Analytical and numerical analyses of wellbore drilled in elastoplastic porous formations

The current wellbore stability analyses for analytical solutions are mostly based on the linear elastic or poroelastic theory. The wellbore is regarded as unstable if the determined elastic/poroelastic stresses anywhere around the wellbore satisfy some failure criteria for the rock formation. Rocks, in general, exhibit nonlinear and plastic (hardening/softening) properties with stiffnesses depending on the stress and strain levels. Ignoring the elastoplastic feature of the rock formations tends to give conservative solutions and thus, a substantial overestimate of the critical mud density necessary for stable wellbores.;When elastoplastic constitutive models are taken into account for the rock formations, the wellbore stability boundary value problem usually can only be solved by numerical techniques such as finite element method, though in the past several attempts were made to obtain approximate analytical solutions based on simplified assumptions. This research is aimed to develop a generic class of rigorous, complete analytical solutions for a cylindrical wellbore drilled in elastoplastic porous formations subject to non-isotropic in situ stresses, under both undrained and drained conditions. To cover a range of rock formations such as sandstones and shlaes, different types of elastoplastic models, including the strain hardening Drucker-Prager and Mohr-Coulomb models as well as the Cam Clay and bounding surface models based on the critical state concept, are considered in this analysis.;The key step in the formulation of the wellbore elastoplastic boundary value problem is to establish an incremental relationship between the effective radial, tangential, and vertical stresses and the corresponding stain components, i.e., the elastoplastic constitutive equations, and then reduce them to a set of differential equations valid for any material point in the plastic zone. For undrained condition, the three stresses can be directly solved from these governing differential equations as an initial value problem, the excess pore pressure then being determined from the radial equilibrium equation. Whereas for the drained condition, the Eulerian radial equilibrium equation must be first transformed into an equivalent one in Lagrangian description, which can be accomplished with the introduction of a suitably chosen auxiliary variable. This transformed equation, together with the aforementioned elastoplastic constitutive relation, again constitute a set of coupled differential equations. The three stress components and the volumetric strain or specific volume thus can be readily solved.;Parametric studies show clearly that the overconsolidation ratio of shale-like formations or soft rocks (the ratio of the maximum stress experienced to the current stress level) has significant influences on the stress and pore pressure distributions as well as on the development and progress of the plastic and failure zones around the wellbore. The computed stress distributions and in particular the stress paths capture well the anticipated elastoplastic failure behaviour of the rocks surrounding the wellbore. The solutions thus are able to contribute to better prediction and design of the wellbore instability problems.;Numerical simulations are also conducted for the drained wellbore problem with the use of ABAQUS, the finite element analysis commercial program. Of importance, a user defined material subroutine (UMAT) for the bounding surface model is developed with FORTRAN following the return mapping algorithm and implemented into the ABAQUS finite element models. The predictions from the analytical solutions and the ABAQUS analyses are generally in excellent agreement for both modified Cam Clay and bounding surface models.