Effect of Stress on Wave Propagation in Fluid-Saturated Porous Thermoelastic Media

Abstract

The effect of stress on wave propagation in fluid-saturated porous thermoelastic media is poorly understood. To fill this gap, we propose the dynamical equations for stressed fluid-saturated porous thermoelastic media based on the poroacoustoelasticity model and porothermoelasticity model to describe the effect of stress on the wave dispersion and attenuation. A plane-wave analysis for dynamical equations formulates stress-dependent velocities of five wave propagation modes, including three longitudinal (P) waves, namely fast P wave, slow P wave and thermal (T) wave, and two shear (S) waves, namely fast S wave and slow S wave. Additional slow P wave and T wave arise due to the Biot and thermal loss mechanisms in porothermoelastic media. The stress-induced rock anisotropy accounts for the S wave splitting phenomenon. Modelling results show that energy dissipations of fast P wave and T wave are induced by the coupling between Biot and thermal loss mechanisms, while the fast and slow S waves, slow P wave are only affected by Biot loss mechanism. The rock permeability and fluid viscosity are mainly related to Biot mechanism, while the thermal conductivity and thermal expansion coefficient for solid phase are related to Biot and thermal mechanisms. In addition, the triaxial stress and confining stress have remarkable effects on the wave velocities as well as attenuation peaks. The predicted wave velocities in water-saturated sandstone and granite behave a reasonable agreement with the laboratory measurements. Our results help to provide better understanding of wave propagation in high-stress high-temperature fields.

Article Highlights

• We propose the dynamical equations for fluid-saturated porous thermoelastic media with the effect of stress.

• Our model predicts five wave propagation modes, namely fast P wave, slow P wave, thermal wave, fast S wave and slow S wave.

• Biot and Thermal loss mechanisms are coupled to describe the stress-dependent dispersions and attenuations for these five wave modes.

Appendices

Appendix A: Stress- and temperature-dependent coefficients

From Eqs. 17 and 18, for the case that the directions of applied stress coincide the principle coordinate axes, and the wave propagates along axis $X_{3}$, the coefficients in Eq. 37 can be given by

$$\Xi_{1313} = \mu_{b} + \left( {\lambda_{b} + \alpha^{2} M + \nu^{\prime}_{2} } \right)e^{s} + 2\left( {\mu_{b} + \nu^{\prime}_{3} } \right)\left( {e_{11}^{s} + e_{33}^{s} } \right) - \left( {\alpha M + {\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\zeta^{s} ,$$

(49)

$$\Xi_{2323} = \mu_{b} + \left( {\lambda_{b} + \alpha^{2} M + \nu^{\prime}_{2} } \right)e^{s} + 2\left( {\mu_{b} + \nu^{\prime}_{3} } \right)\left( {e_{22}^{s} + e_{33}^{s} } \right) - \left( {\alpha M + {\gamma \mathord{\left/ {\vphantom {\gamma 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\zeta^{s} ,$$

(50)

$$\begin{gathered} \Xi_{3333} = \lambda_{b} + 2\mu_{b} + \alpha^{2} M + \left( {\lambda_{b} + \alpha^{2} M + \nu^{\prime}_{1} + 2\nu^{\prime}_{2} } \right)e^{s} \hfill \\ \, + 2\left( {\lambda_{b} + \alpha^{2} M + 3\mu_{b} + 2\nu^{\prime}_{2} + 4\nu^{\prime}_{3} } \right)e_{33}^{s} - \left( {\alpha M - 2\gamma_{2} } \right)\zeta^{s} - \beta_{m} \theta^{s} , \hfill \\ \end{gathered}$$

(51)

$$\Omega_{1} = - \beta_{m} \left( {3 + e^{s} } \right)\theta^{s} ,$$

(52)

$$\Omega_{2} = \beta_{f} \left( {1 + \zeta^{s} } \right),$$

(53)

$$\Upsilon_{33} = \alpha M + \left( {\alpha M + \gamma } \right)e_{33}^{s} - \left( {2\gamma_{2} + \gamma } \right)e^{s} - \left( {2\gamma_{3} + \alpha M} \right)\zeta^{s} ,$$

(54)

$$\Upsilon = M + \left( {2\gamma_{3} + \alpha M} \right)e^{s} - 3\left( {M - 2\gamma_{1} } \right)\zeta^{s} + \beta_{f} \theta^{s} ,$$

(55)

The initial strain tensors for fluid phase and solid phase $e_{ij}^{s}$ and $\zeta^{s}$ can be computed with Eqs. 19 and 20, which comprehensively reflect the magnitudes of applied stresses.

Appendix B: Stress- and temperature-dependent fluid properties

We model the stress- and temperature-dependent fluid (water) properties, such as density, velocity and viscosity, with the BW model (Batzle and Wang 1992). The water density is given by

$$\begin{aligned} \rho_{w} & = 1 + 1 \times 10^{ - 6} \left( { - 80\theta_{t} - 3.3\theta_{t}^{2} + 0.00175\theta_{t}^{3} + 489p_{f} - 2\theta_{t} p_{f} + 0.016\theta_{t}^{2} p_{f} } \right) \\ & + 1 \times 10^{ - 6} \left( { - 1.3 \times 10^{ - 5} \theta_{t}^{3} p_{f} - 0.333p_{f}^{2} - 0.002\theta_{t} p_{f}^{2} } \right), \\ \end{aligned}$$

(56)

where $\rho_{w}$ is water density in g/cm³ at normal pressure and temperature (15.6 °C and atmospheric pressure). $p_{f}$ is pore pressure. $\theta_{t}$ is centigrade temperature and $\theta_{t} = \theta - 273$ where $\theta$ is the corresponding absolute temperature.

The P-wave velocity in the water is

$$V_{w} = \mathop \sum \limits_{i = 0}^{4} \mathop \sum \limits_{j = 0}^{3} w_{ij} \theta_{t}^{i} p_{f}^{j} ,$$

(57)

where constant matrix $w_{ij}$ is given in Table 1 in Batzle and Wang (1992).

For the case of temperature below 250 ℃, the water viscosity can be approximated by

$$\eta_{w} = 0.1 + 0.333x + \left( {1.65 + 91.9x^{3} } \right)e^{{ - \left[ {0.12\left( {x^{0.8} - 0.17} \right)^{2} + 0.045} \right]\theta_{t}^{0.8} }} ,$$

(58)

where $x$ is the weight fraction (ppm) of sodium chloride.

With Eqs. 56–58, the stress- and temperature-dependent density, P-wave velocity and viscosity of water can be computed, and with these properties, the bulk modulus of fluid can be further obtained.

Moreover, detailed laboratory and theoretical studies on the stress and temperature dependences of oil and gas properties can be found in Batzle and Wang (1992).