PSI - Issue 48

Udaya B Sathuvalli et al. / Procedia Structural Integrity 48 (2023) 207–214

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Sathuvalli and Suryanarayana/ Structural Integrity Procedia 00 (2019) 000–000

Assume that the stresses and strains in the plastic region have been determined for an arbitrary location of the elastic-plastic boundary. Next assume that plasticization moves the boundary by a small increment (caused by the load increment). The state of stress for this incremental movement of the elastic-plastic boundary is obtained by integrating the equations (Eqs. (13), (14), (19), (18) and (15) or (17)) in the plastic zone. The initial condition for the integration of the equations is the state of stress at the previous time step. The boundary condition is provided by the contact pressure p EPB at the elastic-plastic boundary, which is yet unknown. This contact pressure is determined by solving the equations in the elastic zone. The equations in the plastic zone are hyperbolic differential equations, and an analytical solution cannot be obtained. They must be integrated numerically. 7. Appendix B: Governing Equations for the Thin-Walled Cylinder Consider an element of casing cross section (in Fig. 5) between r and r + ∆ r , and θ and θ + ∆ θ . Assuming the von Mises yield function, and assuming that the cylindrical coordinates are aligned with the principal stress directions, the governing stress-strain equations for this element during an incremental loading from ( σ r , σ θ , σ z ) to ( σ r + ∆ σ r , σ θ + ∆ σ θ , σ z + ∆ σ z ) can be obtained by discretizing Eqs. (12)-14. These equations are supplemented by Eq. (15) or Eq. (17). Since the radial stresses are small, by setting σ r and ∆ σ r to zero, the incremental equations for a non-hardening material can be written as (Paslay et al. 2006), ( ) ( ) ( ) ( ) ( ) 1 3 0 1 0 1 2 3 1 0 1 2 0 3 0 2 2 0 z r z z z z E E E θ θ θ θ θ ν ν σ σ ε ν σ σ σ ε σ ε ν σ σ λ σ σ σ σ   +   ∆            − − − ∆ − ∆       =       ∆ − ∆       − − −  ∆        − − . (20) The algorithm divides the cross section of the casing into an ( r , θ ) computational grid. For a given increment of radial displacement u o , the increment of tangential strain ∆ ε θ is found from Eq. (9). Next, we assume an increment of axial strain and treat the assumed value as a known quantity. The unknown matrix on the left hand side of equation is then calculated. A search algorithm that ensures global work energy balance determines the correct increment of axial strain. The work done by the external forces equals the deformation work (of bending) in the cross section. The external forces here are the pressure(s) and axial load. The bending work for a grid element equals the bending moment times the curvature change (Eq. 7). The bending moment is known from the hoop stress calculated by solving Eq. (20). The global balance involves summation across all elements of the computational grid. References API TR 5C3 2008. Technical Report on Equations and Calculations for Casing, Tubing, and Line Pipe Used as Casing and Tubing: and Performance Properties Tables for Casing and Tubing, first edition, Washington, DC: API Bickley, M. C., Curry, W. E., 1992. Designing Wells for Subsidence in the Greater Ekofisk Area, European Petroleum Conference. Cannes, France, paper # SPE-24966 MS. Boresi, A. P., Schmidt, R. J., and Sidebottom, O. M., 1993, Advanced Mechanics of Materials , John Wiley and Sons, New York, NY. Bruno, M. S., 2002. Geomechanical and Decision Analyses for Mitigating Compaction-Related Casing Damage. SPE Drill & Compl., 17(3), 179-2002. Geertsma, J., 1973. A Basic Theory of Subsidence due to Reservoir Compaction: The Homogeneous Case. Verhandelingen Koninklijk Nederlandsch Geologisch Mijnbowkundig Gennotschap, 28 (19),43-62. Madhavan, R., Babcock, C. D., Singer, J., 1993. On the Collapse of Long, Thick-Walled Tubes Under External Pressure and Axial Tension. ASME. J. Pressure Vessel Technol. 115(1), 15-26. Maruyama, K., Tsuru, E., Ogasawara, M., Inoue, Y., and Peters, E. J., An Experimental Study of Casing Performance Under Thermal Cycling Conditions. SPE Drilling Engineering, June 1990. Also SPE 18776., Paslay, P. R., Pattillo, P. D., Pattillo II, P. D., Sathuvalli, U. B., Payne, M. L., 2006. A Re-examination of Drillpipe/Slip Mechanics. IADC/SPE Drilling Conference, Miami, FL,USA, paper # SPE-99074-MS. Prager, W. and Hodge, P. G., 1951, Theory of perfectly plastic solids , John Wiley and Sons, New York, NY, pp. 27-31 Pattillo, P. D., 2018. Elements of Oil and Gas Well Tubular Design . Gulf Professional Publishing: An Imprint of Elsevier, Cambridge, MA, USA. Suryanarayana, P. V., Krishnamurthy, R. M., Sathuvalli, U. B., Bowling, J., 2020, A Review of Casing Failures in Geothermal Wells, Proceedings World Geothermal Congress. Reykjavik, Iceland. Timoshenko, S.; Gere, J. M., 1989, Theory of Elastic Stability , second edition, Dover Publications, Mineola, NY, USA Toscano, R. G., and Dvorkin, E. N., 2011. Collapse of Steel Pipes under External pressure and Axial Tension. Journal of Pipeline Engineering, 4th Quarter. z z θ θ  