Recent advances in the modeling of a drillstring inside a curved borehole
Accurate calculation of the position of the contacts between the drillstring and the borehole as well as the determination of the extent of these contacts is the true challenge of a torque-and-drag analysis. Such calculations are required to estimate the loss of torque from the rig to the bit under some assumed friction models, but also to assess the axial force on the bit (the “weight on bit”) given the known axial force at the rig (the “hook load”). Indeed, the weight on bit directly influences the rate of penetration, but also the drilling tendency of bottomhole-assemblies equipped with rotary steerable systems [3, 6]. The accurate and efficient structural analysis of such a slender body, which is constrained to deform inside a borehole, is thus of paramount importance for the modeling of drilling operations. In the late 1990’s, the state-of-the-art in torque-and-drag modeling has shifted from the “soft string model,” based on the assumption that the drillstring coincides with the borehole axis [4], to advanced numerical methods with built-in management of the contacts. Standard finite element techniques have been applied to the torque-and-drag problem, but they require a fine mesh to properly capture the response of the system in the vicinity of contacts [5]. Furthermore, the slenderness of the drillpipe, combined with the limited clearance between the drillstring and the borehole wall are two of the reasons that make the application of such standard techniques rather inefficient [2]. As an alternative, we have developed an approach that segments the drillstring between contacts and treats the series of unconstrained (or classical) elastica problems between successive contacts as a sequence of simpler auxiliary
The problem of a drillstring inside a curved borehole: a perturbed Eulerian approach
This paper is concerned with the calculation of the deformed con guration of a drillstring during drilling, which critically hinges on accurately identifying the contacts between the drillstring and the borehole walls. For this study, we assume that the position of the bit relative to rig is xed and that a known axial force is imposed at the rig. A mathematically related problem is the insertion (or the pulling) of the drillstring into (or out of) the borehole, as the nature of the axial boundary conditions at both ends of the drillstring is exchanged. This subject matter is part of a larger class of problems involving a priori unknown contacts between an elastica and a rigid boundary. These problems are computationally challenging, especially in the context of the drilling applications. Indeed, the large de ections of the drillstring from a stress-free con guration require consideration of a geometrically non-linear model. Furthermore, application of standard numerical tools to this problem results in an ill-conditioned system of equations, owing mainly to the narrowness of the borehole compared to its length, but also to the large exibility of the drillstring and the assumed rigid nature of the borehole walls. We propose here a novel mathematical formulation of this problem, which takes advantage of the extreme slenderness of the borehole and which is based on expressing the deformed con guration of the drillstring as a perturbation of the borehole axis.
Thorium in monazite sand; separation and determination by precipitation from homogeneous solution
A new method is described which consists of three steps: (1) decomposition of the monazite sand with perchloric acid; (2) separation of thorium and the rare earths by hydrolysis of methyl-oxalate in acid solution; and (3) separation of thorium from the rare earths by hydrolysis of urea in the presence of formic acid. The authors consider the method to be accurate and rapid.