Two-dimensional computational simulation of eccentric annular cementing displacements

We consider a two-dimensional Hele–Shaw type model fordisplacement flows occurring in the primary cementing of anoil well. The fluids are visco-plastic and may get stuck inthe annulus if a critical pressure gradient is not exceeded.The model consists of solving a nonlinear elliptic variationalinequality equation for the stream function, coupled to an equationfor interface advection, or alternatively a concentration equationfor the mass fraction of each fluid. The key difficulty is toaccurately compute yielded and unyielded zones of the wellborefluids, which we accomplish by use of an augmented Lagrangianmethod to solve the stream function equation. We validate theaccuracy of our method against analytical solutions for stablesteady-state displacements. We study the convergence of theinterface to the steady state, showing that the apparent meta-stabilityis illusory. We then explore the effects of increasing eccentricity,showing that although the interface may remain stable it becomesunsteady. Initially fully mobile flows are found, but as theeccentricity increases further the narrow side fluids fail tomove in the far field. The narrow side interface can progressslowly through the static fluids by a burrowing motion, butfor still larger eccentricities even the interface becomes staticand a narrow-side mud channel forms.