Analysis of diffusion process in fractured reservoirs using fractional derivative approach

Abstract

The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.

Analytical Solution of Fractional Order Diffusivity Equation With Wellbore Storage and Skin Effects

“…Figures 1, 2 and 3 reveal that the pressure and its logarithmic derivative for FFD are different from traditional Euclidean model that is obtained in the special case of our solution with d mf = 2 and θ = 0. The analytical solutions of three cases are potted in a log-log scale by introducing the following dimensionless variables [18,19]: Figure 1 shows that two characteristic flow regimes can be observed in the response of a well with WS-SE in a Euclidean system: 1. At the initial part of the transient regime early, both the pressure and the derivative curves follow a straight-line with unit slope until the pressure disturbance is in the wellbore.…”

Fractional Order Version of the Hamilton–Jacobi–Bellman Equation

“…Now, resorting to Eqs. (9) and (6), we can replaceẋ by the following relation (component wise): (28) and A(·), B(·), C(·) and W p (·) are defined in (10)- (13). Let…”

“…FC has affected the control engineering discipline in two aspects: getting superior models for the processes, and a robust structure of the closed-loop control system [8,9]. Applications of FC have been explored to various fields of science and engineering, including control engineering [10], chaotic systems [11,12], reservoir engineering [13], diffusive processes [14], and so on [15]. For the design of variableorder fractional proportional-integral-derivative controllers for linear dynamical systems, see [16].…”

Semi-numerical solution to a fractal telegraphic dual-porosity fluid flow model

“…Sometimes, just the short-or long-term system behavior is of interest, and for this purpose asymptotic approaches are commonly used to avoid the inversion needed for complete solutions [10,11]. For solute transport and fluid flow problems in porous media, such as aquifers [3], oil, and geothermal reservoirs, the LT method helps to find semi-analytic solutions [1,2,6,12], exhibiting many advantages over purely numerical procedures. The main example is where subsequent inverse modeling will be implemented in order to retrieve model parameters of interest.…”