**4.2. Wave speeds for open-pore jacketed sample under uniaxial loading**

In this subsection a virtual test is analyzed with uniaxial loading configuration. The uniaxial pressure is *Pu* . The fluid pressure is hold around zero. Biot's constitutive relations are written as <sup>11</sup> 2 *P Ae Ne Q <sup>u</sup>* , <sup>22</sup> 0 2 *Ae Ne Q* , and 0 *Qe R* . Let uniaxial pressure change from 0 to 30MPa, <sup>11</sup> *e* , <sup>22</sup> *e* and are solved out. Substitute <sup>11</sup> *e* , <sup>22</sup> *e* and / 3 into equations (29, 33, 37, 40, 43). Wave center frequency changes from 100 to 1000 KHz.

Figure 3a shows that fast P velocity changes from 2380m/s to 2690m/s along loading direction and changes from 2383m/s to 2389m/s perpendicular to loading direction, while figure 3c shows slow P wave velocity increases slightly in transverse direction and decreases slightly in parallel direction in the loading process. On the same loading level, fast P inverse quality factor is higher in parallel direction than in transverse direction when loading pressure exceeds 5MPa, and both are much lower than in hydrostatic test. The 1 /*Q vs* pressure curve seems to have a trough around 6MPa along loading and around 8MPa perpendicular to loading. The sign of the predicted 1 /*Q* is positive and the value is below <sup>4</sup> 10 .

Numerical results for S waves are shown in figure 3d. All S velocities in three directions are sensitive to the changes of confining pressure. S wave transmitting along loading direction has the highest speed comparing with two other directions, changing from 1177m/s to 1228m/s. S wave whose propagating and vibrating direction are both perpendicular to loading has relatively lower speed changing from 1177m/s to 1224m/s. S wave vibrating along uniaxial loading has the lowest speed changing from 1177m/s to 1184m/s. All three S velocities in uniaxial configuration are lower than the open-pore jacketed hydrostatic test, where S speed changes from 1177m/s to 1280m/s. For all the three directions in uniaxial test and for the unique direction in hydrostatic test, S waves share the same inverse quality factor independent to the pressure. It only changes with wave centre frequency.

As has been shown in figure 1b and figure 3b, the predicted attenuation increases with confining pressure in average, which disagrees with the experimental results in [47]. Local fluid flow, which dominates the wave attenuation and dissipation phenomena in the rocks under low confining pressure, is depressed since the flat throats and soft pores tend to close under higher confining pressure, so that the attenuation in actual experiment generally decreases with effective stress. The mechanism of local fluid flow and its relation to the confining pressure has not been analyzed in this study. The numerical results are reasonable in condition that the rock pores are assumed to be equant and the local fluid flow mechanism can be neglected.

written as

 

<sup>22</sup> *e* and

frequency.

mechanism can be neglected.

from 100 to 1000 KHz.

that local fluid flow can be neglected.

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

In this subsection a virtual test is analyzed with uniaxial loading configuration. The uniaxial pressure is *Pu* . The fluid pressure is hold around zero. Biot's constitutive relations are

Figure 3a shows that fast P velocity changes from 2380m/s to 2690m/s along loading direction and changes from 2383m/s to 2389m/s perpendicular to loading direction, while figure 3c shows slow P wave velocity increases slightly in transverse direction and decreases slightly in parallel direction in the loading process. On the same loading level, fast P inverse quality factor is higher in parallel direction than in transverse direction when loading pressure exceeds 5MPa, and both are much lower than in hydrostatic test. The 1 /*Q vs* pressure curve seems to have a trough around 6MPa along loading and around 8MPa perpendicular to

Numerical results for S waves are shown in figure 3d. All S velocities in three directions are sensitive to the changes of confining pressure. S wave transmitting along loading direction has the highest speed comparing with two other directions, changing from 1177m/s to 1228m/s. S wave whose propagating and vibrating direction are both perpendicular to loading has relatively lower speed changing from 1177m/s to 1224m/s. S wave vibrating along uniaxial loading has the lowest speed changing from 1177m/s to 1184m/s. All three S velocities in uniaxial configuration are lower than the open-pore jacketed hydrostatic test, where S speed changes from 1177m/s to 1280m/s. For all the three directions in uniaxial test and for the unique direction in hydrostatic test, S waves share the same inverse quality factor independent to the pressure. It only changes with wave centre

As has been shown in figure 1b and figure 3b, the predicted attenuation increases with confining pressure in average, which disagrees with the experimental results in [47]. Local fluid flow, which dominates the wave attenuation and dissipation phenomena in the rocks under low confining pressure, is depressed since the flat throats and soft pores tend to close under higher confining pressure, so that the attenuation in actual experiment generally decreases with effective stress. The mechanism of local fluid flow and its relation to the confining pressure has not been analyzed in this study. The numerical results are reasonable in condition that the rock pores are assumed to be equant and the local fluid flow

<sup>22</sup> 0 2 *Ae Ne Q* , and 0 *Qe R*

/ 3 into equations (29, 33, 37, 40, 43). Wave center frequency changes

are solved out. Substitute

. Let uniaxial

<sup>11</sup> *e* ,

**4.2. Wave speeds for open-pore jacketed sample under uniaxial loading** 

loading. The sign of the predicted 1 /*Q* is positive and the value is below <sup>4</sup> 10 .

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

pressure change from 0 to 30MPa, <sup>11</sup> *e* , <sup>22</sup> *e* and

**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* propagating, *Pu* vibrating).

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

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 directions. The six poroelastic constitutive relations are <sup>11</sup> 2 *P Ae Ne Q su* , *P Qe R <sup>f</sup>* , *PP P u su <sup>f</sup>* , <sup>v</sup> <sup>22</sup> 2 *P Ae Ne Q <sup>s</sup>* , 0 *P P sv <sup>f</sup>* and / *P K f f* , where *Psu* and *Ps*<sup>v</sup> respectively denote solid stress components along the loading direction and perpendicular to loading.

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

lower than the open-pore jacketed uniaxial test. As is shown in figure 4b, another particular feature is that fast P inverse quality factors along uniaxial loading decreases with rise of pressure. Fast P inverse quality factor in vertical direction increases with loading and is much higher than in parallel direction ( 0.5~5.5 <sup>6</sup> 10 *vs* 0.5~3.5 <sup>6</sup> 10 ), while in open-pore jacketed test fast P inverse quality factor in vertical direction is lower than in parallel direction ( 5.2~2.4 <sup>7</sup> 10 *vs* 0.5~3.2 <sup>6</sup> 10 ).

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 27

jacketed uniaxial tests. In all test configurations of this paper, S inverse quality factors seem

Two hydrostatic loading tests (one for "open-pore jacketed" configuration, another for "closed-pore jacketed" configuration) are performed on a sandstone sample with moderate porosity (13.26 percent) and low permeability (1.21mD). The sample is collected from a gas reservoir in southwest China. It is from the depth of around 2000 meters from surface. The sandstone sample is mainly constructed by quartz and feldspar. Minor clays and rock fragments reside inside pores and grains. It is moderately sorted. The grain size ranges from below 0.1mm to around 1 mm. The pore size ranges from below 0.1 mm to around 0.4 mm. Most grains contact well, so as to form a rigid solid skeleton. The average grain density is

The experimental setup consists of a digital oscilloscope and a pulse generator. In the test, the rock sample is jacketed with a rubber tubing to isolate it from the confining pressure. The receiving transducer is connected to the digitizing board in the PC through a signal amplifier. A pore fluid inlet in the endplate allows passage of pore fluid through the sample and can help to control the pore pressure inside rocks in experiments. We give a small modification on the original inlet instrument by adding a valve (it is closely connected to the inlet), so that the "closed-pore jacketed" configuration can be realized. The open-pore and closed-pore tests are performed respectively. In each test, the rock is full saturated with water, and both confining pressure and pore pressure is raised to 10 MPa before the P-wave speed measurements. In the "closed-pore jacketed" test, we keep the valve closed and raises the confining pressure from 10 to 62 MPa with an interval of around 4 MPa. The P-wave speed is measured and recorded in each pressure level. In the "open-pore jacketed" test, the valve is kept open in measuring process. The pore pressure is close to atmospheric pressure when confining pressure is not so high (it can be neglected comparing with the confining

The relationship between the measured velocities of P waves and the confining pressure in the "open-pore jacketed" and "closed-pore jacketed" tests are shown in figure 5. In the both cases, measured P velocity increases as the confining pressure gets higher. Comparing with the closed-pore test, the velocity-pressure relationship in the open-pore test has a higher slope in a range of 10-35 MPa if it is fitted with a liner correlation, and the difference of measured velocities between open-pore and closed-pore tests increases as the confining pressure rises. However, when confining pressure increases in a higher range of 40-62 MPa, this difference decreases as the confining pressure rises. The observed velocity-pressure

to share the same value which is only slightly dependent to wave centre frequency.

**5. Experimental data** 

2.659g/cm3. The grain bulk modulus is 39.0 GPa.

**5.2. Discussions on the theory and experimental data** 

results of open-pore test will approach to the results of closed-pore test.

**5.1. Physical setup** 

loading).

**Figure 4.** The closed-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* propagating, *Pu* vibrating).

Numerical results for S waves in closed-pore jacketed sample under uniaxial loading are shown in figure 4d. S wave transmitting along loading has the highest speed ranging from 1177m/s to 1231m/s. S velocities in two vertical directions respectively range from 1177m/s to 1225m/s and from 1177m/s to 1186m/s. S-velocities in all three directions increase as loading rises. S velocities of closed-pore jacketed uniaxial test are very close to open-pore jacketed uniaxial tests. In all test configurations of this paper, S inverse quality factors seem to share the same value which is only slightly dependent to wave centre frequency.
