A new semi-analytical method for calculating well productivity near discrete fractures

Highly permeable discrete fractures are often observed in sandstone and carbonate formations. However, the impacts of these fractures on the productivity index of a vertical well are poorly understood. In this paper, we present a new uniform-flux fracture solution derived by integrating an eccentric point source in a pseudo-steady state within a circular reservoir. Moreover, we establish a new model of fluid flow in a uniform-flux fracture. We then integrate the fracture solution and flow model into a semi-analytical model capable of calculating the productivity index of a vertical well near discrete fractures in a circular reservoir. Finally, we discuss the effect of discrete fractures on the productivity index in detail. Our results show that, for a single-fracture-well system, the well-fracture distance and fracture conductivity exert significant influences on the productivity index of the well. The maximum productivity index of a vertical well can be achieved if the well is drilled on a discrete fracture, and the productivity index decreases with increasing well-fracture distance and falling fracture conductivity. The effect of fracture conductivity on the productivity index of a vertical well can be ignored if the distance between the well and the fracture is greater than a certain value. At large conductivity values, the dimensionless fracture-well distance d_1D has a much greater impact on the productivity index than the fracture conductivity. For a double-fracture-well system, strong interference between fractures will reduce the increment of the productivity index associated with the second fracture if two parallel fractures are very close. If the distance is larger, the effect of fractures on the productivity index can be ignored. The closest fracture to the vertical well contributes the most to the productivity. The second, further fracture contributes relatively little (less than 9% for parallel fractures and less than 25% for orthogonal fractures). This paper provides insight into factors affecting the productivity index of a discrete well-fracture system.