**4.1. Wave speeds for sample under hydrostatic loading**

If the rock sample is open-pore jacketed and subject to a hydrostatic pressure *Ph* , and the fluid pressure is hold on one atmosphere (can be zero in approximation), we have the Biot's constitutive relations as 2 <sup>3</sup> *P Ae Ne Q <sup>h</sup>* and <sup>0</sup> *Qe R* . In the virtual tests, the hydrostatic pressure changes from 0 to 30MPa. The solid and fluid's strains are directly solved. By substituting *e* / 3 and / 3 into equation (19) and equation (24), openpore jacketed wave speeds are calculated.

For closed-pore jacketed rock sample in hydrostatic loading, the Biot's constitutive relations come to 2 <sup>3</sup> *P Ae Ne Q <sup>s</sup>* , *P Qe R <sup>f</sup>* , *P PP h s <sup>f</sup>* and / *P K f f* . Hydrostatic loading changes from 0 to 30MPa. By solving solid and fluid's strain and substituting into equation (19) and equation (24), "closed pore" wave speeds are calculated.

The waves center frequency ranges from 100 to 1000 KHz. The numerical results for openand closed-pore jacketed rock sample under hydrostatic loading are shown in figures 1~2.

Figure 1 shows the relations between wave velocity and attenuation and confining pressure. Fast P wave velocity, Fast P wave inverse quality factor and S wave velocity are sensitive to the change in hydrostatic pressure. Fast P wave attenuation is significantly increased in loading process. Slow P wave's velocity in open-pore jacketed test reaches a maximum value around 13MPa. Moreover, S wave attenuation is entirely not affected by hydrostatic


pressure (it holds on <sup>4</sup> 2.1569 10 at 1MHz), and slow P attenuation has very small change in relation to loading, therefore both have not been drawn in figure 1.

**Table 5.** The coefficients of the virtual rock sample.

20 Wave Processes in Classical and New Solids

The reduction operation leads to

which is identical to equation (1g).

 

 

**4. Numerical tests for poro-acoustoelasticity** 

**4.1. Wave speeds for sample under hydrostatic loading** 

*e* / 3 and

constants are assumed for numerical tests.

constitutive relations as

pore jacketed wave speeds are calculated.

 2 <sup>3</sup> *P Ae Ne Q <sup>s</sup>* ,

solved. By substituting

come to

 2 2 <sup>6</sup> <sup>2</sup> 1 8 1 8 49 <sup>1</sup> 1 33 ( ) (2 ) ( ) 4 24 4 22 2 4 *M M <sup>M</sup> <sup>M</sup> M M M M MM* .

> 

 <sup>2</sup> 2 2 <sup>6</sup> 134 8 9 <sup>1</sup> [( )( 2 ) ( ) ( )( 2 )] <sup>4</sup> 2 44 *M M <sup>M</sup> V MMM M M* (44)

Equations (19, 24, 29, 33, 37, 40, 43) are the central result of this paper, which are applicable

The elastic and density coefficients of a virtual water-saturated porous rock sample are listed in table 5. The 2nd-order elastic constants and the density coefficients come from the actual sample of water-saturated Coldlake sandstone [45]. The seven 3rd-order elastic

If the rock sample is open-pore jacketed and subject to a hydrostatic pressure *Ph* , and the fluid pressure is hold on one atmosphere (can be zero in approximation), we have the Biot's

hydrostatic pressure changes from 0 to 30MPa. The solid and fluid's strains are directly

For closed-pore jacketed rock sample in hydrostatic loading, the Biot's constitutive relations

loading changes from 0 to 30MPa. By solving solid and fluid's strain and substituting into

The waves center frequency ranges from 100 to 1000 KHz. The numerical results for openand closed-pore jacketed rock sample under hydrostatic loading are shown in figures 1~2.

Figure 1 shows the relations between wave velocity and attenuation and confining pressure. Fast P wave velocity, Fast P wave inverse quality factor and S wave velocity are sensitive to the change in hydrostatic pressure. Fast P wave attenuation is significantly increased in loading process. Slow P wave's velocity in open-pore jacketed test reaches a maximum value around 13MPa. Moreover, S wave attenuation is entirely not affected by hydrostatic

*P Qe R <sup>f</sup>* , *P PP h s <sup>f</sup>* and /

<sup>3</sup> *P Ae Ne Q <sup>h</sup>* and <sup>0</sup> *Qe R*

2

 

equation (19) and equation (24), "closed pore" wave speeds are calculated.

 

/ 3 into equation (19) and equation (24), open-

 

 

. In the virtual tests, the

*P K f f* . Hydrostatic

where

in rock tests.

Comparing with open-pore jacketed test, closed-pore jacketed test has lower fast P velocity (2380~2580m/s) and higher S and slow P velocity (1177~1284m/s and 52.5~61.7m/s respectively). Fast P inverse quality factor changes from 0.5 <sup>6</sup> 10 to 4.6 <sup>6</sup> 10 , which are much lower than open-pore jacketed tests (0.5 <sup>6</sup> 10 ~4.5 <sup>5</sup> 10 ). The sign of the predicted 1 /*Q* is positive. The 1 /*Q* value is below <sup>4</sup> 10 , which agrees with the description of dissipation of traditional Biot theory [17, 18] and is obviously lower than the level of the attenuation which can be caused by local fluid flow mechanism. Local fluid flow is not considered in this study, and according to Dvorkin et al. 1994 [46], the magnitude of 1 /*Q* produced by local fluid flow may be 1~2 orders magnitude higher than the one produced by Biot friction. Pressure is mainly confined on solid skeleton in open-pore jacketed configuration, so solid's nonlinear acoustic wave feature dominates the whole rock's speeds. Pore water is more likely to be compressed in closed-pore jacketed test and its finite deformation undertakes part of pressure, therefore water's acoustoelasticity should be considered for closed-pore jacketed case. Actually, water's velocity- pressure slope is much lower than solid, therefore, closed-pore jacketed test will have lower fast P velocity than open-pore jacketed test since part of pressure is consumed by pore water instead of by solid matrix. The higher slow P velocity and lower inverse quality factors physically means solid/fluid coupling effect is strengthened, while Biot dissipation is weakened.

Velocities and inverse quality factors *vs* pressure and attenuation are drawn in figure 2. For both cases, it is obvious that fast P wave and S wave are more sensitive to the changes in confining pressure than the changes in wave frequencies, while slow P waves are opposite. As to inverse quality factors, only fast P wave is sensitive to both the confining pressure and

the wave frequency. It reaches peak value at the highest center frequency and the largest pressure. S wave's inverse quality factor slightly increases when center frequency increases and shows no response to the loading variations.

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 23

**Figure 2.** The velocities and inverse quality factors *vs* pressure and centre frequency in hydrostatic loading (red faces for open-pore jacketed, blue faces for closed-pore jacketed). (a) Fast P wave velocity, (b) Fast P wave inverse quality factor, (c) S wave velocity, (d) S wave inverse quality factor, (e) Slow P

Mechanisms of "local fluid flow" have not been taken into account in this work. The pores in the rock are assumed to be equant when we derive the nonlinear acoustical wave equations for solid/fluid composite, which is also a basic assumption in Biot's theory [35]. In real rocks which contain both equant pores and soft pores, the contributions of wave-induced local fluid flow may be important for the explanation of the observed attenuation in rock tests. The produced attenuation in this study is mainly associated to the traditional Biot friction

wave velocity, (f) Slow P wave inverse quality factor.

**Figure 1.** The results in hydrostatic loading with central frequency at 1M Hz. (a) Fast P wave velocity (dashed line for open-pore jacketed, solid line for closed-pore jacketed), (b) Fast P wave inverse quality factor, (c) S wave velocity, (d) Slow P wave velocity.

Comparing open-pore jacketed results with closed-pore jacketed results shows that S wave inverse quality factors of the two cases are entirely equal and not affected by loading. Slow P inverse quality factors of the two cases have very small difference. S and slow P velocities of closed-pore jacketed test are always slightly higher than open-pore jacketed test. Fast P velocity of open-pore jacketed test is significantly higher than closed-pore jacketed test as confining pressure increases. Fast P inverse quality factor in open-pore jacketed test is lower than closed-pore jacketed test when confining pressure is below 5MPa, but gets much higher than closed-pore jacketed test when pressure reaches 30MPa.

and shows no response to the loading variations.

factor, (c) S wave velocity, (d) Slow P wave velocity.

than closed-pore jacketed test when pressure reaches 30MPa.

the wave frequency. It reaches peak value at the highest center frequency and the largest pressure. S wave's inverse quality factor slightly increases when center frequency increases

**Figure 1.** The results in hydrostatic loading with central frequency at 1M Hz. (a) Fast P wave velocity (dashed line for open-pore jacketed, solid line for closed-pore jacketed), (b) Fast P wave inverse quality

Comparing open-pore jacketed results with closed-pore jacketed results shows that S wave inverse quality factors of the two cases are entirely equal and not affected by loading. Slow P inverse quality factors of the two cases have very small difference. S and slow P velocities of closed-pore jacketed test are always slightly higher than open-pore jacketed test. Fast P velocity of open-pore jacketed test is significantly higher than closed-pore jacketed test as confining pressure increases. Fast P inverse quality factor in open-pore jacketed test is lower than closed-pore jacketed test when confining pressure is below 5MPa, but gets much higher

**Figure 2.** The velocities and inverse quality factors *vs* pressure and centre frequency in hydrostatic loading (red faces for open-pore jacketed, blue faces for closed-pore jacketed). (a) Fast P wave velocity, (b) Fast P wave inverse quality factor, (c) S wave velocity, (d) S wave inverse quality factor, (e) Slow P wave velocity, (f) Slow P wave inverse quality factor.

Mechanisms of "local fluid flow" have not been taken into account in this work. The pores in the rock are assumed to be equant when we derive the nonlinear acoustical wave equations for solid/fluid composite, which is also a basic assumption in Biot's theory [35]. In real rocks which contain both equant pores and soft pores, the contributions of wave-induced local fluid flow may be important for the explanation of the observed attenuation in rock tests. The produced attenuation in this study is mainly associated to the traditional Biot friction and may be underestimated in comparison with an actual laboratory measurement. In the limitation of Biot's theory, the prediction of dissipation can be appropriate only in the case that local fluid flow can be neglected.

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 25

**Figure 3.** The open-pore jacketed results under uniaxial loading with central frequency at 1M Hz. (a) Fast P wave velocity (dashed line for along loading, solid line for perpendicular to loading), (b) Fast P

wave inverse quality factor, (c) Slow P wave velocity, (d) S wave velocity (direction 1: *Pu* propagating, *Pu* vibrating; direction 2: *Pu* propagating, *Pu* vibrating; direction 3: *Pu*

**4.3. Wave speeds for closed-pore jacketed sample under uniaxial loading** 

directions. The six poroelastic constitutive relations are

<sup>v</sup> <sup>22</sup> 2 *P Ae Ne Q <sup>s</sup>* , 0 *P P sv <sup>f</sup>* and /

If water-saturated rock sample is closed-pore jacketed and subject to uniaxial pressure. In process of loading, because compressed pore water is not allowed to flow out from rubber jacket, the fluid phase will impose an extra stress increment on solid matrix in transverse

respectively denote solid stress components along the loading direction and perpendicular

Numerical results for P waves are figured in figures 4a~4c. First particular feature in this configuration is that fast P speed in vertical direction decrease slightly ( from 2385 to 2345 m/s ) when confining pressure increases, while all results in former three configurations show opposite trends. Fast P velocity changes from 2380 to 2660m/s along loading, slightly

<sup>11</sup> 2 *P Ae Ne Q su* ,

*P K f f* , where *Psu* and *Ps*<sup>v</sup>

 *P Qe R <sup>f</sup>* ,

propagating, *Pu* vibrating).

*PP P u su <sup>f</sup>* ,

to loading.
