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

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

Based on the poro-acoustoelasticity theory, 11 elastic constants have to be determined if we want to predict the velocity-pressure relationships in actual rock. The four 2nd order elastic constants can be theoretically calculated with a Biot-Gassmann theory [48]. The dry bulk modulus can be estimated through Pride's frame moduli expression for consolidated sandstone [49] as *KK c d s* (1 ) /(1 ) , where *Kd* , *Ks* and are dry rock modulus, grain bulk modulus and porosity. *c* are the consolidation parameter. We choose *c* 8 since it is a moderate porosity and good consolidated sandstone. According to Table 3, the 4 2nd order elastic constants can be calculated as *M*<sup>1</sup> 37.4*GPa* , *M*<sup>2</sup> -53.6*GPa* , *M*<sup>3</sup> 0.273*GPa* and *M*<sup>4</sup> 1.84*GPa* . As to the 7 3rd-order elastic constants, more prior information about each component's higher-order elastic modulus should be fully realized if we want to give an applicable theoretical estimate. Moreover, some uniaxial loading tests and S velocity measurements should also be performed in the "open/closed-pore jacketed" configurations, so that all the 7 3rd-order elastic constants can be precisely determined and discussed. Nevertheless, these tests also require further improvements on our current experimental instruments. We will leave the issue for a future consideration.

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 29

Based on the numerical approach in Section 4, we perform numerical calculation and draw the velocity-pressure curves in figure 5. They are both in good agreement with open-pore and closed-pore measurements in a low effective pressure range. We choose *M*<sup>5</sup> 1800*GPa* , *M*6= 2500 *GPa* , *M*8=2300*GPa* and *M*10=100*GPa* in calculation. *M*<sup>5</sup> , *M*6 and *M*8 are mostly associated with the solid skeleton's higher order elastic characteristics. The order of magnitude of these constants (KGPa) agrees well with the measurement of 3rd-order elastic constants on rocks performed by Winkler et. al. [15, 16]. As a complement to the pure solid's 3rd-order elastic constant theory, the new 3rd elastic constant *M*10 is related to the higherorder coupling effect between solid and fluid. Its order of magnitude is around 100GPa. It effects on the wave speed expressions for fluid-saturated rocks, and holds a lower magnitude than *M*<sup>5</sup> , *M*6 and *M*<sup>8</sup> , which has been discussed in Section 2. A higher value of

The observed differences between "open-pore jacketed" and "closed-pore jacketed" tests decrease as we still increase the confining pressure in a high pressure range (higher than 40 MPa), which do not agree with our theoretical analysis. Actually, the "open-pore jacketed" measured speeds approach to the "closed-pore jacketed" results in high pressure end. In high confining pressure range, the theoretical results fit well with the "closed-pore" measurement, while give an estimate higher than the actually measured speeds in the "open-pore" tests. As confining pressure is getting higher for an open-pore system, the stress difference between solid skeleton and pore fluid is consequently increased. The effective pressure is also increased. Pore fluid continues to flow out and the soft pores such as grain cracks will tend to close. In a high effective pressure range, the pore system of rock in a "open-pore" test will tend to act like a theoretical "closed-pore" system, where soft pores are closed and fluid stops flowing out. The basic assumption of a theoretical "open-pore"

In a real sandstone, at higher pressure (above 50 MPa), the increase in the values of wave velocities essentially saturates [50, 51]. This saturation effect is caused exclusively by gradual closing of the soft crack-like fraction of the rock-frame porosity. This crack-like fraction strongly dominates the nonlinearity of the material in the pressure range of several tens MPa, whereas visible pores are less affected. The compressibility of crack-like pores strongly depends on the fact whether the filling fluid is confined inside or can be squeezed out. Thus below 50MPa the difference between "open-pore jacketed" and "closed-pore jacketed" tests is obvious until the most part of the cracks become closed. The application of this theory is limited to the pressure range lower than 50MPa. Especially for the "open-pore jacketed" case, the theoretical predictions will be unreliable when the effective pressure

The nonlinear acoustic wave propagation equations are derived in this paper. For hydrostatic and uniaxial loading configurations of rock tests, the analytical deformation-

*M*10 will produce unreasonable results for wave speeds.

configuration breaks down in this case.

exceeds the limitation.

**6. Conclusions** 

**Figure 5.** Comparison between experimental data and theoretical curves.

In this paper, we use 4 3rd order elastic constants ( *M*<sup>5</sup> , *M*<sup>6</sup> , *M*8 and *M*<sup>10</sup> ) to fit the measured results in the open-pore and closed-pore jacketed tests, and neglect other 3 terms, since these four terms are considered as the most important for actual rocks as has been discussed in section 2.

Based on the numerical approach in Section 4, we perform numerical calculation and draw the velocity-pressure curves in figure 5. They are both in good agreement with open-pore and closed-pore measurements in a low effective pressure range. We choose *M*<sup>5</sup> 1800*GPa* , *M*6= 2500 *GPa* , *M*8=2300*GPa* and *M*10=100*GPa* in calculation. *M*<sup>5</sup> , *M*6 and *M*8 are mostly associated with the solid skeleton's higher order elastic characteristics. The order of magnitude of these constants (KGPa) agrees well with the measurement of 3rd-order elastic constants on rocks performed by Winkler et. al. [15, 16]. As a complement to the pure solid's 3rd-order elastic constant theory, the new 3rd elastic constant *M*10 is related to the higherorder coupling effect between solid and fluid. Its order of magnitude is around 100GPa. It effects on the wave speed expressions for fluid-saturated rocks, and holds a lower magnitude than *M*<sup>5</sup> , *M*6 and *M*<sup>8</sup> , which has been discussed in Section 2. A higher value of *M*10 will produce unreasonable results for wave speeds.

The observed differences between "open-pore jacketed" and "closed-pore jacketed" tests decrease as we still increase the confining pressure in a high pressure range (higher than 40 MPa), which do not agree with our theoretical analysis. Actually, the "open-pore jacketed" measured speeds approach to the "closed-pore jacketed" results in high pressure end. In high confining pressure range, the theoretical results fit well with the "closed-pore" measurement, while give an estimate higher than the actually measured speeds in the "open-pore" tests. As confining pressure is getting higher for an open-pore system, the stress difference between solid skeleton and pore fluid is consequently increased. The effective pressure is also increased. Pore fluid continues to flow out and the soft pores such as grain cracks will tend to close. In a high effective pressure range, the pore system of rock in a "open-pore" test will tend to act like a theoretical "closed-pore" system, where soft pores are closed and fluid stops flowing out. The basic assumption of a theoretical "open-pore" configuration breaks down in this case.

In a real sandstone, at higher pressure (above 50 MPa), the increase in the values of wave velocities essentially saturates [50, 51]. This saturation effect is caused exclusively by gradual closing of the soft crack-like fraction of the rock-frame porosity. This crack-like fraction strongly dominates the nonlinearity of the material in the pressure range of several tens MPa, whereas visible pores are less affected. The compressibility of crack-like pores strongly depends on the fact whether the filling fluid is confined inside or can be squeezed out. Thus below 50MPa the difference between "open-pore jacketed" and "closed-pore jacketed" tests is obvious until the most part of the cracks become closed. The application of this theory is limited to the pressure range lower than 50MPa. Especially for the "open-pore jacketed" case, the theoretical predictions will be unreliable when the effective pressure exceeds the limitation.
