Productivity index for Darcy and pre-/post-Darcy flow (analytical approach)

We investigate the impact of nonlinearity of high- and low-velocity flows on the well productivity index (PI). Experi- mental data show the departure from the linear Darcy relation for high and low velocities. High-velocity (post-Darcy) flow occurring near wells and fractures is described by Forchheimer equations and is relatively well-studied. While low-velocity flow receives much less attention, there is multiple evidence suggesting the existence of pre-Darcy effects for slow flows far away from the well. This flow is modeled via pre-Darcy equations. We combine all three flow regimes, pre-Darcy, Darcy, and post-Darcy, under one mathematical formulation subjected to certain critical transitional velocities. This allows one to use our previously developed framework to obtain the analytical formulas for the PI for the cylindrical reservoir. We study the impact of the non-Darcy effect on the PI depending on the well-flux and the parameters of the equations.