Down-hole tubular string buckling is the most classic and complex part of tubular string mechanics in well engineering. Studies of down-hole tubular string buckling not only have theoretical significance in revealing the buckling mechanism but also have prominent practical value in design and control of tubular strings. In this review, the basic principles and applicable scope of three classic research methods (the beam-column model, buckling differential equation, and energy method) are introduced. The critical buckling loads and the post-buckling behavior under different buckling modes in vertical, inclined, horizontal, and curved wellbores from different researchers are presented and compared. The current understanding of the effects of torque, boundary conditions, friction force, and connectors on down-hole tubular string buckling is illustrated. Meanwhile, some unsolved problems and controversial conclusions are discussed. Future research should be focused on sophisticated description of buckling behavior and the coupling effect of multiple factors. In addition, active control of down-hole tubular string buckling behavior needs some attention urgently.

Summary: A new buckling equation in horizontal wells is derived on the basis of the general bending and twisting theory of rods. The boundary conditions of a long tubular string are divided into two categories: the sum of the virtual work of bending moment and shear force at the ends of tubular strings is equal to zero, and the sum of the virtual work of bending moment and shear force at the ends is not equal to zero. Buckling solutions under different boundary conditions are obtained by solving the new buckling model. For the boundary conditions of the first category, the buckling solutions are identical with previous results. For the boundary conditions of the second category, the buckling solutions are different from the results under the boundary conditions of the first category. The results indicate that buckling behaviors depend on both the axial force and the boundary conditions. Compared with previous results, buckling solutions of the new model provide a more comprehensive description of tubular-buckling behaviors.

Summary: Boundary conditions for tubular-string buckling are divided into two kinds on the basis of the virtual work of the nonaxial force on the boundary conditions of the tubular string—namely, the first and the second kinds of boundary conditions. The first kind of boundary conditions means that the virtual work of no-axial force is zero, and both the conventional pinned and fixed ends belong to this kind. The second kind of boundary conditions means that the virtual work of no-axial force is not zero. Previous studies on tubular-string buckling mainly focus on the first kind but ignore the second kind of boundary conditions. In this paper, the effects of the two kinds of boundary conditions on tubular-string buckling are analyzed. The deflection of a long tubular string constrained in a wellbore is divided into the full helical-buckling section and transition section, whereas the transition section is divided further into the no-contact section and perturbed-helix section. The qualitative corresponding relation between boundary conditions and tubular-string buckling in transition and full helical-buckling sections is clarified. To clarify the quantitative relation between boundary conditions and tubular-string buckling, a general-packer model is proposed to depict the two categories of boundary conditions. With the general-packer model, the general potential energy of the tubular string is deduced. According to the minimum-potential-energy principle, the existence and stability of full helical-buckling solutions are given. The deflections of the tubular string in the no-contact and perturbed-helix sections are deduced with buckling differential equation and beam-column model. The bending moment, shear force, and contact force on the tubular string caused by buckling are also analyzed. The results show that boundary conditions, especially the second kind of boundary conditions, are an important factor that makes the tubular-string buckling problem complex, and this paper provides one source for a deeper understanding of boundary conditions.

Summary: Rate of penetration (ROP) is one of the important indicators for evaluating drilling efficiency, which provides the basis for drilling parameter optimization. ROP prediction methods can be divided into two main categories—physical models and machine learning models. The physical models are derived from classical drilling theory and experimental results, with clear physical meanings, good interpretability, and generalization. However, physical models do not result in very accurate predictions due to assumptions and experiences introduced in the modeling process. Machine learning models can effectively learn the intrinsic relationships between data through the training of a lot of data. However, machine learning models are like black boxes and their performances are highly dependent on the quality of drilling data in oil fields, so their interpretability and generalization ability are relatively low. This study mainly focuses on establishing a more accurate model of ROP prediction with clear interpretability. To achieve this goal, two novel categories of hybrid modeling approaches were introduced for horizontal drilling in the China Jimusar oil field, one of which is the error compensation by machine learning and another is the weighted average outputs. In the first category of hybrid model, physical models are taken as the main submodel, and machine learning models are used to predict and counteract the errors caused by physical models. By this method, the physical model can effectively ensure its physical meaning and generalization, and the machine learning model, as a submodel, can effectively compensate for the low-accuracy defects of physical models to improve prediction accuracy. In the second category, combining physical models with machine learning models utilizing ensemble learning, the deficiencies of models are cancelled out by the other models in the ensemble—like a team effector. The paper presents hybrid models with four suggested steps, which include data collection and preprocessing, optimal selection of physical model, optimal selection of machine learning model, and establishment of hybrid model. The performances of physical models, machine learning models, and hybrid models are intercompared. From the view of prediction accuracy, model interpretability, modeling difficulty, and generalization, the hybrid model with error compensation by machine learning is the optimal method for ROP prediction. This study also demonstrates an optimal trade-off between high accuracy and good interpretability.