Rock Static Moduli From Borehole Sonic Data in Stress-Dependent Formations

Abstract

This article describes a novel technique to estimate static Young's modulus of stress-sensitive rocks using dynamic linear and nonlinear constants estimated from borehole sonic data. Two linear and three nonlinear constants are estimated from the transit time of compressional headwaves and inversion of borehole-guided Stoneley and crossdipole dispersions in tectonically stressed formations. A major advantage of this technique is that the rock static Young's modulus is determined from the dynamic elastic constants measured at a chosen reference state that is rather close to the in-situ conditions. These dynamic elastic constants are used in the nonlinear constitutive relations for poroelastic rocks subject to finite deformations. These relations express the second Piola-Kirchhoff axial stresses in terms of elastic constants together with up to quadratic terms in Lagrangian axial strains. Strain derivatives of the second Piola-Kirchhoff stress yield the static Young's modulus as a function of incremental axial strains from a chosen reference state. Consequently, static Young's modulus can be also determined at other depths with different overburden stresses and associated incremental axial strains from a chosen reference state. In contrast, strain derivative of the second Piola-Kirchhoff axial stress expressed in terms of linear elastic constants and only linear terms in axial strain provides the dynamic Young's modulus. Two useful outputs from this workflow are the static stress-strain deformation curves for a core plug; and static Young's modulus under in-situ conditions as a function of logging depth. The proposed technique has been validated with the available experimental stress-strain data from Castlegate and Berea sandstones core plugs. Results have been obtained for the static Young's modulus and finite deformation stress-strain curves for two different stress-sensitive poroelastic formations using borehole sonic data.