| In this thesis we study instabilities and mud channel removal during the primary cementing of an oil well. This process involves displacement of a sequence of non-Newtonian fluids along a narrow eccentric annulus. Previously, this has been modelled as a pseudo-steady process using a Hele-Shaw approximation. In such models, the stream function is governed by a steady elliptic problem and the fluids, (modelled via a concentration equation), simply advect along the annulus. It has been shown that for certain rheological and physical parameters, the displacement front will advances much faster on the wide side of the annulus than on the narrow side. In extreme cases the displacement front does not advance at all on the narrow side of the annulus and a static mud channel results as the finger advances up the wide side. In this thesis we consider whether the interface of a progressively advancing finger will remain stable to small perturbations. There is in fact experimental evidence that interfacial instabilities can occur in this situation. We find that the interface is in fact stable whenever there is a static mud channel on the narrow side of the annulus. Consequently, we also investigate how a mud channel might be removed by pulsation of the flow rate. Study of these two phenomena cannot be undertaken with the pseudo-steady framework. Therefore, we extend this model to flows that are fully transient. The transient model consists of a nonlinear evolution equation for the stream function.; In chapter 3 we show that this transient model is in fact well-posed. In chapter 4 we study stability of multi-layer parallel flows, i.e. long fingers. If both fluids are yielded at the interface, instabilities may arise for different combinations of the 14 dimensionless parameters. These instabilities are related to a jump in tangential velocity at the interface and do not appear to have been identified before. In chapter 5 we investigate the case where a static mud channel develops on the narrow side of the annulus. Our stability theory predicts only linear stability. We therefore study the effects of a finite pulsation of the flow rates via numerical simulation. It seems that if we perturb the flow from the beginning of the displacement, the transient model fully captures the effects of the perturbation and the width of the mud channel is reduced. The pseudo-steady velocity model does not report any significant changes with respect to the results using a constant flow rate. If however, pulsation is applied after the mud channel has already formed, the removal of the mud channel will be unsuccessful. |
