Tempered Fractional Derivative: Application to Linear Flow

Hydraulic fracturing has become the dominant completion method in unconventional shale oil and gas reservoirs. The fluid flow inside unconventional shale reservoirs is different compared to conventional reservoirs. The importance of understanding anomalous diffusion of unconventional reservoir starts to appear. Traditional Darcy’s law is not appropriate to describe sub-diffusion behavior. In order to analytically model the sub-diffusion behavior, continuous time random walk (CTRW) theory is introduced in some literature. Fractional derivative method is used to apply CTRW theory to the flux law and thus modelling sub-diffusion behavior. For flow into fracture in unconventional reservoir, linear flow regime is suitable not only for transient period but also for late-time period. Applying fractional derivative to the flux law successfully describes the sub-diffusion behavior in transient period. However, the flux law using fractional derivative causes inaccurate result for late-time period. In order to ameliorate the problem, we introduce a tempering factor into the fractional derivative. Then, tempering fractional derivative is applied to the flux law. This flux law is applied to linear flow diffusivity equation and transferred into Laplace domain for solution. Real time domain solutions are obtained using GWR numerical inversion. In our study, model for single fracture is successfully created for two different boundary conditions. After verifying our model with numerical model and fractional linear flow model, we are generating type curves for various fractional parameter and tempering factor parameter pairs. Furthermore, we analyze production data from three oil wells in Eagle Ford shale play using our tempered fractional linear flow model