. When K = 10k (k is the conventional permeability when α = 0), the solutions are almost independent of α, and when K = 0.1k the rate of increase in P depends upon α. This effect is enhanced when K = (1-α)k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as α → 1. These results show that the pressure distribution in a porous reservoir is very sensitive to the dependence of the pseudo-diffusivity on the order of the fractional derivative, α. ;copy 2015 Published by NRC Research Press.

AB - In reservoir engineering, an oil reservoir is commonly modeled using Darcys diffusion equation for a porous medium. In this work we propose a fractional diffusion equation to model the pressure distribution, p(x, t), of fluid in a horizontal one-dimensional homogeneous porous reservoir of finite length, L, and uniform thickness. A chief concern in this work is to examine the sensitivity of the pressure distribution, p(x, t), to different forms of pseudo-diffusivity, K, including cases when it depends upon the order of the fractional derivative (α), 0 ≥ α < 1 (e.g., K ∝ (1-α)), which may be more realistic for some types of rock formations. In all cases the systems show a near-linear increase in the pressure difference P(x, t) = (p(x, t)-p in the reservoir for large times, where p = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with α, but there is a crossover at x/L = 0.4 and this trend reverses for x/L > 0.4. When K = 10k (k is the conventional permeability when α = 0), the solutions are almost independent of α, and when K = 0.1k the rate of increase in P depends upon α. This effect is enhanced when K = (1-α)k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as α → 1. These results show that the pressure distribution in a porous reservoir is very sensitive to the dependence of the pseudo-diffusivity on the order of the fractional derivative, α. ;copy 2015 Published by NRC Research Press.

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U2 - 10.1139/cjp-2013-0387

DO - 10.1139/cjp-2013-0387

M3 - Article

VL - 93

SP - 18

EP - 36

JO - Canadian Journal of Physics

JF - Canadian Journal of Physics

SN - 0008-4204

IS - 1

ER -