Elastic potential and pressure dependence of elastic moduli in fluid-saturated rock with double porosity

Abstract

Models of seismic velocity dispersion and attenuation in porous rock are often based on quantitative relations between empirical moduli by Gassmann, Mavko-Jizba, Sayers-Kachanov, and others. All of these relations have a common origin in the concept of elastic potential; nevertheless, this concept itself has been insufficiently used in the context of multiple porosities and squirt flows. Regardless of the microstructure of a rock, knowledge of its macroscopic elastic potential reveals the complete set of parameters that are necessary and sufficient for characterizing its elasticity and obtaining rigorous equations of deformation. For isotropic rock with double (such as stiff and/or compliant) porosity, this parameter set contains six elements of the bulk-modulus matrix and six moduli for shear. The elastic matrices predict all observable elastic properties, such as bulk, shear and Young’s moduli, pore compressibilities, Skempton coefficients, consolidation parameters, Poisson’s ratios, static (Gassmann’s) and ultrasonic (Mavko-Jizba’s) undrained moduli, as well as the hypothetical high-pressure modulus [Formula: see text] and unrelaxed-frame moduli. By using laboratory observations with Westerly granite, all six elements of the elastic matrix for bulk deformation are inverted for exactly; three moduli for shear are determined with uncertainties due to insufficient data. The inverted matrices of the elastic moduli predict all observations (drained and low- and high-frequency undrained moduli) within measurement accuracy, with minor uncertainties for shear deformation. The secondary pores are approximately 50 times more compressible than are the primary ones, which explains their interpretation as “soft.” Most of the inverted elastic moduli vary with the confining pressure. Despite its usually assumed constancy, the high-pressure modulus [Formula: see text] exhibits an approximately 10% decrease with confining pressure from 0 to 100 MPa, which is opposite of the trend of the solid-grain modulus [Formula: see text].