**4.1.1 Phase velocity in the transversely isotropic formation**

For the transversely isotropic media, Hooke's law can be written as

the rock drillability difference between both directions perpendicular and parallel to the bedding plane of the rock sample with a roller bit, calculated by equation (13), and *I*rPDC as

rRB e *a bI I* <sup>+</sup> = ( ) vp

Value −3.418 8.129 −4.286

Value 3.401 -9.993 4.366

Standard error 0.840 1.893 1.491 t-ratio −4.0679 4.295 −2.875 Prob(t) 0.00226 0.00157 0.02068

Standard error 0.914 2.361 1.661 t-ratio 3.721 −4.232 2.628 Prob.(t) 0.00397 0.00174 0.03026

F-ratio 16.92 22.41 8.09

Prob.(F) 0.0021 0.0008 0.0217

Many studies have been made to evaluate the rock drillability anisotropy with the core testing method (Gao & Pan, 2006) and the inversion method (Gao et al, 1994). However, as for the core testing method, its result may not reflect the actual rock drillability anisotropy since the experimental conditions are different from the downhole conditions. Moreover, the profile of rock drillability anisotropy along the hole depth can not be established because of the limitation of the core samples. The inversion method needs to work with a bottom hole assembly (BHA) analysis program and some parameters in the inversion model are not easy to obtain so that its applications are limited to some extent. Thus, the evaluation method will be presented in this section so as to predict rock drillability anisotropy of the formation by

The formation studied here is the transversely isotropic formation which is frequently encountered in drilling for oil & gas. Experimental investigation shows that layered rock has

Correlation coefficient R 0.793 0.832 0.710

**4. Evaluation method based on acoustic wave information** 

dRB e *a bI K* <sup>+</sup> Δ = ( ) vp

rPDC e *a bI I* <sup>+</sup> =

the drillability anisotropy index of the rock sample with a PDC bit.

Regression functions ( ) vp

Table 4. Results of the regression calculations

using the acoustic wave information (Gao et al, 2008).

**4.1.1 Phase velocity in the transversely isotropic formation**  For the transversely isotropic media, Hooke's law can be written as

**4.1 Acoustic wave velocity of the formation** 

the transversely isotropic characteristics.

a

b


Elastodynamic equation of the elastic media can be obtained from the textbook and expressed as

$$\begin{cases} \frac{\partial \sigma\_{xx}}{\partial x} + \frac{\partial \sigma\_{yx}}{\partial y} + \frac{\partial \sigma\_{zx}}{\partial z} + X - \rho \frac{\partial^2 u}{\partial t^2} = 0 \\\\ \frac{\partial \sigma\_{yy}}{\partial y} + \frac{\partial \sigma\_{zy}}{\partial z} + \frac{\partial \sigma\_{xy}}{\partial x} + Y - \rho \frac{\partial^2 v}{\partial t^2} = 0 \\\\ \frac{\partial \sigma\_{zz}}{\partial z} + \frac{\partial \sigma\_{xz}}{\partial x} + \frac{\partial \sigma\_{yz}}{\partial y} + Z - \rho \frac{\partial^2 w}{\partial t^2} = 0 \end{cases} \tag{18}$$

where *X* , *Y* , and *Z* are respectively the body force in directions of *x*, *y* and *z* (Xu, 2011). *u*, *v* and *w* are the corresponding displacements. ρ is the density of the elastic media, <sup>3</sup> g / cm . Substituting equation (17) into equation (18) and solving with geometric equations without considering body force, we can get the following wave equation :

$$\begin{cases} \rho \frac{\partial^2 u}{\partial t^2} = \mathcal{C}\_{11} \frac{\partial^2 u}{\partial x^2} + \mathcal{C}\_{66} \frac{\partial^2 u}{\partial y^2} + \mathcal{C}\_{44} \frac{\partial^2 u}{\partial z^2} + (\mathcal{C}\_{11} - \mathcal{C}\_{66}) \frac{\partial^2 v}{\partial x \partial y} + (\mathcal{C}\_{13} + \mathcal{C}\_{44}) \frac{\partial^2 w}{\partial x \partial z} \\ \rho \frac{\partial^2 v}{\partial t^2} = \mathcal{C}\_{66} \frac{\partial^2 v}{\partial x^2} + \mathcal{C}\_{22} \frac{\partial^2 v}{\partial y^2} + \mathcal{C}\_{44} \frac{\partial^2 v}{\partial z^2} + (\mathcal{C}\_{11} - \mathcal{C}\_{66}) \frac{\partial^2 u}{\partial x \partial y} + (\mathcal{C}\_{13} + \mathcal{C}\_{44}) \frac{\partial^2 w}{\partial y \partial z} \\ \rho \frac{\partial^2 w}{\partial t^2} = \mathcal{C}\_{44} \frac{\partial^2 w}{\partial x^2} + \mathcal{C}\_{44} \frac{\partial^2 w}{\partial y^2} + \mathcal{C}\_{33} \frac{\partial^2 w}{\partial z^2} + (\mathcal{C}\_{13} + \mathcal{C}\_{44}) \frac{\partial^2 u}{\partial x \partial z} + (\mathcal{C}\_{13} + \mathcal{C}\_{44}) \frac{\partial^2 v}{\partial y \partial z} \end{cases} \tag{19}$$

Because of the symmetry of the stress and strain in the direction normal to z direction, the wave equation can be simplified to two dimensions without any loss of generality. In the plane of y=0(that is the xz plane), the wave equation (19) can be written as

$$\begin{cases} \rho \frac{\partial^2 u}{\partial t^2} = \mathbb{C}\_{11} \frac{\partial^2 u}{\partial x^2} + \mathbb{C}\_{44} \frac{\partial^2 u}{\partial z^2} + (\mathbb{C}\_{13} + \mathbb{C}\_{44}) \frac{\partial^2 w}{\partial x \partial z} \\ \rho \frac{\partial^2 v}{\partial t^2} = \mathbb{C}\_{66} \frac{\partial^2 v}{\partial x^2} + \mathbb{C}\_{44} \frac{\partial^2 v}{\partial z^2} \\ \rho \frac{\partial^2 w}{\partial t^2} = \mathbb{C}\_{44} \frac{\partial^2 w}{\partial x^2} + \mathbb{C}\_{33} \frac{\partial^2 w}{\partial z^2} + (\mathbb{C}\_{13} + \mathbb{C}\_{44}) \frac{\partial^2 u}{\partial x \partial z} \end{cases} \tag{20}$$

The solutions of equation (20) are

Evaluation Method for Anisotropic Drilling Characteristics

0 is the vertical P-wave velocity;

where

where

*D*

Letting 2

series at the fixed

Where

θ

π θ

density.

α

ε , γ  and \* δ

β

α

of the Formation by Using Acoustic Wave Information 161

 are rock anisotropy parameters of the formation. Substituting equation (24), (25), (26) and (27) into equation (21), (22) and (23), we can get

2 2 2\* *v D Pa* ( ) [1 sin ( )]

 θ

2 2

εθ

 γ0

θ

=− + <sup>+</sup> <sup>−</sup> − − (31)

( )

 ε

1 2

0.5

0.5

θ

sin cos sin

 θε

and neglecting the second order of small quantity can be expressed as

 θε

> θ

θ

.

( )

 γ

1 2

 θ

 α

 β

θ

 ε

22 2 \* 0 0 0 2 2 0 0

β

2 \* 2 2 \* 2 0 0 2 0 4 1/2 2 2 <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> 0 0 <sup>0</sup> <sup>0</sup> <sup>0</sup> 14 4(1 / ) ( ) (1 ){[1 sin cos sin ] 1} <sup>2</sup> (1 / ) (1 / )

 θ

= and substituting it into equations (28), (29), (30) and (31), we can get

,90 0 ,90 0

= +

α

β

β

\* 22 4

Using equations (33) in equation (28) and (29), expanding *vPa* , *vSVa* and *vSHa* in a Taylor

22 4 *vPa* ( ) (1 sin cos sin )

2

0 *vSVa* ( ) [1 ( )sin cos ] α

<sup>2</sup> *vSHa* ( ) (1 sin )

β

 θ

<sup>0</sup> 2 2 <sup>0</sup> <sup>2</sup>

 γ0

2 (1 / ) δ

β α

 = + <sup>−</sup>

εδ

 δ θ

*Pa SVa SHa*

\*

δ

(1 / )

β α

2 2 0 0

*v v v*

 = 

,90 0

= +

*v D SVa* ( ) [1 sin ( )] α

22 2 *vSHa* ( ) [1 2 sin ]

θα

β

> θ = + β

θ

> δ

 βα

For the case of weak rock anisotropy (i.e. the quantity of

θ α

> θ

β

> θ = + β

1

 ε

δ

expanding equation (31) in the Taylor series at fixed

*D*

small quantity can be simplified as

θ

0 is the vertical SV-wave velocity;

=+ + <sup>0</sup> , (28)

 θ

=+ − (29)

βαε ε

> ε , γ ,θ

 θ

≈ + − (33)

 θ=+ + 0 (34)

> θ

=+ − (35)

− +

β α ρ

(30)

 θ

(32)

and neglecting the second order of

(36)

 and \* δ

is small),

is rock

β

$$\rho \upsilon \upsilon\_{\mathbb{M}}^2(\theta) = \frac{1}{2} [\mathcal{C}\_{33} + \mathcal{C}\_{44} + (\mathcal{C}\_{11} - \mathcal{C}\_{33}) \sin^2 \theta + D(\theta)] \tag{21}$$

$$\rho \rho v\_{\text{SV}4}^2(\theta) = \frac{1}{2} [\mathbf{C}\_{33} + \mathbf{C}\_{44} + (\mathbf{C}\_{11} - \mathbf{C}\_{33}) \sin^2 \theta - D(\theta)] \tag{22}$$

$$
\rho \upsilon\_{\text{SH}}^2(\theta) = \mathbb{C}\_{\theta^6} \sin^2 \theta + \mathbb{C}\_{44} \cos^2 \theta \tag{23}
$$

Where

$$\begin{split} D(\boldsymbol{\theta}) &= \left\{ \left( \mathbf{C}\_{33} - \mathbf{C}\_{44} \right)^{2} + 2 \left[ 2 \left( \mathbf{C}\_{13} + \mathbf{C}\_{44} \right)^{2} - \left( \mathbf{C}\_{33} - \mathbf{C}\_{44} \right) \left( \mathbf{C}\_{11} + \mathbf{C}\_{33} - 2 \mathbf{C}\_{44} \right) \right] \sin^{2} \theta + \\ &+ \left[ \left( \mathbf{C}\_{11} + \mathbf{C}\_{33} - 2 \mathbf{C}\_{44} \right)^{2} - 4 \left( \mathbf{C}\_{13} + \mathbf{C}\_{44} \right)^{2} \right] \sin^{4} \theta \right\}^{1/2} \end{split}$$

where *vPa* is phase velocity of the P-wave; *vSVa* is phase velocity of P-SV wave; *vSHa* is phase velocity of SH-wave; θ is phase angle which is the angle between the wave front normal and the unique (vertical) axis as shown in Fig.4.

Fig. 4. Phase angle and group angle It is defined that

$$
\varepsilon = \frac{\mathbf{C}\_{11} - \mathbf{C}\_{33}}{2\mathbf{C}\_{33}} \tag{24}
$$

$$\mathcal{Y} = \frac{\mathbf{C\_{66}} - \mathbf{C\_{44}}}{2\mathbf{C\_{44}}} \tag{25}$$

$$\boldsymbol{\delta}^\* = \frac{1}{2\mathbf{C}\_{33}^2} \left[ 2(\mathbf{C}\_{13} + \mathbf{C}\_{44})^2 - (\mathbf{C}\_{33} - \mathbf{C}\_{44})(\mathbf{C}\_{11} + \mathbf{C}\_{33} - 2\mathbf{C}\_{44}) \right] \tag{26}$$

and

$$
\alpha\_0 = \sqrt{\mathbb{C}\_{3^3} / \rho} \quad ; \quad \beta\_0 = \sqrt{\mathbb{C}\_{44} / \rho} \tag{27}
$$

where α0 is the vertical P-wave velocity; β0 is the vertical SV-wave velocity; ρ is rock density. ε , γ and \* δare rock anisotropy parameters of the formation.

Substituting equation (24), (25), (26) and (27) into equation (21), (22) and (23), we can get

$$
\psi\_{\mathbb{P}4}(\theta) = \alpha\_0^2 \left[ 1 + \varepsilon \sin^2 \theta + D^"(\theta) \right], \tag{28}
$$

$$v\_{\rm SVa}^2(\theta) = \beta\_0^2 \left[ 1 + \frac{\alpha\_0^2}{\beta\_0^2} \varepsilon \sin^2 \theta - \frac{\alpha\_0^2}{\beta\_0^2} D^\circ(\theta) \right] \tag{29}$$

$$v\_{\rm Sta}^2(\theta) = \beta\_0^2 \left[ 1 + 2\gamma \sin^2 \theta \right] \tag{30}$$

where

160 Acoustic Waves – From Microdevices to Helioseismology

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

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

2 2 <sup>2</sup> 33 44 13 44 33 44 11 33 44

= − + + − − +− + + +− − + ,

θθ

θ

is phase angle which is the angle between the wave front

wavefront

<sup>−</sup> <sup>=</sup> (24)

<sup>−</sup> <sup>=</sup> (25)

0 44 = *C* / (27)

= ++ − −

2 22

*vC C SHa* ( ) sin cos

( ) {( ) 2[2( ) ( )( 2 )]sin

where *vPa* is phase velocity of the P-wave; *vSVa* is phase velocity of P-SV wave; *vSHa* is

O

(source)

g

 11 33 2 <sup>33</sup> *C C C*

\* 2 13 44 33 44 11 33 44 <sup>2</sup>

ρ , β

66 44 2 <sup>44</sup> *C C C*

<sup>1</sup> [2( ) ( )( 2 )] <sup>2</sup> *C C C CC C C*

= + − − +− (26)

 ρ

ε

γ

= ++ − +

 θ

> θ

θ

θ

= + <sup>66</sup> 44 (23)

(21)

(22)

 θ

2 2 33 44 11 33

2 2 33 44 11 33

*v CC CC D Pa*

*v CC CC D SVa*

ρ

ρ

Where

θ

phase velocity of SH-wave;

θ

θ

ρ

[( 2 ) 4( ) ]sin }

*CC C CC*

θ

A

ray

*a v*

33

α

0 33 = *C* /

*C*

δ

*g v*

Fig. 4. Phase angle and group angle

It is defined that

and

normal and the unique (vertical) axis as shown in Fig.4.

θ

θ

01

2 2 <sup>4</sup> 1/2 11 33 44 13 44

*D C C C C C CC C C*

$$D'(\theta) = \frac{1}{2} (1 - \frac{\beta\_0^2}{\alpha\_0^2}) [[1 + \frac{4\delta'}{(1 - \beta\_0^2 \,/\, /\, \alpha\_0^2)^2} \sin^2 \theta \cos^2 \theta + \frac{4(1 - \beta\_0^2 \,/\, /\, \alpha\_0^2 + \varepsilon)\varepsilon}{(1 - \beta\_0^2 \,/\, /\, \alpha\_0^2)^2} \sin^4 \theta]^{1/2} - 1\} \tag{31}$$

Letting 2 π θ= and substituting it into equations (28), (29), (30) and (31), we can get

$$\begin{cases} \upsilon\_{v\_{H,90}} = \alpha\_0 \left( 1 + 2\varepsilon \right)^{0.5} \\\\ \upsilon\_{SVa,90} = \beta\_0 \\\\ \upsilon\_{SHa,90} = \beta\_0 \left( 1 + 2\gamma \right)^{0.5} \end{cases} \tag{32}$$

For the case of weak rock anisotropy (i.e. the quantity of ε , γ ,θ and \* δ is small), expanding equation (31) in the Taylor series at fixed θ and neglecting the second order of small quantity can be simplified as

$$^\*D^\* = \frac{\delta^\*}{(1 - \beta\_0^2 / \alpha\_0^2)} \sin^2 \theta \cos^2 \theta + \varepsilon \sin^4 \theta \tag{33}$$

Using equations (33) in equation (28) and (29), expanding *vPa* , *vSVa* and *vSHa* in a Taylor series at the fixed θand neglecting the second order of small quantity can be expressed as

$$w\_{\mathbb{P}^4}(\theta) = \alpha\_0 (1 + \delta \sin^2 \theta \cos^2 \theta + \varepsilon \sin^4 \theta) \tag{34}$$

$$w\_{\rm SVu}(\theta) = \beta\_0 \left[ 1 + \frac{\alpha\_0^2}{\beta\_0^2} (\varepsilon - \delta) \sin^2 \theta \cos^2 \theta \right] \tag{35}$$

$$w\_{SHa}(\theta) = \beta\_0 (1 + \mathcal{Y} \sin^2 \theta) \tag{36}$$

Where

$$
\delta = \frac{1}{2} \left[ \mathcal{E} + \frac{\delta^\*}{(1 - \beta\_0^2 \,/\,\alpha\_0^2)} \right].
$$

Evaluation Method for Anisotropic Drilling Characteristics

2

*d vR* θα

> ε

*Pa*

 θθ

 θθ

> ε δ

θ

Substituting equations (41) and (42) into equation (38), we can get

*Pa*

Where

Where

θ

θ

φ

*Pa*

direction and rock anisotropy parameters,

φ θ

> θ= 0

In the case of

*v R*

 and θ

θ θ

of the Formation by Using Acoustic Wave Information 163

*dv tt t tR*

( ) <sup>1</sup> 24 2 <sup>2</sup> <sup>2</sup> *R tt t t D t* ( ) 4( 2 2 )sin 4 (2 )sin 2 ( )

 θ

 θ

θ

θ

θ

θ θα

 <sup>=</sup> <sup>90</sup> , ( ) <sup>0</sup> *Pa dv d* θ

velocity of the point can also be made out by using equation (41) and (44).

**4.1.3 Methodology for determining rock anisotropy parameters**  From the above discussion, we know that if rock anisotropy parameters

for measuring the velocity of P-wave or S-wave is illustrated in Fig.5.

θ

θ

−− −− <sup>=</sup> − −

 ε

ε

<sup>0</sup> 2 2 ( ) sin cos 2( 2 2 )sin 2 ( ) ()()

 δ

 δε

<sup>2</sup> 32 4

 δε2 2 *t t* ;

=+ − *tR t* ;

=+ − *tRt* ;

 θ

= +− + − + = + .

{ } <sup>2</sup> 32 1 4 3

*MM M M M MM M*

2 ( ) ( ) 2 sin ( ) 2 ( ) tan tan 2 ( ) ( ) sin ( ) *<sup>g</sup>*

> <sup>2</sup> *M*<sup>1</sup> =− + ε

*M*<sup>2</sup> () 4 () 2

 δεθ

*M*3() 2 ()

Substituting equation (41) and (42) into equation (39), we can get the following equation:

<sup>1</sup> ( ) ( ) ( ) sin cos 2 sin ( ) ()() *Pg g Pa*

*v v R M M*

Where *vPg* is the group velocity of the P-wave; *vPa* is the phase velocity of the P-wave.

 δεθ

<sup>2</sup> *M t tR R* <sup>4</sup> () () 2()

( ) { } <sup>1</sup>

velocity.Thus, when the group angle of the P-wave at the considered point of the formation is given, its phase angle can be calculated by using equation (43). Phase velocity and group

known, the wave velocity at any direction can be calculated by using acoustic wave velocity perpendicular to the bedding of the formation. In other words, the acoustic wave velocity perpendicular to the bedding of the formation can be make out if the wave velocity at any

It is assumed that the formation to be drilled is transversely isotropic with symmetry axis perpendicular to the bedding of the formation and the formation properties do not change significantly from one well to another. Acoustic wave logging provides a way to measure the velocity of P-wave or S-wave in the formation (or slowness time). The schematic figure

δ , ε and γ

 θ

 θθ=− + .

 θ

> θ

 θ

θθθ

 ε

 ε

<sup>2</sup> <sup>2</sup> 4 2 42 2 <sup>2</sup> 0 13

 θ

, are known.

=+ + (44)

<sup>=</sup> + − + −+ (42)

 δε  εθ

> θ

> > (43)

 θθ

> θ

<sup>=</sup> , the phase velocity is equal to the group

 θ

δ , ε and γ, are

### **4.1.2 Phase velocity and group velocity**

The phase velocity is the velocity in the direction of the phase propagation vector, normal to the surface of constant phase, which is also called the wave front velocity since it is the propagation velocity of the wave front along the phase vector. The phase angle is formed between the direction of phase vector and the vertical axis. In contrast, the ray vector points always from the source to the considered point on the wave front. The energy propagates along the ray vector with the group velocity, while the group angle is formed between the propagation direction and the vertical axis. The difference between the phase angle and the ray angle is illustrated in Fig.4.

The relationship between the phase angle and the group angle can be expressed by the following equation:

$$\tan(\phi\_\circ - \theta) = \frac{d\upsilon\_\circ}{\upsilon\_\circ} \tag{37}$$

Where φ*g* is the group angle; *va* is the phase velocity; θ is the phase angle. Expanding equation (37) leads to

$$\tan(\phi\_t(\theta)) = (\tan \theta + \frac{1}{v\_\*} \frac{dv\_\*}{d\theta}) / \left(1 - \frac{\tan \theta}{v\_\*} \frac{dv\_\*}{d\theta}\right) \tag{38}$$

The group velocity ( *vg* ) is related to the phase velocity ( *va* ) as shown by the following formula

$$v\_s^2 \left(\phi\_t(\theta)\right) = v\_s^2(\theta) + \left(\frac{dv\_a}{d\theta}\right)^2\tag{39}$$

Where *vg* is the group velocity.

The following section makes the solution for the relationship between the group velocity and the phase velocity, the group angle and the phase angle of the P-wave at any angle. Another rock anisotropy parameter δis introduced and expressed as

$$\mathcal{S} = \frac{1}{2} \left[ \mathcal{E} + \frac{\mathcal{S}^\*}{\left(1 - \beta\_0^2 \;/\,\alpha\_0^2\right)} \right] = \frac{\left(\mathcal{C}\_{13} + \mathcal{C}\_{44}\right)^2 - \left(\mathcal{C}\_{33} - \mathcal{C}\_{44}\right)^2}{2\mathcal{C}\_{33}\left(\mathcal{C}\_{33} - \mathcal{C}\_{44}\right)} \tag{40}$$

Letting 2 2 *t* = −1 / β0 0 α, equation (38) can be rescaled as

$$\frac{\upsilon\_{\rm Pl}^2(\theta)}{\alpha\_0^2} = 1 + \varepsilon \sin^2 \theta + D(\theta) \tag{41}$$

Where

$$D(\theta) = \frac{1}{2}\sqrt{4(\varepsilon^2 + 2t\varepsilon - 2t\delta)\sin^4\theta + 4t(2\delta - \varepsilon)\sin^2\theta + t^2} - \frac{1}{2}t\theta \dots$$

Making derivation to both sides of equation (41) , we can get

$$\frac{d\upsilon\_{\mathbb{H}}(\theta)}{d\theta} = \frac{a\_{\mathbb{0}}^{2}\sin\theta\cos\theta}{\upsilon\_{\mathbb{H}}(\theta)\mathbb{R}(\theta)} \Big[ 2(\varepsilon^{2} + 2\varepsilon t - 2t\delta)\sin^{2}\theta + 2t\delta - \varepsilon t + \varepsilon \mathbb{R}(\theta) \Big] \tag{42}$$

Where

162 Acoustic Waves – From Microdevices to Helioseismology

The phase velocity is the velocity in the direction of the phase propagation vector, normal to the surface of constant phase, which is also called the wave front velocity since it is the propagation velocity of the wave front along the phase vector. The phase angle is formed between the direction of phase vector and the vertical axis. In contrast, the ray vector points always from the source to the considered point on the wave front. The energy propagates along the ray vector with the group velocity, while the group angle is formed between the propagation direction and the vertical axis. The difference between the phase angle and the

The relationship between the phase angle and the group angle can be expressed by the

1 tan tan( ( )) (tan ) /(1 ) *a a*

The group velocity ( *vg* ) is related to the phase velocity ( *va* ) as shown by the following

( ) <sup>2</sup> 2 2 () () *<sup>a</sup>*

The following section makes the solution for the relationship between the group velocity and the phase velocity, the group angle and the phase angle of the P-wave at any angle.

> 1 ( ) ( ) 2 (1 / ) 2 ( )

 + −− =+ = − −

( ) 1 sin ( ) *vPa <sup>D</sup>*

1 1 24 2 <sup>2</sup> ( ) 4( 2 2 )sin 4 (2 )sin 2 2

 θ

= + − + − +− *t t tt* .

 θ

= +

θ

*a*

− = (37)

is the phase angle. Expanding

(39)

(40)

*dv d v* θ

*a*

*a a*

*dv*

*d*

is introduced and expressed as

\* 22 13 44 33 44

*CC CC CC C*

=+ + (41)

 θ

0 0 33 33 44

2

εθθ

 δε

θ

*dv dv vd v d*

θ

θ

> θ

=+ − (38)

tan( )

φ θ

 θ

*gg a*

*v v*

φθ

δ

δ

β α

, equation (38) can be rescaled as

2 0

α

 ε δ

θ

2

2 2

*g* is the group angle; *va* is the phase velocity;

*g*

φθ

*g*

**4.1.2 Phase velocity and group velocity** 

ray angle is illustrated in Fig.4.

following equation:

φ

equation (37) leads to

Where *vg* is the group velocity.

Another rock anisotropy parameter

Letting 2 2 *t* = −1 / β0 0 α

Where

δ

 ε

*D t*

 ε

Making derivation to both sides of equation (41) , we can get

θ

Where

formula

$$R(\theta) = \left(4(\varepsilon^2 + 2t\varepsilon - 2t\delta)\sin^4\theta + 4t(2\delta - \varepsilon)\sin^2\theta + t^2\right)^{\frac{1}{2}} = 2D(\theta) + t \dots$$

Substituting equations (41) and (42) into equation (38), we can get

$$\tan \phi\_t = \frac{\left[2\left[M\_3(\theta) - M\_2(\theta) - 2M\_1\right]\sin^2 \theta - M\_4(\theta) - 2M\_3(\theta)\right]\tan \theta}{2\left[M\_3(\theta) - M\_2(\theta)\right]\sin^2 \theta - M\_4(\theta)}\tag{43}$$

Where

$$\begin{aligned} M\_1 &= \varepsilon^2 - 2t\delta + 2t\varepsilon \ ; \\\\ M\_2(\theta) &= 4t\delta + \varepsilon R(\theta) - 2t\varepsilon \ ; \\\\ M\_3(\theta) &= 2t\delta + \varepsilon R(\theta) - t\varepsilon \ ; \\\\ M\_4(\theta) &= t^2 - tR(\theta) + 2R(\theta) \ . \end{aligned}$$

Substituting equation (41) and (42) into equation (39), we can get the following equation:

$$\upsilon\_{\mathbb{P}\_{\mathbb{R}}}\left(\phi\_{\mathbb{A}}(\theta)\right) = \frac{1}{\upsilon\_{\mathbb{P}\mathbb{A}}(\theta)\mathbb{R}(\theta)} \left[\upsilon\_{\mathbb{A}}^{4}(\theta)\mathbb{R}^{2}(\theta) + \alpha\_{0}^{4}\sin^{2}\theta\cos^{2}\theta \left[2M\_{1}\sin^{2}\theta + M\_{3}(\theta)\right]^{2}\right]^{\frac{1}{2}}\tag{44}$$

Where *vPg* is the group velocity of the P-wave; *vPa* is the phase velocity of the P-wave. In the case of θ = 0 and θ <sup>=</sup> <sup>90</sup> , ( ) <sup>0</sup> *Pa dv d* θ θ <sup>=</sup> , the phase velocity is equal to the group velocity.Thus, when the group angle of the P-wave at the considered point of the formation is given, its phase angle can be calculated by using equation (43). Phase velocity and group velocity of the point can also be made out by using equation (41) and (44).

### **4.1.3 Methodology for determining rock anisotropy parameters**

From the above discussion, we know that if rock anisotropy parameters δ , ε and γ , are known, the wave velocity at any direction can be calculated by using acoustic wave velocity perpendicular to the bedding of the formation. In other words, the acoustic wave velocity perpendicular to the bedding of the formation can be make out if the wave velocity at any direction and rock anisotropy parameters, δ , ε and γ, are known.

It is assumed that the formation to be drilled is transversely isotropic with symmetry axis perpendicular to the bedding of the formation and the formation properties do not change significantly from one well to another. Acoustic wave logging provides a way to measure the velocity of P-wave or S-wave in the formation (or slowness time). The schematic figure for measuring the velocity of P-wave or S-wave is illustrated in Fig.5.

Evaluation Method for Anisotropic Drilling Characteristics

of the Formation by Using Acoustic Wave Information 165

Fig. 6. Flow chart for the inversion calculations of rock anisotropy parameters

In the figure 5, *S*1 and *S*2 are monopole sonic transducers. *R*1, *R*2, *R*3 and *R*4 are sonic receivers. When the sonic is transmitted from *S*1, time difference between *R*2 and *R*4 is recorded. In the same way, when the sonic is transmitted from *S*2, time difference between *R*1 and *R*3 is recorded. The average of time difference between *R*2 and *R*4 and time difference between *R*1 and *R*3 is the velocity in the formation measured.

Fig. 5. The principle of acoustic wave logging

Since available S-wave velocity is limited in logging data, we restrict ourselves to take consideration of the P-wave only. The frequency of the acoustic wave logging is about 20kHz~25kHz, which has long wave length. Since a monopole sonic transducer has a mini-bulk, a monopole borehole sonic tool may be approximated by a point source in line with an array of point receivers. The group velocity surface is the response from a point source and so the monopole sonic tool response is approximated as a point source coupled with a series of point receivers in an infinite media (neglecting borehole effects). Therefore, we measure group velocity with borehole sonic tools.

Three parameters, the vertical P-wave velocity (α*0*) and the anisotropy parameters ε and δ can be recovered using borehole sonic measurement at different angles relative to the axis of symmetry by following objective function:

$$
\Delta v\_P = \frac{1}{n} \sum\_{i=1}^n \left[ v\_{Pi}(\theta) - v\_{Pi}(\theta) \right]^2 \tag{45}
$$

where *vPmi*( ) θ is the measured P-wave velocity; *vPci*( ) θ is the P-wave velocity calculated by equation (32); *n* is the total number of the measured signals. The goal of the inversion is to find the optimization value of *C*11, *C*13, *C*33 and *C*44, to minimize value of Δ*vP* , by which α*0*, δ and εcan be calculated, as shown in Fig.6 (Gao et al, 2008).

In the figure 5, *S*1 and *S*2 are monopole sonic transducers. *R*1, *R*2, *R*3 and *R*4 are sonic receivers. When the sonic is transmitted from *S*1, time difference between *R*2 and *R*4 is recorded. In the same way, when the sonic is transmitted from *S*2, time difference between *R*1 and *R*3 is recorded. The average of time difference between *R*2 and *R*4 and time difference

Since available S-wave velocity is limited in logging data, we restrict ourselves to take consideration of the P-wave only. The frequency of the acoustic wave logging is about 20kHz~25kHz, which has long wave length. Since a monopole sonic transducer has a mini-bulk, a monopole borehole sonic tool may be approximated by a point source in line with an array of point receivers. The group velocity surface is the response from a point source and so the monopole sonic tool response is approximated as a point source coupled with a series of point receivers in an infinite media (neglecting borehole effects). Therefore,

α

<sup>1</sup> () ()

θ

 θ

θ

Δ= − (45)

can be recovered using borehole sonic measurement at different angles relative to the axis of

equation (32); *n* is the total number of the measured signals. The goal of the inversion is to find the optimization value of *C*11, *C*13, *C*33 and *C*44, to minimize value of Δ*vP* , by which

1

*i v vv*

=

*n*

*n P Pmi Pci*

*0*) and the anisotropy parameters

is the P-wave velocity calculated by

ε and δ

> α*0*, δ

between *R*1 and *R*3 is the velocity in the formation measured.

Fig. 5. The principle of acoustic wave logging

we measure group velocity with borehole sonic tools. Three parameters, the vertical P-wave velocity (

<sup>2</sup>

can be calculated, as shown in Fig.6 (Gao et al, 2008).

is the measured P-wave velocity; *vPci*( )

symmetry by following objective function:

where *vPmi*( )

and ε θ

Fig. 6. Flow chart for the inversion calculations of rock anisotropy parameters

Evaluation Method for Anisotropic Drilling Characteristics

**5. Case study** 

Well

conformation in Qinghai oilfield

of the Formation by Using Acoustic Wave Information 167

Fig. 7. Inversion method of the acoustic wave velocity perpendicular to the bedding plane

Based on some well logging data and drilling information from Qinghai oilfield in west China, the case study is presented in this section to verify the evaluation method for the

Based on these data in table 5, rock drillability anisotropy of the fromation and its anisotropic drilling characteristics can be calculated by using the evaluation method described above. The inversion result of shale anisotropy parameters is shown in table 6.

Well 5 gamma-ray, compensated acoustic wave and compensated density,

Well 6 gamma-ray, compensated acoustic wave and compensated density,

Well 7 gamma-ray, compensated acoustic wave and compensated density,

Table 5. Well drilling & logging information from some completed wells at the Honggouzi

inclinometer data and geologic stratification data

inclinometer data and geologic stratification data

inclinometer data and geologic stratification data, and some other records

anisotropic drilling characteristics of the formation to a certain extent.

Number Well logging information

### **4.2 Prediction model of rock drillability anisotropy**

Based on the previous section 3, a calculation model has been established to predict rock drillability anisotropy of the formation:

$$I\_r = \mathbf{2}^{\mathbf{C}\_1 \mathbb{K}\_{4\nu} \ast \mathbf{C}\_2} \tag{46}$$

$$K\_{\rm dv} = \mathcal{C}\_3 + \mathcal{C}\_4 \ln(\Delta t) \tag{47}$$

Where *K*dv is the rock drillability perpendicular to the bedding plane of the formation; Δ*t* is the time interval of acoustic wave in the same direction, us/m; *Cj* (*j*=1,2,3,4) are the regression coefficients based on the experimental data and the survey data in drilling engineering. For example, by the regression analysis based on some oilfield data in west China, we can get such coefficients as *C*1=0.05246, *C*2=-0.76732, *C*3=32.977, *C*4=-4.950.

### **4.3 Evaluation method of rock drillability anisotropy based on acoustic wave**

From equations (46) and (47), it is shown that the key point for the evaluation of rock drillability anisotropy is how to obtain the rock drillability perpendicular to the bedding plane of the formation which depends on the time interval of acoustic wave in the same direction. Thus, the evaluation of rock drillability anisotropy comes down to determine the time interval of acoustic wave perpendicular to the bedding plane of the formation.

Provided that the formation is of the transversely isotropy and has the symmetry axis perpendicular to the bedding plane of the formation, the angle between hole axis and the formation normal can be calculated by the following formula which is derived from transformation of the formation coordinates to the bottom hole coordinate.

$$\boldsymbol{\alpha} = \arccos\left[\cos\alpha\cos\beta - \sin\alpha\sin\beta\cos(\phi - \phi\_{\uparrow})\right] \tag{48}$$

where ω is the angle between hole axis and normal of the formation; α is hole inclination, degree or radian; φ is azimuth, degree or radian; β is stratigraphic dip, degree or radian; φ*<sup>f</sup>* is azimuth of the formation tendency, degree or radian.

When rock anisotropy parameters of a hole section is known, its acoustic wave velocity perpendicular to the bedding plane of the formation can be calculated by the following procedures:


The flow chart for inversion of the acoustic wave velocity perpendicular to the bedding plane of the formation is shown in figure 7.

The rock drillability can be calculated by equation (47) after obtaining the time interval of the acoustic wave perpendicular to the bedding plane of the formation. Thus, the profile of rock drillability anisotropy index can be established by using equation (46).

Fig. 7. Inversion method of the acoustic wave velocity perpendicular to the bedding plane
