Semi-analytical well model of horizontal wells with multiple hydraulic fractures

A well model is required to relate the well rate to the well pressure and the wellblock pressure while modeling wells in reservoir simulators. The well index is calculated automatically by simulators for conventional wells, but it is generally calculated by the user and supplied in the input while modeling nonconventional wells when the default procedure available in commercial simulators is not adequate. In this paper we describe a new semi-analytical solution for horizontal wells with multiple fractures. The fractures can be rotated at any horizontal angle to the well, and they need not fully penetrate the formation in the vertical direction. This solution for hydraulically fractured wells is obtained by applying Fourier analysis to a 2D solution; therefore, this solution is easy to obtain when the 2D solution is available. The analytical solution provides the well pressure that can be combined with a numerically computed gridblock pressure to obtain the well index (WI). Results of our rigorous solution are compared with empirical approaches currently in use for calculating the well index and the productivity index. Examples are given when horizontal wells with fractures are modeled using our new approach and conventional methods. These results demonstrate the need for computing the correct WI.