SpaceTime Modeling A Wavelet Approach

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Space-Time Modeling A Wavelet Approach

• Shekhar Srinivasan
• Supervisor Dr. Sanjay Srinivasan

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Objective

• Dynamic variables Pressure signals, flow
  response
• Spatially variable attributes
• Prediction of value at any given spatial location
  and time instant
• Modeling spatiotemporal behavior of dynamic
  variables

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Objective

• Pressure P f(u,t)
• Expressing the trend as linear combination
• Correlate the parameters in space
• Develop parameter set at unsampled locations
  using stochastic simulation
• Regenerate pressure or flow response at that
  location

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Highlights

• Principles of Fourier Transforms
• Drawbacks
• Concept of Wavelets and Wavelet Transforms
• Multi-resolution Analysis
• Application to Space-Time modeling
• Results
• Conclusions

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Principles of Fourier Transforms

• Decomposition of a signal into a sine wave of
  different frequencies
• Frequency- vibration about a mean
• Ideal for periodic signals
• Periodic signals-Fast Fourier Transforms
• Aperiodic signals- Discrete Time Fourier
  Transforms

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Fourier transform Concept
Frequency domain
f(t)
Time
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Drawbacks

• Cannot operate simultaneously in both domains
• Need to define a single transform for dual
  representation of the energy density of a signal
• Cannot capture discontinuities in the transient
  signal effectively
• Need large number of coefficients
• Tedious exercise to carry out kriging for large
  number of coefficients

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Wavelets Objectives

• Capture coarse and fine details of a transient
  signal
• Expressing any signal as a linear combination of
  well defined functions
• Scaling translation parameters

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Wavelets Concept

• Discretization
• Box function b(x) 0,1
• b(2kx-n)
• k scaling parameter
• n translation parameter
• Weighting step functions (Haar Wavelet)
• Summing up the weighted functions

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Wavelets Concept

• Coarser Averaging
• Finer Differencing
• Why ?
• Lifting Scheme of wavelets

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Wavelets Binary tree

• Coarse signal at level n
• Coarse at n-1
• Fine at level n-1
• Retain detail
• Split coarse part of signal

Sn
Sn-1
Dn-1
Dn-2
Sn-2
Sn-3
Dn-3
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Geostatistical Nomenclature

• Random variables
• Uncertainty characterized by random function
  Cumulative distribution
• Univariate, Bivariate, Multivariate distributions
• Spatial correlation Variogram
• Techniques - Kriging, Simulation

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Kriging- A Brief Overview

• Generic form of linear regression
• Express estimate as linear combination of data
• Minimize variance subject to constraints
• Matrix of equations containing covariance
• Covariance Inferred from variogram
• Estimate Mean value
• Revisit Discrete Time Fourier transform

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Stochastic Simulation

• Kriging maps are smooth- dampen variability
• Add residual variability to kriged map
• Simulated maps actual variability
• Sequential Simulation
• Define random path to visit all nodes in a
  reservoir
• Univariate Bivariate Multivariate

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Testing on Synfields

• Four well model 1 injector, 3 producers
• Anisotropic permeability field
• Permeability barriers
• Analyze correlation of the wavelet parameters
• Get an idea of hetrogeneity

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Case 1
P2
I1
P3
P1

• P1 I1 oriented perpendicular to anisotropy
• P2 P3 oriented parallel to anisotropy

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Case 2

• P1 I1 oriented perpendicular to anisotropy
• P2 P3 oriented parallel to anisotropy

P1
P3
P2
I1
I1 P2 oriented parellel to anisotropy, no
barrier P1 P3 equidistant from I1, barrier
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Algorithm

• Define a reservoir model
• Develop wavelet coefficients for available wells
• Develop variogram from the permeability field
• Use the same variogram model to generate all
  coefficients using kriging or SGSIM
• Use the generated coefficients to generate
  pressure profile
• Verify by simulating pressure profile at a given
  location

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Ensuring consistency with geology
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Reservoir model

• Producer
• Injector

Simulated node
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Results
SGSIM
Wavelet parameter map

• Permeability field

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Comparison
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Conclusions

• Correlation coefficient depends on the direction
  of orientation
• Depends on the distance between injector and
  producer
• Pressure response is directly correlated to the
  permeability field
• Single variogram model suffices to generate the
  wavelet parameters at unknown locations
• Variogram model can be determined from
  permeability field

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Suggested work

• Extend the exercise to 3D modeling
• Test the exercise on a real model
• Carry out the exercise using other robust wavelet
  schemes like Lifting Scheme

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Acknowledgements

• Dr. John Gilbert
• Dept. of Mathematics
• University of Texas Austin