Primary cementing of oil and gas wells in turbulent and mixed regimes

We present a detailed derivation of a practical two-dimensional model for turbulent and mixed regimes in narrow annular displacement flows, such as are found during the primary cementing of oil and gas wells. Such mixed cross regimes, including those in which different regimes exist in the same annular cross section, are relatively common in primary cementing. The modelling approach considers scaling based on the disparity of length-scales, which allows a narrow-gap averaging approach to be effective. With respect to the momentum equations, the leading-order equations correspond to a turbulent shear flow in the direction of the modified pressure gradient. This leads to a nonlinear elliptic problem that is the natural extension of the laminar displacement model in Bittleston et al. (J Eng Math 43:229-253, 2002). The mass transport equations that model the miscible displacement are however quite different. To leading-order turbulence effectively mixes the fluids. Changes in concentrations within the annular gap arise due to the combined effects of advection with the mean flow, anisotropic Taylor dispersion (along the streamlines) and turbulent diffusivity. The diffusive and dispersive effects are modelled for fully turbulent and transitional flows following Maleki and Frigaard (J Non-Newt Fluid Mech 235:1-19, 2016). The model derived allows the investigation of different well geometries and inclinations, pumping sequences and fluid rheologies, all of which can have importance. A number of computed examples are presented with the aim of demonstrating the complexity of turbulent displacements.