An algorithm for the calculation of material tangent stiffness tensor using extended sandier-rubin cap plasticity model in finite element analysis

Several geomechanical models have been developed to simulate the reservoir response to production and injection from weak, compactible formations. Fracture propagation and closure in a pressure sensitive material is important to many engineering disciplines, including the structural, geotechnical, mining, and petroleum industries. Central to these studies has been the use of sophisticated material models. One of these material models, referred to as the Extended-Sandier-Rubin (ESR) cap model, includes a non-linear shear failure surface and a second yield surface (cap) to account for inelastic compaction at stress state lower than those required to induce shear failure. In numerical analysis schemes involving inelastic rock behavior, loads are applied incrementally. Elemental material tangent stiffness tensors must be evaluated in every increment. The existing traditional analytical formulation in the literature is not suitable for use with the ESR model. This paper presents a careful examination of the derivation of this formulation revealing why. Furthermore, this paper introduces a numerical algorithm to calculate material tangent stiffness tensor and incremental changes in elemental stresses. Through several case studies involving existing experimental rock mechanics test data in the literature, it is demonstrated that using the traditional formulation leads to incorrect estimations of material tangent stiffness tensor. The presented algorithm, however, matches experimental data closely. The case studies include experimental data for the McCormick Ranch Sand and two reservoir formations in the Lost Hills oil field.