Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies |

In the extended formulas for seismic data, the key quantity needed is clearly the value of
the angle . However, this value is quite easily determined since

Therefore,

Thus, is completely determined by the same data used in the standard analysis of reflection seismic data, which determines the various small angle wave speeds and the Thomsen weak anisotropy parameters.

The pertinent fixed factors for use in the formulas are given by

Finally, equation 14 also shows how to determine the extreme value using the same data. Examples of such computations are presented in TABLE 1 for various anisotropic rock types. Data for these examples comes from Thomsen's TABLE 1, and no other information is required.

renewedcommandarraystretch1.2
par
begincenter
sc Table 1. Examples of -- em i.e., the extreme value
-- and its angular location for various rocks
and minerals. The data for , ,
, and are all taken from Table 1 of Thomsen (1986).

par
begintabular|c|c|c|c|c|c|c| hlinehline
em Sample & & & (m/s) & (m/s)
& &

hline
Cotton Valley shale & 0.135 & 0.205 & 4721. & 2890. & -0.1564 & 39.89

Mesaverde sandstone & 0.081 & 0.057 & 3688. & 2774. & 0.0805 & 40.48

Muscovite crystal & 1.12 & -0.235 & 4420. & 2091. & 0.8985 & 26.90

Pierre shale & 0.015 & 0.060 & 2202. & 969. & -0.1076 & 44.48

Taylor sandstone & 0.110 & -0.035 & 3368. & 1829. & 0.3135 & 41.12

Wills Point shale & 0.215 & 0.315 & 1058. & 387. & -0.1543 & 39.27

hlinehline
endtabular
endcenter
par
sectionNORMAL MOVEOUT CORRECTIONS
par
The altered forms of and in equations 26 and 27
suggest that it might also be necessary to alter
the normal moveout (NMO) corrections to the velocities (Tsvankin, 2005, p. 113).
It is easy to see that these corrections are now given by
beginequation
V_NMO,p = v_p(0)sqrt1+2delta,
labeleq:vpnmo
endequation
for the quasi-P-wave, and,
beginequation
V_NMO,sv = v_s(0)sqrt1+2sigma,
labeleq:vsvnmo
endequation
for the quasi-SV-wave, where
beginequation
sigma = left[v^2_p(0)/v^2_s(0)right](epsilon-delta).
labeleq:sigma
endequation
These corrections to the NMO velocities are exactly the same as those for Thomsen's weak anisotropy
approximation because the factor that is pertinent, and that might have potential to alter these
expressions is given, in the small angle limit , by
beginequation
frac2sin^2theta_m1-cos2theta_m equiv 1,
labeleq:factor2
endequation
which holds for any value of (including both and
).
Since Thomsen's formulas accurately approximate all three wave speeds in this limit by design,
the present formulas share this accuracy (and in some cases -- em i.e., larger offsets
-- improves upon it). Therefore, no changes
are needed in short offset (small ) data processing.
par
The NMO correction for the SH-wave
clearly does not change either, since it does not depend on this new factor.
par
sectionRESERVOIRS WITH VERTICALLY ORIENTED FRACTURES
par
To provide some pertinent examples of results for the types of anisotropic media
most interesting in oil and gas reservoirs, two distinct types of reservoirs
having vertical fractures will now be considered. The first case studied will have
vertical fractures that are not preferentially aligned, so the reservoir symmetry is
vertical transverse isotropy (VTI). The second case will also have vertical fractures
but these will be preferentially aligned, so the reservoir symmetry will be horizontal
transverse isotropy (HTI) and, therefore, exhibit azimuthal (angle dependent) anisotropy.
par
These two reservoir fracture models will be built up using results from recent
numerical experiments by Grechka and Kachanov (2006a,b). Those results
were analyzed by Berryman and Grechka (2006) in light of the crack influence
parameter formalism of Kachanov (1980) and Sayers and Kachanov (1991).
The significance of two crack influence parameters -- , for
-- for the case of aligned
horizontal cracks for lower crack densities (where is the
number density of cracks -- being the total number per volume --
and for penny-shaped cracks is the radius of the circular penny crack-face while
is called the aspect ratio) is:
beginequation
Delta S^(1H)_ij =
rho_cleft(beginarraycccccc
0 & 0 & eta_1 & & & cr
0 & 0 & eta_1 & & & cr
eta_1 & eta_1 & 2(eta_1+eta_2) & & & cr
& & & 2eta_2 & & cr
& & & & 2eta_2 & cr
& & & & & 0 cr
endarrayright).
labeleq:number2
endequation
For smaller values of crack density , equation 40 shows how the presence of penny-shaped cracks increases the compliance of the reservoir.
[Note that is usually small and most often negligible, while is
positive and nonnegligible.]
Typical values of crack density for reservoirs are .
The matrix
is the lowest order compliance correction matrix
and should be added to the isotropic compliance matrix
beginequation
Delta S^(0)_ij =
left(beginarraycccccc
1/E & -nu/E & -nu/E & & & cr
-nu/E & 1/E & -nu/E & & & cr
-nu/E & -nu/E & 1/E & & & cr
& & & 1/G & & cr
& & & &1/G & cr
& & & & & 1/G cr
endarrayright),
labeleq:number1
endequation
where
is Poisson's ratio,
is the shear modulus, and is Young's modulus
of the (assumed) isotropic background medium. Summing equations 41 and 40
produces the compliance matrix for a horizontally cracked,
VTI elastic medium. This combined matrix is then used to compute the behavior of
a simple HTI reservoir with aligned vertical cracks using the methods described
in Appendix B.
par
For vertical fractures with randomly oriented axes of symmetry, the
resulting VTI medium has a low crack density correction matrix
of the form
beginequation
Delta S^(1V)_ij =
rho_cleft(beginarraycccccc
(eta_1+eta_2) & eta_1 & eta_1/2 & & & cr
eta_1 & (eta_1+eta_2) & eta_1/2 & & & cr
eta_1/2 & eta_1/2 & 0 & & & cr
& & & eta_2 & & cr
& & & & eta_2 & cr
& & & & & 2eta_2 cr
endarrayright),
labeleq:number3
endequation
in which the 's have the same values as those in equation 40
if the only difference between the cracks in equations 42 and 40
is their orientation.
Note that
is an isotropic
correction matrix for a system having crack density .
Summing equations 41 and 42
produces the compliance
matrix for a vertically cracked VTI elastic medium, in which the crack
normals are randomly and/or uniformly distributed in the horizontal plane.
par
Higher order corrections (em i.e., second order in powers of )
in the Sayers and Kachanov (1991) formulation with the three
crack influence parameters , for , take the form
(again using the Voigt matrix notation):
beginequation
Delta S^(2H)_ij =
rho_c^2left(beginarraycccccc
0 & 0 & eta_4 & & & cr
0 & 0 & eta_4 & & & cr
eta_4 & eta_4 & 2(eta_3+eta_4+eta_5) & & & cr
& & & 2eta_5 & & cr
& & & & 2eta_5 & cr
& & & & & 0 cr
endarrayright)
labeleq:number2plus
endequation
for horizontal fractures -- em i.e., to be combined with equation 40.
Similarly,
beginequation
Delta S^(2V)_ij =
rho_c^2left(beginarraycccccc
(eta_3+eta_4+eta_5) & eta_4 & eta_4/2 & & & cr
eta_4 & (eta_3+eta_4+eta_5) & eta_4/2 & & & cr
eta_4/2 & eta_4/2 & 0 & & & cr
& & & eta_5 & & cr
& & & & eta_5 & cr
& & & & & 2(eta_3+eta_5) cr
endarrayright)
labeleq:number3plus
endequation
for the random vertical fractures producing VTI symmetry
- to be combined with equation 42. par
Examples of values of all five of these crack influence parameters have been
obtained based on the numerical studies of Grechka and Kachanov (2006a,b)
by Berryman and Grechka (2006).
The two models considered have very different Poisson's ratios for the
isotropic background media: (a) and
(b) . We will call these two models,
respectively, the first model and the second model.
The first model has background stiffness matrix values
GPa,
GPa, and
GPa. Bulk modulus for this model
is therefore GPa and shear modulus is GPa.
The purpose of this model is to provide as much contrast as possible with the following
model, while still retaining a physically pertinent value of Poisson's ratio
(for which values typically lie in the range
).
The second model has stiffness matrix values
GPa,
GPa, and
GPa. Bulk modulus for this model
is therefore GPa and shear modulus is GPa.
The second model may be seen to correspond to a background material having
compressional wave speed km/s, shear wave speed km/s,
and mass density
kg/m, and is therefore more typical of
a sandstone reservoir.
Detailed discussion of the method used to obtain the crack influence parameters
is given by Berryman and Grechka (2006), and will not be repeated here.
Results are listed in sc Table 2.
par
In all the following plots, the exact curves (as computed for the model 's)
are plotted first in black; then the Thomsen approximation is plotted in red; and
finally the new approximation is plotted in blue. Thus, in those examples where
red curves appear to be missing, this happens because the blue curves lie
right on top of the red ones (to graphical accuracy). This overlay effect is
expected whenever approaches , which can happen at low crack
densities since the background medium has been taken to be isotropic.
par
1.2

TABLE 2.of five fracture-influence parameters
for the two models considered, from Berryman and Grechka (2006).

Fracture-influence |
First Model |
Second Model |

Parameters |
||

(GPa) | 0.0000 | -0.0192 |

(GPa) | 0.1941 | 0.3994 |

(GPa) | -0.3666 | -1.3750 |

(GPa) | 0.0000 | 0.0000 |

(GPa) | 0.0917 | 0.5500 |

1.2

TABLE 3.of [the extreme value of ]
for the four models considered. The model fracture density is .
The model Poisson ratio for the background medium is .

Model |
for | for | for |

HTI, | 0.0277 | 0.0973 | 0.2943 |

VTI, | 0.0148 | 0.0558 | 0.1965 |

HTI, | 0.0102 | 0.0441 | 0.1595 |

for | for | for | |

VTI, | 0.0011 | 0.0051 | 0.0210 |

For reference purposes, the computed values of are also presented in TABLE 3.

FIG1
For randomly aligned vertical fractures and VTI symmetry:
examples of anisotropic quasi-P compressional wave speed () for
Poisson's ratio of the host medium .
Velocity curves in black are exact for the fracture model discussed
in the text. The Thomsen weak anisotropy velocity curves for the same
fracture model are then overlain in red. Finally, the new curves for the
extended Thomsen approximation valid for stronger anisotropies are overlain
in blue. If any of these curves is not visible, it is because one or
possibly two other curves are covering them.
Figure 1. |
---|

FIG2
Same as Figure 1 for SH shear wave speed ().
Figure 2. |
---|

FIG3
Same as Figure 1 for quasi-SV shear wave speed ().
Figure 3. |
---|

FIG4
Same as Figure 1, for a different background medium
having Poisson's ratio
.
Figure 4. |
---|

FIG5
Same as Figure 2, but the value of
.
Figure 5. |
---|

FIG6
Same as Figure 3, but the value of
.
Figure 6. |
---|

Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies |

2007-09-15