**2. Anisotropic drilling characteristics of the formation**

Although many theories have been proposed to explain the hole deviation since the 1950s (Gao et al, 1994), it is only the rock drillability anisotropy theory (Lubinski & Woods, 1953) that was recognized by petroleum engineers and widely applied to petroleum engineering because it can be used to quantify the anisotropic drilling characteristics of the formation and to explain properly the actual cases of hole deviation encountered in drilling engineering. The theory suggested that since values of rock drillability are not always the same in the directions perpendicular and parallel to the bedding plane of the formation, the formation will bring the bit a considerable force, which may likely cause changes on the original drilling direction and hole deviation.

The orthotropic or the transversely isotropic formations are the typical formations encountered frequently in drilling engineering. The anisotropic effects of the formations (rock drillability) on the well trajectory must be considered in hole deviation control and directional drilling. Based on the rock-bit interaction model, the formation force is defined and modeled in this section to describe quantitatively anisotropic drilling characteristics of the formations to be drilled.

### **2.1 Definition of rock drillability anisotropy**

Because of rock drillability anisotropy, the real drilling direction does not coincide with the resultant force direction of the drill bit (supposed that it is isotropic) on bottom hole. Besides calculating the drill bit force by BHA (bottom hole assembly) analysis, rock drillability anisotropy of the formation must be considered in hole deviation control.

The formation studied here is typical orthotropic one, and the transversely isotropic formation discussed previously is regarded as its particular case. Let **e***<sup>d</sup>* , **<sup>e</sup>***<sup>u</sup>* and **e***s* represent unit vectors in the directions of inner normal, up-dip and strike of the formation respectively, as shown in Fig.1.There are different physical properties along different directions of them. γ in Fig.1 represents dip angle of the formation to be drilled. Rock drillability anisotropy of the formation can be expressed by rock drillability anisotropy index. If the components of penetration rate of the drill bit (isotropic) along inner normal, up-dip and strike of the orthotropic formation are noted as *R*dip , *R*str and *R*n respectively, correspondingly the net applied forces are *F* dip , *F* str and *F* <sup>n</sup> respectively, the rock drillability can be defined as:

$$D\_n = \frac{R\_n}{F\_n} \; \; \; D\_{\text{dip}} = \frac{R\_{\text{dip}}}{F\_{\text{dip}}} \; \; \; \; D\_{\text{str}} = \frac{R\_{\text{str}}}{F\_{\text{str}}} \tag{1}$$

Rock drillability anisotropy of the orthotropic formation may be represented by two indexes ( *I*r1 and *I*r2 ) which are defined as:

$$I\_{r1} = \frac{D\_{\text{dip}}}{D\_{\text{n}}} \; \; \; I\_{r2} = \frac{D\_{\text{str}}}{D\_{\text{n}}} \tag{2}$$

Dip angle and strike of the formation can be obtained from the analysis of well logging and geological structure survey. The values of *I*r1 and *I*r2 for the orthotropic formation can be evaluated by the experimental analysis or using the acoustic wave information.

Evaluation Method for Anisotropic Drilling Characteristics

αγ

αγ

αγ

γ

 ϕ

 ϕ

*a a a a a a*

> 22 23

=

31 13 32 23

*a a a a*

*a a*

*a*

Where Δ= − ϕ ϕψ ; ϕ and α

> γ and ψ

trajectory. Therefore, *G*

mathematically defined as:

αand *GF*

generations of *GF*

It is obviously that the values of *G*

φ

α and *GF*φ

α

and *G*

bottom hole;

Where *GF*

drilled.

( )

αγ

sin sin

= Δ

γ

γ

α

( ) ( )

 αγ

 ϕ

 ϕ

 ϕ

cos sin cos sin cos sin sin

= Δ − Δ

 αγ

 αγ

 αγ

<sup>2</sup> <sup>13</sup>

(cos )

21 12 <sup>2</sup> <sup>22</sup>

α

= Δ − Δ +

cos sin cos sin cos sin sin

= Δ − Δ

sin sin sin sin cos cos cos

= Δ ( ) Δ +

sin cos cos sin cos

=− Δ

( )

 αγ

( )

 ϕ

> 11 12

*c c c c c c c c c c c*

33

α

φ

*c*

<sup>33</sup> sin sin cos cos cos

2

of the Formation by Using Acoustic Wave Information 151

 ϕ

( ) ( )

 γ

 γ

> α

2

 ϕ

α α

2

 ϕ

2

cos sin cos (sin ) sin cos

=− Δ Δ = Δ = = Δ =− Δ Δ = <sup>=</sup> = Δ

= Δ

cos sin sin

= <sup>=</sup> = Δ <sup>+</sup>

α  ϕ

γ

 ϕ

γ

α

α

are respectively azimuth and inclination of well trajectory on the

are not only controlled by rock drillability

(7)

αand *GF*

φare

can be used to describe the anisotropic drilling

are respectively dip angle and up dip azimuth of the formation to be

 ϕ α γ

(6)

 

(5)

cos sin cos sin cos sin sin cos cos cos

γ

 ϕ

( )

ϕ

ϕ α

ϕ

ϕ

ϕ

sin cos

( )

ϕ α

sin sin

 and *G*φ

anisotropy of the formation, but also affected by the formation geometry and the well

characteristics of the formation to be drilled. Thus, the formation force can be

*GF G W GF G W* α α

inclination) and the azimuth force (positive for decreasing the azimuth) of the formation

only an equivalent expression of anisotropic drilling characteristics of the formation and they are completely different from the mechanical action forces of the drill bit on the formation. Rock drillability anisotropy of the formation is the internal cause of the

, while weight on bit is the its external cause.

φ φ

 <sup>=</sup> =

respectively, and *W* ob is weight on bit. It should be pointed out that both *GF*

ob ob

are called as the inclination force (positive for building up the

2

Fig. 1. Descartes coordinates for the formation geometry

### **2.2 The formation force**

Assumed that the drill bit is isotropic for eliminating the effects of its tilt angle on hole deviation, the effects of the orthotropic formation on hole deviation can be presented by the formation force analysis. The two parameter equations related to the formation forces can be derived from the rock-bit interaction model (Gao & Liu, 1989):

$$\begin{cases} \mathbf{G}\_{\alpha} = \frac{t\_{22}t\_{13} - t\_{12}t\_{23}}{t\_{11}t\_{22} - t\_{12}t\_{21}} \\\\ \mathbf{G}\_{\phi} = \frac{t\_{11}t\_{23} - t\_{21}t\_{13}}{t\_{11}t\_{22} - t\_{12}t\_{21}} \end{cases} \tag{3}$$

Where *G*α and *G*φ are called as the building angle parameter (positive for building up the inclination of well trajectory) and the drifting azimuth parameter (positive for left walking of well trajectory) of the formation respectively , and the *tij* (*i*, *j*=1,2,3) can be expressed as (Gao & Liu, 1990):

$$\begin{cases} t\_{\neq} = I\_{r1} \mathcal{S}\_{\neq} + \left( \mathbf{1} - I\_{r1} \right) a\_{\neq} + \left( I\_{r2} - I\_{r1} \right) c\_{\neq} \\\\ \mathcal{S}\_{\neq} = \begin{cases} 0, \text{ i } \neq j \\\\ 1, \text{ i } = j \end{cases} \end{cases} \tag{4}$$

where *a a ij j* = *<sup>i</sup>* , *c c ij j* = *<sup>i</sup>* (*i*, *j*=1, 2, 3) can be calculated by the following equations:

Assumed that the drill bit is isotropic for eliminating the effects of its tilt angle on hole deviation, the effects of the orthotropic formation on hole deviation can be presented by the formation force analysis. The two parameter equations related to the formation forces can be

> *tt tt <sup>G</sup> tt tt*

 <sup>−</sup> <sup>=</sup> <sup>−</sup> <sup>−</sup> <sup>=</sup> −

α

φ

0, 1,

<sup>≠</sup> <sup>=</sup> <sup>=</sup>

δ

*i j i j*

where *a a ij j* = *<sup>i</sup>* , *c c ij j* = *<sup>i</sup>* (*i*, *j*=1, 2, 3) can be calculated by the following equations:

*ij*

δ

*tt tt <sup>G</sup> tt tt*

inclination of well trajectory) and the drifting azimuth parameter (positive for left walking of well trajectory) of the formation respectively , and the *tij* (*i*, *j*=1,2,3) can be expressed as

1 1 21 ( ) 1 ()

*ij r ij r ij r rij*

*t I I a I Ic*

= +− + −

are called as the building angle parameter (positive for building up the

(3)

**dip angle**

**horizontal plane** 

(4)

Fig. 1. Descartes coordinates for the formation geometry

derived from the rock-bit interaction model (Gao & Liu, 1989):

**2.2 The formation force** 

Where *G*

α and *G*φ

(Gao & Liu, 1990):

( ) ( ) ( ) ( ) ( ) ( ) 2 11 12 13 21 12 21 2 22 23 sin cos cos sin cos cos sin cos sin cos sin sin cos sin cos sin cos sin sin cos cos cos cos sin cos sin cos sin sin sin sin sin sin sin sin cos cos cos *a a a a a a a a* αγ αγ ϕ αγ ϕ αγ γ ϕ α γ ϕ α γ α γ ϕ α γ αγ ϕ αγ γ ϕ γ ϕ γ ϕ αγ ϕ α =− Δ = Δ − Δ = Δ − Δ + = Δ − Δ = = Δ = Δ ( ) Δ + ( ) 31 13 32 23 2 <sup>33</sup> sin sin cos cos cos *a a a a a* γ αγ ϕ αγ = <sup>=</sup> = Δ <sup>+</sup> (5) ( ) 2 11 12 <sup>2</sup> <sup>13</sup> 21 12 <sup>2</sup> <sup>22</sup> 23 31 13 32 23 sin cos cos sin cos (sin ) sin cos (cos ) cos sin sin *c c c c c c c c c c c* ϕ α ϕ ϕ α ϕ α α ϕ ϕ ϕ α = Δ =− Δ Δ = Δ = = Δ =− Δ Δ = <sup>=</sup> = Δ (6)

Where Δ= − ϕ ϕψ ; ϕ and α are respectively azimuth and inclination of well trajectory on the bottom hole; γ and ψ are respectively dip angle and up dip azimuth of the formation to be drilled.

( )

ϕ α

sin sin

33

*c*

2

It is obviously that the values of *G*α and *G*φ are not only controlled by rock drillability anisotropy of the formation, but also affected by the formation geometry and the well trajectory. Therefore, *G*α and *G*φ can be used to describe the anisotropic drilling characteristics of the formation to be drilled. Thus, the formation force can be mathematically defined as:

$$\begin{cases} GF\_{\alpha} = G\_{\alpha} \mathcal{W}\_{\text{ob}} \\ GF\_{\theta} = G\_{\theta} \mathcal{W}\_{\text{ob}} \end{cases} \tag{7}$$

Where *GF*α and *GF*φ are called as the inclination force (positive for building up the inclination) and the azimuth force (positive for decreasing the azimuth) of the formation respectively, and *W* ob is weight on bit. It should be pointed out that both *GF*α and *GF*φ are only an equivalent expression of anisotropic drilling characteristics of the formation and they are completely different from the mechanical action forces of the drill bit on the formation. Rock drillability anisotropy of the formation is the internal cause of the generations of *GF*α and *GF*φ, while weight on bit is the its external cause.

Evaluation Method for Anisotropic Drilling Characteristics

1989):

**3.1.2 Rock samples** 

**3.1.3 Testing method** 

standard data.

The diameter of the micro-bit is 31.75 mm.

The rotary speed is 55±1 r/min.

time for each side of the rock sample.

of the Formation by Using Acoustic Wave Information 153

where *T***<sup>v</sup>** and *T*<sup>h</sup> are two parameters representing the drilling time (seconds) in directions perpendicular and parallel to bedding plane of the core samples respectively. The standard definition of rock drillability can be expressed by the following equation (Yin,

where *K*d is the rock drillability and *T* the drilling time. Taking two sides of equation (11)

Fourteen core samples used in laboratory came from the measured depth interval of 48m∼1027 m of the well KZ-1 for scientific drilling in China, which were supplied by the Engineering Center for Chinese Continental Scientific Drilling (CCSD). In the directions perpendicular and parallel to the bedding plane, these core samples were cut into shapes of cube or cuboid and their surfaces of both ends were polished and kept parallel to each other, with an error of less than 0.2 mm. Then, the machined samples were put into an oven with a temperature of 105-110ºC and roasted for 24 h. Finally, all of the samples can be used for the

The rock drillability can be measured with a device for testing the rock drillability (shown in Fig.2). During the measurement, some weight is applied on the micro-bit by the function of a hydraulic pressure tank with the fixed poises, so that the weight on the micro-bit is kept at a constant value. The measured depth to be drilled to is set with the standard indicator, and the drilling time is logged with a stopwatch. Both the roller bit (bit of this kind has three rotating cones and each cone will rotate on its own axis during drilling) drillability and the PDC (the acronym of Polycrystalline Diamond Compact) bit drillability can be tested with the above-mentioned instrument, which is of the following

The total depth to be drilled to is 2.6 mm for the roller bit with a pre-drilled depth of 0.2 mm

During testing the rock drillability, the micro-bit is often checked so that each of the worn micro-bits should be replaced in time to ensure the testing accuracy. The testing points of drilling time for each tested side of a rock sample should be gained as many as possible and their average value is taken as the test value of the side. The grade value of each side drillability of the rock sample can be calculated by equation (16) with the test data of drilling

d

into logarithm to the base 2, we can obtain the following equations:

testing of rock drillability after cooling down to room temperature.

Weight is 90±20 N on the roller bit and 500±20 N on the PDC bit.

and 4 mm for the PDC bit with a pre-drilled depth of 1.0 mm.

*K T* d 2 = log (12)

<sup>r</sup> 2 *<sup>K</sup> I* −Δ = (14)

log log log 2 r 2 v 2 h dv dh *I T TK K K* = − = − = −Δ d (13)

### **2.3 G**α **and** *G*ϕ **of the transversely isotropic formation**

By using equations (5) and (6) and making *III* r1 r2 r = = , equation (3) can be simplified as the following expressions of *G*α and *G*φfor the transversely isotropic formation:

$$\mathbf{G}\_{a} = \frac{\left(1 - I\_{\rm r}\right) \left(\cos\alpha\sin\gamma\cos\Delta\varphi - \sin\alpha\cos\gamma\right) \left(\cos\alpha\cos\gamma + \sin\alpha\sin\gamma\cos\Delta\varphi\right)}{I\_{\rm r} + \left(1 - I\_{\rm r}\right) \left[\left(\sin\gamma\sin\Delta\varphi\right)^{2} + \left(\cos\alpha\sin\gamma\cos\Delta\varphi - \sin\alpha\cos\gamma\right)^{2}\right]} \tag{8}$$

$$G\_{\varphi} = \frac{\left(1 - I\_{\varepsilon}\right)\sin\gamma\sin\Delta\varphi\left(\cos\alpha\cos\gamma + \sin\alpha\sin\gamma\cos\Delta\varphi\right)}{I\_{\varepsilon} + \left(1 - I\_{\varepsilon}\right)\left[\left(\sin\gamma\sin\Delta\varphi\right)^{2} + \left(\cos\alpha\sin\gamma\cos\Delta\varphi - \sin\alpha\cos\gamma\right)^{2}\right]}\tag{9}$$

Where all the symbols here express the same meanings as the previous ones.

### **3. Experiments on rock anisotropy**

Evaluation of rock drillability anisotropy is necessary for hole deviation control in drilling engineering. Many efforts have been made to evaluate rock drillability of the formation through the core testing, the inverse calculation and the acoustic wave. Proposed in this section is an alternative solution by using the acoustic wave to evaluate rock drillability anisotropy of the formation. First, a correlation between the P-wave velocity anisotropy coefficient and the rock drillability anisotropy index of the formation which are calculated according to the core testing data in laboratory, is established by means of mathematical statistics. Then, a mathematical model is obtained for predicting the rock drillability anisotropy index by using the P-wave velocity anisotropy coefficient. Thus, rock drillability anisotropy of the formation can be evaluated conveniently by using the well logging or seismic data (Gao & Pan, 2006).

### **3.1 Rock drillability anisotropy**

### **3.1.1 Definition**

The transversely isotropic formation is a typical anisotropic formation, whose anisotropy can be expressed by a rock drillability anisotropy index:

$$I\_{\rm r} = \frac{D\_{\rm h}}{D\_{\rm v}} \tag{10}$$

where *D VF* v vv = and *D VF* h hh = are respectively rock drillability parameters in the directions perpendicular and parallel to the bedding plane of the transversely isotropic formation; *V*v & *F*<sup>v</sup> and *V*h & *F*h are the corresponding components of the penetration rate & the net applied force of the isotropic bit to the formation.

When the rock drillability is tested in laboratory using the core samples, the weight on the bit and the rotary speed are constant so that rock drillability anisotropy index of the transversely isotropic formation can also be expressed as:

$$I\_{\mathbf{r}} = \frac{T\_{\mathbf{v}}}{T\_{\mathbf{h}}} \tag{11}$$

where *T***<sup>v</sup>** and *T*<sup>h</sup> are two parameters representing the drilling time (seconds) in directions perpendicular and parallel to bedding plane of the core samples respectively. The standard definition of rock drillability can be expressed by the following equation (Yin, 1989):

$$K\_d = \log\_2 T \tag{12}$$

where *K*d is the rock drillability and *T* the drilling time. Taking two sides of equation (11) into logarithm to the base 2, we can obtain the following equations:

$$
\log\_2 I\_r = \log\_2 T\_\mathbf{v} - \log\_2 T\_\mathbf{h} = K\_{\mathrm{dv}} - K\_{\mathrm{dh}} = -\Delta K\_\mathrm{d} \tag{13}
$$

$$I\_{\mathbf{r}} = \mathbf{2}^{-\Delta\mathbb{K}\_{d}} \tag{14}$$

### **3.1.2 Rock samples**

152 Acoustic Waves – From Microdevices to Helioseismology

By using equations (5) and (6) and making *III* r1 r2 r = = , equation (3) can be simplified as the

( )( ) ( ) ( )( ) ( )

> () ( ) ( )( ) ( )

1 sin sin cos cos sin sin cos 1 sin sin cos sin cos sin cos

 αγ

Evaluation of rock drillability anisotropy is necessary for hole deviation control in drilling engineering. Many efforts have been made to evaluate rock drillability of the formation through the core testing, the inverse calculation and the acoustic wave. Proposed in this section is an alternative solution by using the acoustic wave to evaluate rock drillability anisotropy of the formation. First, a correlation between the P-wave velocity anisotropy coefficient and the rock drillability anisotropy index of the formation which are calculated according to the core testing data in laboratory, is established by means of mathematical statistics. Then, a mathematical model is obtained for predicting the rock drillability anisotropy index by using the P-wave velocity anisotropy coefficient. Thus, rock drillability anisotropy of the formation can be evaluated conveniently by using the well logging or

The transversely isotropic formation is a typical anisotropic formation, whose anisotropy

r

h

v *<sup>D</sup> <sup>I</sup>*

where *D VF* v vv = and *D VF* h hh = are respectively rock drillability parameters in the directions perpendicular and parallel to the bedding plane of the transversely isotropic formation; *V*v & *F*<sup>v</sup> and *V*h & *F*h are the corresponding components of the penetration rate

When the rock drillability is tested in laboratory using the core samples, the weight on the bit and the rotary speed are constant so that rock drillability anisotropy index of the

r

h *<sup>T</sup> <sup>I</sup>*

γ

> αγ

 αγ

+− Δ + Δ −

+− Δ + Δ −

1 cos sin cos sin cos cos cos sin sin cos 1 sin sin cos sin cos sin cos

− Δ − +Δ <sup>=</sup>

α

for the transversely isotropic formation:

α

2 2

2 2

 αγ

γ

> ϕ

 ϕ α

> αγ

> > ϕ

 αγ

*D*= (10)

*<sup>T</sup>* <sup>=</sup> **<sup>v</sup>** (11)

γ

 ϕ

(8)

(9)

**2.3 G**α **and** *G*ϕ **of the transversely isotropic formation** 

α and *G*φ

α γ

r

*I*

 ϕ

> ϕ

 ϕ

−Δ + Δ <sup>=</sup>

 ϕ

Where all the symbols here express the same meanings as the previous ones.

γ

γ

γ

r

*I*

r r

r r

*I I*

**3. Experiments on rock anisotropy** 

*I I*

following expressions of *G*

*G*

α

*G*

ϕ

seismic data (Gao & Pan, 2006).

**3.1 Rock drillability anisotropy** 

can be expressed by a rock drillability anisotropy index:

& the net applied force of the isotropic bit to the formation.

transversely isotropic formation can also be expressed as:

**3.1.1 Definition** 

Fourteen core samples used in laboratory came from the measured depth interval of 48m∼1027 m of the well KZ-1 for scientific drilling in China, which were supplied by the Engineering Center for Chinese Continental Scientific Drilling (CCSD). In the directions perpendicular and parallel to the bedding plane, these core samples were cut into shapes of cube or cuboid and their surfaces of both ends were polished and kept parallel to each other, with an error of less than 0.2 mm. Then, the machined samples were put into an oven with a temperature of 105-110ºC and roasted for 24 h. Finally, all of the samples can be used for the testing of rock drillability after cooling down to room temperature.
