Two-dimensional computational simulation of eccentric annular cementing displacements

We consider a two-dimensional Hele–Shaw type model for displacement flows occurring in the primary cementing of an oil well. The fluids are visco-plastic and may get stuck in the annulus if a critical pressure gradient is not exceeded. The model consists of solving a nonlinear elliptic variational inequality equation for the stream function, coupled to an equation for interface advection, or alternatively a concentration equation for the mass fraction of each fluid. The key difficulty is to accurately compute yielded and unyielded zones of the wellbore fluids, which we accomplish by use of an augmented Lagrangian method to solve the stream function equation. We validate the accuracy of our method against analytical solutions for stable steady-state displacements. We study the convergence of the interface to the steady state, showing that the apparent meta-stability is illusory. We then explore the effects of increasing eccentricity, showing that although the interface may remain stable it becomes unsteady. Initially fully mobile flows are found, but as the eccentricity increases further the narrow side fluids fail to move in the far field. The narrow side interface can progress slowly through the static fluids by a burrowing motion, but for still larger eccentricities even the interface becomes static and a narrow-side mud channel forms.