| Particular attention has been focused on studying the mechanical behaviors of drillstrings, because drilling engineering takes high risk and invests heavily capital. Currently, theories and analytical methods for analyzing mechanical behaviors of drillstrings sitting in wellbores of vertical wells, planar wells and curved wells with small curvatures have been well established. It is, however, that theories and analytical methods are far more perfect for analyzing mechanical behaviors of drillstrings sitting in arbitrary 3D boreholes appeared only recently with a certain curvature and tortuosity. Thus, more researches are needed in this area. According to the characteristics of drillstrings, it is assumed that the transverse shear deformation in a beam is neglected and no warping exists. Based on the classical elasticity, differential geometry and matrix methods, a new two-node 3D spatial curved beam finite element with circular cross-sections is established by using non-polynomial displacement functions in spatial natural (curvilinear) coordinate system. There are six degrees of freedom at each node, namely, three translations and three rotations. In obtaining the element non-polynomial displacement functions, the curvature change in the deformed element state is assumed lineally varying along the element centerline, while curvature and tortuosity are constant with the element. The non-polynomial displacement functions, including all rigid modes and constant strain states, are obtained by directly integrating the geometrical relations. A rotational pseudovector is introduced for conveniently describing the finite rotations of cross-sections. Based on the generalized Green Strain definition, the nonlinear geometrical relationships under natural coordinate system are derived by employing the differential rule of vectors. Using proposed beam element, buckling problems of curved beams subjected to both compressive axial loads and torques have been solved by finite element method. To determine the wellbore axis reasonably, the so-called helix interpolation method is proposed. With this method, the wellbore axis with consistent well depth is obtained based on the field measured well depths, azimuth angles and oblique angles. Define the deformed state of drillstrings coincident with the wellbore axis the drillstring's reference configuration, thus, the nonlinear problem caused by the large displacement and large rotation of the drillstring from its initial straight state to the reference configuration is simplified to the computations of the initial deformation. Thus the initial stress state of the drillstring is determined based on the classical elasticity and on the curvature and tortuosity of the drillstring at its reference configuration,thus the consistent initial load vector can be obtained for the subsequent finite element analysis. Meanwhile, a new contact model in accordance with the contact characteristics between drillstring and borehole wall is proposed. The model can ensure that the contact segments of the drillstring closely contact with the wellbore wall. The procedures for the contact nonlinear analysis, including the coupling effects among the deformation, contact position and contact forces, are given. Based on the proposed beam element, algorithms and computational strategy, a Fortran program is developed for double nonlinear finite element analyses of drillstrings sitting in arbitrary 3D wellbores. Numerical examples are performed. The results are compared with the existing analytical or/and numerical data to verity the correctness of the program as well as the effectiveness, correctness and rationality of the proposed beam element, contact model, developed formulae, and computational strategy. Meanwhile, three practical examples, taking several field factors into considerations, are analyzed by using the developed program. Reasonable results are obtained. Thus, the validity and feasibility of the double nonlinear finite element modeling for the whole drillstrings are verified.
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