This paper presents a novel approach to solving the 3D flow and displacement of completion fluids in the annulus formed by the gap between the outer wall of the casing and the wellbore rock face. Completion fluids displace each-other and follow a complex path promoted by rheology and density contrasts, casing movement and by the shape and orientation of the annulus. Muds and cement slurries often exhibit a yield stress, an additional challenge for optimal mud removal and cement coverage. This work extends previously published 2D models, to now capture fluid distribution and velocity profiles across the gap width in the 3D axial-azimuthal-radial space, at a lower computer cost than with conventional 3D CFD approaches. This gain is obtained by solving a 2D-only pressure equation for calculating the 3D annular flow, under the so-called narrow-gap approximation.