In this paper, a set of partial differential equations considering the dynamic effects of the coiled tubing (CT) is established based on the bending theory of slender beams considering the axial loads. The analytical solution of sinusoidal deformation with the time term is obtained. The critical load of coiled tubing during sinusoidal buckling and the change of half-wave number during sinusoidal post-buckling are studied through the introduction of the necessary conditions, solution and the separation constant. The contact force of coiled tubing with sinusoidal post-buckling on the horizontal well wall is analyzed. The results show that the critical load for sinusoidal buckling of fixed-size CT is related to the section angular acceleration coefficient. The half-wave number produced by the sinusoidal post-buckling bending of the CT gradually decreases during compression. The contact force of the deformed CT to the borehole wall is related to the compression speed of the CT. By introducing the dynamic term and the separation constant, this research model can provide a theoretical basis for studying the transformation of the CT from sinusoidal buckling to helical buckling.

Coiled tubing is widely used in oil and gas operations. However, the long coiled tubing may undergo sinusoidal or helical buckling deformations, and even lock-up, leading to property damage and serious accidents. To study boundary conditions effects on helical post-buckling of coiled tubing in horizontal wellbores, firstly, the mathematical model containing dynamic terms for nonlinear buckling deformation of coiled tubing in horizontal wells is established in this paper. Secondly, using the improved perturbation method, the third-order perturbation solution of the relationship between pitch and time during helical post-buckling of coiled tubing in horizontal wells is obtained, and the expression of the contact force between coiled tubing and wellbore wall with the time terms is derived. Finally, the effects of gravity, the change velocity of the pitch, and the axial load on the contact force are analyzed separately. The results show that when considering the effect of gravity, the axial internal stress in each micro-element segment of coiled tubing undergoing helical post-buckling exhibits periodic variation. The greater the compression velocity, the greater the contact force between coiled tubing and the wellbore wall. The discriminative conditions for the occurrence of lock-up in coiled tubing undergoing helical post-buckling are proposed, and increasing the axial load can delay the occurrence of lock-up. The mathematical model established in this paper innovatively incorporates time-dependent dynamics, and breaking through the limitations of the traditional quasi-static assumptions of buckling analysis. This model provides a new method and theoretical basis for further study of coiled tubing helical post-buckling process.

In the process of horizontal well construction, the working environment in the well is very complicated. It is difficult to determine the actual deformation and force of the coiled tubing in the wellbore from the load change at the wellhead along. In addition, the coiled tubing wrapped around the reel and entering the wellbore through the injection head may cause an initial residual bending. In this paper, the mathematical model of the coiled tubing under axial load during buckling deformation is established, and the analytical solution with the time term is derived by reasonable simplification of the model. The effect of residual bending on the post-sinusoidal buckling of the coiled tubing in the horizontal well was investigated using the separation variable method. The effect of residual bending variations on the well wall contact forces is analyzed. The research shows that the initial parameter Г is the main factor influencing the post-sinusoidal buckling of the coiled tubing with residual bending. The larger the parameter value Г, the greater the effect of the residual bending on the post-sinusoidal buckling deformation of the coiled tubing, and the earlier the critical point of the mixed sinusoidal-helical buckling of the coiled tubing appears. The compression velocity of the coiled tubing has a significant effect on the well wall contact force. The faster the compression, the greater the contact force. By introducing the time term and the separation variable, this paper provides a new method and theoretical basis for further study of the process of entering sinusoidal-helical buckling of the coiled tubing with initial residual bending.

Irreversible erosion damage of the hearth lining determines the campaign life of a blast furnace (BF). Among the factors involved, structural thermal stress resulting from both internal and external temperature differences and external constraints is a key mechanism in the damage to the hearth lining. Based on a thermal–fluid–structural coupling model that accounts for molten iron flow and solidification, this study, building on thermal stress analysis of the hearth lining, proposes a method to determine the critical strength-based erosion boundary of the lining, using the compressive strength of carbon bricks as the criterion. It also investigates the influence of factors such as dead iron layer depth, tapping productivity, and molten iron temperature on the thermal stress-driven erosion boundary. The findings reveal that the depth of the dead iron layer determines the morphology of the hearth lining’s erosion. With increasing depth, the erosion pattern transitions from an elephant foot profile to a wide-face profile, while the radial erosion depth first increases and then decreases. Both increased tapping productivity and elevated molten iron temperature do not change the erosion shape but aggravate the erosion degree and induce axial displacement of the erosion zone. The research findings are of great significance for deepening the understanding of thermal stress damage in the hearth lining and provide an effective reference for long-term hearth design. Subsequent validation with a large amount of industrial data will further enhance the practical applicability of the proposed method.