Third-order elastic solution of the stress field around a wellbore

摘要

Within a certain range of strain, consolidated granular materials may be characterized as nonlinear elastic solids. The nonlinearity can be easily observed by examining the effect of stress on the acoustical properties of the material. Ignoring damage evolution and failure that occur in higher strains and the hysteretic behavior due to intercyranular friction, the material can be modeled as a nonlinear hyperelastic solid. A simple example of such a model is formulating the strain energy as a third-order polynomial of the strain invariants. This model is limited in the sense that the material is assumed to be isotropic with respect to the stress free state, and that the mechanical response of the material is described by only five material constants. Nevertheless, this model is appealing because it naturally exhibits stress dependent stiffness and stress induced anisotropy, and it allows a different mechanical response to positive and negative volume changes. In this work, this model is used to calculate the stress field around a wellbore. Many well logging tools use acoustics (e.g., tube, surface, torsion, and flexural waves) to detect pore fluids and ore in the surrounding granular rock. By modeling the rock as an isotropic third-order elastic material the effects of the inhomogeneous stiffness and the stress induced anisotropy may be examined. Analysis of the tangential stress around a wellbore in an isotropic third-order elastic (TOE) material yields different results than the same analysis in the related isotropic linear elastic (LE) material (i.e., both materials have the same stiffness tensor at the stress free state). This difference modifies the far-field stress that is interpreted of from hydraulic fracturing data. The analysis in the present work is static and pore fluid effects are ignored.