Quantifying error introduced by finite order discretization of a hydraulic well model

A model of a hydraulic transmission line used to model the pressure fluctuations in a drilling well caused by vertical movement of the drill string (heave) is presented. Using the Laplace transform and appropriate boundary conditions the transfer function of the model is derived. The model uses the heave disturbance and controlled flow into the drilling well as inputs, and the measured topside pressure and controlled downhole pressure as output. A rational approximation of the system using spatial discretization is obtained using the Control Volume method. The error introduced by the discretization is analysed in the frequency domain. Then, the discretized models with varying number of control volumes are used for Kalman filter and LQG design. The performance of the Kalman filter and LQG when the well is subjected to the heave disturbance is compared over different number of control volumes and well lengths. Finally, it is shown how the robust performance of the LQGs can be checked w.r.t. the plant uncertainty introduced by the discretization.