**6. Conclusions**

28 Wave Processes in Classical and New Solids

sandstone [49] as *KK c d s* (1 ) /(1 )

instruments. We will leave the issue for a future consideration.

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

discussed in section 2.

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

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

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

, where *Kd* , *Ks* and

are dry rock modulus, grain

The nonlinear acoustic wave propagation equations are derived in this paper. For hydrostatic and uniaxial loading configurations of rock tests, the analytical deformationdependent P- and S- speed expressions are presented. We discuss the reductions from current theoretical approach to the traditional works of acoustoelasticity in pure solid material. Neglecting solid/fluid differences both in a large static part of strain induced by confining pressure and in a small dynamic part of strain induced by wave's propagation and vibration, the velocity equations in this paper (equations 21, 26, 31, 34, 38, 41 & 44) are completely compatible with the expressions of acoustoelasticity theory in pure solid material (equations 1a~g).

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 31

fluid distributions will induce local fluid flow if elastic waves squeeze the porous rock, which surely will effect on the waves' propagation velocities and intrinsic attenuation [23, 36, 50, 52]. However, to include the mechanism of local fluid flow in a poro-acoustoelasticity theory will produce too complicated expressions to be handled on the present stage, and there probably will be several additional 2nd order terms [52, 53] and much more 3rd order terms in the strain energy expression. The present paper is limited to the area of applicability of the Biot approach, whereas rocks in nature usually contain the stiff and the soft pores. It may be important and essential that the local fluid flow mechanism should be analyzed in a poro-acoustoelasticity context, since the effects of confining pressure on pore shapes are also believed to be relevant to the local fluid flow and will consequently change the observed waves' velocities and attenuation in laboratory, which can be described in detail by

introducing additional 2nd and 3rd order terms. This extension will be a future study.

*Geophysical Department, Research Institute of Petroleum Exploration and Development,* 

*School of Geosciences, China University of Petroleum (East China), Qingdao, China* 

Discussions with Winkler K. W. were helpful. Wu X. Y. and Hao Z. B. helped in designing the experimental setup for a "closed-pore jacket" rock test and performing the measurements. This research was sponsored by the Major State Basic Research Development Program of China (973 Program, 2007CB209505), the CNPC 12-5 Basic Research Plan (2011A-3601) the Natural Science Foundation of China (41074087, 41104066) and the RIPED

[1] Truesdell C (1965) Continuum Mechanics IV: Problems of Non-linear Elasticity. Gordon

[2] Murnaghan F D (1937) Finite deformations of an elastic solid. Amer. J. Math., 59: 235-260. [3] Murnaghan F D (1951) Finite deformation of an elastic solid. John Wiley & Sons, Inc.,

[4] Hearmon R F S (1953) 'Third-order' elastic coefficients. Acta Crystallographica, 6: 331-

[5] Hughs D S, Kelly J L (1953) Second-order elastic deformation of solids. Phys. Rev., 92:

[6] Landau L D, Lifshitz E M (1959) Theory of Elasticity. Pergamon Oxford, London.

**Author details** 

Qizhen Du

Jing Ba and Hong Cao

*PetroChina, Beijing, China* 

**Acknowledgement** 

**7. References** 

New York.

1145-1149.

340.

& Breach, New York.

*Key Laboratory of Geophysics, PetroChina, Beijing, China* 

Youth Innovation Foundation (2010-A-26-01).

We design a virtual rock sample with seven assumed 3rd-order elastic constants. We perform four numerical tests on this virtual rock sample with different considerations: openpore jacketed sample under hydrostatic loading, open-pore jacketed sample under uniaxial loading, closed-pore jacketed sample under hydrostatic loading and closed-pore jacketed sample under uniaxial loading. Numerical results show that fast P wave's inverse quality factor is both sensitive to the confining pressure and wave centre frequency, which agrees with the laboratory observations of our former studies [47] in the aspect that P wave's inverse quality factors are more sensitive to pressure than S wave's. For the virtual rock sample, velocities of fast P waves and S waves seem to be more sensitive to pressure change than to changes in wave centre frequency, while slow P wave shows an opposite feature. In closed-pore jacketed uniaxial loading configuration, fast P wave's velocity in transverse direction is in decreasing trend with the rise of loading pressure, which is completely contrary to other three configurations. In all test configurations of this paper, S inverse quality factors share the same value which is independent to confining loading. S inverse quality factor can be influenced by wave centre frequency.

We perform the "open-pore jacketed" and "closed-pore jacketed" P-velocity measurements on a sandstone sample under hydrostatic loading. The observed P-velocity differences between the "open-pore jacketed" and "closed-pore jacketed" tests are getting larger as the confining pressure is increased in the range of 10~35 MPa, which agrees well with theoretical prediction, while in a higher confining loading range (>40MPa), the tested "openpore jacketed" velocity approaches to the "closed-pore jacketed" velocity. When hydrostatic confining pressure is increased and pore pressure keeps constant in an "open-pore" experiment, fluid will continue to flow out and microscopic connections between pores tend to close. When effective pressure reaches a high level, the rock in an "open-pore" test will approach to the theoretical configuration of the "closed-pore" system, where the pore fluid will actually stop flowing out under loading. The basic assumptions of the theoretical "open-pore jacketed" configuration is violated. A rock under high effective pressure in an "open-pore jacketed" test is actually "closed". In a high confining loading, the crack-like fraction will become gradually closed while the visible pores remain almost affected. Mostly the equant pores remain survived for which the compressibility weakly depend on the presence of the fluid (does not matter "closed-" or "open-pore"). The theory presented in this paper can give reasonable predictions for wave speeds in a confining pressure range lower than 50 MPa.

Moreover, in this work, the mechanism of "local fluid flow" have been neglected when we theoretically derive the nonlinear wave equations. The heterogeneity in pore structures and fluid distributions will induce local fluid flow if elastic waves squeeze the porous rock, which surely will effect on the waves' propagation velocities and intrinsic attenuation [23, 36, 50, 52]. However, to include the mechanism of local fluid flow in a poro-acoustoelasticity theory will produce too complicated expressions to be handled on the present stage, and there probably will be several additional 2nd order terms [52, 53] and much more 3rd order terms in the strain energy expression. The present paper is limited to the area of applicability of the Biot approach, whereas rocks in nature usually contain the stiff and the soft pores. It may be important and essential that the local fluid flow mechanism should be analyzed in a poro-acoustoelasticity context, since the effects of confining pressure on pore shapes are also believed to be relevant to the local fluid flow and will consequently change the observed waves' velocities and attenuation in laboratory, which can be described in detail by introducing additional 2nd and 3rd order terms. This extension will be a future study.
