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
Eulerian Formulation of the Torque and Drag Problem
The petroleum industry relies on several kilometer long drillstrings to transmit the axial force and torque necessary to drill the rock formations and reach deep hydrocarbon reservoirs. The assessment of the energy loss along the drillstring, known as the torque and drag problem, plays an essential role in well planning and drilling as the friction appearing at the contacts between the drillstring and the borehole may dramatically increase the costs or, even, be a limiting factor in some configurations [1]. The identification of the number of contacts as well as their positions and extents constitutes the central concern of this question. The nonlinearities associated with the large deflection of the drillstring and the non-penetration condition as well as the a priori unknown number of contacts, however, make the use of conventional numerical tools rather inefficient. Additionally, the commonly adopted division of the problem in segments bounded by two contacts [2, 3] requires to solve the governing equations on domains which are initially unknown and, therefore, leads to the establishment of integral constraints on the unknown length of a rod forced to go through fixed points in space. Finally, the assessment of the unilateral contact condition, which requires in principle the comparison of two curves parameterized by distinct curvilinear coordinates (e.g. the drillstring and borehole axes), prove to be a rather intensive computational task. For inextensible drillstrings constrained in the vertical plane, the abovementioned difficulties have been circumvented by reformulating the problem within the Eulerian framework associated with the borehole and by describing the drillstring deformed configuration by means of its transverse position relative to the borehole axis [4, 5, 6]. Generalizing these concepts to three-dimensional configurations, the drillstring is modeled by means of the special Cosserat rod
The Inca Occupation of the South Coast of Peru
like the others, however; it forks into several nearly parallel ravines of which the largest are Huayuri, Rio Grande-Palpa, Ingenio (called Collao in ancient times), Nasca proper (the ancient Caxamarca), Taruga, and Poroma. Due to peculiarities of the local topography, the irrigated areas in Ica and the ravines of Nasca are located inland, at the foot of the mountains, rather than at the mouths of the rivers as is more usual in Peruvian coastal valleys. No sharp natural or cultural boundary separates the valley of Chincha from the valleys of Cafiete (ancient Huarco and Runahuana), Asia, and Mala, the next valleys going up the coast to the northwest. At the other end of the area we are considering, however, there is a sharp break; beyond Yauca the Andean foothills reach out to the sea, natural conditions are quite different, and there are important differences in the archaeological picture as well. The historical sources which mention the south coast are not particularly helpful, except in so far as they provide us with a few dates. To begin with, no chronicler gives an explicit account of the political organization of the south coast at the time it came into contact with the Incas. It can, perhaps, be inferred from Cabello Balboa's statements about common policy toward the Inca invaders that there was some sort of political connection between the valleys of Mala, Cafiete, and Chincha, but the connection need have been no more than a temporary military alliance. Cabello makes no mention at all of Pisco. The valleys of Ica, Nasca, and 125