Evaluation of reservoir parameters through well test and decline curve analysis is a current practice used to estimate formation parameters and to forecast production decline identifying different flow regimes, respectively. From practical experience, it has been observed that certain cases exhibit different wellbore pressure and production behavior from those presented in previous studies. The reason for this difference is not understood completely but it can be found in the distribution of fractures within a Naturally Fractured Reservoir (NFR). Currently, most of these reservoirs are studied by means of Euclidean models, which implicitly assume a uniform distribution of fractures and that all fractures are interconnected. However, evidences from outcrops, well logging, and production behavior studies, and in general, the dynamic behavior observed in these systems, indicate that the above assumptions are not representative for these systems. Thus, the fractal theory can contribute to explain the above.
The objective of this paper is to investigate the production decline behavior in NFR exhibiting single and double porosity with fractal networks of fractures. The diffusion equations used in this work are a fractal continuity expression presented in previous studies in the literature, and a more recent generalization of this equation which includes a temporal fractional derivative. The second objective is to present a combined analysis methodology which uses transient well test and boundary-dominated decline production data to characterize NFR exhibiting fractures depending on scale.
Several analytical solutions for different diffusion equations in fractal systems are presented in the Laplace space for both constant wellbore pressure and variable pressure-variable rate inner boundary conditions. Both single and dual-porosity systems are considered. For the case of single porosity reservoirs analytical solutions for different diffusion equations in fractal systems are presented. For the dual-porosity case, an approximate analytical solution, which uses a pseudo-steady state matrix-to-fractal fracture transfer function, is introduced. This solution is compared with a finite difference solution and good agreement is found for both rate and cumulative production. Short- and long-time approximations are used in order to obtain practical procedures in time for determining some fractal parameters. Thus, this paper demonstrates the importance of analyzing both transient and boundary-dominated flow rate data for a single-well situation in order to fully characterize a NFR exhibiting fractal geometry.
Synthetic and field examples are presented to illustrate the methodology proposed in this work and to demonstrate that the fractal formulation explains consistently the peculiar behavior observed in some real production decline curves.