Based on the premises of anomalous diffusion models in fractal porous media, an alternative to dual-porosity based formulations of flow in fractured unconventional reservoirs is presented. The new formulation is implemented in the trilinear flow idealization for a fractured horizontal well in a tight formation and verified by using the asymptotic cases. The results of the new model are compared with the dual-porosity based trilinear flow formulation and the differences and similarities are delineated. A discussion of the characteristics of the pressure and derivative responses obtained from the trilinear anomalous-diffusion model is provided and related to the fractal nature of fractured media. Physical interpretations are also assigned to fractional derivatives and the phenomenological coefficient of the fractional flux law. It is shown that the anomalous diffusion formulation does not require explicit references to the intrinsic properties of the matrix and fracture media and thus relaxes the stringent requirements used in dual-porosity idealizations to couple matrix and fracture flows. The trilinear anomalous-diffusion model should be useful for performance predictions and pressure- and rate-transient analysis of fractured horizontal wells in tight unconventional reservoirs.
Two common approaches to model naturally fractured media are the discrete fracture network (DFN) models and dual-porosity idealizations (Fig. 1). In discrete fracture network (DFN) models (Fig. 1A), it is possible to consider the details of each fracture and the distribution and connectivity of the fracture network. However, DFN models require extensive characterization studies and also lead to computationally inefficient models. In general, the level of detail that can be utilized from the DFN model is limited by the capabilities of the flow model, which will use the DFN model. Therefore, despite their potential, the DFN models are not the tool of choice for most routine engineering applications.