The goal of the proposed research is to improve the accuracy, stability, and computational efficiency of numerical petroleum reservoir simulations, particularly targeting unconventional reservoirs in tight-gas and gas and oil shale plays. This will be achieved through development and demonstration of a partition-of-unity finite-element framework that will allow accurate descriptions of near-fracture/wellbore physics and geometry by using both analytic and computer generated enrichments to finite-element approximations. The phrase “partition-of-unity” refers to the generic and unifying idea behind recent computational method developments such as the eXtended Finite Element Method and Generalized Finite-Element Method. The basic idea is that standard finite element approximations can be enriched to include higher-order or even discontinuous solution fields within the elements. The proposed framework will be validated and analyzed for accuracy and efficiency, and demonstrated on problems of interest specific to unconventional reservoirs. Single- and multi-phase flow in fractured reservoirs and capillary wicking during production and drying of gas wells will be analyzed and compared with results from other popular modeling techniques.