**3.3 Correlations between** *I***r and** *I***v**

With the experimental data in table 1 to table 3 and the corresponding calculations, it can be found that the rock drillability anisotropy is inherently related to the acoustic anisotropy of the rock samples. Therefore, exponential function, logarithmic function, polynomial function, and linear function, are used to make a regression analysis of the data obtained by experiments. With the matching & extrapolating effects of these regression functions comprehensively considered, exponential function is finally selected as the regression model of correlation between *I*<sup>r</sup> and *I*<sup>v</sup> . The results of regression calculations for the correlations are listed in Table 4.

In Table 4, *I*rRB is denoted as the drillability anisotropy index of the rock sample with a roller bit, *I*vp as the acoustic anisotropy index of P-wave through the rock sample, Δ*K*dRB as

Evaluation Method for Anisotropic Drilling Characteristics

σ

σ

σ

σ

σ

σ

expressed as

of the Formation by Using Acoustic Wave Information 159

*xx xx yy yy zz zz yz yz xz xz xy xy*

2 0 0 0

0 0 0 00 0 0 00 0

0 0 000

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

*xyz t*

 ∂∂∂ ∂ + + +− = ∂∂∂ ∂

*yzx t*

 ∂∂∂ ∂ + + +− = ∂∂∂ ∂

*zxy t*

where *X* , *Y* , and *Z* are respectively the body force in directions of *x*, *y* and *z* (Xu, 2011). *u*,

Substituting equation (17) into equation (18) and solving with geometric equations without

2222 2 2 11 66 44 11 66 13 44 2 222 2222 2 2 66 22 44 11 66 13 44 2 222

∂∂∂∂ ∂ ∂ = + + +− ++ ∂∂∂∂ ∂ ∂ ∂ ∂ ∂∂∂∂ ∂ ∂ = + + +− ++ ∂∂∂∂ ∂ ∂ ∂ ∂

*uuuu v w C C C CC CC t xyz x y x z*

*vvvv u w C C C CC CC t xyz x y y z*

ρ

∂∂∂ ∂ + + +− = ∂∂∂ ∂

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

44

*C*

2 000

44

0

0

0

( )( )

( )( )

( )(

( )

( )

*C*

*<sup>u</sup> <sup>X</sup>*

ρ

*<sup>v</sup> <sup>Y</sup>*

ρ

*<sup>w</sup> <sup>Z</sup>*

ρ

000

66

 ε

 ε

 ε

(17)

(18)

2 3 44 ) *<sup>v</sup> <sup>C</sup>* (19)

*y z*

(20)

 ε

 ε

 ε

is the density of the elastic media, <sup>3</sup> g / cm .

*C*

11 11 66 13 11 66 11 13 13 13 33

*C C CC CC C C C CC*

*xx yx zx*

σσσ

*yy zy xy*

σσσ

considering body force, we can get the following wave equation :

*v* and *w* are the corresponding displacements.

ρ

 

ρ

ρ

*zz xz yz*

σσσ

2 222 2

∂∂∂∂ ∂

plane of y=0(that is the xz plane), the wave equation (19) can be written as

ρ

ρ

ρ

The solutions of equation (20) are

222 66 44 2 22

 ∂∂∂ = + ∂∂∂

*vvv C C txz*

44 44 33 13 44 1 2 2 22

<sup>∂</sup> <sup>+</sup> ∂ ∂

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

> 222 2 11 44 13 44 2 22

 ∂∂∂ ∂ = + ++ ∂∂∂ ∂ ∂

*uuu w C C CC txz x z*

2 22 2 44 33 13 44 2 22

∂∂∂ ∂ = + ++ ∂∂∂ ∂ ∂

*www u C C CC t xz x z*

*wwww u C C C CC C t xyz x z*

= + + ++ + ∂∂∂∂ ∂ ∂


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

Table 4. Results of the regression calculations
