QUANTITATIVE IMPLEMENTATION OF PREISACH-MAYERGOYZ SPACE TO FIND STATIC AND DYNAMIC ELASTIC-MODULI IN ROCK

In this paper we describe the analysis of quasi-static stress-strain d ata using a Preisach-Mayergoyz (PM) [after Preisach, 1935; Mayergoyz, 1985] space picture for the elastic behavior of rock. In contrast to t he traditional analytic approach to stress strain (an energy density a s a function of the strain invariants), the PM space picture reproduce s hysteresis and discrete memory seen in the data. In addition, the PM space picture establishes a relationship between experimental data an d a number density rho of microscopic mechanical units within the rock . The density rho allows us to make quantitative predictions of dynami c elastic properties. Determining rho 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: s imulated annealing, normal modes, and exponential decay. All three met hods are tested on a Berea sandstone data set and found to give an exc ellent description of stress versus strain. Choosing one method, the n ormal mode method, we analyze quasi-static stress-strain curves on two additional sandstones, namely, another sample of Berea and a sample o f Castlegate sandstone. From the density rho for each sample we predic t the dynamic modulus as a function of pressure and the nonlinear elas tic constants. For each of these cases the agreement between the predi ctions based on rho 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 method s.