Azimuthal p analysis in anisotropic media

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Azimuthal -p analysis in anisotropic media

• Samik Sil and mrinal k. sen
• Edger Forum, Feb-27,2007

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Talk overview

• VTI and HTI media.
• NMO Correction and parameter estimation processes
  in VTI and HTI media (x-t domain).
• Tau-p transform and its importance.
• Development of new tau-p equation for HTI medium.
• Simulated case studies.
• Conclusions.

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VTI and HTI media

• Anisotropy due to layering
• Most common anisotropy in reservoir
• Found in shale formations
• Vertical axis of symmetry

VTI

• Anisotropy due to vertical cracks
• Common anisotropy, found in fractured medium.
• Horizontal axis of symmetry
• Wave velocity varies with azimuth

HTI
The stiffness matrix of VTI and HTI media have
only 5 independent elements.
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VTI and HTI Media parameters

• Thomsen (1986) and Tsvankin (1998) introduced 5
  parameters related to stiffness matrix elements
  to characterize a VTI and HTI media, respectively

Vertical P wave velocity Vertical S wave
Velocity Fractional difference between
horizontal and vertical P wave velocities
Second derivative of P wave velocity at vertical
incidence
Fractional difference between horizontal and
vertical S wave velocities
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Travel time analysis NMO and parameter
estimation in x-t domain
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NMO in anisotropic Media

• (x,t) NMO Equation in Isotropic Media

(1)

• (x,t) NMO Equation in VTI and HTI Media
  (Thomsen, Tsvankin, Alkhalifah)

For VTI
(2)
For HTI
where
NMO correction and parameter estimation is done
by curve fitting.
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Remarks

• Equation for traveltime is approximate and uses
  rms
• parameters.
• Truncated non hyperbolic terms may be required
  even for isotropic media to model traveltimes
  at large offsets.
• When a ray transmits through an overlying
  anisotropic layer, x-t curve fitting method,
  results in misinterpretation.
• These problems can be solved by converting x-t
  domain data in tau-p domain.

HTI
ISO
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x-t to ?-p conversion
X
?
T
event
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Slant Stacking
Why is it called slant stacking? - It sums
amplitudes along a line of constant slope on
(x,t) space
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An example of tau-p converted seismogram
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( t , p ) NMO Equation in Anisotropic media
In general
In an isotropic or VTI,HTI medium
i.e. for two layer case, tau-p curve of the 2nd
layer will be obtain by summing with the tau
values from the 1st layer
Therefore
This helps to introduce layer striping approach
of NMO correction and parameter estimation
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And also the exact equation

• Van der Baan and Kendall (2003) obtain exact
  equation for tau-p domain both for VTI and HTI
  medium.

• VTI Equation

• HTI Equation

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Problem with exact equation

• many parameters are not constrained by data.
• Inherently non-unique, due to large number of
  parameters.
• Difficult to implement.
• Solution Approximate equations with reduced
  parameters

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Previous solutions (Approximate equations)
Example of approximate equation (VTI)
Sen and Mukherjee, 2003, Number of reduced
parameters2
Example of approximate equation (HTI)
Baan and Kendall, 2003, Number of reduced
parameters4
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Results from approximate equations
VTI Medium
HTI Medium
Approximate tau-p equation performed better curve
matching (thus NMO correction and parameter
estimation) for HTI medium.
Baan and Kendall, 2003
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Drawback of approximated tau-p equation for HTI
medium

• The existing equation has 4 parameters that are
  not well constrained from single azimuthal
  measurement.
• Fit with exact data is not good enough for
  performing better NMO correction.
• Different equations for different medium is not
  desirable.

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New azimuthal tau-p equation for HTI medium

• So we develop the following 2 parameter equation
  for HTI media

Where
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Advantages from the new equation

• Has only 2 parameters, so easy to implement.
• Reduction in uncertainty space.
• One equation for all the TI and Isotropic media.

3 different azimuth data are sufficient to
characterize the reservoir.
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Simulated Case studies
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Simulated Case studies

• We use 3 models

2.4 km
2.4 km
2.4 km
ISO
ISO
ISO
1 km
1 km
HTI
HTI
HTI
1 km
1 km
ISO
ISO
VTI
1 km
infinite
ISO
ISO
infinite
Model 1
Model 2
Model 3
All the anisotropic parameters used here for
standard anisotropy, based on the work of Xu and
Tsvankin, 2006 and Sen and Mukherjee, 2003
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Seismograms for the model 1
Plots of x-t and tau-p seismograms for zero
degree azimuth.
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Seismograms for the model 1
Plots of x-t and tau-p seismograms for azimuth45
deg
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Seismograms for the model 1
Plots of x-t and tau-p seismograms for azimuth90
deg.
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Error analysis

• Since the number of parameters are less, grid
  search can be employed to find the parameters.
• Non-uniqueness of the error space has reduced a
  lot too.
• Example of grid searching (model 1, 0 degree
  azimuth).

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NMO correction
NMO corrected result using our equation and
isotropic equation for model 1 for azimuth0
degree.
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NMO correction
NMO corrected result using our equation and
isotropic equation for model 1 for azimuth45
degree.
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NMO correction
NMO corrected result using our equation and
isotropic equation for model 1 for azimuth90
degree.
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Seismograms for model 2
Plots of x-t and tau-p seismograms for azimuth0
degree. Tau-p method can fit the curve from
isotopic layer very well.
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Seismograms for model 2
Plots of x-t and tau-p seismograms for azimuth45
degree. Tau-p method can fit the curve from
isotopic layer very well.
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Seismograms for model 2
Plots of x-t and tau-p seismograms for azimuth90
degree. Tau-p method can fit the curve from
isotopic layer very well.
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Seismograms for model 3
Plots of x-t and tau-p seismograms for azimuth0
degree. Tau-p method can fit the curve from the
VTI layer very well.
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Seismograms for model 3
Plots of x-t and tau-p seismograms for azimuth45
degree. Tau-p method can fit the curve from the
VTI layer very well.
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Seismograms for model 3
Plots of x-t and tau-p seismograms for azimuth90
degree. Tau-p method can fit the curve from the
VTI layer very well.
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NMO correction results

• NMO correction using our equation in tau-p
  domain.

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Conclusions

• The Tau-p transformation is suitable to perform
  moveout correction for multi anisotropic layer.
• We have developed a 2 parameter approximate
  equation for HTI medium, suitable for VTI and
  isotropic too.
• Our equation performs better than the published
  equation by reduction of error space for
  parameter estimation and by performing better NMO
  correction.
• The equation is developed for horizontal layer
  case only.
• The equation is developed considering weak
  anisotropy, but still can perform satisfactory
  NMO correction of standard HTI medium.
• The mismatch tends to increase for higher values
  of p and lower value of azimuth angle.
• We have developed an interactive code for
  velocity picking, using our formula, for real
  life case.

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Acknowledgement

• Dr. Mrinal Sen, Advisor
• Sponsors of the EDGER forum
• Rishi Dev Bansal
• Dr. Robert Ferguson