Calculation of Stretching and Bending Stress in a Casing String Installed in a Well with a Complex Profile
The casing column in the curvilinear well is represented as a long continuous elastic rod. It has a vertically acting weight, which is evenly distributed along the length and creates variable axial tensile forces in the column body. At the same time, it is influenced by the reaction forces of the borehole walls, which, together with the weight, bend the column of initially straight pipes. Here is assumed that the casing axis replicates the axis of the bent borehole, and the walls reaction is continuously distributed along the length according to a certain law, which, together with the weight, bends the column of the initially straight pipes. A system of differential equilibrium equations of internal and external forces and moments was composed, which was supplemented down to a closed form with a differential equation of curvature. This system describes large deformations of a long elastic rod in one plane. An introduction of the distributed weights, wall reactions and resistance forces into the calculation makes it non-uniform. Its feature is the need to solve the inverse problem. Here are known the external load and rod deformations, which are defined by the well shape in the form of an inclinometric data table. The unknown internal forces and the function of the walls reaction, which creates its predetermined shape, must be determined. It is established that this function depends on the distribution of axial forces caused by weight and resistance forces. As a result, the system was reduced to a linear inhomogeneous differential equation with variable coefficients of the first order with respect to the axial force and its solution was obtained as a sum of integrals. It is shown that one of them can be found in quadratures only in the case of a constant radius of curvature of the well. This necessitated the use of numerical integration methods. Formulas for the distribution of axial forces and bending moments in the body of the column, as well as the reactions of the walls leading the column to the actual well profile are obtained from the solution of the basic equation. To calculate these force factors, a method for numerical integration of inclinometric measurements data and software for numerical analysis of a real well are developed. This technique allows to detect the areas of local increase of the curvature and difficult passage of the curvilinear wellbore and calculate the main parameters of the stress-strain state of the casing column in it.
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