Poisson's ratio, porosity and aspect ratio as geomechanical index of shale brittleness

Elastic constant (stiffness or compliance) of rock has major applications in seismic analysis, rock physics, and geomechanics. However, the controversy of the elastic constant of isotropic solids is a long standing problem which challenge basic assumptions. In the early years of "mathematical theory of elasticity", only one elastic constant is used to characterize elastically isotropic solid. Much later, two independent elastic constants were introduced, as compliance coefficients (Young's Modulus E, Poisson's Ratio) and stiffness coefficients (Lame Parameters). In recent years, the combination of elastic constants to is of paramount importance to understand brittle behaviour of the constitutive response of engineering and geological materials. In connection with shale characterization, brittleness prediction serves as a good indicator to delineate susceptible to hydraulic fracturing. However, controversy exist among researchers on what brittleness formula to apply. For the purpose of quantitative interpretation of brittleness, we consider the classical Hooke's law and Griffith's crack theory. Based on Eshelby formulation of the elastic state of the crack inclusion and matrix and the consideration of hydrostatic compression a simple and elegant mathematical expression is derived. Experimental data is then generated to check the prediction of this relation.