Order Reduction of a Transient Model of Viscous Pressure Loss Gradient in Laminar Flow for non-Newtonian Fluid Flowing in Circular Pipes

When circulating non-Newtonian fluids in pipes, the flowrate may vary because of inherent fluctuations of the mud-pumps, swab and surge due to axial displacement of the drill-string, variations caused by a positive displacement motor (PDM) while drilling. Steady state estimations of viscous pressure loss gradients are not anymore valid under accelerated flowrate conditions. It is possible to precisely calculate the dynamic response in transient conditions, but such numerical calculations tend to be slow and incompatible with real-time constraints when used during a drilling operation. It is therefore desirable to derive a reduced order model that is fast, yet accurate. Numerical schemes utilized to estimate viscous pressure gradients under fluctuating flowrate conditions need to solve a larger problem than just the evolution of the pressure drop. Indeed, they need to calculate the whole fluid velocity field in a cross-section. However, when only the pressure gradient is of interest, a reduced order model would be sufficient. To find a possible reduced order dynamic model that mimics the dynamic response of the advanced numerical model, the method of sparse identification of non-linear dynamic (SINDy) is used on synthetic datasets generated using the advanced model. The SINDy method allows to discover the non-linear ordinary differential equation (ODE) that models precisely the dynamic that is observable from the synthetic dataset. Nevertheless, the calibrated parameters of the discovered non-linear ODE are only valid for a specific pipe diameter, fluid density and rheological behavior. Therefore, the method is applied multiple times on various combinations of pipe sizes and fluid rheological behavior parameters, and an interpolator function is generated. The interpolator is built using radial basis functions. The interpolator can then be used to find the correct parameters of the non-linear ODE as a function of the current rheological behavior of the fluid that flows in a pipe section and the dynamic pressure gradients can be estimated as a function of the fluctuating flowrate by solving a simple ODE, for example by using the Runge-Kuta method. The reduced order model is very efficient, yet accurate, and can therefore be used in a real-time context. This is possible because many high-quality simulations are made upfront and used by machine learning methods (SINDy and radial basis functions) to discover a non-linear ODE and its associated parameter interpolator. The application of these two machine learning methods results in a solution that has good generalization capabilities. The reason for providing good estimation of the dynamic response outside the domain of the training examples is due to the ability of the SINDy method to extract ODEs that represent a good approximation of the real physical behavior of the system.