In reservoir engineering, an oil reservoir is commonly modeled using Darcy's diffusion equation f... more In reservoir engineering, an oil reservoir is commonly modeled using Darcy's 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 (␣), 0 ≤ ␣ < 1 (e.g., K ϰ (1 -␣)), 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 i )/p i in the reservoir for large times, where p i = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with ␣, 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 ␣ = 0), the solutions are almost independent of ␣, and when K = 0.1k the rate of increase in P depends upon ␣. This effect is enhanced when K = (1 -␣)k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as ␣ ¡ 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, ␣.
A diffusion problem involving time derivative acting on two time scales represented by two fracti... more A diffusion problem involving time derivative acting on two time scales represented by two fractional derivatives is investigated. The orders of the fractional derivatives are both between 0 and 1 and therefore the problem corresponds to the subdiffusion case. It is considered on a semi-infinite axis and the forcing term and the initial data are assumed compactly supported. To reduce the problem to that support there is a risk of being lead to an "infected" problem due to the reflected waves on the new settled boundary. To avoid this undesirable effect of reflected waves on standard boundaries, we establish artificial boundaries and find the appropriate artificial boundary conditions. Then, using properties of fractional derivatives, a generalized version of the Mittag-Leffler function and some adequate manipulations of inverse Laplace transforms we find the explicit solution of the reduced problem.
In reservoir engineering, an oil reservoir is commonly modeled using Darcy's diffusion equation f... more In reservoir engineering, an oil reservoir is commonly modeled using Darcy's 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 (␣), 0 ≤ ␣ < 1 (e.g., K ϰ (1 -␣)), 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 i )/p i in the reservoir for large times, where p i = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with ␣, 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 ␣ = 0), the solutions are almost independent of ␣, and when K = 0.1k the rate of increase in P depends upon ␣. This effect is enhanced when K = (1 -␣)k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as ␣ ¡ 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, ␣.
A diffusion problem involving time derivative acting on two time scales represented by two fracti... more A diffusion problem involving time derivative acting on two time scales represented by two fractional derivatives is investigated. The orders of the fractional derivatives are both between 0 and 1 and therefore the problem corresponds to the subdiffusion case. It is considered on a semi-infinite axis and the forcing term and the initial data are assumed compactly supported. To reduce the problem to that support there is a risk of being lead to an "infected" problem due to the reflected waves on the new settled boundary. To avoid this undesirable effect of reflected waves on standard boundaries, we establish artificial boundaries and find the appropriate artificial boundary conditions. Then, using properties of fractional derivatives, a generalized version of the Mittag-Leffler function and some adequate manipulations of inverse Laplace transforms we find the explicit solution of the reduced problem.
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Papers by R. Ghanam