This work is an analysis of non-Newtonian fluid flow in an annulus of an eccentricity with inner pipe motion, for typical petroleum drilling operations. The annular flow geometry and the two-dimensional laminar equation of motion are not approximations, but rather a finite difference method is used to develop a computer model which calculates the velocity profile and flow rate for a given frictional pressure loss. Very accurate velocity profiles of non-Newtonian (Bingham plastic and power law) fluids with inner pipe motion are obtained using exact geometries. Contour plots of the resulting velocity profiles in annuli of varying eccentricities help one visualize the effects of eccentricity as well as the effects of inner pipe movement and rheological properties of a fluid on surge pressures. Example cases show a significant reduction in surge pressure as eccentricity increases. The surge pressures are as much as 50 percent less than a concentric calculation when the inner pipe lays to one side of the hole. Therefore, previous concentric models are found to overpredict surge pressures in directional and horizontal wells. A correlation of frictional pressure is developed based on this model's results. This correlation, permitting a quick calculation of surge pressures in eccentric annuli, can easily be incorporated into existing concentric models (steady state or unsteady state) to accurately calculate surge/swab pressures at any pipe eccentricity.