PATENT ABSTRACT
A method of TI formation evaluation is disclosed. The method comprises receiving a plurality of borehole measurements; deriving a correlation between a first TI stiffness parameter and other TI stiffness parameters where the first and other TI stiffness parameters representing mechanical behavior of the TI formation; and computing the first and other TI stiffness parameters based on the borehole measurements and the derived correlation. The method further comprises evaluating TI formation elastic properties based on the computed first and other TI stiffness parameters. The method further comprises assuming that the shear modulus parallel to TI symmetric axis can be approximated from other moduli.

PATENT DESCRIPTION
FIELD OF THE INVENTION 
   The present invention relates generally to measurement and analysis of formation. More particularly, the present invention relates to a method of evaluating elastic properties of a transversely isotropic formation. 
   BACKGROUND OF THE INVENTION 
   It is well known that the laminated formation rock presents anisotropic mechanical properties. This anisotropic behavior is due to formation&#39;s sedimentary structures, such as the fine layers, oriented fissures/fractures, or anisotropy fibers/grains. The effects of this anisotropy on seismic shear anisotropy have been well documented since the 1970&#39;s, e.g. “Weak Elastic Anisotropy” by Leon Thomsen (Geophysics, Vol. 51, 1986). One common form of the anisotropy model, the Transversely Isotropic (TI) model, has been widely used in geophysical and geomechanical applications, e.g. “A model for bedding related formation failure” by Atkinson, C. and Bradford, 2001 (I.D.R.: OFSR/RN/2001/005/RDV/C). However, one of the major difficulties constraining the applications of the anisotropy model is how to determine the elastic constants from seismic or borehole sonic measurements. This constraint effects even the simplest anisotropy models, such as the TI model, with five independent elastic constants. 
   Recently, with the development of measuring tools, the borehole&#39;s four velocities (a compressional velocity V P , a tube wave velocity V T , a shear horizontal wave velocity V SH , and a shear vertical wave velocity V SV ) can be measured with more accuracy, for example, via Schlumberger&#39;s new sonic tool SONIC SCANNER. However, it is still impossible to determine the TI properties directly from the measured four velocities. 
   SUMMARY OF THE INVENTION 
   The current invention provides methods and apparatus for the determination of the transversely isotropic (TI) formation elastic properties directly from the borehole measurements. In accord with the objects of the invention which will be discussed in more detail below, a method of TI formation evaluation comprises receiving a plurality of borehole measurements; deriving a correlation between a first TI stiffness parameter and other TI stiffness parameters where the first and other TI stiffness parameters representing mechanical behavior of the TI formation; and computing the first and other TI stiffness parameters based on the borehole measurements and the derived correlation. The method further comprises evaluating TI formation elastic properties based on the computed first and other TI stiffness parameters. The method further comprises assuming that the shear modulus parallel to TI symmetric axis can be approximated from other moduli. The method further comprises assuming the Shear Modulus G′ (parallel to TI symmetric axis) is proportional to the shear module in the plane that inclined to TI symmetric axis with about 45 degree. 
   Additional objects and advantages of the invention will become apparent to those skilled in the art upon reference to the detailed description taken in conjunction with the provided figures. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention is illustrated by way of example and not intended to be limited by the figures of the accompanying drawings in which like references indicate similar elements and in which: 
       FIG. 1  is a diagram used to illustrate the transversely isotropic (TI) medium with a coordinate system; 
       FIG. 2  is a diagram used to illustrate elastic moduli of TI medium with the same coordinate system; 
       FIG. 3  is a flowchart showing steps associated with the present method, apparatus, and article of manufacture; 
       FIG. 4  is a diagram used to illustrate a borehole with a coordinate system relative to the TI formation coordinate system; 
       FIG. 5   a  is a diagram used to illustrate an embodiment of a correlation between two shear moduli G′ and G′ 45  of sand formation; 
       FIG. 5   b  is a diagram used to illustrate another embodiment of a correlation between two shear moduli G′ and G′ 45  of shale formation; 
       FIG. 6   a  is a diagram of a sample of four velocities measured from sonic tool of wireline logging device; 
       FIG. 6   b  is a diagram of TI formation elastic moduli calculated from the measurements form the borehole after applying the present invention; and 
       FIG. 7  is a schematic illustration of computer hardware associated with the apparatus and article of manufacture. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   In the early part of the 20th century anisotropy was more a topic of scientific research than a property used in engineering design. Nye gave an excellent introduction to anisotropy in crystals from a material scientist perspective and it was in Lekhnitski&#39;s paper “Theory of Elasticity of an Anisotropic Body” [Lekhnitski, 1963], that the mechanical properties of anisotropy material was first addressed in engineering design. 
   We start by reviewing the classical TI theory and the relations of stiffness tensor (c) and the compliance tensor (a). This is followed by giving the variation of elastic moduli (Young&#39; Modulus and Shear Modulus) along a specific line versus the inclination of this line to TI symmetric axis. From elastic theory, the deformation constitution of elastic medium can be described with the generalized Hook&#39; law as:
 
σ ij=c   ijklε   kl  (i, j, k=1,2,3)  (1)
 
where σ ij  and ε kl  are stress and elastic strain tensor respectively, and c ijkl  is the fourth order (3×3×3×3) elastic stiffness tensor.
 
   With the consideration that stress and strain tensors are symmetric tensors (σ ij =σ ji , ε ij =ε ji ), the above relation can be represented with compacted indices, following Voigt&#39;s recipe:
 
{σ}=[ c]{ε}   (2)
 
where
 
{σ}={σ 11 ,σ 22 ,σ 33 ,τ 23 ,τ 31 ,τ 12 } T   (3)
 
{ε}={ε 11 ,ε 22 ,ε 33 ,γ 23 ,γ 31 ,γ 12 }  (4)
 
here c is the compacted 2 nd  order stiffness tensor (6×6).
 
   For the TI medium, without loss of generality, we assume the symmetric axis is parallel to X 3  axis, as shown in  FIG. 1 . The full form of c can be represented as: 
   
     
       
         
           
             
               
                 
                   [ 
                   c 
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           c 
                           11 
                         
                       
                       
                         
                           c 
                           12 
                         
                       
                       
                         
                           c 
                           13 
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           c 
                           12 
                         
                       
                       
                         
                           c 
                           11 
                         
                       
                       
                         
                           c 
                           13 
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           c 
                           13 
                         
                       
                       
                         
                           c 
                           13 
                         
                       
                       
                         
                           c 
                           33 
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           c 
                           44 
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           c 
                           44 
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           c 
                           66 
                         
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   The five independent constants are c 11 , c 33 , c 44 , c 66  and c 13 , and c 12  is a dependent constant where c 12 =c 11 −2c 66 . All these parameters have been well documented in the geophysical area. 
   It is more convenient to rewrite the TI medium&#39;s stress and strain relation (equation (2)) as:
 
{ε}=[ a]{σ}   (6)
 
where a is defined as elastic compliance tensor, it relates to the stiffness tensor c as:
 
[ a]=[c]   −1   (7)
 
   Therefore, the elements in compliance tensor a can be represented by the elastic moduli of the medium as: 
                         [   a   ]     =       ⁢     [           a   11           a   12           a   13         0       0       0             a   12           a   11           a   13         0       0       0             a   13           a   13           a   33         0       0       0           0       0       0         a   44         0       0           0       0       0       0         a   44         0           0       0       0       0       0         a   66           ]                 =       ⁢     [           1   /   E             -   ν     /   E             -     ν   ′       /     E   ′           0       0       0               -   ν     /   E           1   /   E             -     ν   ′       /     E   ′           0       0       0               -     ν   ′       /     E   ′               -     ν   ′       /     E   ′             1   /     E   ′           0       0       0           0       0       0         1   /     G   ′           0       0           0       0       0       0         1   /     G   ′           0           0       0       0       0       0         1   /   G           ]                   [   8   ]               
where: E is Young&#39;s modulus in the plane orthogonal to TI symmetric axis;
         E′ is Young&#39;s modulus parallel to TI symmetric axis;   υ is Poisson&#39;s ratio in the plane orthogonal to TI symmetric axis;   υ′ is Poisson&#39;s ratio parallel to TI symmetric axis;   G is Shear modulus in the plane orthogonal to TI symmetric axis; and   G′ is Shear modulus parallel to TI symmetric axis.       
   The physical meanings of these elastic constants are shown in  FIG. 2 . The five independent elastic moduli are E, E′, υ, υ′ and G′. The other module G is not an independent parameter, it can be expressed as: 
   
     
       
         
           
             
               
                 G 
                 = 
                 
                   E 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         1 
                         + 
                         υ 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 9 
                 ) 
               
             
           
         
       
     
   
   From equations (7) and (8), we can derive the relations of C ij  and elastic moduli as: 
   
     
       
         
           
             
               
                 E 
                 = 
                 
                   
                     
                       
                         ( 
                         
                           
                             c 
                             11 
                           
                           - 
                           
                             
                               c 
                               13 
                               2 
                             
                             / 
                             
                               c 
                               33 
                             
                           
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             c 
                             12 
                           
                           - 
                           
                             
                               c 
                               13 
                               2 
                             
                             / 
                             
                               c 
                               33 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                   
                     ( 
                     
                       
                         c 
                         11 
                       
                       - 
                       
                         
                           c 
                           13 
                           2 
                         
                         / 
                         
                           c 
                           33 
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 10 
                 ) 
               
             
           
           
             
               
                 υ 
                 = 
                 
                   
                     ( 
                     
                       
                         c 
                         12 
                       
                       - 
                       
                         
                           c 
                           13 
                           2 
                         
                         / 
                         
                           c 
                           33 
                         
                       
                     
                     ) 
                   
                   
                     ( 
                     
                       
                         c 
                         11 
                       
                       - 
                       
                         
                           c 
                           13 
                           2 
                         
                         / 
                         
                           c 
                           33 
                         
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 11 
                 ) 
               
             
           
           
             
               
                 
                   E 
                   ′ 
                 
                 = 
                 
                   
                     c 
                     33 
                   
                   - 
                   
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         c 
                         13 
                         2 
                       
                     
                     
                       ( 
                       
                         
                           c 
                           11 
                         
                         + 
                         
                           c 
                           12 
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 12 
                 ) 
               
             
           
           
             
               
                 
                   υ 
                   ′ 
                 
                 = 
                 
                   
                     c 
                     13 
                   
                   
                     ( 
                     
                       
                         c 
                         11 
                       
                       + 
                       
                         c 
                         12 
                       
                     
                     ) 
                   
                 
               
             
             
               
                 ( 
                 13 
                 ) 
               
             
           
           
             
               
                 
                   G 
                   ′ 
                 
                 = 
                 
                   c 
                   44 
                 
               
             
             
               
                 ( 
                 14 
                 ) 
               
             
           
         
       
     
   
   Once the stiffness parameters C ij  have been derived from sonic or seismic measurements, the elastic moduli E, E′, υ, υ′ and G′ can be computed from the above equations (10) to (14). It should be noted that the above equations are defined at the material Cartesian coordinate system (X 3  axis parallel to the material symmetric axis). The current invention is related to a method to derive the stiffness parameters C ij  from sonic or seismic measurements thus further to evaluate the elastic moduli E, E′, υ, υ′ and G′. 
     FIG. 3  shows several steps associated with the present method, apparatus and article of manufacture and provides a general overview of the invention. In the Receive Borehole Measurements Step  30 , a plurality of measurements can be obtained from a deviated well. In one embodiment, the borehole measurements include borehole sonic measurements from a tool using acoustic technology. In another embodiment, the borehole measurements include borehole seismic measurements. Still in another embodiment, the borehole measurements include a formation compressional velocity (V P ), a tube wave velocity (V T ), a shear horizontal wave velocity (V SH ), and a shear vertical wave velocity (V SV ). Specifically, the four velocities are: (a) Formation compressional velocity V P  from a monopole source; (b) Tube wave velocity V T  from a low-frequency Stoneley dispersion; (c) Shear-horizontal (SH) wave velocity V SH  from the low-frequency extrapolation of horizontal-polarized flexural wave dispersion; and (d) Shear-vertical (SV) wave velocity V SV  from the low-frequency extrapolation of quasi vertical-polarized flexural wave dispersion. Still in another embodiment, the borehole measurements include an angle (θ) that the borehole is deviated from formation isotropic axis, as shown in  FIG. 4 . Still in another embodiment, the borehole measurements include rock mass density ρ, borehole fluid density ρ f  and fluid velocity V f . In fact, the borehole measurement can be any sonic data or seismic data that represents the mechanics behavior of formation rock. 
   Some research has been done to build equations between borehole measurements and the TI stiffness parameters C ij . For example, Sinha and Norris [1993] gave the equations of these four velocities with the TI elastic constants as: 
   
     
       
         
           
             
               
                 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     p 
                     2 
                   
                 
                 = 
                 
                   
                     
                       c 
                       11 
                     
                     ⁢ 
                     
                       sin 
                       4 
                     
                     ⁢ 
                     θ 
                   
                   + 
                   
                     
                       c 
                       33 
                     
                     ⁢ 
                     
                       cos 
                       4 
                     
                     ⁢ 
                     θ 
                   
                   + 
                   
                     0.5 
                     ⁢ 
                     
                       ( 
                       
                         
                           c 
                           13 
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             c 
                             44 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       sin 
                       2 
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                 
               
             
             
               
                 ( 
                 15 
                 ) 
               
             
           
           
             
               
                 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     sh 
                     2 
                   
                 
                 = 
                 
                   
                     
                       c 
                       44 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       c 
                       66 
                     
                     ⁢ 
                     
                       sin 
                       2 
                     
                     ⁢ 
                     θ 
                   
                 
               
             
             
               
                 ( 
                 16 
                 ) 
               
             
           
           
             
               
                 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     SV 
                     2 
                   
                 
                 = 
                 
                   
                     
                       c 
                       44 
                     
                     ⁢ 
                     
                       cos 
                       2 
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                   + 
                   
                     0.25 
                     ⁢ 
                     
                       ( 
                       
                         
                           c 
                           11 
                         
                         + 
                         
                           c 
                           33 
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                             c 
                             13 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       sin 
                       2 
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     θ 
                   
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       ρ 
                       f 
                     
                     ⁢ 
                     
                       V 
                       f 
                       2 
                     
                     ⁢ 
                     
                       V 
                       T 
                       2 
                     
                   
                   
                     ( 
                     
                       
                         V 
                         f 
                         2 
                       
                       - 
                       
                         V 
                         T 
                         2 
                       
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       c 
                       44 
                     
                     ⁢ 
                     
                       sin 
                       2 
                     
                     ⁢ 
                     θ 
                   
                   + 
                   
                     
                       c 
                       66 
                     
                     ⁢ 
                     
                       cos 
                       2 
                     
                     ⁢ 
                     θ 
                   
                   + 
                   
                     
                       1 
                       8 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               
                                 c 
                                 11 
                               
                               + 
                               
                                 c 
                                 33 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   c 
                                   13 
                                 
                               
                               - 
                               
                                 4 
                                 ⁢ 
                                 
                                   c 
                                   44 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       sin 
                       4 
                     
                     ⁢ 
                     θ 
                   
                 
               
             
             
               
                 ( 
                 18 
                 ) 
               
             
           
         
       
     
   
   The five unknown independent constants are c 11 , c 33 , c 44 , c 66  and c 13 . What can be obtained from the borehole measurements are V P , V T , V SH , V SV , θ, ρ, V f  and ρ f . Thus the four equations (15) to (18) are not enough to solve the five stiffness parameters c 11 , c 33 , c 44 , c 66  and c 13  and further the elastic moduli E, E′, υ, υ′ and G′ based on borehole measurements. 
   In the Derive a Correlation Step  32 , we derive a correlation between the five stiffness parameters c 11 , c 33 , c 44 , c 66  and c 13  based on core data. In one embodiment, which will be detailed later, we derive an equation between the five stiffness parameters c 11 , c 33 , c 44 , c 66  and c 13  as: 
                   1       f   gain     ⁢     c   44         =         (       c   33     ⁢     c   11       )     -       (       c   13     -     2   ⁢     c   66         )     2     +     4   ⁢     c   11     ⁢     c   66           4   ⁢       c   66     ⁡     (         (       c   11     -     c   66       )     ⁢     c   33       -     c     1   ⁢           ⁢   3     2       )                   (   19   )               
where f gain  is a gain factor which will be detailed and verified later.
 
   The details of the Derive a Correlation Step  32  will be explained later. At this stage, the five equations (15) to (19) are enough to solve the five stiffness parameters c 11 , c 33 , c 44 , c 66  and c 13 . Thus, in the following Compute Stiffness Parameters Step  34 , the five stiffness parameters c 11 , c 33 , c 44 , c 66  and c 13  are computed based on the borehole measurements by applying the five equations (15) to (19). Further, in the Calculate Elastic Moduli Step  36 , the elastic moduli E, E′, υ, υ′ and G′ are computed by applying the equations (10) to (14). 
   Now we turn to details of the Derive a Correlation Step  32 . We start with examining the variation of these elastic moduli along any specific direction inclined to the TI symmetric axis, which is corresponding to the case of wellbore penetration in TI formation, as shown in  FIG. 4 . 
   Since both stress and strain are second order tensors and they transform as second order tensors, the compliance tensor a (and also the stiffness tensor c) must transform as the fourth order tensor. The transformation equation is:
 
a′ ijkl =b ip b jq b kr b ls a pqrs   (20)
 
where a pqrs  and a′ ijkl  are the fourth order compliance tensors before and after the coordination system transformation; and b ip  is the transformation tensor.
 
   By rotating the coordinate space along X 1  axis with an angle of θ to the borehole coordinate system, we get the new coordinate system X′ 1 X′ 2 X′ 3  where X′ 3  is the borehole axis, as shown in  FIG. 4 . The transformation tensor is given by: 
   
     
       
         
           
             
               
                 
                   [ 
                   b 
                   ] 
                 
                 = 
                 
                   
                     [ 
                     
                       
                         
                           
                             b 
                             11 
                           
                         
                         
                           
                             b 
                             12 
                           
                         
                         
                           
                             b 
                             13 
                           
                         
                       
                       
                         
                           
                             b 
                             21 
                           
                         
                         
                           
                             b 
                             22 
                           
                         
                         
                           
                             b 
                             23 
                           
                         
                       
                       
                         
                           
                             b 
                             31 
                           
                         
                         
                           
                             b 
                             32 
                           
                         
                         
                           
                             b 
                             33 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                     
                     ] 
                   
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
         
       
     
   
   Combing equations (20) and (21) and compacting the indices as described previously (equations (2) and (6)), we can get the transformation equation, as given by Lekhniskii [1963]:
 
a′ ij =q mi q nj a mn  (i,j,m,n=1,2,3,4,5,6)  (22)
 
where q ij  takes the form:
 
   
     
       
         
           
             
               
                 
                   [ 
                   q 
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           
                             cos 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         
                           
                             - 
                             cos 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θsin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           
                             sin 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         
                           
                             cos 
                             2 
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θsin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         
                           
                             - 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                     
                       
                         
                           0 
                           ⁢ 
                           
                               
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           
                             - 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                       
                         
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 23 
                 ) 
               
             
           
         
       
     
   
   This gives the TI formation elastic compliance tensor in X′ 1 X′ 2 X′ 3  coordinate system (borehole coordinate system) as: 
                   [     a   ′     ]     =     [           a   11   ′           a   12   ′           a   13   ′           a   14   ′         0       0             a   12   ′           a   22   ′           a   23   ′           a   24   ′                   ⁢   0         0             a   13   ′           a   23   ′           a   33   ′           a   34   ′         0       0             a   14   ′           a   24   ′           a   34   ′           a   44   ′         0       0           0       0       0       0         a   55   ′           a   56   ′             0       0       0       0         a   56   ′           a   66   ′           ]             (   24   )               
with:
 a′ 11 =a 11   (25)   a′   12 =cos 2    θ·a   12 +sin 2    θ·a   13   (26)   a′   13 =sin 2    θ·a   12 +cos 2    θ·a   13   (27)   a′   14 =0.5 sin 2θ(− a   12   +a   13 )  (28)   a′   22 =cos 4    θ·a   11 +0.5 sin 2  2θ· a   13 +sin 4    θ·a   33 +sin 2  2θ· a   44   (29)   a′   23 =0.25 sin 2  2θ( a   11   +a   33 )+(sin 4  θ+cos 4  θ)· a   13 −sin 2  2θ· a   44   (30)   a′   24 =−sin θ cos 3    θ·a   11 +0.25 sin 4θ· a   13 +sin 3  θ cosθ· a   33 +0.5 sin 4θ· a   44   (31)   a′   33 =sin 4    θ·a   11 +0.5 sin 2  2θ· a   13 +cos 4    θ·a   33  +sin 2  2θ· a   44   (32)   a′   34 =−sin 3  θ cos θ· a   11 −0.25 sin 4θ· a   13 +sin θ cos 3  θ· a   33 −0.5 sin 4θ· a   44    (33)   a′   44 =cos 2  2θ· a   44 +sin 2  2θ·( a   11   +a   33 −2 a   13 )  (34)   a′   55 =cos 2  θ· a   44 +sin 2    θ·a   66    (35)   a′   56 =0.5 cos 2θ·( a   44   −a   66 )  (36)   a′   66 =sin 2    θ·a   44 +cos 2    θ·a   66    (37) 
   The non-zero a′ 14 , a′ 24  and a′ 34  illustrate that normal stress can induce not only normal strains, but also shear strains, and versa visa. The non-zero a′ 56  represents that applying a shear stress in one direction can also induce shear strain in another direction. This kind of complexity can explain clearly the complexity of seismic/sonic waves transmitting in layered rock formation. Parameter a′ 44  represents the shear compliance in the plane of X′ 2 X′ 3  along the X′ 3  axis (borehole axis). By comparing with other components of a′ ij , we find that variation of a′ 44  versus θ is relatively small. Specifically, in one embodiment, it takes the value between a 44  (where θ=0 or θ=90 in equation 34) and (a 11 +a 33 −2a 13 ) (where θ=45 in equation 34). 
   As an approximation, we propose the assumption:
 
 a   44   =f   gain ( a   11   +a   33 −2 a   13 )  (38)
 
where f gain  is the gain factor which can be derived with published core and field test data. From equation (34), equation (38) assumes the Shear Modulus G′ (parallel to TI symmetric axis) is proportional to the shear module in the direction that inclined to TI symmetric axis with an angle of about 45 degree in the plane perpendicular to formation isotropy plane.
 
   From Equations (7), (10), (11) and (12), Equation (38) can be rewritten as: 
                   1       f   gain     ⁢     G   ′         =       1   E     +       1   +     2   ⁢           ⁢     υ   ′           E   ′                 (   39   )               
or equation (19) as we stated before:
 
   
     
       
         
           
             
               
                 
                   1 
                   
                     
                       f 
                       gain 
                     
                     ⁢ 
                     
                       c 
                       44 
                     
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           c 
                           33 
                         
                         ⁢ 
                         
                           c 
                           11 
                         
                       
                       ) 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             c 
                             13 
                           
                           - 
                           
                             2 
                             ⁢ 
                             
                               c 
                               66 
                             
                           
                         
                         ) 
                       
                       2 
                     
                     + 
                     
                       4 
                       ⁢ 
                       
                         c 
                         11 
                       
                       ⁢ 
                       
                         c 
                         66 
                       
                     
                   
                   
                     4 
                     ⁢ 
                     
                       
                         c 
                         66 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               ( 
                               
                                 
                                   c 
                                   11 
                                 
                                 - 
                                 
                                   c 
                                   66 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               c 
                               33 
                             
                           
                           - 
                           
                             c 
                             
                               1 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               3 
                             
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 19 
                 ) 
               
             
           
         
       
     
   
   Thus, with the derived equation (19) together with equations (15)-(18), we can solve the five stiffness parameters c 11 , c 33 , c 44 , c 66  and c 13 , based on the borehole measurements V P , V T , V SH , V SV , θ, ρ, V f  and ρ f  , with the assumption that shear module parallel to symmetric axis has a correlation with other moduli. Therefore, the invention proposes a method to evaluate the TI formation elastic properties directly from sonic measurement. The model is based on mechanical deformation analysis and it assumes that the shear modulus parallel to TI symmetric axis can be approximated from other moduli. 
   The following part checks the accuracy of the assumption and further derives the value of the gain factor f gain  from the statistical analysis of a variety of published core data and field measurements. Here we define G′ 45  as the shear modulus in the plane that inclined to TI symmetric axis with the angle of about 45 degree. From equation (34), we will have: 
   
     
       
         
           
             
               
                 
                   1 
                   
                     G 
                     45 
                     ′ 
                   
                 
                 = 
                 
                   
                     1 
                     E 
                   
                   + 
                   
                     
                       1 
                       + 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           υ 
                           ′ 
                         
                       
                     
                     
                       E 
                       ′ 
                     
                   
                 
               
             
             
               
                 ( 
                 40 
                 ) 
               
             
           
         
       
     
   
   Comparing with equation (39) with equation (40), we understand that the gain factor f gain  can be calculated from:
 
G′ 45 =f gain  G′  (41)
 
   Three groups of data have been collected and used for check: core data published by Zhijing Wang, in the article “Seismic anisotropy in sedimentary rocks” (Geophysics, Vol. 67, NO. 5, 2002); field and core data published by Leon Thomsen, in the article “Weak Elastic Anisotropy” (Geophysics, Vol. 51, 1986); and core data published by Lev Vernik and Xingzhou Liu, in the article “Velocity anisotropy in shales: A petrophysical study” (Geophysics. Vol. 62, No. 2, 1997). 
   Table 1 a  shows measured formation rock TI stiffness tensor C, elastic moduli, and shear modulus G′ 45 , on sand and shale formation published by Zhijing Wang, where Lith=1 and 2 represents the sand formation and shale formation respectively. 
   
     
       
             
             
             
             
             
             
             
             
             
             
             
             
             
           
             
             
             
             
             
             
             
             
             
             
             
             
             
           
         
             
               TABLE 1a 
             
             
                 
             
             
                 
               G Density/ 
               C33 
               C44 
               C11 
               C66 
               C13 
               E 
                 
               E′ 
                 
               G′ 
               G45′ 
             
             
               Lith 
               cc 
               GPa 
               GPa 
               GPa 
               GPa 
               GPa 
               GPa 
               υ 
               GPa 
               υ′ 
               GPa 
               GPa 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
               2 
               2.408 
               24.64 
               6.19 
               30.62 
               7.89 
               13.37 
               20.903 
               0.3246 
               16.7756 
               0.2941 
               6.19 
               7.016852 
             
             
               2 
               2.408 
               27.2 
               6.85 
               32.22 
               8.66 
               14.45 
               22.417 
               0.2943 
               18.3374 
               0.30666 
               6.85 
               7.542153 
             
             
               2 
               2.408 
               28.53 
               7.37 
               33.17 
               9.06 
               14.64 
               23.443 
               0.2938 
               19.6403 
               0.30361 
               7.37 
               8.032859 
             
             
               2 
               2.511 
               24.41 
               5.89 
               39.9 
               11.93 
               11.25 
               31.321 
               0.3127 
               19.8851 
               0.20111 
               5.89 
               9.761457 
             
             
               2 
               2.511 
               26.28 
               6.21 
               41.84 
               12.76 
               12.78 
               32.759 
               0.2837 
               20.6635 
               0.21974 
               6.21 
               9.98113 
             
             
               2 
               2.511 
               27.58 
               6.59 
               43.25 
               13.31 
               13.45 
               33.927 
               0.2745 
               21.5378 
               0.22462 
               6.59 
               10.33452 
             
             
               2 
               2.499 
               26.8 
               6.85 
               38.87 
               12.55 
               17 
               27.769 
               0.1063 
               15.8198 
               0.32295 
               6.85 
               7.140199 
             
             
               2 
               2.499 
               28.54 
               7.21 
               40.39 
               13.21 
               18.67 
               28.067 
               0.0623 
               15.7155 
               0.34345 
               7.21 
               6.994542 
             
             
               2 
               2.499 
               29.68 
               7.83 
               41.42 
               13.63 
               18.74 
               29.404 
               0.0787 
               17.0428 
               0.33717 
               7.83 
               7.561329 
             
             
               2 
               2.574 
               35.95 
               10.8 
               49.22 
               15.96 
               13.25 
               40.859 
               0.2801 
               30.6715 
               0.19919 
               10.8 
               14.2722 
             
             
               2 
               2.574 
               38.22 
               11.69 
               51.54 
               17.22 
               14.14 
               43.267 
               0.2563 
               32.3943 
               0.206 
               11.69 
               14.99239 
             
             
               2 
               2.574 
               39.39 
               12.12 
               52.52 
               17.65 
               14.87 
               44.035 
               0.2474 
               33.0488 
               0.21322 
               12.12 
               15.18117 
             
             
               2 
               2.535 
               26.33 
               6.54 
               43.5 
               13.76 
               18.57 
               30.13 
               0.0948 
               14.7347 
               0.31221 
               6.54 
               6.971843 
             
             
               2 
               2.535 
               28.02 
               7.05 
               45.2 
               14.44 
               19.69 
               31.167 
               0.0792 
               15.4161 
               0.32006 
               7.05 
               7.221509 
             
             
               2 
               2.535 
               28.86 
               7.32 
               46.09 
               14.77 
               20.34 
               31.6 
               0.0697 
               15.6507 
               0.32471 
               7.32 
               7.297393 
             
             
               2 
               2.319 
               12.17 
               4.17 
               16.91 
               5.43 
               3.66 
               14.26 
               0.3131 
               11.0031 
               0.15941 
               4.17 
               5.263572 
             
             
               2 
               2.319 
               12.89 
               4.39 
               17.77 
               5.69 
               3.77 
               14.99 
               0.3172 
               11.7134 
               0.15604 
               4.39 
               5.595146 
             
             
               2 
               2.319 
               13.46 
               4.56 
               18.42 
               5.89 
               4.22 
               15.443 
               0.311 
               12.0387 
               0.1684 
               4.56 
               5.6885 
             
             
               2 
               2.319 
               13.91 
               4.68 
               18.9 
               6.03 
               4.55 
               15.767 
               0.3074 
               12.3014 
               0.17677 
               4.68 
               5.765172 
             
             
               2 
               2.354 
               13.59 
               4.93 
               20.28 
               6.75 
               4.01 
               17.457 
               0.2931 
               12.4015 
               0.14819 
               4.93 
               6.179742 
             
             
               2 
               2.354 
               13.4 
               5.09 
               21.66 
               7.01 
               4.57 
               18.262 
               0.3025 
               11.9744 
               0.15597 
               5.09 
               6.085604 
             
             
               2 
               2.354 
               14.82 
               5.21 
               22.29 
               7.14 
               4.91 
               18.691 
               0.3089 
               13.2287 
               0.16205 
               5.21 
               6.510713 
             
             
               2 
               2.354 
               15.07 
               5.26 
               22.59 
               7.28 
               5.1 
               18.959 
               0.3021 
               13.3711 
               0.16656 
               5.26 
               6.559709 
             
             
               2 
               2.439 
               23.41 
               4.43 
               27.94 
               5.78 
               16.49 
               14.934 
               0.2919 
               11.1392 
               0.37207 
               4.43 
               4.473524 
             
             
               2 
               2.439 
               25.21 
               5.22 
               30.51 
               6.62 
               17.21 
               17.136 
               0.2943 
               12.8122 
               0.36019 
               5.22 
               5.19123 
             
             
               2 
               2.439 
               26.36 
               5.66 
               31.64 
               7.06 
               17.52 
               18.269 
               0.2938 
               13.8722 
               0.35639 
               5.66 
               5.611492 
             
             
               2 
               2.439 
               27.43 
               6.08 
               32.73 
               7.46 
               17.75 
               19.361 
               0.2977 
               14.9622 
               0.35121 
               6.08 
               6.044836 
             
             
               2 
               2.44 
               18.33 
               4.64 
               26.33 
               5.77 
               12.22 
               15.756 
               0.3654 
               11.0669 
               0.29718 
               4.64 
               4.818533 
             
             
               2 
               2.44 
               19.31 
               4.75 
               27.36 
               5.92 
               13.21 
               16.029 
               0.3538 
               11.1708 
               0.30807 
               4.75 
               4.829492 
             
             
               2 
               2.44 
               20.07 
               4.85 
               28.56 
               6.28 
               13.87 
               16.806 
               0.3381 
               11.4355 
               0.31127 
               4.85 
               4.965544 
             
             
               2 
               2.44 
               21.11 
               4.95 
               29.67 
               6.68 
               15 
               17.332 
               0.2973 
               11.3231 
               0.32623 
               4.95 
               4.910753 
             
             
               2 
               2.44 
               22.02 
               5.02 
               30.92 
               7.03 
               15.89 
               17.958 
               0.2773 
               11.4511 
               0.33257 
               5.02 
               4.9727 
             
             
               2 
               2.605 
               36.18 
               14.73 
               54.42 
               20.23 
               7.94 
               49.844 
               0.2319 
               34.3361 
               0.11612 
               14.73 
               17.8731 
             
             
               2 
               2.605 
               37.4 
               14.95 
               55.32 
               20.36 
               8.35 
               50.421 
               0.2382 
               35.4056 
               0.11942 
               14.95 
               18.24057 
             
             
               2 
               2.605 
               38.4 
               15.12 
               56.09 
               20.48 
               8.71 
               50.917 
               0.2431 
               36.2696 
               0.1223 
               15.12 
               18.53396 
             
             
               2 
               2.605 
               39.67 
               15.34 
               56.98 
               20.63 
               9.3 
               51.454 
               0.2471 
               37.2906 
               0.12792 
               15.34 
               18.82816 
             
             
               2 
               2.627 
               39.01 
               17.48 
               52.52 
               19.22 
               5.04 
               48.392 
               0.2589 
               38.2472 
               0.07568 
               17.48 
               19.69767 
             
             
               2 
               2.627 
               41.32 
               18.14 
               53.76 
               19.44 
               5.66 
               49.23 
               0.2662 
               40.3866 
               0.08246 
               18.14 
               20.34297 
             
             
               2 
               2.627 
               43.46 
               18.53 
               54.88 
               19.58 
               6.6 
               49.857 
               0.2732 
               42.226 
               0.09348 
               18.53 
               20.76104 
             
             
               2 
               2.627 
               46.82 
               18.71 
               56 
               19.65 
               9.03 
               50.135 
               0.2757 
               44.5768 
               0.12421 
               18.71 
               20.85404 
             
             
               2 
               2.626 
               36.51 
               16.36 
               54.93 
               20.82 
               7.97 
               50.682 
               0.2171 
               34.6478 
               0.11683 
               16.36 
               18.07126 
             
             
               2 
               2.626 
               37.42 
               16.53 
               55.73 
               20.91 
               8.39 
               51.162 
               0.2234 
               35.3984 
               0.12048 
               16.53 
               18.31415 
             
             
               2 
               2.626 
               38.3 
               16.69 
               56.54 
               21.01 
               8.57 
               51.715 
               0.2307 
               36.2329 
               0.1206 
               16.69 
               18.6591 
             
             
               2 
               2.626 
               39.34 
               16.88 
               57.39 
               21.12 
               8.96 
               52.244 
               0.2368 
               37.1266 
               0.12352 
               16.88 
               18.96468 
             
             
               2 
               2.56 
               32.98 
               12.97 
               52.59 
               17.7 
               10.55 
               45.337 
               0.2807 
               29.7899 
               0.15119 
               12.97 
               15.20316 
             
             
               2 
               2.56 
               35.11 
               13.13 
               53.71 
               17.84 
               12.11 
               45.659 
               0.2797 
               31.0216 
               0.1688 
               13.13 
               15.37982 
             
             
               2 
               2.56 
               36.83 
               13.27 
               54.53 
               17.96 
               13.56 
               45.794 
               0.2749 
               31.802 
               0.1854 
               13.27 
               15.39861 
             
             
               2 
               2.56 
               38.53 
               13.41 
               55.27 
               18.09 
               15.5 
               45.665 
               0.2622 
               32.0682 
               0.20845 
               13.41 
               15.13262 
             
             
               2 
               2.597 
               45.23 
               18.08 
               58.37 
               20.63 
               5.81 
               52.977 
               0.284 
               44.3356 
               0.07697 
               18.08 
               22.26984 
             
             
               2 
               2.597 
               47.26 
               18.38 
               59.62 
               20.96 
               7.37 
               53.786 
               0.2831 
               45.855 
               0.09532 
               18.38 
               22.44292 
             
             
               2 
               2.597 
               48.78 
               18.59 
               60.58 
               21.16 
               8.7 
               54.299 
               0.2831 
               46.8599 
               0.11035 
               18.59 
               22.48881 
             
             
               2 
               2.597 
               50.48 
               18.76 
               61.63 
               21.31 
               10.71 
               54.638 
               0.282 
               47.6352 
               0.13281 
               18.76 
               22.2859 
             
             
               2 
               1.841 
               13.54 
               4.04 
               15.19 
               4.4 
               5.71 
               11.541 
               0.3115 
               10.5183 
               0.2646 
               4.04 
               4.309825 
             
             
               2 
               1.841 
               13.68 
               4.08 
               15.29 
               4.47 
               5.81 
               11.647 
               0.3028 
               10.5602 
               0.26848 
               4.08 
               4.321463 
             
             
               2 
               1.841 
               13.81 
               4.12 
               15.37 
               4.52 
               5.91 
               11.716 
               0.296 
               10.5908 
               0.27235 
               4.12 
               4.325116 
             
             
               2 
               1.841 
               13.9 
               4.15 
               15.45 
               4.56 
               6 
               11.772 
               0.2908 
               10.5942 
               0.27548 
               4.15 
               4.322604 
             
             
               2 
               1.949 
               17.46 
               5.22 
               18.24 
               5.34 
               3.56 
               14.847 
               0.3902 
               16.4776 
               0.13798 
               5.22 
               6.90662 
             
             
               2 
               1.949 
               17.75 
               5.32 
               18.4 
               5.42 
               3.73 
               15.01 
               0.3847 
               16.6781 
               0.14368 
               5.32 
               6.953497 
             
             
               2 
               1.949 
               17.98 
               5.38 
               18.52 
               5.48 
               3.9 
               15.124 
               0.3799 
               16.8136 
               0.14954 
               5.38 
               6.974186 
             
             
               2 
               1.949 
               18.2 
               5.44 
               18.61 
               5.53 
               4.01 
               15.219 
               0.3761 
               16.9706 
               0.15329 
               5.44 
               7.007904 
             
             
               2 
               1.783 
               12.79 
               3.82 
               23.43 
               6.76 
               4.93 
               18.55 
               0.372 
               11.332 
               0.14787 
               3.82 
               5.943453 
             
             
               2 
               1.783 
               13.29 
               3.89 
               23.95 
               6.85 
               5.16 
               18.848 
               0.3758 
               11.7329 
               0.15088 
               3.89 
               6.097374 
             
             
               2 
               1.783 
               13.58 
               3.92 
               24.27 
               6.88 
               5.39 
               18.965 
               0.3782 
               11.9094 
               0.15497 
               3.92 
               6.145416 
             
             
               2 
               1.783 
               13.91 
               3.98 
               24.64 
               6.95 
               5.44 
               19.218 
               0.3826 
               12.2371 
               0.15376 
               3.98 
               6.293892 
             
             
               2 
               1.725 
               10.91 
               3.71 
               22.08 
               6.86 
               1.25 
               18.859 
               0.3746 
               10.8073 
               0.04106 
               3.71 
               6.529376 
             
             
               2 
               1.725 
               11.75 
               3.82 
               22.63 
               6.96 
               2.58 
               19.058 
               0.3691 
               11.3252 
               0.08232 
               3.82 
               6.438796 
             
             
               2 
               1.725 
               12.23 
               3.89 
               23 
               6.98 
               3.2 
               19.127 
               0.3701 
               11.5908 
               0.09988 
               3.89 
               6.418833 
             
             
               2 
               1.725 
               12.63 
               3.94 
               23.43 
               6.98 
               3.54 
               19.235 
               0.3778 
               11.8682 
               0.1076 
               3.94 
               6.477492 
             
             
               1 
               2.337 
               24.38 
               6.54 
               24.27 
               6.41 
               11.24 
               17.03 
               0.3284 
               17.3062 
               0.31467 
               6.54 
               6.541576 
             
             
               1 
               2.337 
               26.8 
               6.75 
               26.4 
               6.65 
               13.53 
               17.561 
               0.3204 
               17.5311 
               0.34253 
               6.75 
               6.533244 
             
             
               1 
               2.337 
               27.3 
               7.04 
               27.16 
               6.91 
               13.66 
               18.243 
               0.32 
               18.0854 
               0.33728 
               7.04 
               6.783913 
             
             
               1 
               2.307 
               28.73 
               7.49 
               28.45 
               6.99 
               13.91 
               18.96 
               0.3562 
               19.7138 
               0.32409 
               7.49 
               7.334143 
             
             
               1 
               2.307 
               32.46 
               9.32 
               33 
               8.86 
               14.74 
               23.504 
               0.3264 
               23.4597 
               0.3053 
               9.32 
               8.992782 
             
             
               1 
               2.307 
               34.11 
               10.11 
               34.28 
               9.56 
               14.62 
               25.19 
               0.3175 
               25.4634 
               0.29571 
               10.11 
               9.785062 
             
             
               1 
               2.281 
               31.78 
               10.19 
               32.34 
               10.41 
               8.91 
               27.114 
               0.3023 
               28.1599 
               0.20315 
               10.19 
               11.51805 
             
             
               1 
               2.281 
               35.65 
               11.81 
               35.78 
               12.01 
               9.9 
               30.573 
               0.2728 
               31.5267 
               0.20825 
               11.81 
               12.88015 
             
             
               1 
               2.281 
               37.19 
               12.63 
               36.65 
               12.5 
               9.82 
               31.648 
               0.2659 
               33.1969 
               0.20331 
               12.63 
               13.51913 
             
             
               1 
               2.639 
               66.06 
               20.49 
               70.82 
               22.87 
               24.01 
               57.786 
               0.2634 
               54.0375 
               0.25036 
               20.49 
               22.1842 
             
             
               1 
               2.639 
               67.93 
               21.15 
               72 
               23.42 
               24.28 
               59.032 
               0.2603 
               55.795 
               0.2499 
               21.15 
               22.82038 
             
             
               1 
               2.639 
               69.41 
               21.59 
               72.76 
               23.82 
               24.85 
               59.742 
               0.254 
               56.792 
               0.25388 
               21.59 
               23.10136 
             
             
               1 
               2.639 
               70.41 
               21.91 
               73.72 
               24.1 
               24.95 
               60.591 
               0.2571 
               57.8646 
               0.25141 
               21.91 
               23.54303 
             
             
               1 
               2.639 
               65.96 
               23.99 
               78.21 
               25.86 
               24.61 
               64.688 
               0.2507 
               54.3907 
               0.23505 
               23.99 
               23.5364 
             
             
               1 
               2.639 
               68.66 
               24.37 
               79.1 
               26.02 
               25.64 
               65.128 
               0.2515 
               56.2747 
               0.24152 
               24.37 
               23.97617 
             
             
               1 
               2.639 
               69.87 
               24.6 
               79.48 
               26.16 
               25.89 
               65.471 
               0.2514 
               57.2989 
               0.24278 
               24.6 
               24.2716 
             
             
               1 
               2.639 
               71.24 
               24.9 
               79.97 
               26.39 
               25.91 
               66.072 
               0.2518 
               58.7105 
               0.24179 
               24.9 
               24.74985 
             
             
               1 
               2.622 
               57.05 
               24.79 
               69.64 
               26.83 
               11.27 
               64.608 
               0.204 
               54.0831 
               0.13163 
               24.79 
               25.74948 
             
             
               1 
               2.622 
               59.01 
               25.17 
               71.01 
               26.86 
               11.64 
               65.442 
               0.2182 
               55.9412 
               0.13182 
               25.17 
               26.40646 
             
             
               1 
               2.622 
               60.43 
               25.48 
               71.82 
               27.01 
               12.35 
               65.929 
               0.2204 
               57.0262 
               0.1378 
               25.48 
               26.64057 
             
             
               1 
               2.622 
               61.82 
               25.75 
               72.32 
               27.12 
               13.71 
               66.015 
               0.2171 
               57.6615 
               0.15166 
               25.75 
               26.48931 
             
             
               1 
               2.751 
               53 
               18.53 
               59 
               19.89 
               16.1 
               50.315 
               0.2648 
               46.3723 
               0.20583 
               18.53 
               19.87406 
             
             
               1 
               2.751 
               54.97 
               19.39 
               60.23 
               20.64 
               16.67 
               51.676 
               0.2518 
               47.9508 
               0.21053 
               19.39 
               20.41342 
             
             
               1 
               2.751 
               56.23 
               19.95 
               61 
               21.14 
               16.68 
               52.668 
               0.2457 
               49.25 
               0.20923 
               19.95 
               20.92572 
             
             
               1 
               2.751 
               57.24 
               20.38 
               61.58 
               21.55 
               17.15 
               53.288 
               0.2364 
               49.8924 
               0.21421 
               20.38 
               21.09877 
             
             
               1 
               2.698 
               54.92 
               18.3 
               57.92 
               18.87 
               17.1 
               48.4 
               0.2825 
               47.4319 
               0.21895 
               18.3 
               19.61695 
             
             
               1 
               2.698 
               57.08 
               19.14 
               59.17 
               19.72 
               18.29 
               49.701 
               0.2602 
               48.6003 
               0.23181 
               19.14 
               19.9061 
             
             
               1 
               2.698 
               58.38 
               19.66 
               60 
               20.24 
               18.5 
               50.692 
               0.2523 
               49.7721 
               0.23265 
               19.66 
               20.33886 
             
             
               1 
               2.698 
               59.37 
               20.03 
               60.68 
               20.63 
               18.75 
               51.431 
               0.2465 
               50.5919 
               0.23408 
               20.03 
               20.63417 
             
             
               1 
               2.074 
               16.13 
               7.04 
               17.42 
               6.67 
               2.63 
               16.207 
               0.2149 
               15.4866 
               0.12233 
               7.04 
               7.038634 
             
             
               1 
               2.074 
               22.73 
               8.9 
               22.77 
               8.77 
               4.88 
               20.917 
               0.1925 
               21.029 
               0.17429 
               8.9 
               8.933589 
             
             
               1 
               2.074 
               25.74 
               10.02 
               25.72 
               9.85 
               5.89 
               23.477 
               0.1917 
               23.554 
               0.18557 
               10.02 
               9.919808 
             
             
               1 
               2.049 
               21.77 
               9.84 
               23.75 
               10.38 
               3.32 
               22.978 
               0.1069 
               20.9456 
               0.12416 
               9.84 
               9.697682 
             
             
               1 
               2.049 
               27.38 
               11.65 
               28.7 
               11.92 
               4.93 
               27.245 
               0.1428 
               25.9316 
               0.1469 
               11.65 
               11.54774 
             
             
               1 
               2.049 
               30.26 
               12.45 
               30.95 
               12.53 
               5.72 
               29.095 
               0.161 
               28.4838 
               0.15527 
               12.45 
               12.44083 
             
             
               1 
               2.598 
               43.42 
               18.79 
               52.11 
               21.27 
               7.72 
               49.413 
               0.1616 
               41.4875 
               0.12516 
               18.79 
               19.85113 
             
             
               1 
               2.598 
               48.92 
               20 
               56.94 
               22.65 
               10.72 
               53.01 
               0.1702 
               45.5686 
               0.15631 
               20 
               20.97756 
             
             
               1 
               2.598 
               51.6 
               20.55 
               59 
               23.22 
               11.55 
               54.651 
               0.1768 
               47.8716 
               0.1614 
               20.55 
               21.77213 
             
             
               1 
               2.598 
               54.15 
               21.05 
               60.93 
               23.7 
               12.87 
               55.977 
               0.1809 
               49.701 
               0.17284 
               21.05 
               22.25172 
             
             
               1 
               2.616 
               49.82 
               23.29 
               65.15 
               25.45 
               10.28 
               60.695 
               0.1924 
               47.1581 
               0.12947 
               23.29 
               23.16312 
             
             
               1 
               2.616 
               54.1 
               23.91 
               67.14 
               25.92 
               12.61 
               61.821 
               0.1925 
               50.2424 
               0.15296 
               23.91 
               23.71458 
             
             
               1 
               2.616 
               56.91 
               24.39 
               68.48 
               26.18 
               14.18 
               62.507 
               0.1938 
               52.1565 
               0.16761 
               24.39 
               24.03937 
             
             
               1 
               2.616 
               59.37 
               24.86 
               69.61 
               26.44 
               15.58 
               63.082 
               0.1929 
               53.7472 
               0.18045 
               24.86 
               24.28798 
             
             
               1 
               2.6 
               52.61 
               24.04 
               65.67 
               26.5 
               10.98 
               61.679 
               0.1638 
               49.5321 
               0.14016 
               24.04 
               23.77488 
             
             
               1 
               2.6 
               56.27 
               24.77 
               67.97 
               26.85 
               12.06 
               63.297 
               0.1787 
               52.7329 
               0.14664 
               24.77 
               24.79926 
             
             
               1 
               2.6 
               58.68 
               24.99 
               69.47 
               27.12 
               14.82 
               63.72 
               0.1748 
               53.4939 
               0.17497 
               24.99 
               24.43243 
             
             
               1 
               2.6 
               60.69 
               25.7 
               70.56 
               27.34 
               14.64 
               64.754 
               0.1842 
               55.731 
               0.16937 
               25.7 
               25.33922 
             
             
               1 
               2.504 
               33.89 
               17.15 
               47.31 
               19.74 
               5.29 
               45.429 
               0.1507 
               32.875 
               0.09594 
               17.15 
               17.16231 
             
             
               1 
               2.504 
               39.94 
               18.92 
               52.3 
               21.14 
               8.23 
               49.235 
               0.1645 
               37.7663 
               0.13206 
               18.92 
               18.59323 
             
             
               1 
               2.504 
               44.03 
               20.11 
               54.36 
               21.96 
               9.02 
               51.106 
               0.1636 
               41.5189 
               0.1392 
               20.11 
               19.85791 
             
             
               1 
               2.504 
               46.9 
               21.13 
               54.95 
               22.62 
               8.02 
               52.281 
               0.1556 
               44.9105 
               0.12403 
               21.13 
               21.31398 
             
             
               1 
               2.609 
               37.67 
               17.75 
               47.66 
               19.81 
               6.66 
               45.469 
               0.1476 
               36.0773 
               0.11957 
               17.75 
               17.74952 
             
             
               1 
               2.609 
               43.63 
               19.41 
               52.95 
               21.22 
               8.16 
               49.854 
               0.1747 
               41.5315 
               0.12858 
               19.41 
               19.86938 
             
             
               1 
               2.609 
               47.2 
               20.37 
               54.95 
               22.02 
               9.52 
               51.506 
               0.1695 
               44.4478 
               0.14455 
               20.37 
               20.65356 
             
             
               1 
               2.609 
               50.5 
               21.11 
               56.57 
               22.67 
               10.42 
               52.905 
               0.1668 
               47.2972 
               0.15369 
               21.11 
               21.48526 
             
             
                 
             
           
        
       
     
   
   Table 1b shows measured formation rock TI stiffness tensor C, elastic moduli and shear modulus G′ 45 , on sand and shale at various condition based on test data from Leon Thomsen, where Lith=1 and 2 represents the sand formation and shale formation respectively. 
   
     
       
             
             
             
             
             
             
             
             
             
             
             
             
             
           
             
             
             
             
             
             
             
             
             
             
             
             
             
           
         
             
               TABLE 1b 
             
             
                 
             
             
                 
               Density 
               C33 
               C44 
               C11 
               C66 
               C13 
               E 
                 
               E′ 
                 
               G′ 
               G45′ 
             
             
               Lith 
               G/cc 
               GPa 
               GPa 
               GPa 
               GPa 
               GPa 
               GPa 
               υ 
               GPa 
               υ′ 
               GPa 
               GPa 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
               1 
               2.6 
               77.5102 
               26.9411 
               77.5102 
               26.5639 
               −4.8578 
               69.697 
               0.3119 
               77.047 
               −0.04768 
               26.9411 
               38.32971 
             
             
               1 
               2.5 
               28.3586 
               8.3631 
               34.5974 
               12.6283 
               10.6139 
               29.684 
               0.1753 
               23.2307 
               0.24156 
               8.3631 
               10.25309 
             
             
               1 
               2.87 
               45.0517 
               24.5714 
               50.0073 
               26.5863 
               −8.5963 
               47.89 
               −0.099 
               41.8965 
               −0.1835 
               24.5714 
               27.78611 
             
             
               1 
               2.51 
               48.7041 
               16.219 
               56.4968 
               19.2358 
               16.1196 
               48.014 
               0.248 
               41.7305 
               0.21631 
               16.219 
               18.12995 
             
             
               1 
               2.45 
               41.1644 
               13.4841 
               47.5037 
               15.264 
               14.6048 
               39.035 
               0.2787 
               34.5483 
               0.2265 
               13.4841 
               14.77649 
             
             
               1 
               2.69 
               48.2913 
               24.5014 
               51.7683 
               25.9715 
               −2.6759 
               51.618 
               −0.006 
               48.0138 
               −0.0518 
               24.5014 
               26.28811 
             
             
               1 
               2.47 
               53.9141 
               19.8239 
               56.3942 
               20.3394 
               14.3739 
               49.875 
               0.2261 
               48.1837 
               0.19933 
               19.8239 
               20.37582 
             
             
               1 
               2.43 
               61.4567 
               21.6809 
               60.8422 
               21.8977 
               17.1622 
               53.37 
               0.2186 
               53.8936 
               0.22034 
               21.6809 
               21.99302 
             
             
               1 
               2.73 
               37.1317 
               21.0076 
               43.147 
               21.0076 
               −2.8901 
               42.903 
               0.0211 
               36.7544 
               −0.0653 
               21.0076 
               21.29275 
             
             
               1 
               2.5 
               50.0864 
               19.7965 
               59.8032 
               21.8157 
               14.752 
               52.936 
               0.2133 
               44.3577 
               0.19417 
               19.7965 
               19.92454 
             
             
               1 
               2.03 
               4.03012 
               1.23505 
               4.20745 
               1.24493 
               1.63164 
               3.2319 
               0.298 
               3.13147 
               0.27538 
               1.23505 
               1.242797 
             
             
               1 
               2.71 
               58.1693 
               28.2907 
               55.1445 
               30.271 
               −0.3976 
               54.613 
               −0.098 
               58.163 
               −0.0080 
               28.2907 
               28.38576 
             
             
               1 
               2.5 
               59.2679 
               21.1848 
               63.1796 
               20.3798 
               19.1995 
               52.352 
               0.2844 
               50.6552 
               0.22429 
               21.1848 
               20.96508 
             
             
               1 
               2.16 
               31.3548 
               12.112 
               33.2361 
               12.8388 
               8.49342 
               30.042 
               0.17 
               27.8181 
               0.2082 
               12.112 
               11.876 
             
             
               1 
               2.14 
               37.8575 
               15.1874 
               38.009 
               15.3392 
               8.22773 
               35.373 
               0.153 
               34.8714 
               0.18147 
               15.1874 
               14.84668 
             
             
               1 
               2.46 
               46.528 
               16.2607 
               54.996 
               19.6754 
               20.249 
               45.173 
               0.1479 
               34.9194 
               0.28665 
               16.2607 
               14.8827 
             
             
               1 
               2.48 
               51.0943 
               18.1596 
               57.2256 
               19.794 
               21.4126 
               46.696 
               0.1796 
               38.8453 
               0.28602 
               18.1596 
               16.15916 
             
             
               1 
               2.45 
               47.8209 
               16.3968 
               52.8899 
               20.7583 
               21.8437 
               42.867 
               0.0325 
               32.9711 
               0.33991 
               16.3968 
               13.46322 
             
             
               2 
               2.075 
               22.4873 
               6.48608 
               31.2574 
               8.82108 
               3.39909 
               25.16 
               0.4262 
               21.9724 
               0.07575 
               6.48608 
               10.85167 
             
             
               2 
               2.42 
               27.598 
               5.37264 
               38.6372 
               10.8527 
               14.6763 
               28.131 
               0.296 
               19.8457 
               0.26411 
               5.37264 
               8.884666 
             
             
               2 
               2.42 
               22.4825 
               5.37264 
               33.9486 
               10.5304 
               10.5736 
               26.814 
               0.2732 
               17.7085 
               0.22575 
               5.37264 
               8.38496 
             
             
               2 
               2 
               7.03125 
               1.36455 
               10.1953 
               2.30609 
               4.96635 
               6.0435 
               0.3103 
               3.90488 
               0.31476 
               1.36455 
               1.715946 
             
             
               2 
               2.44 
               26.6683 
               8.07338 
               35.6822 
               12.4491 
               10.5216 
               30.136 
               0.2104 
               21.9034 
               0.22644 
               8.07338 
               10.04885 
             
             
               2 
               2.44 
               26.6683 
               8.07338 
               33.8154 
               10.5923 
               10.5216 
               27.24 
               0.2859 
               21.9014 
               0.22653 
               8.07338 
               9.703456 
             
             
               2 
               2.81 
               53.3925 
               26.1058 
               71.7595 
               34.303 
               1.18097 
               71.597 
               0.0436 
               53.3552 
               0.01576 
               26.1058 
               30.02977 
             
             
               2 
               2.64 
               40.175 
               18.9898 
               51.1829 
               19.9773 
               1.70759 
               48.675 
               0.2183 
               40.0816 
               0.02736 
               18.9898 
               21.3408 
             
             
               2 
               2.34 
               17.632 
               5.32131 
               21.2641 
               8.99301 
               6.97169 
               18.493 
               0.0282 
               13.6711 
               0.28407 
               5.32131 
               5.924879 
             
             
               2 
               2.34 
               21.1865 
               6.40158 
               23.6866 
               8.48849 
               8.3622 
               19.816 
               0.1672 
               16.5855 
               0.27511 
               6.40158 
               6.94772 
             
             
               2 
               2.69 
               63.1712 
               27.0315 
               71.1308 
               28.5453 
               9.61 
               67.398 
               0.1805 
               61.0026 
               0.11283 
               27.0315 
               28.6293 
             
             
               2 
               2.92 
               41.0406 
               14.9274 
               51.547 
               17.9129 
               14.2117 
               44.124 
               0.2316 
               35.0356 
               0.21127 
               14.9274 
               15.8063 
             
             
               2 
               2.31 
               40.1106 
               13.6628 
               43.3194 
               14.4825 
               13.1831 
               36.411 
               0.257 
               34.0838 
               0.22858 
               13.6628 
               14.24157 
             
             
               2 
               2.25 
               9.67832 
               1.69911 
               11.8076 
               2.25982 
               7.10818 
               5.9381 
               0.3139 
               4.38637 
               0.37224 
               1.69911 
               1.766442 
             
             
               2 
               2.34 
               21.1865 
               6.40158 
               21.7374 
               6.84969 
               8.3622 
               17.22 
               0.257 
               16.4896 
               0.28084 
               6.40158 
               6.545334 
             
             
               2 
               2.25 
               9.97928 
               1.77023 
               13.8712 
               2.83237 
               8.03082 
               6.998 
               0.2354 
               4.1368 
               0.36375 
               1.77023 
               1.784145 
             
             
               2 
               2.35 
               26.895 
               13.968 
               30.3914 
               15.9515 
               0.45875 
               30.308 
               −0.05 
               26.8804 
               0.01588 
               13.968 
               14.00978 
             
             
               2 
               2.68 
               68.9707 
               24.0879 
               70.3501 
               23.847 
               21.6151 
               59.608 
               0.2498 
               58.9237 
               0.23241 
               24.0879 
               24.01793 
             
             
               2 
               2.56 
               36.8498 
               11.0118 
               50.779 
               14.8659 
               21.4854 
               36.354 
               0.2227 
               23.9959 
               0.29913 
               11.0118 
               10.62553 
             
             
               2 
               2.31 
               44.8029 
               15.4001 
               47.0431 
               16.0161 
               16.3714 
               39.076 
               0.2199 
               36.1645 
               0.26383 
               15.4001 
               14.74206 
             
             
               2 
               2.66 
               49.0919 
               16.2415 
               57.0448 
               17.8007 
               22.4266 
               44.12 
               0.2393 
               36.276 
               0.28573 
               16.2415 
               15.15495 
             
             
               2 
               2.64 
               45.0302 
               14.954 
               52.6854 
               20.487 
               20.1119 
               43.532 
               0.0624 
               32.4678 
               0.31231 
               14.954 
               13.69686 
             
             
               2 
               2.64 
               58.8399 
               22.0495 
               74.7267 
               29.9874 
               25.2904 
               63.62 
               0.0608 
               44.5436 
               0.28264 
               22.0495 
               19.66231 
             
             
               2 
               2.25 
               10.9098 
               2.11266 
               11.2371 
               2.23942 
               7.31638 
               5.7889 
               0.2925 
               4.96057 
               0.40657 
               2.11266 
               1.857857 
             
             
               2 
               2.52 
               51.6898 
               18.4116 
               55.2047 
               20.1055 
               24.4059 
               43.406 
               0.0794 
               34.7195 
               0.34767 
               18.4116 
               13.91438 
             
             
               2 
               2.37 
               42.4262 
               72.7128 
               59.3967 
               93.7996 
               −47.017 
               −4451 
               −24.72 
               106.682 
               0.68333 
               72.7128 
               45.5383 
             
             
               2 
               1.8 
               2.01486 
               0.26958 
               2.88124 
               0.42055 
               2.02418 
               0.8476 
               0.0078 
               0.34976 
               0.4113 
               0.26958 
               0.156474 
             
             
               2 
               2.59 
               39.9616 
               10.9376 
               66.6559 
               23.5159 
               39.4187 
               14.417 
               −0.693 
               3.94319 
               0.45687 
               10.9376 
               1.802814 
             
             
                 
             
           
        
       
     
   
   Table 1c shows measured formation rock TI stiffness tensor C, elastic moduli, and shear modulus G′ 45 , on sand and shale at various condition based on test data from Lev Vernik and Xingzhou Liu, where Lith=1 and 2 represents the sand formation and shale formation respectively. 
   
     
       
             
             
             
             
             
             
             
             
             
             
             
             
             
           
             
             
             
             
             
             
             
             
             
             
             
             
             
           
         
             
               TABLE 1c 
             
             
                 
             
             
                 
               Density 
               C33 
               C44 
               C11 
               C66 
               C13 
               E 
                 
               E′ 
                 
               G′ 
               G45′ 
             
             
               Lith 
               G/cc 
               GPa 
               GPa 
               GPa 
               GPa 
               GPa 
               GPa 
               υ 
               GPa 
               υ′ 
               GPa 
               GPa 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
               2 
               2.21 
               25.6981 
               9.469629 
               38.24538 
               13.92322 
               9.594818 
               33.3225 
               0.196652 
               21.91305 
               0.197244 
               9.469629 
               10.67839 
             
             
               2 
               2.22 
               25.36217 
               9.977568 
               38.97454 
               14.66288 
               6.390109 
               35.63503 
               0.215144 
               23.68258 
               0.131421 
               9.977568 
               12.28714 
             
             
               2 
               2.33 
               27.89383 
               9.32 
               46.97303 
               15.99405 
               11.36532 
               39.8103 
               0.244535 
               23.72421 
               0.183436 
               9.32 
               12.0869 
             
             
               2 
               2.34 
               28.82903 
               9.642906 
               45.71518 
               17.69625 
               14.17368 
               38.45637 
               0.086568 
               21.65912 
               0.25293 
               9.642906 
               10.46803 
             
             
               2 
               2.35 
               25.43664 
               8.13006 
               51.25092 
               17.64286 
               15.61126 
               40.69172 
               0.153206 
               18.18506 
               0.232255 
               8.13006 
               9.513961 
             
             
               2 
               2.55 
               45.19646 
               16.19352 
               62.9873 
               21.59366 
               13.69968 
               54.67321 
               0.265955 
               40.6624 
               0.16548 
               16.19352 
               19.5992 
             
             
               2 
               2.38 
               32.93539 
               12.04875 
               45.24285 
               15.96528 
               9.803627 
               39.77208 
               0.245581 
               29.65264 
               0.167426 
               12.04875 
               14.25323 
             
             
               2 
               2.48 
               36.7598 
               14.2848 
               56.19085 
               19.30457 
               12.54674 
               48.50106 
               0.256207 
               32.49207 
               0.170073 
               14.2848 
               16.16463 
             
             
               2 
               2.44 
               27.21966 
               11.59586 
               48.3181 
               19.1296 
               7.461435 
               44.88498 
               0.173181 
               25.3123 
               0.127815 
               11.59586 
               13.91117 
             
             
               2 
               2.32 
               22.43927 
               9.003688 
               46.14851 
               17.545 
               3.740951 
               43.13305 
               0.229212 
               21.95001 
               0.065393 
               9.003688 
               13.38679 
             
             
               2 
               2.34 
               25.17466 
               10.3194 
               45.50855 
               17.82518 
               9.497451 
               40.9863 
               0.149674 
               21.91633 
               0.171537 
               10.3194 
               11.67129 
             
             
               2 
               2.5 
               27.225 
               13.225 
               58.564 
               23.56225 
               15.69441 
               49.40108 
               0.04831 
               20.1878 
               0.224195 
               13.225 
               10.87095 
             
             
               2 
               2.57 
               37.50247 
               16.84275 
               61.7057 
               24.38005 
               11.41894 
               56.68908 
               0.162612 
               34.0091 
               0.152964 
               16.84275 
               17.84458 
             
             
               2 
               2.43 
               28.25628 
               12.63211 
               55.75416 
               21.72444 
               6.999206 
               51.95163 
               0.195695 
               26.81669 
               0.10284 
               12.63211 
               15.57421 
             
             
               2 
               2.49 
               32.2704 
               15.3145 
               58.32974 
               23.62114 
               8.140698 
               54.82595 
               0.160527 
               30.36105 
               0.117272 
               15.3145 
               16.97746 
             
             
               2 
               2.46 
               33.49561 
               15.74621 
               56.20706 
               23.33654 
               8.266962 
               53.13002 
               0.138344 
               31.41646 
               0.12575 
               15.74621 
               17.04809 
             
             
               2 
               2.15 
               28.17446 
               11.07874 
               39.20074 
               12.80024 
               4.233491 
               34.2065 
               0.336167 
               27.49559 
               0.080178 
               11.07874 
               13.99859 
             
             
               2 
               1.56 
               10.38398 
               4.455516 
               13.1196 
               4.942704 
               1.472952 
               12.20178 
               0.234322 
               10.11865 
               0.090068 
               4.455516 
               5.035627 
             
             
               2 
               1.44 
               8.2944 
               3.283344 
               10.26562 
               3.779136 
               2.639997 
               9.055492 
               0.19809 
               7.219921 
               0.2035 
               3.283344 
               3.275384 
             
             
               2 
               1.56 
               11.20454 
               4.998396 
               16.1747 
               6.5559 
               1.744747 
               15.41312 
               0.175515 
               10.88807 
               0.090695 
               4.998396 
               5.76758 
             
             
               2 
               2.23 
               43.96133 
               15.77859 
               48.63385 
               19.40658 
               13.27008 
               43.87048 
               0.130299 
               37.93631 
               0.227015 
               15.77859 
               16.36057 
             
             
               2 
               1.66 
               10.20966 
               4.741126 
               13.9606 
               5.867104 
               2.789096 
               13.03618 
               0.110955 
               9.248515 
               0.172305 
               4.741126 
               4.502555 
             
             
               2 
               2.56 
               52.30182 
               20.93978 
               66.32474 
               22.4297 
               13.41736 
               57.71689 
               0.286618 
               48.20055 
               0.152835 
               20.93978 
               22.51531 
             
             
               2 
               2.6 
               53.59016 
               19.94954 
               65.26026 
               24.50474 
               22.6739 
               54.87077 
               0.119595 
               40.97578 
               0.27817 
               19.94954 
               17.79151 
             
             
               2 
               1.93 
               27.28557 
               11.96619 
               29.80866 
               13.34962 
               3.889505 
               29.03107 
               0.087337 
               26.36642 
               0.118157 
               11.96619 
               12.29474 
             
             
               2 
               1.99 
               23.68598 
               8.943856 
               29.19111 
               10.80351 
               5.798263 
               26.40329 
               0.221977 
               21.85758 
               0.157668 
               8.943856 
               10.19871 
             
             
               2 
               1.6 
               10.48576 
               3.99424 
               18.06336 
               7.056 
               3.5632 
               16.40687 
               0.162619 
               9.332314 
               0.161855 
               3.99424 
               4.931168 
             
             
               2 
               2.02 
               18.3014 
               7.2922 
               26.76419 
               9.250792 
               4.765187 
               23.59164 
               0.275115 
               17.00485 
               0.136044 
               7.2922 
               8.532769 
             
             
               2 
               2.14 
               29.2966 
               12.63649 
               36.32522 
               15.02815 
               3.72804 
               34.91426 
               0.161629 
               28.64401 
               0.087525 
               12.63649 
               14.35459 
             
             
               2 
               2.62 
               54.24055 
               19.95811 
               76.1165 
               26.32812 
               11.4956 
               67.68118 
               0.28534 
               51.58634 
               0.115445 
               19.95811 
               25.88266 
             
             
               2 
               2.57 
               47.74058 
               16.0625 
               64.76503 
               22.21405 
               17.47087 
               55.04079 
               0.238873 
               40.56727 
               0.205293 
               16.0625 
               18.88935 
             
             
               2 
               2.5 
               41.209 
               13.924 
               58.32225 
               21.17025 
               14.17309 
               51.13944 
               0.207814 
               35.80211 
               0.190745 
               13.924 
               17.19951 
             
             
               2 
               2.6 
               46.52154 
               15.35274 
               63.96416 
               21.866 
               15.81606 
               54.82052 
               0.253556 
               40.57953 
               0.187847 
               15.35274 
               19.1782 
             
             
               2 
               2.55 
               49.368 
               16.71168 
               68.1587 
               23.25702 
               10.56468 
               60.19612 
               0.29415 
               46.88229 
               0.117642 
               16.71168 
               23.27692 
             
             
               2 
               2.6 
               52.88426 
               18.67424 
               64.22234 
               22.78016 
               24.00629 
               52.19433 
               0.145609 
               38.9781 
               0.289636 
               18.67424 
               16.75713 
             
             
               2 
               2.51 
               37.398 
               12.37028 
               55.4459 
               17.75976 
               13.39453 
               46.12938 
               0.298705 
               32.63727 
               0.177712 
               12.37028 
               15.82076 
             
             
               2 
               2.42 
               24.7808 
               9.68 
               39.10817 
               13.70785 
               9.561464 
               33.61054 
               0.22596 
               21.18157 
               0.188215 
               9.68 
               10.55576 
             
             
               2 
               1.93 
               16.79583 
               7.642993 
               22.04909 
               9.172132 
               2.913788 
               21.06848 
               0.148505 
               16.1365 
               0.11314 
               7.642993 
               8.099894 
             
             
               2 
               2.73 
               78.13943 
               22.9593 
               98.93629 
               30.09115 
               32.22083 
               78.07729 
               0.297346 
               63.05947 
               0.234009 
               22.9593 
               27.71025 
             
             
               2 
               2.69 
               87.09171 
               28.9401 
               103.0703 
               31.0964 
               25.61674 
               83.89862 
               0.349009 
               77.97428 
               0.177959 
               28.9401 
               34.11987 
             
             
               2 
               2.63 
               60.34298 
               20.03429 
               79.26846 
               28.29459 
               17.78374 
               69.91958 
               0.235564 
               54.1386 
               0.17444 
               20.03429 
               25.49886 
             
             
               2 
               2.6 
               89.89344 
               26.45786 
               106.1635 
               29.87946 
               34.22102 
               81.1747 
               0.35837 
               74.54188 
               0.2243 
               26.45786 
               31.4936 
             
             
               2 
               2.52 
               35.62675 
               12.7575 
               60.5052 
               23.75075 
               13.74249 
               54.12946 
               0.139532 
               30.48844 
               0.18695 
               12.7575 
               15.73881 
             
             
                 
             
           
        
       
     
   
   Results of G′ and G′ 45  are plotted in  FIGS. 5   a  and  5   b  for data of sand formation and shale formation respectively. It can be concluded from  FIGS. 5   a  and  5   b  that the value of the gain factor f gain  is around 1.0. Therefore, it verifies the assumption that the shear modulus G′ (parallel to TI symmetric axis) and shear module G′ 45  in the plane that inclined to TI symmetric axis with about 45 degree are very close. 
   The statistical study has been carried out to build the correlation gain factor f gain  versus formation type and Thomsen&#39;s parameter γ. The results are shown in Table 2. 
   
     
       
             
             
             
             
           
         
             
               TABLE 2 
             
             
                 
             
             
                 
               Anisotropy 
               Gain 
               Uncertainty (%) 
             
             
               Formation 
               parameter γ 
               factor f gain   
               (Standard Deviation) 
             
             
                 
             
           
           
             
               Sand 
                 0~0.05 
               1.01 
               +/−4  
             
             
                 
               0.05~0.25 
               1.03 
               +/−8  
             
             
               Shale 
                 0~0.05 
               1.08 
               +/−8  
             
             
                 
               0.05~0.10 
               1.08 
               +/−10 
             
             
                 
               0.10~0.15 
               1.08 
               +/−12 
             
             
                 
               0.15~0.20 
               1.15 
               +/−16 
             
             
                 
               0.20~0.60 
               1.25 
               +/−25 
             
             
                 
             
           
        
       
     
   
   A case study has been attached to demonstrate the process of determining the TI formation elastic properties using the above invention. Specifically, we used the field measurement of sonic data from SONIC SCANNER of Schlumberger (as shown in  FIG. 6   a ) and applied equations (15)-(19) to derive the TI elastic constants, and further use equations (10) to (14) to derive the elastic moduli E, E′, υ, υ′ (Poisson ratios PR and PR′ here) and G′ (as shown in  FIG. 6   b ). 
     FIG. 7  schematically illustrates computer hardware that may be used to implement the inventive method. Computer  70  has a media reading device, such as a CD-ROM Reader  72 , a floppy disk device, or a ZIP drive. The media reading device may also be capable of recording the output of the program the computer  70  is running. A user of the computer  70  may enter commands using a user input device, such as a keyboard  74  or a mouse, may view output of the program code on a visual display device, such as monitor  76 , and may make hardcopies of output using an output device, such as printer  78 . When properly configured, computer  70  (and its associated peripheral devices) is an apparatus for outputting the elastic moduli E, E′, υ, υ′ and G′ directly from borehole measurement in accordance with the present invention. Computer media, such as a CD-ROM  79 , a floppy disk, or a ZIP disk, may have computer readable program code that allows the computer  70  to output the elastic moduli E, E′, υ, υ′ and G′ directly from borehole measurement in accordance with the inventive method. 
   The foregoing description of the preferred and alternate embodiments of the present invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or limit the invention to the precise examples described. Many modifications and variations will be apparent to those skilled in the art. The embodiments were chosen and described in order to best explain the principles of the invention and its practical application, thereby enabling others skilled in the art to understand the invention for various embodiments and with various modifications as are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the accompanying claims and their equivalents. 
   
     
       
             
             
             
           
         
             
                 
                 
             
             
                 
               Symbol 
               Definition 
             
             
                 
                 
             
           
           
             
                 
               E and E′ 
               Young&#39;s moduli 
             
             
                 
               υ and υ′ 
               Poisson ratio 
             
             
                 
               G and G′ 
               shear modulus 
             
             
                 
               σ ij   
               stress 
             
             
                 
               ε kl   
               elastic strain 
             
             
                 
               C ij   
               stiffness parameter 
             
             
                 
               V P   
               formation compressional velocity 
             
             
                 
               V T   
               tube wave velocity 
             
             
                 
               V SH   
               shear horizontal wave velocity 
             
             
                 
               V SV   
               shear vertical wave velocity 
             
             
                 
               θ 
               angle 
             
             
                 
               ρ 
               rock mass density 
             
             
                 
               ρ f   
               borehole fluid density 
             
             
                 
               a 
               compliance tensor 
             
             
                 
               c 
               stiffness tensor 
             
             
                 
               f gain   
               gain factor 
             
             
                 
               b 
               transformation tensor 
             
             
                 
               γ 
               Thomsen&#39;s parameter 
             
             
                 
               X′ 1 X′ 2 X′ 3   
               borehole coordinate system axis