Error estimates of elastic components in stress-dependent VTI media

Abstract

Abstract This work examines the ranges of physically acceptable elastic components for a vertical transversely isotropic (VTI) laboratory shale data set. A stochastic rock-physics approach combined with physically based acceptance and rejection criteria determined the ranges. The importance of this work is to demonstrate that multiple constrained models explain independently calculated measurement error bars. The data set consisted of pressure- and directional-dependent velocity measurements conducted on a low porosity, brine-saturated hard shale. Error bars were calculated for all five elastic stiffnesses and compliances as a function of pressure. The rock physics model is pressure dependent and represents simultaneously five elastic compliances for a VTI medium. A non-linear least squares fitting routine established a best-fit model to the five compliances at all pressures. Perturbations of the best-fit model provided the statistical parameter space. Twelve physical constraints or data-set-specific conditions comprised the acceptance/rejection criteria. These constraints and conditions included strain-energy requirements, inequalities among stiffnesses and anisotropy parameters, and rates of change of moduli with pressure. The largest number of rejected models resulted from violating a criterion relating a compressional and shear stiffness. Minimum misfits between the accepted models and the data illustrate that a fraction of the accepted models best explain the data. The misfits between these accepted models and data explain the error in the data and/or inhomogeneities at the measurement scale. The ranges of acceptable elastic component values and the corresponding uncertainty estimates could be incorporated into seismic-inversion, imaging, and velocity-modeling schemes.