A Multiresolution Analysis of the Relationship between Spatial Distribution of Reservoir Parameters

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A Multiresolution Analysis of the Relationship
between Spatial Distribution of Reservoir
Parameters and Time Distribution of Data
Measurements

• Abeeb A. Awotunde
• Roland N. Horne
• Stanford University

SPE 115795
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Outline

• Objective
• Time wavelets
• Space wavelets
• Demonstrative examples
• Field example
• Conclusion

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Objective

• To investigate the relationship between
  time-dependent wavelet transforms of pressure
  response and spatially-dependent wavelet
  transforms of reservoir parameters

time
space
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Time wavelets Wt

• Time series can be decomposed into averages and
  differences at multiple resolutions

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Parameterizations
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Four choices of regression
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1. p-k model

• Vary reservoir parameters to match measured data

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2. p-wk model

• Reparameterization of model space using wavelet
  transform by Lu and Horne SPE 62985

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3. wp-k model

• First consider time-dependent wavelets
• Establish the relationship between regression in
  the time domain and regression in the wavelet
  domain.
• Provided that all the wavelet coefficients are
  used, regression in the time domain is exactly
  the same as regression in the wavelet domain.

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3. wp-k model

• Transform the data space
• Compute wavelet sensitivity coefficient
• Threshold the sensitivity matrix and data space
• Solve the inverse problem

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3. wp-k model
Full wavelet sensitivity matrix
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3. wp-k model
Thresholded wavelet sensitivity matrix
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4. wp-wk model

• Nonlinear regression in wavelet domain
• Transform the model space and data into space and
  time wavelet domains
• Compute sensitivity of transformed data to
  transformed model parameters
• Use this wavelet sensitivity matrix to threshold
  the model space, the data space and the wavelet
  sensitivity matrix (itself)
• Solve the inverse problem

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4. wp-wk model

• Solution of inverse problem in the two wavelet
  domains

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4. wp-wk model
Full wavelet sensitivity matrix
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4. wp-wk model
Thresholded wavelet sensitivity matrix
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4. wp-wk model

• The thresholding scheme is
• data-or-sensitivity-based for data space
• sensitivity-based for parameter space
• parameter-dependent
• regression adaptive

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Example 32 rings
Match to observed data
dp, t.dp/dt, psia
t, hrs
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Example 32 rings
Modeled permeability distributions
k, md
ring number
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Example 32 rings
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Example 32 rings
wp-wk
No. of wavelets used as model parameters
p-wk
iteration count
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Example 32 rings
wp-k
No. of wavelets used as regression data
wp-wk
iteration count
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Field Example (Fall-off Test)

• Water injection followed by shut-in
• 16,384 measured data
• We model the k/µ in 16 rings, s and C
• Initial guesses
• Homogeneous mobility of 353 md/cp
• Skin -2
• Wellbore storage coefficient 1 bbl/psi

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Field Example (Fall-off Test)
Match to observed data
dp, t.dp/dt, psia
t, hrs
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Field Example (Fall-off Test)
mobility, md/cp
Modeled mobility distributions
radius, ft
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Field Example (Fall-off Test)
No of wavelets used as regression data
(Recall 16,384 measured data)
iteration count
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2D Reservoir

• Available data
• BHP from 2 injection wells
• BHP from 3 producers
• Water cut from 3 producers
• Multiobjective function minimization

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2D Reservoir
Match to BHP
BHP at Injectors
BHP, psia
BHP at Producers
t, days
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2D Reservoir
Match to Water cut at Producers
water cut
t, days
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2D Reservoir
Decay of residual
residual
p-k, p-wk
wp-k, wp-wk
iteration number
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2D Reservoir
Decay of residual
p-wk
residual
reduced number of coefficients
wp-wk
iteration number
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2D Reservoir
residual norm of k
model type
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Ongoing work

• Multiobjective function Analysis
• Minimizing the Frobenius norm of the residual
  matrix
• Results will be presented at SUPRID Annual meeting

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Conclusion

• Determines the appropriate resolution of the
  problem both for the data and for the reservoir
  description.
• Also improves the computational efficiency of
  nonlinear regression.

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• Thanks