Decline-curve analysis of fractured reservoirs with fractal geometry

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, evidence from outcrops, well logging, production-behavior studies, and the dynamic behavior observed in these systems, in general, indicate the above assumptions are not representative of these systems. Fractal theory considers a nonuniform distribution of fractures and the presence of fractures at different scales; thus, it can contribute to explain the behavior of many fractured reservoirs. The objective of this paper is to investigate the production-decline behavior in an NFR exhibiting single and double porosity with fractal networks of fractures. The diffusion equations used in this work are a fractal-continuity expressions presented in previous studies in the literature and a more recent generalization of these equations, 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 an NFR exhibiting fractures, depending on scale. Several analytical solutions for different diffusion equations in fractal systems are presented in Laplace space for both constant-wellbore-pres sure and 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 pseudosteady-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 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 to fully characterize an 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 consistently explains the peculiar behavior observed in some real production-decline curves.