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When wellbores are exposed to loads of geomechanical origin, the outer casings can become vulnerable to the transverse load components. These loads are usually non-uniform in character (Veeken et al. 1994). The collapse resistance of tubulars to non-uniform loads is substantially lower than their resistance to uniform loads. In the face of such reduced tubular strength, the well engineer uses thick walled high strength casings or a system of concentric cemented casings (Clegg 1971; Pattillo and Rankin 1981; Pattillo et al. 1995). In either case, the designer has to determine a quantitative measure of well integrity i.e., a safety factor (SF) of the casing for the non-uniform loading. In our industry, well designers have used finite element analyses (FEA) to assess the mechanical response of the casings subjected to such loads (Pattillo et al. 1995; Li et al. 2003). The simplest stress analysis problem in such situations consists of at least three cylinders, the innermost casing, the cement sheath and the formation up to the farfield boundary. Depending on the numerical methods employed in the geomechanical analysis, the farfield geomechanical loads are presented as displacements or tractions. In this paper, we present an analytical procedure to determine the mechanical response of a system of nested concentric cylinders exposed to an arbitrary traction or displacement on the outer radius of the outermost cylinder. We use the solution to quantify the effect of the loads on the concentric casings and the intervening cement sheaths, and to assess the effect of the formation. To this end we use well-known methods employed in the theory of elasticity to derive our solution. The analytical solution presented in the mathematical appendices can be implemented in a programmable spreadsheet. |
