Generalized Bottom-Hole Pressure with Fractality and Analyses of Three-Dimensional Anisotropic Fractal Reservoirs

Abstract

There have been some limitations in the application of fractal diffusion theory for the analysis of fractally fractured reservoirs. These include the lack of generality of the acquired solution and assumptions of no wellbore storage and skin. To overcome the limitations, new mathematical procedures are derived following novel theories regarding the fractal diffusion in the area of physics. The wellbore storage and skin effects are characterized by a new bottom-hole pressure solution in fractal reservoirs. The equation obtained here is the most generalized version of the solution for bottom-hole pressure. It reduces to the classical equation by Ramey and Agarwal for a Euclidean case. Furthermore, when dynamic fractal dimension is equal to 2, it reduces to Chang and Yortsos' result. The sensitivity analysis shows that less pressure drop arises for larger dynamic fractal dimension at early time. Following the multiscaling transport theory of fractal, a general governing equation and its solution in three-dimensional anisotropic fractal reservoirs are investigated. The result is an extension of classical three-dimensional anisotropic solution by Raghavan. This research will be useful for characterizing fractal reservoirs. Combining the information obtained from each procedure, it is possible to determine the parameters, such as fractal dimensions, wellbore storage, and skin, more precisely than conventional approaches available in the literature.