Quantitative implementation of Preisach‐Mayergoyz space to find static and dynamic elastic moduli in rock

Abstract

In this paper we describe the analysis of quasi‐static stress‐strain data using a Preisach‐Mayergoyz (PM) [after Preisach, 1935; Mayergoyz, 1985] space picture for the elastic behavior of rock. In contrast to the traditional analytic approach to stress strain (an energy density as a function of the strain invariants), the PM space picture reproduces hysteresis and discrete memory seen in the data. In addition, the PM space picture establishes a relationship between experimental data and a number density ρ of microscopic mechanical units within the rock. The density ρ allows us to make quantitative predictions of dynamic elastic properties. Determining ρ from quasi‐static stress‐strain data requires us to solve a highly underdetermined inverse problem. We explore the following three methods of solving the inverse problem: simulated annealing, normal modes, and exponential decay. All three methods are tested on a Berea sandstone data set and found to give an excellent description of stress versus strain. Choosing one method, the normal mode method, we analyze quasi‐static stress‐strain curves on two additional sandstones, namely, another sample of Berea and a sample of Castlegate sandstone. From the density ρ for each sample we predict the dynamic modulus as a function of pressure and the nonlinear elastic constants. For each of these cases the agreement between the predictions based on ρ and experiment is quite good. We establish that PM space provides a quantitative description of the elastic response of a rock and that PM space may be found by a variety of inversion methods.