1 USING THE HYPERGEOMETRIC MODEL TO ANALYZE THE BUCKLING OF DRILLSTRINGS IN CURVED BOREHOLES Jorge H. B. Sampaio Jr. 1 PETROBRAS - EP/GERPRO/GETEP Rio de Janeiro, Brazil Alfred W. Eustes III 2 Colorado School of Mines - Pet. Eng. Dept. Golden, CO 1 jsampaio@ep.petrobras.com.br 2 aeustes@mines.edu ABSTRACT Current methodologies for analytically determining the onset of buckling of drillstrings within curved boreholes are limited. In this paper, the Hypergeometric Model is shown to be an effective model to determine drillstring buckling within curved boreholes. With the Hypergeometric Model, the analysis of drillstring buckling results in curves expressing the local buckling force versus the angle of inclination. The local buckling force alone, however, does not contain all the information required for a practical analysis. From the local buckling force curve, the positional buckling force is derived. The positional buckling force considers the distributed weight of the drillstring and the friction between the drillstring and the borehole wall. From this curve, the point of minimum resistance to buckling of the drillstring is determined. Using the local and positional buckling force curves, experimental results and simulations are presented. When multiple configurations exist (for example tapered drillstrings, tapered boreholes, multi-curved boreholes, or any combination of these), the analysis procedure uses superposition of two or more single configuration curves and a graphical algorithm. The Hypergeometric Model permits the optimization of the position of the crossing points (cross-over positioning, casing-shoe positioning, and change of curvature) to achieve extended reach with less risk and cost. The procedure for this model and examples are presented in the paper. NOMENCLATURE X Dimensionless axial coordinate X1 Dimensionless coordinate of the lower end of a inclined beam-column X2 Dimensionless coordinate of the upper end of a inclined beam-column b i Constants (b1, b2, b3, b4) x Axial coordinate, ft y Lateral displacement coordinate, ft  Cross-sectional beam-column constant, in INTRODUCTION The Hypergeometric Model is a fully analytical model that describes the buckling of drillstrings within curved sections of boreholes. The model hypotheses and mathematical details, has been discussed in the literature (Sampaio, 1996 and 1998). The kernel of the Hypergeometric Model is the mathematical solution for an inclined beam-column, whose differential equation and boundary condition for pinned-ends are:              y Xy X b yX yX y X y X  2 0 3 1 2 1 2 , ( ) ( ) ( ) ( ) The variable X represents a dimensionless axial coordinate expressed in terms of physical dimensions of the column, and X 1 and X 2 , correspond to the lower and upper ends of the column. A closed solution of the differential equation is y(X) = b1 M1(X) + b2 M2(X) + b3 M3(X) + b4 - X where M1, M2,and M3 are special functions written in terms of generalized hypergeometric functions. In the process of determining the buckling force, the boundaries X 1 and X 2 of the column are adjusted until the buckled column fits inside the curved constraints of the borehole. Then the buckling force can be calculated. The advantage of the Hypergeometric Model is its mathematical basis. It is independent of either experimental results to adjust fudge factors or correcting factors to adapt models developed for straight