One of the most challenging aspects in Drilling Automation is to have fast and accurate enough prediction of the output of interest (such as pressure in the wellbore). The typical model used for multi-phase hydraulic modelling in such applications is called Drift Flux Model (DFM). It is a system of multiscale non-linear Partial Differential Equations, whose convective subset is conditionally hyperbolic. We will discuss about the numerical framework we have developed to obtain a high-fidelity scheme for solving the hydraulic model. We will demonstrate few numerical experiments (such as multi-phase shock-tube test case and a representative case to model propagation of fast/ slow transients) to highlight the representative features of such convection dominated problems.

Classical numerical techniques, used to discretize such a distributed system, yield high fidelity models, which are highly complicated and need to be reduced in order to speed up the required computational time. Hence, we will discuss about the Reduced Order Modelling (ROM) framework to capture the distributed non-linearities of the model and the key wave propagation effects. As a first step, we apply standard projection based approaches in conjunction with hyper-reduction techniques on few benchmark problems. We will demonstrate the results, discuss the limitations of these classical reduced-order representations and motivate the need of deriving more efficient alternative reduced-order models. We invoke the idea of the method of freezing, which serves as a pre-processing ingredient in the envisioned ROM framework. We will then show some preliminary results obtained using this pre-processing step. Finally, we will briefly discuss about the open challenges and perspectives in combining the method of freezing with non-linear reduced basis approximations.