**3. Wave speeds for specific experimental configurations**

#### **3.1. The 3rd-order elastic constants for solid/fluid composite**

There are four 2nd-order and seven 3rd-order elastic constants in poro-acoustoelasticity theory. The determination of four 2nd-order elastic constants has been discussed by [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 solid/fluid systems.

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 rocks.

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 seven 3rd-order elastic constants.

Five gedanken experiments are designed.

10 Wave Processes in Classical and New Solids

*d W <sup>f</sup> dx <sup>u</sup>*

*d W <sup>F</sup> dx U*

*i*

*i*

 ( ) ( )

 ( ) ( )

*x*

*i j j*

> 

1 1 00

1,1 1,2 1,3 1,1 2,1 2,2 2,3 2,2

*uu u U u uu U*

we only give simplified schemes for some particular experimental configurations.

**3. Wave speeds for specific experimental configurations** 

**3.1. The 3rd-order elastic constants for solid/fluid composite** 

, , 11 12 , , 12 22

**J J K K**

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

 

 

*ik j kj ik kj j i i i i ik j kj ik kj j i i i i*

*a a u U bu U b b u U bu U*

 

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,

There are four 2nd-order and seven 3rd-order elastic constants in poro-acoustoelasticity theory. The determination of four 2nd-order elastic constants has been discussed by

*uu u U*

3,1 3,2 3,3 3,3

1 , 01 0 1 0 01

**J K** (10)

 , , , 1,2,3 *ij ij ij ij*

*x*

,

.

*W W a b ij* (9)

T T

( ) (a) ( ), (b)

(11)

*i j j*

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

where *ui*, *<sup>j</sup>* designates

medium are written as

 <sup>2</sup> *b*

*k* .

 *i j u x* .

Moreover,

and

where


$$a\_{ii} = P\_h = K\_b e + l\_b e^2 + \frac{1}{9} n\_b e^2 \,\text{\AA} \tag{12}$$
 
$$b\_{ii} = 0 \,\text{\AA} \tag{b}$$

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

For isotropic rocks under hydrostatic loading, 11 22 33 . By substituting equations (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 *bl* , *mb* , *nb* .

3. The fluid-saturated rock sample surrounded by a flexible rubber is subject to a uniaxial pressure *Pu* along axis 1, fluid being allowed to squirt out.

The solid and fluid pressures are expressed as

$$\begin{aligned} a\_{11} = P\_u &= (\mathcal{K}\_b + \frac{4}{3}\mu\_b)e - 2\,\mu\_b(\mathcal{E}\_{22} + \mathcal{E}\_{33}) + (l\_b + 2m\_b)e^2 \\ &- 2m\_b[e(\mathcal{E}\_{22} + \mathcal{E}\_{33}) + \mathcal{E}\_{22}\mathcal{E}\_{33} + \mathcal{E}\_{11}\mathcal{E}\_{33} + \mathcal{E}\_{11}\mathcal{E}\_{22}] + n\_b\varepsilon\_{22}\varepsilon\_{33} \end{aligned} \tag{a}$$

$$a\_{22} = a\_{33} = 0 = (\mathcal{K}\_b + \frac{4}{3}\mu\_b)e - 2\,\mu\_b(\varepsilon\_{11} + \varepsilon\_{22}) + (l\_b + 2m\_b)e^2 \tag{13}$$

$$-2m\_b[\varepsilon(\varepsilon\_{11}+\varepsilon\_{33})+\varepsilon\_{22}\varepsilon\_{33}+\varepsilon\_{11}\varepsilon\_{33}+\varepsilon\_{11}\varepsilon\_{22}]+n\_b\varepsilon\_{11}\varepsilon\_{33}\prime\_{\prime}\tag{b}$$

 0 (c) *ii b*

where solid strain in two transverse directions are identical as 22 33 .

By substituting equations (13a~c) into equations (9), we get one more relation.

4. Let the open-pore jacketed porous sample be subjected to a uniform hydrostatic pressure *Pf* .

In the acoustoelasticity of solid grains and fluid, strains can be solved through

$$\begin{aligned} P\_f &= K\_s e + K\_{s3} e^2, & \quad \text{(a)}\\ P\_f &= K\_f \zeta + K\_{f^3 3} \zeta^2 & \quad \text{(b)}\\ a\_{ii} &= (1 - \phi) P\_{f^\*}, & \quad \text{(c)}\\ b\_{ii} &= \phi P\_f & \quad \text{(d)} \end{aligned} \tag{14}$$

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 13

 11 11 11 ,

( ), (a) ( ) (b)

(16)

*ij* denote solid strains induced by pore pressure and effective

 22 22 22 ,

The total strain in solid frame is then calculated with

By substituting equations (15a~g) into equations (9), we get the last two relations.

**3.2. Wave velocity expressions under hydrostatic confining pressure** 

measurements for fluid-saturated porous solids, especially in rock experiments.

The total seven relations in gedanken experiments (1~5) are well-posed to solve out the

Based on poro-acoustoelasitcity theory, relations between acoustic wave velocity and loading pressure will be derived, which can be directly applicable to the actual

In this paper we only consider the case of rocks being subject to hydrostatic or uniaxial loading in experiments. No transverse stress is loaded on samples therefore both shear

For simplicity let us consider the plane P-waves propagating along the axis ''1". Wave

 

> 

> >

is a component of the solid's deformation in the 3D space under hydrostatic

 

*ij ij* (17)

 

(18)

and *k* denote angle

 11,1 11,1 1,1 11 1,11 11 1 12 1 1 1 11,1 11,1 1,1 11 1,11 12 1 22 1 1 1

*a a u a u u U bu U b b U b U u U bu U*

is a component of the fluid's deformation.

 

*ij ij*

where *S* and *S* are wave amplitudes in solid and fluid phases.

The small dynamic strain induced by the plane P-waves' vibration along axis 1 is

*Se S e*

 

For hydrostatic loading, the "large" part of the static strain in "small-on-large" constitutive

00 00 0 0, 0 0 00 00

 

By substituting equations (17~18) into equations (16a~b), neglecting all terms whose power orders are higher than 1 and eliminating *S* and *S* , equations (16a~b) will be simplified to

*i t kx i t kx*

 1 1 [ (1 )] [ (1 )] 0 0 0 0 0 00 , 0 00 0 00 0 00

 *ij* and 

seven unknown 3rd-order elastic constants.

deformation of solid and fluid can be neglected.

relations induced by static confining pressure is expressed as

equations (11a~b) are reduced to

frequency and wave number.

 33 33 33 , where

where

loading, while

pressure respectively.

where *Ks* and *Kf* are the solid grain and fluid's bulk modulus. *Ks*3 and *Kf* <sup>3</sup> indicate the solid grain and fluid's 3rd-order moduli [38].

is the porosity.

By substituting equations (14a~d) into equations (9), we get another two relations.

5. The closed-pore jacketed porous rock is subjected to a uniaxial pressure *Pu* along axis 1, while at the same time the pore pressure is kept with *Pp* . Let *P P <sup>u</sup> <sup>p</sup>* , we derive the last two relations as below.

$$P\_p = K\_s(\varepsilon\_{11}' + \varepsilon\_{22}' + \varepsilon\_{33}') + K\_{s3}(\varepsilon\_{11}' + \varepsilon\_{22}' + \varepsilon\_{33}')^2,\tag{a}$$

$$P\_p = \mathbf{K}\_f \boldsymbol{\zeta} + \mathbf{K}\_{f3} \boldsymbol{\zeta}^2 \tag{b}$$

$$\begin{aligned} 0 = (K\_b + \frac{4}{3}\mu\_b)e'' - 2\mu\_b(\varepsilon\_{22}'' + \varepsilon\_{33}'') + (l\_b + 2m\_b)e''^2 \\ -2m\_b[e''(\varepsilon\_{22}'' + \varepsilon\_{33}'') + \varepsilon\_{22}''\varepsilon\_{33}'' + \varepsilon\_{11}''\varepsilon\_{33}'' + \varepsilon\_{11}''\varepsilon\_{22}''] + n\_b\varepsilon\_{22}''\varepsilon\_{33}'' \\ 4. \end{aligned} \tag{c}$$

$$-P\_p = (K\_b + \frac{4}{3}\mu\_b)e'' - 2\mu\_b(\varepsilon\_{11}'' + \varepsilon\_{22}'') + (l\_b + 2m\_b)e''^2 \tag{15}$$

$$-2m\_b[\varepsilon''(\varepsilon''\_{11}+\varepsilon''\_{33})+\varepsilon''\_{22}\varepsilon''\_{33}+\varepsilon''\_{11}\varepsilon''\_{33}+\varepsilon''\_{11}\varepsilon''\_{22}]+n\_b\varepsilon''\_{11}\varepsilon''\_{33}.\tag{\bf d}$$
  $a\_{11} = (1-\phi)P\_n$ 

$$a\_{11} - (1 - \varphi)a\_{\varphi} \tag{f}$$

$$a\_{22} = a\_{33} = -\phi P\_p \tag{f}$$

$$
\Phi\_{ii} = \phi P\_p
\tag{g}
$$

The total strain in solid frame is then calculated with 11 11 11 , 22 22 22 , 33 33 33 , where *ij* and *ij* denote solid strains induced by pore pressure and effective pressure respectively.

By substituting equations (15a~g) into equations (9), we get the last two relations.

12 Wave Processes in Classical and New Solids

pressure *Pf* .

is the porosity.

two relations as below.

*pf f*

*PK K*

*ii*

*b*

11 22 33

 

*aPK e l me*

*ub b b b b*

 

*aa K e l me*

where solid strain in two transverse directions are identical as

 

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

 

<sup>4</sup> 0 ( ) 2 ( )( 2 ) <sup>3</sup>

*b b b bb*

22 33 11 22

solid grain and fluid's 3rd-order moduli [38].

2 3

*p s s*

 

*P K K*

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

*pb b b*

*p p*

*PK e*

11

22 33

*a P aa P*

 

*ii p*

*b P*

11 22 33 3 11 22 33

<sup>4</sup> 0( ) 2 ( )( 2 ) <sup>3</sup>

 

 

 

*b b b bb*

<sup>2</sup>

 

 

*K e l me*

22 33

 

11 22

 

> 

By substituting equations (13a~c) into equations (9), we get one more relation.

In the acoustoelasticity of solid grains and fluid, strains can be solved through

 

*fs s ff f ii f ii f*

*P Ke K e PK K a P b P*

By substituting equations (14a~d) into equations (9), we get another two relations.

 

 

*b b*

*m e n*

22 33 22 33 11 33 11 22 22 33

4. Let the open-pore jacketed porous sample be subjected to a uniform hydrostatic

 

where *Ks* and *Kf* are the solid grain and fluid's bulk modulus. *Ks*3 and *Kf* <sup>3</sup> indicate the

5. The closed-pore jacketed porous rock is subjected to a uniaxial pressure *Pu* along axis 1, while at the same time the pore pressure is kept with *Pp* . Let *P P <sup>u</sup> <sup>p</sup>* , we derive the last

2

( )( ) , (a)

2

2 [( ) ] , (c)

   

> 

22 33 22 33 11 33 11 22 22 33

11 33 22 33 11 33 11 22 11 33

2 [( ) ] . (d)

 

*b b*

 

*b b*

*l me*

(1 ) (e)

 

)( 2 )

*b b*

*m e n*

*m e n*

 

> 

(1 ) , (c)

0 (c)

   

2 [( ) ] , (a)

11 33 22 33 11 33 11 22 11 33

2

2 [( ) ] , (b)

2

 

> 22 33 .

 

, (a)

(b)

(d)

*b b*

*m e n*

 

> 

(14)

(b)

(f) (g) (15)

(13)

The total seven relations in gedanken experiments (1~5) are well-posed to solve out the seven unknown 3rd-order elastic constants.

#### **3.2. Wave velocity expressions under hydrostatic confining pressure**

Based on poro-acoustoelasitcity theory, relations between acoustic wave velocity and loading pressure will be derived, which can be directly applicable to the actual measurements for fluid-saturated porous solids, especially in rock experiments.

In this paper we only consider the case of rocks being subject to hydrostatic or uniaxial loading in experiments. No transverse stress is loaded on samples therefore both shear deformation of solid and fluid can be neglected.

For simplicity let us consider the plane P-waves propagating along the axis ''1". Wave equations (11a~b) are reduced to

$$\begin{aligned} a\_{11,1} + a\_{11,1}\mu\_{1,1} + a\_{11}\mu\_{1,11} &= \rho\_{11}\ddot{\mu}\_{1} + \rho\_{12}\ddot{\mathcal{U}}\_{1} + b(\dot{\mu}\_{1} - \dot{\mathcal{U}}\_{1}) & \text{(a)}\\ b\_{11,1} + b\_{11,1}\mathcal{U}\_{1,1} + b\_{11}\mathcal{U}\_{1,11} &= \rho\_{12}\ddot{\mu}\_{1} + \rho\_{22}\ddot{\mathcal{U}}\_{1} - b(\dot{\mu}\_{1} - \dot{\mathcal{U}}\_{1}) & \text{(b)}\end{aligned} \tag{16}$$

For hydrostatic loading, the "large" part of the static strain in "small-on-large" constitutive relations induced by static confining pressure is expressed as

$$
\varepsilon\_{ij} = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix}, \ \zeta\_{ij} = \begin{pmatrix} \beta & 0 & 0 \\ 0 & \beta & 0 \\ 0 & 0 & \beta \end{pmatrix} \tag{17}
$$

where is a component of the solid's deformation in the 3D space under hydrostatic loading, while is a component of the fluid's deformation.

The small dynamic strain induced by the plane P-waves' vibration along axis 1 is

$$
\boldsymbol{\varepsilon}\_{ij} = \begin{pmatrix}
\mathrm{Se}^{i[\mathrm{tot} - k\mathbf{x}\_1(1+\alpha)]} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{pmatrix}, \quad \boldsymbol{\zeta}\_{ij} = \begin{pmatrix}
\mathrm{S}'e^{i[\mathrm{tot} - k\mathbf{x}\_1(1+\beta)]} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{pmatrix} \tag{18}
$$

where *S* and *S* are wave amplitudes in solid and fluid phases. and *k* denote angle frequency and wave number.

By substituting equations (17~18) into equations (16a~b), neglecting all terms whose power orders are higher than 1 and eliminating *S* and *S* , equations (16a~b) will be simplified to

$$\begin{vmatrix} \Upsilon\_1 \mathbf{k}^2 - \rho\_{11} a \mathbf{o}^2 + iboo & (\Upsilon\_2 + M\_4 \beta) \mathbf{k}^2 - \rho\_{12} o \mathbf{o}^2 - iboo \\ (\Upsilon\_2 + M\_4 a) \mathbf{k}^2 - \rho\_{12} o^2 - iboo & \Upsilon\_3 \mathbf{k}^2 - \rho\_{22} o^2 + iboo \end{vmatrix} = \mathbf{0},\tag{19}$$

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 15

Murnaghan [2] Hughs & Kelly [5]

is also used in equation 5.9 in [26] by replacing


2

*l m* - -

( ) (a)

  Biot [26] Grinfeld &

*C* <sup>2</sup>

Norris [33]

2

2

2

2

2

2

2

(23)

has used a different invariant for fluid phase, the constant relations between [26] and this

*M*<sup>1</sup> *A* 2*N* , *P* - <sup>1</sup> 2*C* <sup>11</sup> 2 *M*<sup>3</sup> *<sup>R</sup>* - <sup>6</sup> 2*C*

*M*<sup>4</sup> 2*Q* - <sup>8</sup> 2*C* <sup>1</sup>

*M*<sup>5</sup> - - <sup>3</sup> 2*C* <sup>111</sup> 2 *M*<sup>6</sup> - 2*n* <sup>5</sup> 2*C* <sup>3</sup> 2 *M*<sup>7</sup> - - <sup>7</sup> 2*C*

*M*<sup>8</sup> - - <sup>4</sup> 2*C* <sup>12</sup> 2 *M*<sup>9</sup> - - <sup>11</sup> 2*C* <sup>2</sup>

*M*<sup>10</sup> - - <sup>9</sup> 2*C* <sup>11</sup>

*M*<sup>11</sup> - - <sup>10</sup> 2*C* <sup>1</sup>

*M M* - 2*<sup>m</sup>* - -

**Table 4.** The relations between elastic constants as used in different studies. (The relation between column 1 and column 3 holds when poro-acoustoelasticity theory is reduced to acoustoelasticity

*i t kx i t kx*

0 ( ) (b)

 

 2 2 [ (1 )] [ (1 )] 0 0 0 0 0 00 , 0 00 0 00 0 00

12,2 12,2 1,1 22 1,22 11 1 12 1 1 1

*a a u a u u U bu U*

 

 (22)

For plane S-waves propagating along 2 axis and vibrating along 1 axis,

3

 2

2

, the constants can be listed in the column 4 of table 4.)

paper are very complicated. If we assume

This paper Biot [17, 18]

*M*<sup>134</sup> *M M A* 2 2 *N QR*

*<sup>M</sup> <sup>N</sup>*

*MMM M* - <sup>2</sup>

*m* with

2 4

5 7 10 11 2

> 8 9 2

The small dynamic strain induced by S-waves is

 

12 1 22 1 1 1

*u U bu U*

 

*Se S e*

*ij ij*

theory.)

where

$$\begin{aligned} \Upsilon\_1 &= M\_1 + (\Upsilon M\_1 + M\_2 + 9M\_5 + 2M\_8)\alpha + (\frac{3}{2}M\_4 + 3M\_{10})\beta\_{10} \\\\ \Upsilon\_2 &= \frac{1}{2}M\_4 + (\frac{1}{2}M\_4 + M\_9 + 3M\_{10})\alpha + (3M\_{11} + \frac{1}{2}M\_4)\beta\_{10} \end{aligned}$$

and

$$\Upsilon\_3 = M\_3 + (\frac{3}{2}M\_4 + 3M\_{11})\alpha + (9M\_7 + 7M\_3)\beta.$$

Frequency-dependent P velocity can be solved through *V k* / . The fast and slow P-wave velocities respectively correspond to the two solutions of the quadratic equation.

Similar to the reduction process from Biot equations to Gassmann equations [44] in zero frequency limit, if we neglect the relative motion between solid matrix and pore fluid, which means fluid particles have the same vibration amplitudes as solid in wave propagation (that is, *S S* ), equation (19) will be reduced to

$$
\rho V^2 = \Upsilon\_1 + 2\Upsilon\_2 + \Upsilon\_3 + M\_4(\alpha + \beta). \tag{20}
$$

where 11 12 22 2 denotes the average density [35].

Equation (20) is also a nonlinear extension of Gassmann theory. The difference between solid and fluid's "small" dynamic strain induced by wave vibration is neglected, while solid and fluid's "large" static strain induced by loading are still different with .

Another necessary condition must be satisfied to reduce equation (20) to equation (1a). By assuming pore fluid have the same static deformation as solid skeleton under hydrostatic loading, which means solid and fluid share the same constitutive relation, both in "small" dynamic and "large" static part, and then substituting in equation (20),

$$\begin{aligned} \rho V^2 &= M\_1 + M\_3 + M\_4 \\ &+ [7(M\_1 + M\_3 + M\_4) + M\_2 + 9(M\_5 + M\_7 + M\_{10} + M\_{11}) + 2(M\_8 + M\_9)] \mathbf{a} \end{aligned} \tag{21}$$

Equation (21) is identical to equation (1a) if we take the average body deformation of 3 2 *<sup>P</sup>* for . The constant relations between different studies are listed in table 4 (As to the elastic potential *W* in equation 3, a similar notation in equation 5.9 in [26] has been defined as a function of the relative variation of fluid content *m* , <sup>1</sup> *m I* ( ) . Since [26]



where

and

where  ( )

 

   

 

1 1 1 2 5 8 4 10

2 4 4 9 10

3 3 4 11

velocities respectively correspond to the two solutions of the quadratic equation.

2

Frequency-dependent P velocity can be solved through *V k* /

dynamic and "large" static part, and then substituting

134

*V MMM*

11 12 22 2 denotes the average density [35].

and fluid's "large" static strain induced by loading are still different with

defined as a function of the relative variation of fluid content *m* ,

is, *S S* ), equation (19) will be reduced to

 

 3 2

*<sup>P</sup>* for

2

 

*M MM M M M M*

 1 1 <sup>1</sup> ( 3 ) (3 ) , 2 2 <sup>2</sup> *M MM M M M*

*M MM MM*

Similar to the reduction process from Biot equations to Gassmann equations [44] in zero frequency limit, if we neglect the relative motion between solid matrix and pore fluid, which means fluid particles have the same vibration amplitudes as solid in wave propagation (that

2

Equation (20) is also a nonlinear extension of Gassmann theory. The difference between solid and fluid's "small" dynamic strain induced by wave vibration is neglected, while solid

Another necessary condition must be satisfied to reduce equation (20) to equation (1a). By assuming pore fluid have the same static deformation as solid skeleton under hydrostatic loading, which means solid and fluid share the same constitutive relation, both in "small"

1 3 4 2 5 7 10 11 8 9 [7( ) 9 ) 2( )]

Equation (21) is identical to equation (1a) if we take the average body deformation of

the elastic potential *W* in equation 3, a similar notation in equation 5.9 in [26] has been

 3 ( 3 ) (9 7 ) .

2 4 12 3 22

22 22

 2 2 2 2 1 11 2 4 12

*k ib M k ib M k ib k ib*

 

( ) 0,

 

11 4

7 3

 

 

*MMM M MMM M MM* (21)

. The constant relations between different studies are listed in table 4 (As to

1 23 4 *V M* 2 ( ). (20)

<sup>3</sup> (7 9 2 ) ( 3 ) , <sup>2</sup>

 

(19)

 

. The fast and slow P-wave

 .

  <sup>1</sup> *m I* ( ) . Since [26]

in equation (20),

**Table 4.** The relations between elastic constants as used in different studies. (The relation between column 1 and column 3 holds when poro-acoustoelasticity theory is reduced to acoustoelasticity theory.)

For plane S-waves propagating along 2 axis and vibrating along 1 axis,

$$a\_{12,2} + a\_{12,2}
\mu\_{1,1} + a\_{22}
\mu\_{1,22} = \rho\_{11}
\ddot{\boldsymbol{u}}\_1 + \rho\_{12}
\ddot{\boldsymbol{U}}\_1 + b(\dot{\boldsymbol{u}}\_1 - \dot{\boldsymbol{U}}\_1) \tag{22}$$

$$0 = \rho\_{12}
\ddot{\boldsymbol{u}}\_1 + \rho\_{22}
\ddot{\boldsymbol{U}}\_1 - b(\dot{\boldsymbol{u}}\_1 - \dot{\boldsymbol{U}}\_1) \tag{3}$$

The small dynamic strain induced by S-waves is

$$
\boldsymbol{\omega}\_{\dot{\boldsymbol{\eta}}} = \begin{pmatrix} \mathrm{S}e^{i[\boldsymbol{\alpha}t - k\mathbf{x}\_2(1+\alpha)]} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{pmatrix}, \boldsymbol{\zeta}\_{\dot{\boldsymbol{\eta}}} = \begin{pmatrix} \mathrm{S}e^{i[\boldsymbol{\alpha}t - k\mathbf{x}\_2(1+\beta)]} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{pmatrix} \tag{23}
$$

By substituting equation (17) and equation (23) into equations (22a~b),

$$
\begin{vmatrix}
\Psi k^2 - \rho\_{11} o^2 + ibo & -\rho\_{12} o^2 - ibo \\ 
\end{vmatrix} = 0\tag{24}
$$

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 17

 

( ) 0,

 

 

1 23 4 *V M* 2 ( ). (30)

 

> 

   

(29)

 

is satisfied.

with the

(31)

(32)

 

 

 

along axis 2 and axis 3 (the two

 

 

 

 1 1 1 5 1 2 5 8 4 10 <sup>3</sup> (5 3 ) (2 6 2 ) ( 3 ) <sup>2</sup> *M M M MM M M M M* ,

 2 4 4 10 9 10 11 4 1 1 <sup>1</sup> ( ) ( 2 ) (3 ) 2 2 <sup>2</sup> *M MM M M M M*

3 3 4 11 4 11 7 3

The two frequency-dependent P velocities are solved out from equation (29). The equation

( ) ( 2 ) (9 7 )

*M MM M M M M* .

2 4 12 3 22

 

 

If the difference between solid and fluid deformation is neglected for uniaxial loading,

(19) to (1a) in hydrostatic case. In equation (30), if we replace the fluid strain

1 3 4 5 7 10 11

[5( ) 3( )]

*MMM MMM M*

 

 

> 

*ij ij*

2

Equation (30) can not be directly reduced to equation (1c) like the reduction process from

orthogonal directions perpendicular to axis 1), we can derive an analytical scheme which is

*MMM M MMM M MM*

[2( ) 6( ) 2( )]

 

With the same "large" static strain, for transverse waves propagating perpendicular to

 

*i t kx i t kx*

*Se S e*

 2 2 [ (1 )] [ (1 )] 0 0 0 0 0 00 , 0 00 , 0 00 0 00

 ( ) , (3 2 ) 2 (3 2 ) *P P u u* .

1 3 4 2 5 7 10 11 8 9

with

22 22

 2 2 2 2 1 11 2 4 12

*k ib M k ib M k ib k ib*

1

2

along axis 1 and replace

( )

where

and

solid strain

where

Then

2

completely compatible with equation (1c).

uniaxial loading and vibrating along loading,

*V MMM*

134

 

 

will be reduced to equation (19) in hydrostatic loading configuration if

where

$$\Psi = -\frac{M\_2}{4} + (\Im M\_1 - \frac{M\_6}{4} - \frac{3}{4}M\_8)\alpha + (\frac{3}{2}M\_4 - \frac{3}{4}M\_9)\beta \ .$$

Frequency-dependent S-velocity can be solved through equation (24).

By assuming pore fluid has the same static deformation as solid skeleton, equation (24) comes to

$$
\rho V^2 = -\frac{M\_2}{4} + (\Im M\_1 + \frac{\Im}{2} M\_4 - \frac{M\_6}{4} - \frac{\Im}{4} M\_8 - \frac{\Im}{4} M\_9) \alpha. \tag{25}
$$

Equation (25) can not be directly reduced to equation (1b), because low viscous fluid's shear constitutive functions has been neglected in Biot theory. Return to pure solid's nonlinear acoustic theory needs to regard pore fluid as solid grains with the same shear constitutive behaviors. Therefore, neglecting solid/fluid relative deformation and adding the neglected term of fluid's shear deformation, S-velocity expression is reduced to

$$
\rho V^2 = -\frac{M\_2}{4} + (3M\_1 + 3M\_4 + 3M\_3 - \frac{M\_6}{4} - \frac{3}{4}M\_8 - \frac{3}{4}M\_9)a,\tag{26}
$$

which is identical to equation (1b).

#### **3.3. Velocity expressions under Uniaxial confining pressure**

For uniaxial loading along axis 1, the "large" static strain in "small-on-large" constitutive relations induced by confining pressure is expressed as

$$
\mathcal{L}\_{ij} = \begin{pmatrix} a & 0 & 0 \\ 0 & \alpha' & 0 \\ 0 & 0 & \alpha' \end{pmatrix}, \mathcal{L}\_{ij} = \begin{pmatrix} \mathcal{J} & 0 & 0 \\ 0 & \mathcal{J} & 0 \\ 0 & 0 & \mathcal{J} \end{pmatrix} \tag{27}
$$

The small dynamic strain induced by plane P-waves vibration along axis 1 is

$$
\boldsymbol{\varepsilon}\_{ij} = \begin{pmatrix}
\mathrm{Se}^{i[\mathrm{lat}-k\mathbf{x}\_1(1+\alpha)]} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{pmatrix}, \ \boldsymbol{\zeta}\_{ij} = \begin{pmatrix}
\mathrm{S}^{\prime} e^{i[\mathrm{lat}-k\mathbf{x}\_1(1+\beta)]} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{pmatrix} \tag{28}
$$

By substituting equations (27~28) into equations (16a~b),

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 17

$$\begin{vmatrix} \Upsilon\_1 k^2 - \rho\_{11} o^2 + ibo & (\Upsilon\_2 + M\_4 \beta) k^2 - \rho\_{12} o^2 - ibo \\ (\Upsilon\_2 + M\_4 \alpha) k^2 - \rho\_{12} o^2 - ibo & \Upsilon\_3 k^2 - \rho\_{22} o^2 + ibo \end{vmatrix} = 0,\tag{29}$$

where

16 Wave Processes in Classical and New Solids

where

comes to

By substituting equation (17) and equation (23) into equations (22a~b),

Frequency-dependent S-velocity can be solved through equation (24).

term of fluid's shear deformation, S-velocity expression is reduced to

**3.3. Velocity expressions under Uniaxial confining pressure** 

The small dynamic strain induced by plane P-waves vibration along axis 1 is

 

*ij ij*

 

*Se S e*

*i t kx i t kx*

 

 1 1 [ (1 )] [ (1 )] 0 0 0 0 0 00 , 0 00 0 00 0 00

relations induced by confining pressure is expressed as

By substituting equations (27~28) into equations (16a~b),

which is identical to equation (1b).

 

*k ib ib*

 2 2 2 11 12 2 2 12 22

By assuming pore fluid has the same static deformation as solid skeleton, equation (24)

<sup>2</sup> <sup>2</sup> <sup>6</sup>

Equation (25) can not be directly reduced to equation (1b), because low viscous fluid's shear constitutive functions has been neglected in Biot theory. Return to pure solid's nonlinear acoustic theory needs to regard pore fluid as solid grains with the same shear constitutive behaviors. Therefore, neglecting solid/fluid relative deformation and adding the neglected

<sup>2</sup> <sup>2</sup> <sup>6</sup>

For uniaxial loading along axis 1, the "large" static strain in "small-on-large" constitutive

 

00 00 0 0, 0 0 00 00

 

 

 

14 89 3 33 (3 ) . 4 2 44 4

143 8 9 3 3 (3 3 3 ) , <sup>4</sup> 44 4

> 

> >

*ij ij* (27)

 

(28)

*<sup>M</sup> <sup>M</sup> V MMM M M* (26)

*<sup>M</sup> <sup>M</sup> V MM MM* (25)

*ib ib*

 2 6 1 8 49 3 33 (3 ) ( ) 4 44 2 4 *<sup>M</sup> <sup>M</sup> M M MM* .

 

0

(24)

 

$$\begin{aligned} \Upsilon\_1 &= M\_1 + (5M\_1 + 3M\_5)\alpha + (2M\_1 + M\_2 + 6M\_5 + 2M\_8)\alpha' + (\frac{3}{2}M\_4 + 3M\_{10})\beta', \\\\ \Upsilon\_2 &= \frac{1}{2}M\_4 + (\frac{1}{2}M\_4 + M\_{10})\alpha + (M\_9 + 2M\_{10})\alpha' + (3M\_{11} + \frac{1}{2}M\_4)\beta' \end{aligned}$$

and

$$\Upsilon\_3 = M\_3 + (\frac{1}{2}M\_4 + M\_{11})\alpha + (M\_4 + 2M\_{11})\alpha' + (9M\_7 + 7M\_3)\beta' \dots$$

The two frequency-dependent P velocities are solved out from equation (29). The equation will be reduced to equation (19) in hydrostatic loading configuration if is satisfied.

If the difference between solid and fluid deformation is neglected for uniaxial loading,

$$
\rho V^2 = \mathbf{Y}\_1' + 2\mathbf{Y}\_2' + \mathbf{Y}\_3' + M\_4(\alpha + \beta). \tag{30}
$$

Equation (30) can not be directly reduced to equation (1c) like the reduction process from (19) to (1a) in hydrostatic case. In equation (30), if we replace the fluid strain with the solid strain along axis 1 and replace with along axis 2 and axis 3 (the two orthogonal directions perpendicular to axis 1), we can derive an analytical scheme which is completely compatible with equation (1c).

$$\begin{aligned} \rho V^2 &= M\_1 + M\_3 + M\_4 \\ &+ [\mathbb{E}(M\_1 + M\_3 + M\_4) + \mathbb{E}(M\_5 + M\_7 + M\_{10} + M\_{11})] \alpha \\ &+ [\mathbb{E}(M\_1 + M\_3 + M\_4) + M\_2 + \mathbb{E}(M\_5 + M\_7 + M\_{10} + M\_{11}) + \mathbb{E}(M\_8 + M\_9)] \alpha' \end{aligned} \tag{31}$$

where

$$\alpha \approx -\frac{(\lambda + \mu)P\_u}{\mu(\Im \lambda + 2\,\mu)}, \quad \alpha' \approx \frac{\lambda P\_u}{2\,\mu(\Im \lambda + 2\,\mu)}.$$

With the same "large" static strain, for transverse waves propagating perpendicular to uniaxial loading and vibrating along loading,

$$
\varepsilon\_{ij} = \begin{pmatrix}
Se^{i[\alpha t - kx\_2(1+\alpha^\*)]} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix},
\zeta\_{ij} = \begin{pmatrix}
S'e^{i[\alpha t - kx\_2(1+\beta)]} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix},
\tag{32}
$$

Then

$$
\begin{vmatrix}
\Psi k^2 - \rho\_{11}o\rho^2 + ibo & -\rho\_{12}o\rho^2 - ibo \\
\end{vmatrix} = 0,\tag{33}
$$

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 19

 

   

0,

 

> 

> > 0,

 with and ,

(38)

(39)

(40)

 

 

(42)

(43)

If solid/fluid relative motion is neglected, replace fluid strain

*MMM M MMM M MM*

For transverse waves propagating along uniaxial loading and vibrating perpendicular to

 

*i t kx i t kx*

 

*k ib ib*

 2 2 2 11 12 2 2 12 22

 

 

A similar operation for the reduction to pure solid's acoustoelasticity leads to

 

perpendicular to uniaxial loading, the dynamic strains are expressed as

 

*ij ij*

 2 2 [ (1 )] [ (1 )] 0 0 0 0 0 00 , 0 00 0 00 0 00

> 

 

*ib ib*

 2 2 <sup>2</sup> <sup>6</sup> 1 81 8 4 9 <sup>1</sup> 1 33 ( ) (2 ) ( ) 4 24 22 4 2 4 *MM M <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* (41)

For transverse waves propagating perpendicular to uniaxial loading and vibrating

 

*i t kx i t kx*

 

*k ib ib*

 2 2 2 11 12 2 2 12 22

 

*Se S e*

 3 3 [ (1 )] [ (1 )] 0 0 0 0 0 00 , 0 00 0 00 0 00

> 

 

*ib ib*

 

 

loading, the static strain is expressed as equation (35) and the dynamic strain is

*Se S e*

 

*ij ij*

1 3 4 2 5 7 10 11 8 9

<sup>1</sup> [6( ) 6( ) ( )] . <sup>2</sup>

1 3 4 2 5 7 10 11 8 9

<sup>1</sup> [( ) 3( ) ( )] <sup>2</sup>

*MMM M MMM M MM*

Then

where

Therefore

which is identical to equation (1e).

2

Equation (38) is identical to equation (1d).

*V MMM*

134

where

$$\Psi = -\frac{M\_2}{4} + (M\_1 - \frac{1}{4}M\_8)\alpha + (2M\_1 - \frac{1}{2}M\_8 - \frac{M\_6}{4})\alpha' + (\frac{3}{2}M\_4 - \frac{3}{4}M\_9)\beta' \dots$$

By replacing with and and include fluid's shear constitutive relations, equation (33) will be reduced to an identical scheme of equation (1f).

$$
\rho V^2 = -\frac{M\_2}{4} + \left[ (M\_1 + M\_3 + M\_4)(a + 2a') - \frac{M\_6}{4}a' - \frac{1}{4}(M\_8 + M\_9)(a + 2a') \right] \tag{34}
$$

For longitudinal waves propagating perpendicular to uniaxial loading, assume rock sample is subject to a uniaxial pressure along axis 2 and the P-waves transmit along axis 1. The "large" strains are expressed as

$$
\omega\_{ij} = \begin{pmatrix} \alpha' & 0 & 0 \\ 0 & \alpha & 0 \\ 0 & 0 & \alpha' \end{pmatrix}, \quad \zeta\_{ij} = \begin{pmatrix} \beta & 0 & 0 \\ 0 & \beta & 0 \\ 0 & 0 & \beta \end{pmatrix} \tag{35}
$$

The small dynamic strain induced by plane P-waves propagating along axis 1 is

$$
\varepsilon\_{ij} = \begin{pmatrix} S e^{i[\alpha t - kx\_1(1+\alpha')]} & 0 & 0 \\ & 0 & 0 & 0 \\ & & 0 & 0 \end{pmatrix}, \quad \zeta\_{ij} = \begin{pmatrix} S' e^{i[\alpha t - kx\_1(1+\beta)]} & 0 & 0 \\ & 0 & 0 & 0 \\ & & 0 & 0 \end{pmatrix} \tag{36}$$

Therefore,

$$\begin{vmatrix} \Upsilon\_1 k^2 - \rho\_{11} o^2 + ibo & (\Upsilon\_2 + M\_4 \beta) k^2 - \rho\_{12} o^2 - ibo \\ (\Upsilon\_2 + M\_4 \alpha') k^2 - \rho\_{12} o^2 - ibo & \Upsilon\_3 k^2 - \rho\_{22} o^2 + ibo \end{vmatrix} = 0 \tag{37}$$

where

$$\begin{aligned} \Upsilon\_1 &= M\_1 + (M\_1 + \frac{1}{2}M\_2 + 3M\_5 + M\_8)\alpha + (6M\_1 + \frac{1}{2}M\_2 + 6M\_5 + M\_8)\alpha' + (\frac{3}{2}M\_4 + 3M\_{10})\beta', \\\\ \Upsilon\_2 &= \frac{1}{2}M\_4 + (\frac{1}{2}M\_9 + M\_{10})\alpha + (\frac{1}{2}M\_4 + \frac{1}{2}M\_9 + 2M\_{10})\alpha' + (3M\_{11} + \frac{1}{2}M\_4)\beta' \end{aligned}$$

and

$$\Upsilon\_3 = M\_3 + (\frac{1}{2}M\_4 + M\_{11})\alpha + (M\_4 + 2M\_{11})\alpha' + (9M\_7 + 7M\_3)\beta' \dots$$

The two frequency-dependent P velocities can be solved from equation (37).

If solid/fluid relative motion is neglected, replace fluid strain with and ,

$$\begin{aligned} \rho V^2 &= M\_1 + M\_3 + M\_4 \\ &+ [(M\_1 + M\_3 + M\_4) + \frac{1}{2}M\_2 + 3(M\_5 + M\_7 + M\_{10} + M\_{11}) + (M\_8 + M\_9)] \alpha \\ &+ [6(M\_1 + M\_3 + M\_4) + \frac{1}{2}M\_2 + 6(M\_5 + M\_7 + M\_{10} + M\_{11}) + (M\_8 + M\_9)] \alpha'. \end{aligned} \tag{38}$$

Equation (38) is identical to equation (1d).

For transverse waves propagating along uniaxial loading and vibrating perpendicular to loading, the static strain is expressed as equation (35) and the dynamic strain is

$$
\boldsymbol{\varepsilon}\_{ij} = \begin{pmatrix}
\mathcal{S}e^{i[\boldsymbol{\alpha}t - k\mathbf{x}\_2(1+\alpha)]} & \mathbf{0} & \mathbf{0} \\
& \mathbf{0} & \mathbf{0} & \mathbf{0} \\
& \mathbf{0} & \mathbf{0} & \mathbf{0}
\end{pmatrix}, \ \boldsymbol{\zeta}\_{ij} = \begin{pmatrix}
\mathcal{S}'e^{i[\boldsymbol{\alpha}t - k\mathbf{x}\_2(1+\beta)]} & \mathbf{0} & \mathbf{0} \\
& \mathbf{0} & \mathbf{0} & \mathbf{0} \\
& \mathbf{0} & \mathbf{0} & \mathbf{0}
\end{pmatrix} \tag{39}
$$

Then

18 Wave Processes in Classical and New Solids

 with and 

"large" strains are expressed as

where

By replacing

Therefore,

where

and

(33) will be reduced to an identical scheme of equation (1f).

 

*k ib ib*

 2 2 2 11 12 2 2 12 22

 

 

 

 

The small dynamic strain induced by plane P-waves propagating along axis 1 is

*Se S e*

 

> 

 

The two frequency-dependent P velocities can be solved from equation (37).

*ij ij*

*i t kx i t kx*

1

2

( )

 <sup>2</sup> <sup>2</sup> <sup>6</sup> 134 8 9 <sup>1</sup> [( )( 2 ) ( )( 2 )] <sup>4</sup> 4 4 *<sup>M</sup> <sup>M</sup> V MMM M M* (34)

For longitudinal waves propagating perpendicular to uniaxial loading, assume rock sample is subject to a uniaxial pressure along axis 2 and the P-waves transmit along axis 1. The

> 

> >

 

 1 1 1 2 5 8 1 2 5 8 4 10 1 13 ( 3 ) (6 6 )( 3) 2 22 *M M M MM M M MM M M* ,

 2 4 9 10 4 9 10 11 4 1 1 11 <sup>1</sup> ( )( 2 ) (3 ) 2 2 22 <sup>2</sup> *M MM M M M M M*

3 3 4 11 4 11 7 3

( ) ( 2 ) (9 7 )

*M MM M M M M* .

22 22

 2 2 2 2 1 11 2 4 12

*k ib M k ib M k ib k ib*

2 4 12 3 22

 

 

 1 1 [ (1 )] [ (1 )] 0 0 0 0 0 00 , 0 00 0 00 0 00

> 

00 00 0 0, 0 0 00 00

 

 

 <sup>2</sup> <sup>6</sup> 18 18 4 9 1 1 33 ( ) (2 ) ( ) 4 4 2 4 24 *<sup>M</sup> <sup>M</sup> MM MM MM* .

*ib ib*

 

0,

(33)

 

(36)

(37)

   

 

 

 

 

> 

 

( ) <sup>0</sup>

 

*ij ij* (35)

 

> 

> >

 

and include fluid's shear constitutive relations, equation

$$
\begin{vmatrix}
\Psi k^2 - \rho\_{11}o\rho^2 + ibo & -\rho\_{12}o^2 - ibo \\
\end{vmatrix} = 0,\tag{40}
$$

where

$$\Psi = -\frac{M\_2}{4} + (M\_1 - \frac{M\_2}{2} - \frac{1}{4}M\_8)\alpha + (2M\_1 + \frac{M\_2}{2} - \frac{1}{2}M\_8 - \frac{M\_6}{4})\alpha' + (\frac{3}{2}M\_4 - \frac{3}{4}M\_9)\beta' - \frac{1}{4}M\_{12}\alpha'$$

A similar operation for the reduction to pure solid's acoustoelasticity leads to

$$\rho V^2 = -\frac{M\_2}{4} + \left[ (M\_1 + M\_3 + M\_4)(a + 2a') - \frac{M\_2}{2}(a - a') - \frac{M\_6}{4}a' - \frac{1}{4}(M\_8 + M\_9)(a + 2a') \right] \tag{41}$$

which is identical to equation (1e).

For transverse waves propagating perpendicular to uniaxial loading and vibrating perpendicular to uniaxial loading, the dynamic strains are expressed as

$$
\varepsilon\_{ij} = \begin{pmatrix} S e^{i[\alpha t - k x\_3 (1 + \alpha')]} & 0 & 0 \\ & 0 & 0 & 0 \\ & & 0 & 0 \end{pmatrix}, \quad \zeta\_{ij} = \begin{pmatrix} S e^{i[\alpha t - k x\_3 (1 + \beta)]} & 0 & 0 \\ & 0 & 0 & 0 \\ & & 0 & 0 \end{pmatrix} \tag{42}$$

Therefore

$$
\begin{vmatrix}
\Psi k^2 - \rho\_{11} o^2 + ibo & -\rho\_{12} o^2 - ibo \\ 
\end{vmatrix} = 0,\tag{43}
$$

where

$$\Psi = -\frac{M\_2}{4} + (M\_1 + \frac{M\_2}{2} - \frac{1}{4}M\_8 - \frac{M\_6}{4})\alpha + (2M\_1 - \frac{M\_2}{2} - \frac{1}{2}M\_8)\alpha' + (\frac{3}{2}M\_4 - \frac{3}{4}M\_9)\beta' - \frac{1}{4}$$

Nonlinear Acoustic Waves in Fluid-Saturated Porous Rocks – Poro-Acoustoelasticity Theory 21

<sup>11</sup> 2111 Kg/ m3

<sup>12</sup> -348 Kg/ m3

<sup>22</sup> 697 Kg/ m3

0.335

pressure (it holds on <sup>4</sup> 2.1569 10 at 1MHz), and slow P attenuation has very small change

*M*<sup>5</sup> 244.3 GPa *k* 1×10-12 m2 *M*<sup>6</sup> 131.2 GPa *Kb* 2.5 GPa

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

in relation to loading, therefore both have not been drawn in figure 1.

*M*<sup>1</sup> 8.566 GPa

*M*<sup>2</sup> -11.704 GPa

*M*<sup>3</sup> 0.7285 GPa

*M*<sup>4</sup> 2.7 GPa

*M*<sup>7</sup> 7.0 GPa *M*<sup>8</sup> -449.8 GPa *M*<sup>9</sup> -149.9 GPa *M*<sup>10</sup> 65.1 GPa *M*<sup>11</sup> 24.1 GPa

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

The reduction operation leads to

$$\rho V^2 = -\frac{M\_2}{4} + \left[ (M\_1 + M\_3 + M\_4)(\alpha + 2\alpha') + \frac{M\_2}{2}(\alpha - \alpha') - \frac{M\_6}{4}\alpha' - \frac{1}{4}(M\_8 + M\_9)(\alpha + 2\alpha') \right] \tag{44}$$

which is identical to equation (1g).

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