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

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 (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) – pi)/pi in the reservoir for large times, where pi = 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.