Simplified Explicit Flow Equations for Herschel-Bulkley Fluids in Couette-Poiseuille Flow—For Real-Time Surge and Swab Modeling in Drilling

Abstract

Summary Simplified flow equations are developed for Herschel-Bulkley (HB) fluids in laminar Couette-Poiseuille (CP) flow. Such flow problems are encountered in drilling when the drillstring is moved longitudinally (surge and swab operations). The new equations give the frictional-pressure gradient explicitly as a function of the bulk-flow rate. This makes them very well-suited for applications in fast, robust dynamic models for real-time applications. Incorporating these equations in a dynamic model based on ordinary differential equations (ODEs) enables coupling with system identification and control-system theory. This is a great advantage in drilling applications where there are many uncertain parameters and few measurements. Thorough analysis of relevant analytical solutions is performed to replace the most complex (implicit) parts of the solution with simpler approximations. The accuracy of the new equations, in comparison with numerical simulations and available analytical solutions, is shown to be acceptable for most practical applications.