Simulating quartz resonators at high temperature and pressure: Limitations regarding lack of temperature derivatives of third-order elastic coefficients

It is known that the currently available anisotropic material properties of quartz allow for finite element simulations that closely match experimental resonator frequency response with respect to both changes in pressure near ambient temperatures and changes in temperature near ambient pressures. However, as observed in the current work, such models can be shown to deviate from experimental frequency values when high temperatures and pressures are applied simultaneously to the resonator. A three-dimensional finite element model was developed based on the linear field equations for superposed small vibrations onto nonlinear thermoelastic stressed media given by Lee and Yong [1]. The frequency response of the model was then benchmarked to experimental data from a commercially available quartz pressure sensor with temperature ranging from 50°C to 200°C and pressure from 14 psi to 20,000 psi. Such conditions directly correspond to current uses of quartz resonators as temperature and pressure sensors for the oil and gas industry. The normalized frequency response to the change in external pressure matched very well with experimental data for lower temperatures, having a maximum deviation of only 7.5% at 20,000 psi, when assuming constant 50°C temperature. However, the same deviation grew to about 25.7% at 20,000 psi assuming a higher 200°C constant temperature. Similarly, the temperature-frequency response from 50°C to 200°C matched the experimental trend well for lower pressures, but this agreement deteriorated as pressure increased. It is hypothesized that changes in the nonlinear elastic coefficients with temperature yield the primary source of error at high combined temperature and pressure. These changes, which are quantified in the temperature derivatives of the third-order elastic coefficients, are not currently available in literature and thus are not supported by the model.