Post on 22-Dec-2015
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Upscaling and History Matching of Fractured Reservoirs
Pål Næverlid SævikDepartment of Mathematics
University of Bergen
Modeling and Inversion of Geophysical Data (Uni CIPR)
March 26, 2015
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What makes fractures special?
• Dual porosity behaviour• Scale separation issues• Heterogeneities are larger than lab scale• Prior information on fracture geometry may
be available
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Large and small fractures
The distinction between «large» and «small» fractures is determined by the size of the computational cell
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Prior fracture information
• Core samples• Well logs• Outcrop analogues• Well testing• Seismic data• EM data (?)
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Fracture parametersRoughnessAperture (thickness)Filler material
Connectivity
Fracture density
Clustering
ShapeSize
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Common assumptionsFisher distribution of
orientationsPower-law size
distribution
Cubic transmissitivity law:
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Numerical upscaling
• Flexible formulation• Accurate solution• Slow• Gridding difficulties• May not have sufficient
data to utilize the flexible formulation
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Analytical upscaling
• Idealized geometry• Fast solution• Easy to obtain
derivatives• Requires statistical
homogeneity• Difficult to link idealized
and true fracture geometry
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Percolation theory
• Assumption: All fractures are polygons of equal shape, distributed randomly in space
• Percolation theory tells us that:
– (percolation threshold)
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Connectivity prediction
• Mourzenko, V. V., J.-F. Thovert, and P. M. Adler (2011)• is calculated from fracture shape, size and
orientation distribution• is slightly shape-dependent
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Summary: Input and output parameters
Connectivity f
Density A
Transmissitivity τ
Density A
Aperture aFiller material
Permeability K
Porosity φ
Transfer coefficient σ
Orientation
ShapeSize
Clustering
Roughness
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Integrated upscaling and history matching
Fracture parame tersAd
just
ups
cale
d pa
ram
eter
s
Permea bility, shape factor
Press ure , flow rates
Mismatch
Upscaling
Simulation
Real data
Adju
st fr
actu
re
par
amet
ers
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Ensemble Kalman Filter update
• Based on Bayes’ formula:
• All distributions are approximated by a Gaussian distribution, and the covariance is defined using the ensemble
• Update formula:
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Test problem: Permeability measurement
• Single grid cell• Measured permeability: 200 mD ± 20 mD• Expected aperture: 0.2 mm ± 0.02 mm• Expected density: 1 m-1 ± 0.2 m-1
• Randomly oriented, infinitely extending fractures
• Cubic law for transmissitivity
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Linear fracture upscaling
• Lognormal fracture parameters– Expected log aperture (mm): – Expected log density (m-1):
• Logarithm of the upscaling equation
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Partially connected fractures
• We set fracture size to R = 5 m• Connectivity is computed as
• Connectivity is then a monotonically increasing function of fracture density
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Field case: PUNQ-S3
• Three-phase reservoir• 6 production wells• 0 injection wells (but
strong aquifer support)• Dual continuum
extension with capillary pressure
• Constant production rate
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Field case: PUNQ-S3
• 2 years of production• 2 years of prediction• Data sampling every
100 days• Data used– GOR– WCT– BHP
• Assimilation using LM-EnRML
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Data match summaryNumber of LM-EnRML iterations
0 1 2 3 4
Fracture parameters as primary variables
BHP 11.07 3.15 0.99 0.46 0.44
GOR 11.16 5.54 1.38 0.13 0.35
WCT 3.60 0.91 0.90 0.41 0.40
Total 9.31 3.71 1.11 0.37 0.40
Upscaled parameters as primary variables
BHP 11.07 4.78 4.15 4.32 4.42
GOR 11.16 10.26 9.74 9.62 9.65
WCT 3.60 1.26 1.23 1.16 1.21
Total 9.31 6.57 6.15 6.12 6.17