Advances in Swab-Surge Modeling for Managed Pressure Drilling
Authors: Chin Wilson C. , Zhuang Xiaoying 
Published: May 2011

Abstract
Non-Newtonian flows in highly eccentric annuli with cuttings beds, washouts and fractures, encountered in cementing and managed pressure (and underbalanced) drilling, are solved without crude slot flow and hydraulic radius approximations. The nonlinear partial differential equations, written to customized, boundary-conforming, curvilinear coordinate grid systems providing high physical resolution in tight spaces, are solved exactly with no-slip conditions, and detailed velocity, apparent viscosity, shear rate and viscous stress fields are computed for pressure drop, hole cleaning and other applications. For fluids with yield stress, well known uncertainties related to plug zone size and shape are fully resolved using Herschel-Bulkley relations applicable across transition boundaries (determined iteratively as part of the solution) reaching into and across the plug. Two-dimensional, single-phase, steady flow simulations, solved rapidly using finite difference methods, provide detailed numbers and color displays for all physical quantities within seconds, with excellent numerical stability for all fluid types with and without yield stress. Formulations for steady-state casing or drillpipe longitudinal translation and rotation are presented, and extensions to model transient incompressible effects associated with starting, stopping and periodic movement, important in evaluating cement-mud displacement efficiency, axial-helical cuttings transport, swab-surge, and jarring remedies for freeing stuck pipe, are developed. Practical problems are presented and the advantages over existing models are described. In particular, extensive calculation methods and new modeling capabilities are presented for job planning and swab-surge predictive analysis in modern managed pressure drilling applications.

Introduction
Annular flow modeling in boreholes, important to both drilling and cementing, is as old as petroleum engineering itself. In the simplest case, flow configurations are represented by concentric circles through which steady, two-dimensional, Newtonian and power law fluids flow; in these limits, exact analytical or numerical solutions of the flow equations provide useful tools for operational applications. For more complicated problems, e.g., eccentric annuli, non-ideal geometric irregularities, non-Newtonian yield stress fluids, pipe translation and rotation, however, numerous mathematical obstacles arise, which unfortunately introduce inefficiencies into field practices. We discuss these problems next.
Geometric complications. In deviated and horizontal wells, heavy pipe and drill collar weight implies eccentric positioning within the borehole, as shown in Fig. 1a, leading to difficulties in geometric description and solution. High eccentricities are often accompanied by non-symmetrical washouts, thick and irregularly formed cuttings beds, and possibly, fracture indentations. Early in petroleum engineering, the notion of a simple " mean hydraulic radius?? permitting representation as an equivalent circular pipe flow, as depicted in Fig. 1b, was widely employed; this approach, however, was not useful since what is meant by " mean?? is not obvious and certainly not generally applicable from one situation to the next. Later " slot flow?? models " unwrapped?? the eccentric annulus, with the result as illustrated in Fig. 1c, and then, further discretized the resulting slot into local parallel plate elements, each of which is approximately modeled by simple solutions for fluid flow between ideal parallel plates. While somewhat reasonable, this approach applied strictly to very narrow annuli, but even then, curvature terms in the general governing momentum equations are always neglected. Thus, inertial effects are never properly modeled even in the limit of very narrow elements.