Control-oriented modeling and model order reduction for managed pressure drilling systems

Managed Pressure Drilling (MPD) is a method for fast and accurate pressure control in drilling operations. It has been introduced to overcome drawbacks of conventional pressure control methods, such as the incapability of rejecting transient pressure fluctuations [14]. In MPD, the annulus is sealed off at the top with a rotating control device and the mud is circulated out of the well through a choke valve. This combination provides a surface back pressure that can be controlled by manipulating the choke. In automated MPD systems, the surface pressure, and thereby the Bottom-Hole Pressure (BHP), is controlled by an automatic control system [13, 6]. The achievable accuracy and efficiency by this control system is dependent not only on the control design method, but also on the hydraulics model used for designing the control system. This model should be accurate enough to capture the essential dynamics of the system and, at the same time, the complexity of the model should be limited to allow for the use of established system-theoretic analysis and design techniques. In principle, hydraulics models for MPD systems come in the form of highfidelity Partial Differential Equations (PDE). From a control perspective, these models are too complex to be used directly for model-based controller design and, thus, further simplification/reduction is needed. Complexity-reduction approaches for these systems may be split into three groups. Firstly, the approximation of high-fidelity models by low-order Ordinary Differential Equation (ODE)/PDE models based on a time-scale separation has been proposed in [6, 2]. Such models are, however, incapable of capturing essential transients such as the propagation of pressure waves. Ignoring such phenomena in modeling and control design can cause a failure in the accomplishment of control objectives, such as the effective rejection of transient pressure fluctuations. It can even cause instability, which is especially probable in the case of long wells [10]. Secondly, model reduction can also be achieved by using low-resolution discretization of the PDE model based on, for example, a staggered-grid approach [10]. This, however, suffers from the lack of a quantitative measure on the achieved accuracy. Thirdly, model reduction can be performed based on a combination of a fine discretization, reducing the PDE model into a high-order ODE model, and automatic model order reduction techniques. Such an idea has been used in [12] for complexity-reduction of an MPD control system, and in [9] for deriving a control-oriented model for MPD systems. However, the high-complexity models in both cases were linear and the reductions were performed without providing any guarantee on accuracy. In this paper, the third type of approach is pursued to obtain a controloriented model for a nonlinear single-phase MPD system. Given 1) the spatially discretized ODE model combined with 2) (local) nonlinear boundary conditions, the resulting model is a nonlinear system comprising high-order linear dynamics with local nonlinearities. For this class of systems, a model order reduction procedure has been recently developed in [3]. This method, unlike many other model order reduction methods for nonlinear systems, preserves key system properties (such as L2 stability, a form of input-output stability). Moreover, it provides a computable error bound on the error induced by the reduction. The remainder of this paper is organized as follows. Section 2 is devoted to the mathematical modeling of the system. In Section 3, the nonlinear model order reduction procedure is described. Illustrative simulation results are presented in Section 4 and, finally, conclusions are presented in Section