Boundary Control and Estimation for Under-Balanced Drilling with Uncertain Reservoir Parameters

In underbalanced drilling (UBD), the bottom hole pressure is kept below pore pressure, causing pressure-dependent influx of reservoir gas into the wellbore that makes the system unstable at low drawdowns. In this article, we propose a feedback controller which stabilizes the system, represented by the “reduced drift flux model (DFM),” around an arbitrary pressure setpoint, while using only topside measurement and assuming unknown reservoir parameters. A particular challenge with this problem is the distributed and highly nonlinear nature of the system dynamics, where the “reduced DFM” models gas–liquid flow as a nonlinear transport equation with a nonlocal integral source term. An observer estimates the distributed gas concentration, downhole pressure, and reservoir parameters by solving the system dynamics backward relative to how the gas rises in the well. The control inputs are then constructed by designing target states over the next sampling period and again solving reversed dynamics to obtain the required topside pressures. The resulting controller is implemented with a 2-min zero-order hold to accommodate the actuation limitation situation on an actual drilling rig. Finally, the results are illustrated in simulations with an industry standard drift flux formulation as the plant model.