By Tarek H. Ahmed
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36 Newtonian mixtures, no rotation (microscopic properties). requires inlet and outlet properties and also pressure gradient inputs for the different flow rates assumed. 41 and apply accordingly as the flows are concentric, rotating, Herschel-Bulkley, and so on. 40. 43. Their underlying math models and functions are described elsewhere in this book. 37 Newtonian mixtures, no rotation (microscopic). 38 Non-Newtonian mixtures, rotating (microscopic properties). 39 Non-Newtonian mixtures, rotating (microscopic, main menu).
When transient effects are to be considered, the 28 CHAPTER 1 Fluid Mechanics Challenges and Technology Overview density-dependent inertia terms “ρ @u/@t 1 Á Á Á ” are important and must be included in the analysis. This completely changes the nature of the mathematics, so the nonlinear partial differential equation, originally one for a single elliptic equation, is now controlled by parabolic or diffusive effects. The parabolic system can become quite complicated. When rotation exists, an additional coupled equation for azimuthal momentum appears, which must be solved together with the axial equation.
If so, the locations of all fluid interfaces can be determined kinematically on a volume basis alone without rheological considerations. Then, at any given time tn, with the volumetric flow rate Q(tn) known, either of the simulators “Steady 2D” and “Transient 2D” or any of the many specialized flow solvers available in their respective utilities menus can be used to determine the pressure gradient within any fluid slug. Since the lengths of all slugs are also available kinematically, the available set of pressure gradients can be integrated spatially, starting with the pressure value known at the surface choke to provide the pressure distribution along the borehole at that instant in time.