Comparison of effective stiffness and compliance for characterizing cracked rocks

Abstract

The noninteraction approximation (NIA) is commonly used for prediction of the anisotropic elastic stiffnesses of cracked rocks. At large crack density, the NIA has a desirable nonlinear stiffness behavior; however, this is inconsistent with the dilute crack assumption. The nonlinear behavior of stiffness predicted by the NIA at high crack density is produced by defining compliance to be a linear function of crack density and then inverting the compliance tensor to stiffness. The linear behavior of compliance is strictly valid only when there is no crack interaction (at low crack density), so the resulting nonlinear stiffnesses at high crack density are unconstrained extrapolations. Comparison of results from the NIA method, an effective stiffness (T-matrix) method, and numerical modeling shows that: (1) first-order compliance methods are not better than first-order stiffness methods; they are equivalent and valid only under the noninteraction (or dilute crack) assumption; and (2) it is misleading to use the NIA at high crack density; crack interactions (shielding and amplification) should not be assumed to cancel for all crack distributions, so they require explicit consideration, for example, with a high-order T-matrix formulation. Examples of unreliable predictions of the NIA at high crack density include nonzero stiffnesses at 100% porosity for a model with dry cracks, and errors in relative stiffnesses in the isotropic limit.