The unsteady laminar couette flow of non-Newtonian power-law fluids in a concentric annulus is a flow phenomenon that can be applied to predict the surge or swab pressure encountered when running or pulling pipes in a liquid-filled borehole. It is similar to a moving cylinder, concentrically placed in a liquid-filled outer cylinder. This flow phenomenon has a wide variety of applications in the solution of problems in petroleum, chemical, mechanical, and ceramic industries. Several numerical models have been proposed by several authors for surge or swab pressure predictions. Though previous investigators had reported that the magnitude of the surge or swab pressure is determined by the annular geometry, the moving pipe velocity, and the drilling fluid properties, there has been no reported simple analytical presentation of the application of the unsteady couette flow phenomenon to predict either the surge or swab pressure which occurs when the casing pipes are in motion in a liquid-filled borehole. In this study, the motion equations are analytically solved for power-law fluids by the perturbation method. The solutions of the resulting pressure gradient equations are presented in both dimensionless and graphical forms and for different pipe/borehole diameter ratios and power-law index values. These allow for a more general application in annular fluid flow analysis and for predicting the surge or swab pressure during the unsteady motion of casing strings.