Third-Order Padé Thermoelastic Constants of Solid Rocks

The physical and mechanical properties of rocks are highly susceptible to significant changes in temperature. For example, temperature-induced variations in elastic properties generally exhibit strong nonlinearity at low temperatures attributed to the differences in the thermal expansion coefficient of the rocks. When Taylor series of the Helmholtz free energy density is used in the formulation of the classical theory of thermoelasticity, the resulting second- and third-order thermoelastic constants can be used to characterize temperature-dependent rock properties, though with certain insufficiencies.

For most rock crystals, the associated temperature-elastic constants are generally linear and weak at low temperatures but gain prominence with temperature increment. This results in higher-order thermoelastic constants that hold potential applications in studying the impact of temperature changes on rock properties. While studying higher-order thermoelastic constants could provide more insights into the nonlinear behavior of rocks at varying temperatures, it is a challenging approach due to the involvement of several unknown parameters. In addition, the available Taylor-type thermoelastic models have several drawbacks limiting their practical applications.

To this note, China University of Petroleum (East China) researchers led by Professor Li-Yun Fu proposed the use of third-order Padé approximation based on Helmholtz free energy density as a suitable replacement of Taylor series for modeling thermoelasticity for solid rocks. Padé rational function is best suited for functions that exhibit a decreasing trend from higher-order to lower-order terms and is applicable for functions that tend to diverge when expanded in power series like the ones in this case. The feasibility of this approach was validated through application with laboratory measurements to predict the temperature-dependent velocities of elastic waves. Their work is currently published in the research Journal of Geophysical Research: Solid Earth.

The authors showed that the resulting Padé approximation thermoelastic model gave reasonable theoretical predictions for acoustic solid rock velocities at higher temperatures (up to 1000°C for solid rocks and 1500 K for polycrystals). The prediction accuracy and precision were higher and comparable to that third-order Taylor series, respectively, attributed to the slow convergence of the Taylor-type thermoelastic models. The relationship between the third-order Padé thermoelastic constants and associated higher-order Taylor thermoelastic constants was formulated with the same accuracy, and the obtained Padé coefficients corresponded to the second-, third-, and fourth-order Taylor thermoelastic constants associated with the Brugger’s constants. It was also relevant for thermally induced microcrack deformation and rock heterogeneities-induced thermal expansion mismatch. The results were consistent with those obtained by fitting the experimental data of polycrystalline material.

Furthermore, a close correlation between the Padé coefficients and the nonlinearity changes in elastic wave velocities was observed. The coefficients characterizing Padé thermoelastic constants were successfully used to describe rock heterogeneity in both microstructures and lithologies. It was worth noting that the Padé coefficients depended on several parameters, including the composition, geometry and heterogeneity of the materials. For example, an increase in heterogeneities resulted in a significant increase in the Padé coefficients. Due to its high accuracy compared to conventional alternatives, the proposed model is relevant and can be used in several fields, including seismic attenuation, earthquake seismology and geothermal and hydrocarbon exploration.

In summary, this is the first comprehensive analysis of Padé thermoelasticity rational function theory and its application to predicting nonlinear temperature dependence of elastic wave velocities of solid rocks. Based on the results, the third-order Padé thermoelastic model outperformed the conventional third-order Taylor thermoelastic model in characterizing and accurately predicting thermally induced velocity changes in solid rocks at high temperatures. In a statement to Advances in Engineering, Professor Li-Yun Fu stated that Padé approximation is a promising universal model for studying thermally-induced velocity changes for polycrystals and solid rocks.