A Semianalytical Productivity Model for a Vertically Fractured Well With Arbitrary Fracture Length Under Complex Boundary Conditions

Summary Vertically fractured wells are used widely to achieve efficient development of low-permeability reservoirs and unconventional tight reservoirs. Currently, there are some commonly adopted productivity models for a fractured vertical well, including the Giger (1985) model, the Economides model (Economides et al. 1989), the Joshi (1991) model, the Guo model (Guo and Evans 1993), the Zhang (1999) model, and others. However, these models are only suitable for either short fractures or long fractures, and thus cannot satisfy the need for arbitrary-length fracture selection. In addition, these models have certain requirements for boundary conditions, and few models consider the nonuniform flow in actual hydraulic factures. Therefore, it is desirable to establish a productivity model for vertically fractured wells with arbitrary fracture length under complex boundary conditions. In this paper, first, we evaluate the existing fractured-vertical-well productivity models in detail. After that, on the basis of the potential superposition principle and mirror-image method, we develop the productivity model for a fractured vertical well with arbitrary fracture length under different boundary conditions. Grounded in this model, further considering the frictional pressure drop caused by fracture-flow nonuniformity, we establish the productivity equation of a finite-conductivity-fracture vertical well with nonuniform flow in the fracture. Finally, after comparing with the existing model under certain conditions, the reliability of the proposed model is verified successfully. Results show that the Giger model assumes that the boundary in the fracture-extension direction is impermeable, and the pressure wave reaches the boundary earlier than the constant-pressure boundary, which is perpendicular to the extension direction of the fracture, whereas both the Economides model and the Joshi model assume that the boundary in the fracture-extension direction is far, and is always farther than the boundary perpendicular to the direction of the fracture extension. The results forecast by the previous models will deviate greatly from actual well-production performance, if the real field case cannot meet the requirements of these models. The proposed model is not only in good agreement with the previous models at given conditions, but is also suitable for fractured vertical wells under arbitrary-length fractures and complex boundary conditions. Compared with the existing models, the correctness and reasonableness of the model are shown, and the application of the proposed model is broadened. For the development of low-permeability and unconventional tight oil and gas reservoirs with fractured vertical wells or fractured horizontal wells, the establishment and application of this model have great theoretical significance and application value for production prediction. This model can be a foundation or reference for productivity prediction of low-permeability and unconventional tight oil and gas reservoirs with fractured vertical wells or fractured horizontal wells.