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.

Description:
FIELD OF THE INVENTION 
       [0001]    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 
       [0002]    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. 
         [0003]    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 
       [0004]    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. 
         [0005]    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  
         [0006]    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: 
           [0007]      FIG. 1  is a diagram used to illustrate the transversely isotropic (TI) medium with a coordinate system; 
           [0008]      FIG. 2  is a diagram used to illustrate elastic moduli of TI medium with the same coordinate system; 
           [0009]      FIG. 3  is a flowchart showing steps associated with the present method, apparatus, and article of manufacture; 
           [0010]      FIG. 4  is a diagram used to illustrate a borehole with a coordinate system relative to the TI formation coordinate system; 
           [0011]      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; 
           [0012]      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; 
           [0013]      FIG. 6   a  is a diagram of a sample of four velocities measured from sonic tool of wireline logging device; 
           [0014]      FIG. 6   b  is a diagram of TI formation elastic moduli calculated from the measurements form the borehole after applying the present invention; and 
           [0015]      FIG. 7  is a schematic illustration of computer hardware associated with the apparatus and article of manufacture. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0016]    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. 
         [0017]    We start by reviewing the classical TI theory and the relations of stiffness tensor (c) and the compliance tensor (α). 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: 
         [0000]      σ ij   =c   ijkl ε kl  ( i, j, k= 1,2,3)   (1) 
         [0000]    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. 
         [0018]    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: 
         [0000]      {σ}=[c]{ε}  (2) 
         [0000]    where 
         [0000]      {σ}={σ 11 ,σ 22 ,σ 33 ,τ 23 ,τ 31 ,τ 12 } T   (3) 
         [0000]      {ε}={ε 11 ,ε 22 ,ε 33 ,γ 23 ,γ 31 ,γ 12 }  (4) 
         [0000]    here c is the compacted 2 nd  order stiffness tensor (6×6). 
         [0019]    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: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     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 
                   ) 
                 
               
             
           
         
       
     
         [0020]    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. 
         [0021]    It is more convenient to rewrite the TI medium&#39;s stress and strain relation (equation (2)) as: 
         [0000]      {ε}=[α]{σ}  (6) 
         [0000]    where α is defined as elastic compliance tensor, it relates to the stiffness tensor c as: 
         [0000]      [α]=[ c]   −1   (7) 
         [0022]    Therefore, the elements in compliance tensor α can be represented by the elastic moduli of the medium as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           [ 
                           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 
                   ] 
                 
               
             
           
         
       
     
         [0000]    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.       
 
         [0029]    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: 
         [0000]    
       
         
           
             
               
                 
                   G 
                   = 
                   
                     E 
                     
                       2 
                        
                       
                         ( 
                         
                           1 
                           + 
                           υ 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0030]    From equations (7) and (8), we can derive the relations of C ij  and elastic moduli as: 
         [0000]    
       
         
           
             
               
                 
                   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 
                   ) 
                 
               
             
           
         
       
     
         [0031]    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′. 
         [0032]      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 ρƒ 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. 
         [0033]    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: 
         [0000]    
       
         
           
             
               
                 
                   
                     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 
                   ) 
                 
               
             
           
         
       
     
         [0034]    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 ρƒ. 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. 
         [0035]    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: 
         [0000]    
       
         
           
             
               
                 
                   
                     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 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where f gain  is a gain factor which will be detailed and verified later. 
         [0036]    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). 
         [0037]    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 . 
         [0038]    Since both stress and strain are second order tensors and they transform as second order tensors, the compliance tensor α (and also the stiffness tensor c) must transform as the fourth order tensor. The transformation equation is: 
         [0000]      α′ ijkl =b ip b jq b kr b ls α pqrs   (20) 
         [0000]    where  α     pqrs    and  α′     ijkl    are the fourth order compliance tensors before and after the coordination system transformation; and  b     ip    is the transformation tensor. 
         [0039]    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: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     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 
                   ) 
                 
               
             
           
         
       
     
         [0040]    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]: 
         [0000]      α′ ij=q   mi   q   nj α mn  ( i,j,m,n= 1,2,3,4,5,6)  (22) 
         [0000]    where q ij  takes the form: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     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 
                   ) 
                 
               
             
           
         
       
     
         [0041]    This gives the TI formation elastic compliance tensor in X′ 1 X′ 2 X′ 3  coordinate system (borehole coordinate system) as: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       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 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with: 
         [0000]      α′ 11 =α 11   (25) 
         [0000]      α′ 12 =cos 2  θ·α 12 +sin 2  θ·α 13    (26) 
         [0000]      α′ 13 =sin 2  θ·α 12 +cos 2  θ·α 13   (27) 
         [0000]      α′ 14 =0.5 sin 2θ(−α 12 +α 13 )  (28) 
         [0000]      α′ 22 =cos 4  θ·α 11 +0.5 sin 2  2θ·α 13 +sin 4  θ·α 33 +sin 2  2θ·α 44    (29) 
         [0000]      α′ 23 =0.25 sin 2  2θ(α 11 +α 33 )+(sin 4  θ+cos 4  θ)·α 13 −sin 2  2θ·α 44    (30) 
         [0000]      α′ 24 =−sin θ cos 3  θ·α 11 +0.25 sin 4θ·α 13 +sin 3  θ cosθ·α 33 +0.5 sin 4θ·α 44   (31) 
         [0000]      α′ 33 =sin 4  θ·α 11 +0.5 sin 2  2θ·α 13 +cos 4  θ·α 33  +sin 2  2θ·α 44    (32) 
         [0000]      α′ 34 =−sin 3  θ cos θ·α 11 −0.25 sin 4θ·α 13 +sin θ cos 3  θ·α 33 −0.5 sin 4θ·α a   44    (33) 
         [0000]      α′ 44 =cos 2  2θ·α 44 +sin 2  2θ·(α 11 +α 33 −2α 13 )  (34) 
         [0000]      α′ 55 =cos 2  θ·α 44 +sin 2  θ·α 66    (35) 
         [0000]      α′ 56 =0.5 cos 2θ·(α 44 −α 66 )  (36) 
         [0000]      α′ 66 =sin 2  θ·α 44 +cos 2  θ·α 66    (37) 
         [0042]    The non-zero a′ 14 , α′ 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). 
         [0043]    As an approximation, we propose the assumption: 
         [0000]      α 44 =ƒ gain (α 11 +α 33 −2α 13 )   (38) 
         [0000]    where ƒ 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. 
         [0044]    From Equations (7), (10), (11) and (12), Equation (38) can be rewritten as: 
         [0000]    
       
         
           
             
               
                 
                   
                     1 
                     
                       
                         f 
                         gain 
                       
                        
                       
                         G 
                         ′ 
                       
                     
                   
                   = 
                   
                     
                       1 
                       E 
                     
                     + 
                     
                       
                         1 
                         + 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             υ 
                             ′ 
                           
                         
                       
                       
                         E 
                         ′ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    or equation (19) as we stated before: 
         [0000]    
       
         
           
             
               
                 
                   
                     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 
                   ) 
                 
               
             
           
         
       
     
         [0045]    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 71  , 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. 
         [0046]    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: 
         [0000]    
       
         
           
             
               
                 
                   
                     1 
                     
                       G 
                       45 
                       ′ 
                     
                   
                   = 
                   
                     
                       1 
                       E 
                     
                     + 
                     
                       
                         1 
                         + 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             υ 
                             ′ 
                           
                         
                       
                       
                         E 
                         ′ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0047]    Comparing with equation (39) with equation (40), we understand that the gain factor f gain  can be calculated from: 
         [0000]      G′ 45 =f gain  G′  (41) 
         [0048]    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). 
         [0049]    Table 1a 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. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 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 
               
               
                   
               
             
          
         
       
     
         [0050]    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. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 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 
               
               
                   
               
             
          
         
       
     
         [0051]    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. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 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 
               
               
                   
               
             
          
         
       
     
         [0052]    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. 
         [0053]    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. 
         [0000]    
       
         
               
               
               
               
             
           
               
                 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 
               
               
                   
               
             
          
         
       
     
         [0054]    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 ). 
         [0055]      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. 
         [0056]    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. 
         [0000]    
       
         
               
               
               
             
           
               
                   
                   
               
               
                   
                 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