The oscillations of an aspherical droplet subjected to different external forces are considered. For an arbitrary shape deformation, it is shown that the frequency spectrum splits into (2l−1) peaks for a mode l oscillation, and the splitting of the frequency spectrum is calculated for mode 2, 3, and 4 oscillations. The deformation is then treated as a consequence of a general external force, and the frequency split is obtained in terms of the external force parameters. Droplets levitated by acoustic, electromagnetic, and combined acoustic-electromagnetic forces are considered in particular, and it is shown that the effects of asphericity adequately explain the splitting of the frequency spectrum observed commonly in experiments. The interpretation of spectra with regard to accurate surface tension measurement using the oscillations of levitated droplets is discussed, and the results applied to some previous experimental results. It is shown that the accuracy of surface tension measurements can improve if the asphericity caused by the levitating force, and the resulting frequency split, are taken into account.

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Buckling and post-buckling lock-up place a limit on the reach of extended-reach and horizontal wells. Although buckling has received considerable theoretical attention in the past, no serious attempt has been made to study this process experimentally. This paper describes results from experiments on buckling, unbuckling and post-buckling behavior of rods laterally constrained in a cylindrical enclosure, with particular emphasis on the effects of friction. The experimental apparatus, procedures, and uncertainty analysis are described. Results indicate that friction significantly delays the onset of buckling (both sinusoidal and helical buckling), and causes noticeable hysterisis in the post-buckling behavior, As a result of this hysterisis, the unbuckling loads are always less than the corresponding buckling loads. Mitigation of friction reduces the hysterisis. Friction is also a cause of post-buckling snapping and reversals in the direction of the helix. As expected, the effects of friction become less significant as the inclination decreases. For inclinations (from vertical) less than 15 degrees, the effects of friction are negligible for the initiation of sinusoidal buckling, but are significant once the rod has buckled into a full helix. Curvature also delays the onset of buckling, and both curvature and friction have a stabilizing effect on the tubulars. Current theory is re-visited and re-interpreted irr the light of these experimental observations, and its limitations are discussed with an emphasis toward operational practices and field applications. It is shown that currently used theory actually predicts unbuckling and not buckling. When friction is significant, current theory underestimates the compressive loads at which sinusoidal and helical buckling of tubulars occur in straight wellbore intervals. Ignoring friction limits the weight on bit well below the safe load that can be used in many drilling and completion operations in extended reach or horizontal wells. Moreover the hysterisis effect of friction means that once buckling has occurred, the compressive loads must be reduced to values much below the buckling initiation loads to fully straighten the buckled pipe. INTRODUCTION Buckling of tubulars such as drill string, casing, or tubing places operational limits on extended reach wells. In particular, buckling and subsequent "lock-up" are limiting factors to the reach of horizontal wells. Buckling occurs when the compressive load on a tubular exceeds some critical value. This critical load is the load beyond which the equilibrium shape is no longer stable, and small increases in load lead to large lateral deflections. Typically, compressive load is provided by the weight on bit (WOB) or the slack-off friction. However, unlike Euler buckling of bars, the lateral deflection of buckled tubulars in wells is limited by the outer constraint of the wellbore. As a result, buckling progresses differently. When the compressive load reaches the critical value, the straight shape of the string is no longer stable. A slight increase in the load from this value causes the string to deflect (buckle) into a sinusoidal shape along the lower portion of the hole. This is referred to as sinusoids/buckling.

The effects of friction and curvature on buckling, post-buckling, and unbuckling behavior of rods laterally constrained within an enclosure are studied experimentally. The experimental apparatus, measurement procedures, and uncertainty analysis are described. Results indicate that friction significantly delays the onset of buckling, and causes noticeable hysteresis in the post-buckling behavior. As a result, the unbuckling loads are always less than the corresponding buckling loads. The drag-related loss, which eventually leads to lock-up, is also measured and reported in this work. Friction is also a cause of post-buckling snapping and reversals in the direction of wrap of the helix. As expected, the effects of friction become less significant as the inclination decreases. It is shown that predictions of current theory agree with experimentally measured unbuckling rather than buckling loads. When friction is significant, current theory underestimates the compressive loads at which buckling occurs in straight or curved wellbores. Ignoring friction or curvature limits the estimated weight on bit well below the safe load that can be used in many drilling and completion operations in extended reach or horizontal wells. Moreover, the hysteresis effect of friction means that once buckling has occurred, the compressive loads must be reduced to values much below the buckling initiation loads to fully straighten the buckled pipe.

We present a procedure that uses nonlinear optimization theory to plan complex, three-dimensional well paths and path corrections while drilling. The problem of hitting a 3-D target is posed as seeking a profile that optimizes some well-defined objective function (the optimality criterion) subject to equality and inequality constraints. The well path is idealized to contain a finite combination of turn and straight sections. Operational restrictions translate into inequality constraints, and target restrictions translate into equality constraints. Several optimality criteria may be chosen, and appropriate choices are discussed. In this work, we choose optimization with respect to user preferred parameters as the criterion. The resulting nonlinear optimization problem is solved using a sequential gradient-restoration algorithm (SGRA), with scaling and optimal step-size selection. The optimization problem formulation and the solution procedure are described. The procedure is robust, efficient, and clearly superior to trial-and-error heuristic techniques that are commonly used to plan well paths today. A computer program based on this technique has been developed and successfully used. Two examples are included to illustrate the procedure. It is concluded that nonlinear optimization is a powerful and versatile mathematical tool that can be used for planning better, optimal well paths, and can be extended to several other drilling and production problems.