Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs

In the oil industry, many reservoirs produce from partially penetrated wells, either to postpone the arrival of undesirable fluids or to avoid problems during drilling operations. The majority of these reservoirs are heterogeneous and anisotropic, such as naturally fractured reservoirs. The analysis of pressure-transient tests is a very useful method to dynamically characterize both the heterogeneity and anisotropy existing in the reservoir. In this paper, a new analytical solution for a partially penetrated well based on a fractal approach to capture the distribution and connectivity of the fracture network is presented. This solution represents the complexity of the flow lines better than the traditional Euclidean flow models for single-porosity fractured reservoirs, i.e., for a tight matrix. The proposed solution takes into consideration the variations in fracture density throughout the reservoir, which have a direct influence on the porosity, permeability, and the size distribution of the matrix blocks as a result of the fracturing process. This solution generalizes previous solutions to model the pressure-transient behavior of partially penetrated wells as proposed in the technical literature for the classical Euclidean formulation, which considers a uniform distribution of fractures that are fully connected. Several synthetic cases obtained with the proposed solution are shown to illustrate the influence of different variables, including fractal parameters.