Analytical Solution of Stress State Wellbore Instability Due to Collapse Washout and Induced Fractures
Authors: Haoge Liu , Rahman Md Motiur , Lu Jing 
Published: June 2021

Abstract
Wellbore suffers from washout, partial collapse and induced fractures, making it more likely to be elliptical rather than circular. The classic stress analysis of a wellbore, which regards the wellbore as circular, is actually not so suitable. Based on the linear elasticity, this paper introduces the 2D analytical solution of stress state for wellbore instability. Analytical results, compared to numerical, show how the wellbore pressure would be maintained to prevent wellbore instability while drilling.
Analytical solution of stress analysis is proposed for complicated models where the wellbore is regarded as non-circular. Stress problems of a 2D linear elastic model are simplified with mathematical equations. Muskhelishvili theory is used for complex analyses for stress analysis in elasticity. Then the stress state of the formation is obtained, where finding a suitable conformal transformation to map the formation area into a unit circle is crucial in the process. Finite Element Method (FEM) is also applied for the same case. Finally, the analytical result is compared to the numerical result, considering collapse, washout and induced fractures in wellbore.
Typical data set are used for a vertical wellbore. By comparing to the numerical results (FEM), the simplicity for calculation and the correctness of the analytical solution is established whereas it is found that the intrinsic error of numerical solution cannot be eliminated. Results show that with larger boundary sizes, the FEM result become closer to the analytical result. A conformal transformation for the wellbore mapping with fractures was found. Trials have been done to the fractured wellbore, which can be regarded as a stress cage model, while two mathematical problems in solving the stress state analytically on the vertical wellbore with fractures were encountered. Trial and suggestions towards solving these two problems with results are introduced in details. Stress state of the formation has been calculated and plotted by using the analytical solution. Results show the stress contours plotted by analytical solution in Mathematica™, and the ones plotted by FEM with boundary size set as 2000 mm in Solidworks™. The stress states calculated by these two methods match quite well, which means the proposed analytical solution is correct. An insightful sensitivity analysis (with elliptical factor of wellbore and anisotropic factor of the tectonic stresses) will be presented.
For decades, numerical method for stress analysis has been applied, ignoring the development of analytical method. It's the complexity of analytical solution that makes it difficult to handle. However, based on the same simplifications for an engineering problem, analytical method is always faster and more accurate compared to the numerical solution in many cases. The analytical solution provides the possibility for process control in real-time technologies and this can be applied to wellbore instability case of collapse, washout and fractures.