A model for annular displacements of wellbore completion fluids involving casing movement

This paper presents a mathematical and a numerical model for solving the flow and displacement of completion fluids in the annular space formed by the gap between the outer wall of the casing and the rock face. Such flows occur during mud circulation and cementing operations and may involve casing rotation and reciprocation. Most completion fluids have a shear-dependent apparent viscosity. Additionally, muds and cement slurries often exhibit a yield stress and a gel strength. The displacement patterns and the final placement of the cement depend on injection rate and casing movement histories, rheology contrasts, density contrasts as well as the actual shape and orientation of the annular space which may vary along the wellbore axis. All the above listed phenomena are included in the model. The mathematical model is derived using the lubrication approach and the narrow-slot approximation for the momentum balance equations. These methods provide a (2+1)D-averaged model where the radial dimension is not neglected but averaged across the gap. The numerical model is developed in the goal of minimizing computational time. It takes advantage of multiprocessor architectures, first to pre-compute the closure equations linking local flow velocity to local pressure gradient, prior to running the displacement simulation, and second, to solve the non-linear pressure equation by sampling multiple choices of relaxation parameters.