Third-Order Pade Thermoelastic Constants of Solid Rocks

Classical third-order thermoelastic constants are generally derived from the theory of small-amplitude acoustic waves in isotropic materials during heat treatments. Investigating higher-order thermoelastic constants for higher temperatures is challenging owing to the involvement of the number of unknown parameters. These Taylor-type thermoelastic constants from the classical thermoelasticity theory are formulated based on the Taylor series of the Helmholtz free energy density for preheated crystals. However, these Taylor-type thermoelastic models are limited even at low temperatures in characterizing the temperature-dependent velocities of elastic waves in solid rocks as a polycrystal compound of different mineral lithologies. Thus, we propose using the Pade rational function to the total thermal strain energy function. The resulting Pade thermoelastic model gives a reasonable theoretical prediction for acoustic velocities of solid rocks at a higher temperature. We formulate the relationship between the third-order Pade thermoelastic constants and the corresponding higher-order Taylor thermoelastic constants with the same accuracy. Two additional Pade coefficients alpha 1 ${\mathit{\alpha }}_{1}$ and alpha 2 ${\mathit{\alpha }}_{2}$ can be calculated using the second-, third-, and fourth-order Taylor thermoelastic constants associated with the Brugger's constants, which are consistent with those obtained by fitting the experimental data of polycrystalline material. The third-order Pade thermoelastic model (with four constants) is validated by the fourth-order Taylor thermoelastic prediction (with six constants) with ultrasonic measurements for polycrystals (olivine samples) and solid rocks (sandstone, granite, and shale). The results demonstrate that the third-order Pade thermoelastic model can characterize thermally induced velocity changes more accurately than the conventional third-order Taylor thermoelastic prediction (with four constants), especially for solid rocks at high temperatures. The Pade approximation could be considered a more accurate and universal model in describing thermally induced velocity changes for polycrystals and solid rocks.

Automated summary

This study proposes a thermoelastic model using the Pade rational function to characterize the temperature-induced velocity changes in solid rocks. The results show that the Pade model provides a more accurate description of the velocity changes, especially for solid rocks at high temperatures, compared to the conventional Taylor thermoelastic prediction.