**1. Introduction**

The two elastic constants of bulk and shear moduli are insufficient to describe the nonlinear acoustic nature of solid materials under higher confining pressure. The theory of third-order elastic constants (nonlinear acoustics) has long been developed to analyze the velocity *vs* stress relationships in theoretical and experimental research for hyperelastic solid materials [1-7].

The 3rd order constants theory in solid material was completed with the papers by Toupin and Bernstein, Jones and Kobett and Truesdell [8-10]. Truesdell used four 3rd-order elastic constants in his general theory for isotropic solid. Brugger [11] gave the thermodynamic definition of higher order elastic constants. In 1973, Green reviewed 3rd-order constants measurements of various crystals, and gave the relations between the 3rd-order constant notations in isotropic solids by different authors [12].

From late 1980s to the present, 3rd-order nonlinear elasticity theory has been applied to rock experiments [13, 14]. For the special case of porous solid materials, Winkler et al. [15] measured third order elastic constants based on pure solid's acoustoelasticity theory in a variety of dry rocks and found that pure solid's third-order elasticity could successfully describe dry rock's velocity-stress relationships. A similar approach performed on watersaturated rocks showed that traditional 3rd-order elasticity theory for the isotropic solid could not fully describe the stress dependence of velocities in water-saturated rocks [16]. For fluid-saturated porous material, as the confining pressure increases, the unrelaxed fluid's effect must be taken into account when probing into the quantitative relationships between two-phase rock's velocities and confining pressures.

For the fluid/solid composite, Biot derived the two-phase wave equations on the basis of the linear elasticity, in which the coupling motion of solid and fluid was first analyzed. Based

© 2012 Ba et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

on the same linear assumption, some new expansions on Biot's theory have included local fluid flow, dynamic permeability and multi-scale heterogeneity in last two decades [17-23]. Around linear poroelasticity, the research interests of recent years are focused on the frequency-dependent P- and S- waves' velocity and attenuation features which are influenced by patchy saturation, pore distribution and rock microstructure.

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 5

In this paper, we derive the wave propagation equations by substituting the 11-term potential function and the Biot kinetic energy function into Lagrange equations. Finite strain is included, and the velocities and quality factors of P- and S-waves are analytically expressed as functions of components' strains, confining pressure, pore fluid pressure and rock's 2nd- and 3rd-order moduli in cases of hydrostatic and uniaxial loading conditions. The reduction of this theory to a pure solid's acoustoelasticity theory is discussed. Numerical examples for velocity and attenuation are given which are corresponding to the open-pore jacketed and closed-pore jacketed rock tests, respectively. In the last section, we perform ultrasonic P-velocity measurements in the "open-pore jacketed" and "close-pore jacketed" tests with hydrostatic loading. The comparisons between theory and rock

3rd-order elastic constants theory has long been developed to describe the wave propagation and mechanical deformation in pure solid material. The strain energy function for isotropic solid was given with three independent 3rd-order elastic constants (see Table1). For nonlinear waves propagating in an isotropic solid media, relations between velocity

> 

( 2 ) (7 10 6 4 ), (a) 3 2

   

( 2 ). (g) 32 2 *Tc*

transverse waves. The subscript *P* denotes hydrostatic loading. The subscript *a* denotes waves propagating parallel to uniaxial pressure. In waves propagating perpendicular to uniaxial pressure, subscript *b* and *c* respectively denote waves' vibrating parallel and

4 4 , (e) 3 2 <sup>4</sup>

( 2 ), (f) 32 4

are 2nd-order elastic constants. *l* , *m* and *n* are 3rd-order elastic constants.

<sup>0</sup> is density. The subscripts *L* and *T* denote longitudinal and

( 2 ) [ (4 10 4 ) 2 ], (c) 3 2

<sup>2</sup> ( 2 ) [2 ( 2 )], (d) 3 2

<sup>1</sup> (3 6 3 ), (b) 3 2 <sup>2</sup>

 

(1)

experimental data are then given.

2 0

*LP*

2 0

*TP*

2 0

*La*

2 0

*Lb*

2 0

*Ta*

2 0

*Tb*

2 0

perpendicular to uniaxial pressure.

*V*

*P* is confining pressure.

 

 

> 

> 

 

 

Here, and 

**2. Nonlinear wave equations for poroelasticity** 

square and confining pressure was derived as [3, 5, 12, 34].

 

 

*<sup>P</sup> V mn*

 

*<sup>P</sup> <sup>V</sup> m n*

 

*P n V m*

 

*P n m*

 

 

*<sup>P</sup> <sup>V</sup> l m*

 

 

> 

 

*<sup>P</sup> V lm*

 

**2.1. Review on nonlinear acoustoelasticity for pure solid material** 

 

 

 

*<sup>P</sup> <sup>V</sup> m l*

 

According to Biot theory, an isotropic poroelastic medium has four independent static moduli (2nd-order elastic constants). Biot [24] developed semilinear mechanics of porous solids, in which total seven physical constants describes the semilinear properties of an isotropic media, four to characterize the linear behavior and three to characterize the nonlinear behavior. The theory is adapted to solid/fluid composite systems by Norris, Sinha and Kosteck in 1994 [25]. Biot [26] also gave eleven-constant elastic potential function for fluid-saturated porous media, in which seven 3rd-order elastic constants describe the nonlinear features of the two-phase system. Drumheller and Bedford chose a Eulerian reference frame to derive nonlinear equations for wave propagation through porous elastic media in 1980 [27], while Berryman and Thigpen chose a Lagrange reference frame in their theory [28].

Donskoy, Khashanah and McKee [29] derived the nonlinear acoustic wave equations for porous media which established a correlation between the measurable effective nonlinear parameter and structural parameters of the porous medium. Their work is based on the semilinear approximation of Biot's poroelasticity theory. The assumption of semilinearity implies a linear relation between the volume change of the solid matrix and the effective stress, therefore a modified strain (*Equ. 7* in [29], *Equ.* 54 in [24]) is used and the effective stress is included, so that the nonlinear acoustoelasticity equations can be simplified and only three independent 3rd-order elastic constants are needed to be considered in application. Dazel and Tournat [30] considered one-dimensional problems following this theoretical approach and derived the solutions for the second harmonic Biot waves. The wave velocity dispersion and dissipation were analyzed in a semi-infinite medium. Zaitsev, Kolpakov and Nazarov [31, 32] theoretically analyzed the experimental results of the propagation of low-frequency video-pulse signals in dry and water-saturated river sand and demonstrated the nonlinearity of such a loose granular media can be non-quadratic in amplitude.

Grinfeld and Norris [33] derived wave speed formulas in closed-pore jacketed and openpore jacketed configurations to find a complete set of seven 3rd-order elastic moduli for poroelastic medium. Nevertheless, solid and fluid's finite strain is not included in Grinfeld and Norris's work, so that 2nd-order elastic constants will not appear in the velocity square *vs* compression ratio slope relations for both transverse and longitudinal waves. This means there will be no compatibility between their poro-acoustoelasticity theory and the pure solid material's acoustoelasticity theory, within which 2nd-order elastic constants do affect the velocity square *vs* compression ratio slopes (*i.e.* the slope in equation (1a) is 7 10 6 4 *l m* , where and are 2nd-order elastic constants, and *l* are *m* the 3rdorder elastic constants.).

In this paper, we derive the wave propagation equations by substituting the 11-term potential function and the Biot kinetic energy function into Lagrange equations. Finite strain is included, and the velocities and quality factors of P- and S-waves are analytically expressed as functions of components' strains, confining pressure, pore fluid pressure and rock's 2nd- and 3rd-order moduli in cases of hydrostatic and uniaxial loading conditions. The reduction of this theory to a pure solid's acoustoelasticity theory is discussed. Numerical examples for velocity and attenuation are given which are corresponding to the open-pore jacketed and closed-pore jacketed rock tests, respectively. In the last section, we perform ultrasonic P-velocity measurements in the "open-pore jacketed" and "close-pore jacketed" tests with hydrostatic loading. The comparisons between theory and rock experimental data are then given.

## **2. Nonlinear wave equations for poroelasticity**

4 Wave Processes in Classical and New Solids

theory [28].

amplitude.

7 10 6 4

*l m* , where

order elastic constants.).

 and 

 

on the same linear assumption, some new expansions on Biot's theory have included local fluid flow, dynamic permeability and multi-scale heterogeneity in last two decades [17-23]. Around linear poroelasticity, the research interests of recent years are focused on the frequency-dependent P- and S- waves' velocity and attenuation features which are

According to Biot theory, an isotropic poroelastic medium has four independent static moduli (2nd-order elastic constants). Biot [24] developed semilinear mechanics of porous solids, in which total seven physical constants describes the semilinear properties of an isotropic media, four to characterize the linear behavior and three to characterize the nonlinear behavior. The theory is adapted to solid/fluid composite systems by Norris, Sinha and Kosteck in 1994 [25]. Biot [26] also gave eleven-constant elastic potential function for fluid-saturated porous media, in which seven 3rd-order elastic constants describe the nonlinear features of the two-phase system. Drumheller and Bedford chose a Eulerian reference frame to derive nonlinear equations for wave propagation through porous elastic media in 1980 [27], while Berryman and Thigpen chose a Lagrange reference frame in their

Donskoy, Khashanah and McKee [29] derived the nonlinear acoustic wave equations for porous media which established a correlation between the measurable effective nonlinear parameter and structural parameters of the porous medium. Their work is based on the semilinear approximation of Biot's poroelasticity theory. The assumption of semilinearity implies a linear relation between the volume change of the solid matrix and the effective stress, therefore a modified strain (*Equ. 7* in [29], *Equ.* 54 in [24]) is used and the effective stress is included, so that the nonlinear acoustoelasticity equations can be simplified and only three independent 3rd-order elastic constants are needed to be considered in application. Dazel and Tournat [30] considered one-dimensional problems following this theoretical approach and derived the solutions for the second harmonic Biot waves. The wave velocity dispersion and dissipation were analyzed in a semi-infinite medium. Zaitsev, Kolpakov and Nazarov [31, 32] theoretically analyzed the experimental results of the propagation of low-frequency video-pulse signals in dry and water-saturated river sand and demonstrated the nonlinearity of such a loose granular media can be non-quadratic in

Grinfeld and Norris [33] derived wave speed formulas in closed-pore jacketed and openpore jacketed configurations to find a complete set of seven 3rd-order elastic moduli for poroelastic medium. Nevertheless, solid and fluid's finite strain is not included in Grinfeld and Norris's work, so that 2nd-order elastic constants will not appear in the velocity square *vs* compression ratio slope relations for both transverse and longitudinal waves. This means there will be no compatibility between their poro-acoustoelasticity theory and the pure solid material's acoustoelasticity theory, within which 2nd-order elastic constants do affect the velocity square *vs* compression ratio slopes (*i.e.* the slope in equation (1a) is

are 2nd-order elastic constants, and *l* are *m* the 3rd-

influenced by patchy saturation, pore distribution and rock microstructure.

#### **2.1. Review on nonlinear acoustoelasticity for pure solid material**

3rd-order elastic constants theory has long been developed to describe the wave propagation and mechanical deformation in pure solid material. The strain energy function for isotropic solid was given with three independent 3rd-order elastic constants (see Table1).

For nonlinear waves propagating in an isotropic solid media, relations between velocity square and confining pressure was derived as [3, 5, 12, 34].

$$
\rho\_0 V\_{LP}^2 = (\lambda + 2\,\mu) - \frac{P}{3\lambda + 2\,\mu} (7\lambda + 10\,\mu + 6l + 4m),
\tag{a}
$$

$$
\rho\_0 \rho\_0 V\_{\rm TP}^2 = \mu - \frac{P}{3\lambda + 2\mu} (3\lambda + 6\mu + 3m - \frac{1}{2}n)\_\prime \tag{b}
$$

$$
\rho\_0 V\_{La}^2 = (\lambda + 2\,\mu) - \frac{P}{3\lambda + 2\,\mu} [\frac{\lambda + \mu}{\mu} (4\lambda + 10\,\mu + 4\,m) + \lambda + 21] \tag{c}
$$

$$
\rho\_0 V\_{Lb}^2 = (\lambda + 2\mu) - \frac{P}{3\lambda + 2\mu} [2l - \frac{2\lambda}{\mu}(\lambda + 2\mu + m)],\tag{1}
$$

$$
\rho\_0 V\_{Ta}^2 = \mu - \frac{P}{3\lambda + 2\mu} \left( 4\lambda + 4\mu + m + \frac{3n}{4\mu} \right) \tag{e}
$$

$$
\rho\_0 V\_{lb}^2 = \mu - \frac{P}{3\lambda + 2\mu} (m + \frac{\lambda n}{4\mu} + \lambda + 2\mu),
\tag{f}
$$

$$
\rho\_0 V\_{Tc}^2 = \mu - \frac{P}{3\lambda + 2\mu} (m - \frac{\lambda + \mu}{2\mu} n - 2\lambda). \tag{\text{g}}
$$

Here, and are 2nd-order elastic constants. *l* , *m* and *n* are 3rd-order elastic constants. *P* is confining pressure. <sup>0</sup> is density. The subscripts *L* and *T* denote longitudinal and transverse waves. The subscript *P* denotes hydrostatic loading. The subscript *a* denotes waves propagating parallel to uniaxial pressure. In waves propagating perpendicular to uniaxial pressure, subscript *b* and *c* respectively denote waves' vibrating parallel and perpendicular to uniaxial pressure.


**Table 1.** Brief relations between 3rd-order elastic constants by different authors [12]

#### **2.2. Review on linear, semilinear and nonlinear mechanics for poroelasticity**

In an isotropic medium, the strain energy is a function of four variables (see *equ.* 3.1 in [35]), the three invariants <sup>1</sup>*I* , <sup>2</sup>*I* , <sup>3</sup>*I* of solid strain components and the fluid displacement divergence .

$$\mathcal{W} = \mathcal{W}(I\_1, I\_2, I\_3, \mathcal{L}), \tag{2}$$

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 7

Berryman & Thigpen [28]

*M*4 1*I* and

Bemer, Boutéca, Vincké, Hoteit & Ozanam [39]

*K*<sup>0</sup> , *Ks* , *Kfl* , 0

*D* , *F*

*M*4 1*I* is generally more

Grinfeld & Norris [33]

<sup>11</sup> , <sup>2</sup> , <sup>1</sup>

11 , <sup>2</sup>,

*<sup>M</sup>*9 2*<sup>I</sup>* <sup>3</sup> *M*5 1*I* , *<sup>M</sup>*6 3*<sup>I</sup>* , *<sup>M</sup>*<sup>812</sup> *I I*

<sup>3</sup> *M*<sup>7</sup>

> *M*9 2*I* ,

<sup>2</sup> *M I* 11 1

<sup>3</sup> *M*<sup>7</sup> ,

<sup>2</sup> *M I* 11 1 are related to

<sup>2</sup> *M*10 1*I* ,

<sup>111</sup> , <sup>12</sup> , <sup>3</sup> , <sup>1</sup>,

,

*M*4 1*I* 's effects will get weaker with a high-porosity and

For a gas-saturated rock (in this case pore fluid has a very low elastic constant),

fluid/solid composite. For the rocks saturated with water or light oil,

Biot [24]; Donskoy, Khashanah & McKee [29]; Dazel & Tournat [30]

<sup>2</sup> *M*3 tend to zero, and <sup>2</sup> *M*1 1*I* and *M*2 2*I* dominate the effective elastic nature of the

softer solid skeleton. For the high-porosity poor-consolidated sand/sandstone, the term of

*M*4 1*I* .

Pierce [37]; Norris, Sinha & Kostek [25, 38]

, *M A* , *N*<sup>1</sup> *a* , *b* , *c* , *d*

*D* , *F* , *G B e* , *f* , *g* , *m* , *n*

<sup>2</sup> *M I* 11 1

<sup>2</sup> *M*10 1*I*

*<sup>M</sup>*<sup>812</sup> *I I* - - -

**Table 3.** The significance of relative-magnitude of the 7 3rd-order terms in the nonlinear elastic

(1st, the most significant level; 2nd, the 2nd significant level; 3rd, the 3rd significant level; 4th, the 4th significant level;

As to the 7 3rd-order terms in equation (3), <sup>3</sup> *M*5 1*I* , *M*6 3*I* and *M*<sup>812</sup> *I I* are related to solid

the fluid-solid coupling effect. The significance of the relative magnitude of these terms are listed in Table 3. As to the actual rocks in nature, the terms of <sup>3</sup> *M*5 1*I* , *M*6 3*I* and *M*<sup>812</sup> *I I* are

*M*9 2*I* ,

1st 2nd 3rd 4 th 5th

<sup>2</sup> *M*10 1*I*

*M*9 2*I* <sup>2</sup> *M I* 11 1

<sup>2</sup> *M*10 1*I* and

**Table 2.** Different notations of 2nd and 3rd order elastic constants for poroelasticity

2nd

3rd -

Soft solid skeleton saturated with extremely highmodulus fluid

Rock saturated with water/oil

Rock saturated with gas

potential expression

skeleton,

significant than

Biot [17, 18]; Ba [23, 36]

> *A* , *N* , *Q* , *R*

<sup>2</sup> *M*3 may become more significant than

Biot [26]

*C*<sup>1</sup> ,*C*<sup>2</sup> , *C*<sup>6</sup> ,*C*<sup>8</sup>

*C*<sup>3</sup> ,*C*<sup>4</sup> , *C*<sup>5</sup> ,*C*<sup>7</sup> , *C*<sup>9</sup> ,*C*<sup>10</sup> , *C*<sup>11</sup>

 , ,

<sup>3</sup> *M*<sup>7</sup>

<sup>3</sup> *M*5 1*I* , *<sup>M</sup>*6 3*<sup>I</sup>* , *M*<sup>812</sup> *I I*

<sup>3</sup> *M*5 1*I* , *<sup>M</sup>*6 3*<sup>I</sup>* ,

5th, the lowest significant level, whose actual effect can be neglected.)

<sup>3</sup> *M*7 is related to pore fluid, while

<sup>2</sup> *M*<sup>3</sup> , nevertheless,

Biot demonstrated that the expansion to the third degree of equation (2) leads to eleven elastic constants, and he only considered the linear relations to derive the quadratic form for *W* in [35]. The semilinear mechanics for porous media was developed by Biot in [24].

In nonlinear investigations for poroelasticity, perform Taylor expansion on equation (2) to the 3rd order so that we derive the power series of four independent variables <sup>1</sup>*I* , <sup>2</sup>*I* <sup>3</sup>*I* and (see *equ.* 5.9 in [26]).

$$\begin{split} 2\mathcal{V} &= M\_1 I\_1^2 + M\_2 I\_2 + M\_3 \zeta^2 + M\_4 \zeta I\_1 + M\_5 I\_1^3 + M\_6 I\_3 + M\_7 \zeta^3 \\ &+ M\_8 I\_1 I\_2 + M\_9 \zeta I\_2 + M\_{10} I\_1^2 \zeta + M\_{11} I\_1 \zeta^2. \end{split} \tag{3}$$

Equation (3) is an elastic potential expression with four 2nd-order elastic constants and seven 3rd-order elastic constants.

We list in table 2 the different notations for 2nd- and 3rd-order elastic constants which have been used in literatures of poroelasticity.

There are 4 2nd-order terms in equation (3). <sup>2</sup> *M*1 1*I* (the dilatation term) and *M*2 2*I* (the shear term) are mainly dependent on the elastic characteristics of the solid skeleton, <sup>2</sup> *M*3 is related to the pore fluid, and *M*4 1*I* is associated with the coupling effect between solid and fluid. If pore fluid has a extremely high bulk modulus (fluid's 2nd-order elastic constants) while solid grain has relatively lower bulk modulus for a specific porous material, <sup>2</sup> *M*3 may become the most significant term. But as to the natural rocks, solid grain's bulk modulus is much higher than pore fluid (generally solid modulus is one magnitude higher than fluid), therefore, <sup>2</sup> *M*1 1*I* and *M*2 2*I* are the most significant, *M*4 1*I* is the moderate, and <sup>2</sup> *M*3 is the least significant. For a gas-saturated rock (in this case pore fluid has a very low elastic constant), *M*4 1*I* and <sup>2</sup> *M*3 tend to zero, and <sup>2</sup> *M*1 1*I* and *M*2 2*I* dominate the effective elastic nature of the fluid/solid composite. For the rocks saturated with water or light oil, *M*4 1*I* is generally more significant than <sup>2</sup> *M*<sup>3</sup> , nevertheless, *M*4 1*I* 's effects will get weaker with a high-porosity and softer solid skeleton. For the high-porosity poor-consolidated sand/sandstone, the term of <sup>2</sup> *M*3 may become more significant than *M*4 1*I* .

6 Wave Processes in Classical and New Solids

<sup>456</sup> 4*C*

*C*<sup>144</sup>

divergence

.

(see *equ.* 5.9 in [26]).

to the pore fluid, and

2

seven 3rd-order elastic constants.

and *M*2 2*I* are the most significant,

been used in literatures of poroelasticity.

Brugger [11] Truesdell [10] Toupin &

 <sup>6</sup> 4 4

 <sup>4</sup> 2

 

 3 4 2

**Table 1.** Brief relations between 3rd-order elastic constants by different authors [12]

Bernstein [8]

 1 1 2

**2.2. Review on linear, semilinear and nonlinear mechanics for poroelasticity** 

In an isotropic medium, the strain energy is a function of four variables (see *equ.* 3.1 in [35]), the three invariants <sup>1</sup>*I* , <sup>2</sup>*I* , <sup>3</sup>*I* of solid strain components and the fluid displacement

Biot demonstrated that the expansion to the third degree of equation (2) leads to eleven elastic constants, and he only considered the linear relations to derive the quadratic form for *W* in [35]. The semilinear mechanics for porous media was developed by Biot in [24].

In nonlinear investigations for poroelasticity, perform Taylor expansion on equation (2) to the 3rd order so that we derive the power series of four independent variables <sup>1</sup>*I* , <sup>2</sup>*I* <sup>3</sup>*I* and

22 33 11 22 3 4 1 51 63 7

.

*M*4 1*I* is associated with the coupling effect between solid and fluid. If

2 2

Equation (3) is an elastic potential expression with four 2nd-order elastic constants and

We list in table 2 the different notations for 2nd- and 3rd-order elastic constants which have

There are 4 2nd-order terms in equation (3). <sup>2</sup> *M*1 1*I* (the dilatation term) and *M*2 2*I* (the shear

pore fluid has a extremely high bulk modulus (fluid's 2nd-order elastic constants) while solid

most significant term. But as to the natural rocks, solid grain's bulk modulus is much higher than pore fluid (generally solid modulus is one magnitude higher than fluid), therefore, <sup>2</sup> *M*1 1*I*

*M*4 1*I* is the moderate, and

8 1 2 9 2 10 1 11 1

*MII M I M I M I*

*W MI MI M M I MI MI M*

term) are mainly dependent on the elastic characteristics of the solid skeleton,

grain has relatively lower bulk modulus for a specific porous material,

Landau & Lifshitz [6], Goldberg [7]

<sup>3</sup> 4 *A n*

<sup>2</sup> *<sup>B</sup>* <sup>1</sup>

<sup>123</sup> *W WI I I* ( , , , ), (2)

 

(3)

<sup>2</sup> *M*3 is related

<sup>2</sup> *M*3 may become the

<sup>2</sup> *M*3 is the least significant.

*C*

Murnaghan [2, 3], Hughs & Kelly [5]

> 2 *m n*

1 2 *lm n*


**Table 2.** Different notations of 2nd and 3rd order elastic constants for poroelasticity


(1st, the most significant level; 2nd, the 2nd significant level; 3rd, the 3rd significant level; 4th, the 4th significant level; 5th, the lowest significant level, whose actual effect can be neglected.)

**Table 3.** The significance of relative-magnitude of the 7 3rd-order terms in the nonlinear elastic potential expression

As to the 7 3rd-order terms in equation (3), <sup>3</sup> *M*5 1*I* , *M*6 3*I* and *M*<sup>812</sup> *I I* are related to solid skeleton, <sup>3</sup> *M*7 is related to pore fluid, while *M*9 2*I* , <sup>2</sup> *M*10 1*I* and <sup>2</sup> *M I* 11 1 are related to the fluid-solid coupling effect. The significance of the relative magnitude of these terms are listed in Table 3. As to the actual rocks in nature, the terms of <sup>3</sup> *M*5 1*I* , *M*6 3*I* and *M*<sup>812</sup> *I I* are

in the most significant level. In gas-saturated rocks, <sup>3</sup> *M*<sup>7</sup> , *M*9 2*I* , <sup>2</sup> *M*10 1*I* and <sup>2</sup> *M I* 11 1 can be neglected in comparison with <sup>3</sup> *M*5 1*I* , *M*6 3*I* and *M*<sup>812</sup> *I I* , which means fluid's effect can be completely neglected. Therefore, the acoustoelasticity theory for a pure solid can be sufficient to describe the nonlinear wave phenomena in gas-saturated or dry rock samples. But as the fluid's elastic modulus increases, the terms of *M*9 2*I* and <sup>2</sup> *M*10 1*I* will play an important role in its nonlinear elastic effects on the whole solid/fluid system. In this case, the pure solid's three 3rd-order elastic constants may be not enough to give a full description for nonlinear phenomena in a water/oil-saturated rock, which has been discussed in the experimental studies by Winkler et al [15, 16].

#### **2.3. Finite strain in solid/fluid system**

If the fluid-saturated rock is loaded under high confining pressure, the infinitesimal strain expression is insufficient to describe solid's and fluid's microscale finite deformation. On this occasion, the Lagrangian strain tensor has much higher precision than infinitesimal strain for solid material. It can be written

$$\varepsilon\_{ij} = \frac{1}{2} (\frac{\partial u\_j}{\partial \mathbf{x}\_i} + \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_l}{\partial \mathbf{x}\_i} \frac{\partial u\_l}{\partial \mathbf{x}\_j}), \ i, j, l = 1, 2, 3 \tag{4}$$

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 9

difficult to realize in practice. The "open-pore jacketed" system [33, 42] corresponds to the constancy of fluid pressure, which means the fluid/solid composite are closed-pore jacketed with a thin impermeable jacket and the inside of the jacket is made to communicate with the atmosphere through a tube, so that in experiment pore fluid will flow out under confining

Pore fluid will not be finitely deformed like solid if the porous solid skeleton is "open-pore jacketed", especially in case of highly permeable rocks being saturated with low viscous fluid. In the open-pore jacketed porous media, fluid particles would rather flow out the matrix through connected pores and throats when being subject to pressure gradient, rather than get compressed into a finite deformation state. On the other hand, if the porous structure is "closed-pore jacketed" or the whole fluid system could be regarded as a closed

Several wave speed formulas for specific experimental configurations are derived by Grinfeld and Norris [33]. However, finite strain (equations 4~5) has not been considered in their derivation, therefore their poro-acoustoelasticity velocity expressions can not be reduced to pure solid's acoustoelasticity expressions (equations 1a~1g) if we perform a

In this paper, we use a theoretical approach similar to Biot's [35] and Norris's [33], but in our development finite strain is used instead of infinitesimal strain, so that this newly developed theory will be more appropriate to describe wave phenomena in the finite-deformed

The dissipation function and the kinetic energy of a unit volume for the isotropic fluid-solid

By applying Lagrange's equations and take *ui* and *Ui* as generalized coordinates,

( ) , (a)

( ) (b)

 

*dT D <sup>f</sup> dt u u*

*dT D <sup>F</sup> dt U U*

*i i*

*i i*

2 222 11 22 33 2 (( ) ( ) ( ) ) *D uU uU uU*

 222 222 11 1 2 3 12 1 1 2 2 3 3 22 1 2 3 2 ( )2 ( *T u u u uU uU uU U U U* )( ) (7)

22 have been defined by [17,18].

(6)

 , 

(8)

and *k* are the

 

gedanken experiment by replacing pore fluid with solid in a solid/fluid composite.

**2.4. Nonlinear wave equations in fluid-saturated solid structure** 

*k*

*i*

*i*

constants of porosity, fluid viscosity and permeability.

generalized forces of solid and fluid phase are derived.

solid/fluid composed system.

Mass parameters

composed system [35] is given by

<sup>11</sup> , 12 and

pressure. The "open-pore jacketed" configuration is very common in actual rock tests.

system, finite deformation of the inclosed pore fluid will happen.

where *ui* denotes solid displacement in *i* direction. The convention of summation over repeated indices is adopted.

Different from Lagrangian description in solid material, the nonlinear problems of the acoustics of fluids are usually formulated in terms of an Eulerian description of wave motion [38, 40, 41]. Kostek, Sinha and Norris [38] gave the explicit relations for the two descriptions in inviscid fluid. In dealing with nonlinear problems involving both fluid and solid, the unified treatment of Lagrangian variables is used in this paper. If an assumption is reasonably made that fluid viscosity's effect on rock's shear deformation can be neglected so that fluid's shear deformation is completely ignored, the fluid's finite strain under high confining pressure can be approximately written

$$\mathcal{L}\_{li} = \frac{\partial \mathcal{U}\_i}{\partial \mathbf{x}\_i} + \frac{1}{2} \frac{\partial \mathcal{U}\_i}{\partial \mathbf{x}\_i} \frac{\partial \mathcal{U}\_i}{\partial \mathbf{x}\_i}, \qquad i = 1, 2, 3 \tag{5}$$

where *Ui* denotes fluid displacement in *i* direction. The convention of summation over repeated indices is not adopted in equation (5). Here we only take into account the longitudinal finite deformation for those low viscous fluid like water and gas. For some non-Newtonian fluid such as bitumen and heavy oil, finite shear deformation need be considered.

In this paper, two types of rock experimental configurations, "open-pore jacketed" and "closedpore jacketed", are considered. The "closed-pore jacketed" system [33] corresponds to constancy of pore fluid mass, which means the fluid/solid composite are closed-pore jacketed with a impervious deformable jacket under confining pressure so that pore fluid cannot flow out solid matrix in experiments. The "closed-pore jacketed" system is the simplest and rather difficult to realize in practice. The "open-pore jacketed" system [33, 42] corresponds to the constancy of fluid pressure, which means the fluid/solid composite are closed-pore jacketed with a thin impermeable jacket and the inside of the jacket is made to communicate with the atmosphere through a tube, so that in experiment pore fluid will flow out under confining pressure. The "open-pore jacketed" configuration is very common in actual rock tests.

8 Wave Processes in Classical and New Solids

in the most significant level. In gas-saturated rocks,

But as the fluid's elastic modulus increases, the terms of

experimental studies by Winkler et al [15, 16].

**2.3. Finite strain in solid/fluid system** 

strain for solid material. It can be written

repeated indices is adopted.

confining pressure can be approximately written

*ii*

*ij*

<sup>3</sup> *M*<sup>7</sup> ,

can be neglected in comparison with <sup>3</sup> *M*5 1*I* , *M*6 3*I* and *M*<sup>812</sup> *I I* , which means fluid's effect can be completely neglected. Therefore, the acoustoelasticity theory for a pure solid can be sufficient to describe the nonlinear wave phenomena in gas-saturated or dry rock samples.

important role in its nonlinear elastic effects on the whole solid/fluid system. In this case, the pure solid's three 3rd-order elastic constants may be not enough to give a full description for nonlinear phenomena in a water/oil-saturated rock, which has been discussed in the

If the fluid-saturated rock is loaded under high confining pressure, the infinitesimal strain expression is insufficient to describe solid's and fluid's microscale finite deformation. On this occasion, the Lagrangian strain tensor has much higher precision than infinitesimal

> <sup>1</sup> ( ), , , 1, 2, 3 <sup>2</sup> *j i ll*

 

*i j ij <sup>u</sup> u uu ijl x x xx*

where *ui* denotes solid displacement in *i* direction. The convention of summation over

Different from Lagrangian description in solid material, the nonlinear problems of the acoustics of fluids are usually formulated in terms of an Eulerian description of wave motion [38, 40, 41]. Kostek, Sinha and Norris [38] gave the explicit relations for the two descriptions in inviscid fluid. In dealing with nonlinear problems involving both fluid and solid, the unified treatment of Lagrangian variables is used in this paper. If an assumption is reasonably made that fluid viscosity's effect on rock's shear deformation can be neglected so that fluid's shear deformation is completely ignored, the fluid's finite strain under high

> 

where *Ui* denotes fluid displacement in *i* direction. The convention of summation over repeated indices is not adopted in equation (5). Here we only take into account the longitudinal finite deformation for those low viscous fluid like water and gas. For some non-Newtonian fluid such as bitumen and heavy oil, finite shear deformation need be considered. In this paper, two types of rock experimental configurations, "open-pore jacketed" and "closedpore jacketed", are considered. The "closed-pore jacketed" system [33] corresponds to constancy of pore fluid mass, which means the fluid/solid composite are closed-pore jacketed with a impervious deformable jacket under confining pressure so that pore fluid cannot flow out solid matrix in experiments. The "closed-pore jacketed" system is the simplest and rather

*i ii*

*i ii U UU <sup>i</sup>*

*x xx*

<sup>1</sup> , 1,2,3 <sup>2</sup>

*M*9 2*I* ,

*M*9 2*I* and

<sup>2</sup> *M*10 1*I* and

> <sup>2</sup> *M*10 1*I* will play an

<sup>2</sup> *M I* 11 1

(4)

(5)

Pore fluid will not be finitely deformed like solid if the porous solid skeleton is "open-pore jacketed", especially in case of highly permeable rocks being saturated with low viscous fluid. In the open-pore jacketed porous media, fluid particles would rather flow out the matrix through connected pores and throats when being subject to pressure gradient, rather than get compressed into a finite deformation state. On the other hand, if the porous structure is "closed-pore jacketed" or the whole fluid system could be regarded as a closed system, finite deformation of the inclosed pore fluid will happen.

Several wave speed formulas for specific experimental configurations are derived by Grinfeld and Norris [33]. However, finite strain (equations 4~5) has not been considered in their derivation, therefore their poro-acoustoelasticity velocity expressions can not be reduced to pure solid's acoustoelasticity expressions (equations 1a~1g) if we perform a gedanken experiment by replacing pore fluid with solid in a solid/fluid composite.

#### **2.4. Nonlinear wave equations in fluid-saturated solid structure**

In this paper, we use a theoretical approach similar to Biot's [35] and Norris's [33], but in our development finite strain is used instead of infinitesimal strain, so that this newly developed theory will be more appropriate to describe wave phenomena in the finite-deformed solid/fluid composed system.

The dissipation function and the kinetic energy of a unit volume for the isotropic fluid-solid composed system [35] is given by

$$\text{V}\Sigma D = \phi^2 \frac{\eta}{k} \{ (\dot{\mu}\_1 - \dot{\mathcal{U}}\_1)^2 + (\dot{\mu}\_2 - \dot{\mathcal{U}}\_2)^2 + (\dot{\mu}\_3 - \dot{\mathcal{U}}\_3)^2 \} \tag{6}$$

$$\text{2T} = \rho\_{11}(\dot{\mu}\_1^2 + \dot{\mu}\_2^2 + \dot{\mu}\_3^2) + \text{2}\,\rho\_{12}(\dot{\mu}\_1\dot{\mathcal{U}}\_1 + \dot{\mu}\_2\dot{\mathcal{U}}\_2 + \dot{\mu}\_3\dot{\mathcal{U}}\_3) + \rho\_{22}(\dot{\mathcal{U}}\_1^2 + \dot{\mathcal{U}}\_2^2 + \dot{\mathcal{U}}\_3^2) \tag{7}$$

Mass parameters <sup>11</sup> , 12 and 22 have been defined by [17,18]. , and *k* are the constants of porosity, fluid viscosity and permeability.

By applying Lagrange's equations and take *ui* and *Ui* as generalized coordinates, generalized forces of solid and fluid phase are derived.

$$f\_i = \frac{d}{dt}(\frac{\partial T}{\partial \dot{u}\_i}) + \frac{\partial D}{\partial \dot{u}\_i},\tag{8}$$

$$F\_i = \frac{d}{dt}(\frac{\partial T}{\partial \dot{M}\_i}) + \frac{\partial D}{\partial \dot{M}\_i} \tag{b}$$

where *<sup>i</sup> f* and *Fi* satisfy

$$f\_i = \frac{d}{d\mathbf{x}\_j} (\frac{\partial W}{\partial(\frac{\partial \mathbf{u}\_i}{\partial \mathbf{x}\_j})}).$$

$$F\_i = \frac{d}{d\mathbf{x}\_j} (\frac{\partial W}{\partial(\frac{\partial U\_i}{\partial \mathbf{x}\_j})}).$$

Moreover,

$$a\_{i\dot{j}} = \frac{\partial \mathcal{W}}{\partial \mathcal{L}\_{i\dot{j}}}, \ b\_{i\dot{j}} = \frac{\partial \mathcal{W}}{\partial \mathcal{L}\_{i\dot{j}}}, \ \text{i.} \ j = 1, 2, 3 \tag{9}$$

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 11

[42]. In this subsection methods of measurement are described for the determination of the seven 3rd-order elastic constants. With the determination of all the 11 elastic constants, the poro-acoustoelasticity theory will be directly applicable to nonlinear

Studies of Winkler et al. [16] showed that traditional 3rd-order elastic constants theory of pure solid material gives much better description for observed velocity-pressure relations in dry rocks than in saturated ones. Therefore we assume the three 3rd-order elastic constants of *bl* , *mb* and *nb* are sufficient for describing nonlinear acoustic features in dry

Three gedanken experiments have been discussed to determine the four 2nd-order elastic constants for poroelasticity. Similar approach was also adopted on the determination of the six 2nd-order elastic constants in double-porosity models [43]. For the case of poroacoustoelasticity, the problem comes to much more complicated for determining the total

1. Because the fluid viscosity effect has been neglected and it will not directly contribute to the whole structure's higher-order distortion, the 3rd-order elastic coefficient of 3*I* in

2. If the fluid-saturated rock sample surrounded by a flexible rubber is subject to a hydrostatic pressure *Ph* and the fluid is allowed to squirt out, all the outside pressure will be confined to the solid skeleton because fluid will squirt out when rock sample is squeezed. For the generalized stress tensors for solid and fluid phase *ij a* and *ij b* , because the rock sample only responds to frame stiffness and fluid pressure is nearly

2 2 <sup>1</sup> , (a) <sup>9</sup>

 11 22 33 

(12)

. By substituting equations

0, (b)

(12a~b) into equations (9), we get one relation between *M*<sup>5</sup> , *M*<sup>7</sup> , *M*<sup>8</sup> , *M*<sup>9</sup> , *M*<sup>10</sup> , *M*11 and

3. The fluid-saturated rock sample surrounded by a flexible rubber is subject to a uniaxial

solid/fluid systems.

seven 3rd-order elastic constants.

Five gedanken experiments are designed.

equation (3) will satisfy <sup>6</sup> 2 *M nb* .

For isotropic rocks under hydrostatic loading,

The solid and fluid pressures are expressed as

*ii*

*b*

*ii h b b b*

11 22 33 *e* . *Kb* is the bulk modulus of solid matrix.

pressure *Pu* along axis 1, fluid being allowed to squirt out.

*a P Ke le ne*

rocks.

zero,

where 

*bl* , *mb* , *nb* .

 

and

$$\mathbf{J} = \begin{bmatrix} 1 + u\_{1,1} & u\_{1,2} & u\_{1,3} \\ & u\_{2,1} & 1 + u\_{2,2} & u\_{2,3} \\ & u\_{3,1} & u\_{3,2} & 1 + u\_{3,3} \end{bmatrix}^{\mathrm{T}}, \ \mathbf{K} = \begin{bmatrix} 1 + \mathcal{U}\_{1,1} & 0 & 0 \\ 0 & 1 + \mathcal{U}\_{2,2} & 0 \\ 0 & 0 & 1 + \mathcal{U}\_{3,3} \end{bmatrix}^{\mathrm{T}} \tag{10}$$

where *ui*, *<sup>j</sup>* designates *i j u x* .

By substituting equations (9~10) into equations (8a~b), nonlinear wave equations in 2-phase medium are written as

$$\begin{aligned} a\_{ik,j}\mathbf{J}\_{kj} + a\_{ik}\mathbf{J}\_{kj,j} &= \rho\_{11}\ddot{u}\_i + \rho\_{12}\ddot{\mathbf{U}}\_i + b(\dot{u}\_i - \dot{\mathbf{U}}\_i) & \quad \text{(a)}\\ b\_{ik,j}\mathbf{K}\_{kj} + b\_{ik}\mathbf{K}\_{kj,j} &= \rho\_{12}\ddot{u}\_i + \rho\_{22}\ddot{\mathbf{U}}\_i - b(\dot{u}\_i - \dot{\mathbf{U}}\_i)\_\prime & \quad \text{(b)} \end{aligned} \tag{11}$$

where <sup>2</sup> *b k* .

By substituting equations (4~5) into equation (3), substituting equation (3) into equation (9), then solving *ij a* and *ij b* , and finally substituting into (11a~b), we derive the nonlinear acoustic wave

propagation equations in 2-phase medium in which both solid and fluid's finite strain are considered. The full expansion scheme for equations (11a~b) is very complicated. In this paper, we only give simplified schemes for some particular experimental configurations.
