Upscaling and History Matching
of Fractured Reservoirs
Pål Næverlid Sævik
Department of Mathematics
University of Bergen
Porous media seminar at MI, UiB
May 06, 2015
Outline
EnKF
Simulation
2
Fractured Rocks
What makes fractures different from other
heterogeneities?
3
What makes fractures special?
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Dual porosity behavior
Scale separation issues
Heterogeneities are larger than lab scale
Prior information on fracture geometry may
be available
4
Dual porosity behavior
5
Scale separation issues
Large faults and fractures
may be impossible to upscale
6
Large and small fractures
The distinction between «large» and «small» fractures
is determined by the size of the computational cell
7
Prior fracture information
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Core samples
Well logs
Outcrop analogues
Well testing
Seismic data
EM data (?)
8
Fracture parameters
Roughness
Aperture (thickness)
Filler material
Shape
Size
Orientation
Connectivity
Fracture density
Clustering
9
Common assumptions
Fisher distribution of
orientations
Power-law size
distribution
Cubic transmissitivity law: 𝑇 =
1 3
𝑎
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Numerical upscaling
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•
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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
12
Effective permeability
K1
pin
pin
pout
K2
φ1K1 + φ 2K2
= A·a·K1 + K2
= A·τ + K2
pout
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Several fracture sets
• Single frac: 𝑲𝑒 = 𝑲𝑚𝑎𝑡 + 𝐴𝜏 𝑰 − 𝒏𝒏
• Extension: 𝑲𝑒 = 𝑲𝑚𝑎𝑡 + 𝐴𝑖 𝜏𝑖 𝑰 − 𝒏𝒊 𝒏𝒊
14
Partially connected fractures
• Snow (1969): 𝑲𝑒 = 𝑲𝑚𝑎𝑡 + 𝐴𝑖 𝜏𝑖 𝑰 − 𝒏𝒊 𝒏𝒊
• Oda (1985): 𝑲𝑒 = 𝑲𝑚𝑎𝑡 + 𝑓 𝐴𝑖 𝜏𝑖 𝑰 − 𝒏𝒊 𝒏𝒊
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Percolation theory
• 𝑲𝑒 = 𝑓 𝐴𝑖 𝜏𝑖 𝑰 − 𝒏𝒊 𝒏𝒊
• Assumption: All fractures are polygons of
equal shape, distributed randomly in space
• Percolation theory tells us that:
–𝑓
–𝑓
–𝑓
–𝑓
∈ 0, 1
= 0 𝑖𝑓 𝐴 < 𝐴𝑐 (percolation threshold)
→ 1 𝑎𝑠 𝐴 → ∞
∝ 𝐴 − 𝐴𝑐 2 𝑤ℎ𝑒𝑛 𝐴 ≈ 𝐴𝑐
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Connectivity prediction
• Mourzenko, V. V., J.-F. Thovert, and P. M.
Adler (2011)
• 𝑲𝑒 = 𝑓 𝐴𝑖 𝜏𝑖 𝑰 − 𝒏𝒊 𝒏𝒊
• 𝑓=
𝐴−𝐴𝑐 2
𝐴 𝐴−𝐴𝑐 +𝛽/𝑅
• 𝐴𝑐 is calculated from fracture shape, size and
orientation distribution
• 𝛽 is slightly shape-dependent
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Connectivity and spacing
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Transfer coefficient
• Kazemi (1976):
𝜎 = 4 𝐴2𝑥 + 𝐴2𝑦 + 𝐴2𝑧
• Generalization:
𝜎 = 4 Tr 𝑨⊤ 𝑨 , where
𝑨 = 𝐴 𝑖 𝒏𝑖 𝒏𝑖
⊤
• Other alternatives also exists
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Summary: Input and
output parameters
Clustering
Shape
Size
Density A
Connectivity f
Density A
Orientation
Aperture a
Filler material
Roughness
Transmissitivity τ
Permeability K
Transfer
coefficient σ
Porosity φ
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History matching of fractured
reservoirs
Simulation
EnKF
Simulation
21
Fracture parameters
Upscaling
Permeability, shape factor
Simulation
Pressure, flow rates
Real data
Mismatch
Adjust fracture parameters
Adjust upscaled parameters
Integrated upscaling and history
matching
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Ensemble Kalman Filter update
• Based on Bayes’ formula:
𝑃 𝐷|𝑀 𝑃 𝑀
𝑃 𝑀|𝐷 =
𝑃 𝐷
• All distributions are approximated by a
Gaussian distribution, and the covariance is
defined using the ensemble
• Update formula:
𝑀𝑑𝑖𝑓𝑓 = Δ𝑀Δ𝐷⊤ 𝐶𝐷 + Δ𝐷Δ𝐷⊤ −1 ⋅ (𝐷𝑜𝑏𝑠 − 𝐷 + 𝜖)
Δ𝑀 = 𝑀 − 𝑀 / 𝑁 − 1
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Test problem: Permeability
measurement
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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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Test problem: Permeability
measurement
• Resulting upscaling equations:
1
𝐾=
𝑓𝑎3 𝐴
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𝜙 = 𝑎𝐴
4 2
𝜎= 𝐴
3
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Predicted fracture porosity
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Predicted transfer coefficient
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Inverse relation and connectivity
• Inverse relation of the upscaling equations
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𝑓=
𝐾𝜎𝜙 −3
2
𝐴=
3𝜎/4
𝑎 = 𝜙/ 3𝜎/4
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Predicted connectivity
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Linear fracture upscaling
• Lognormal fracture parameters
– Expected log aperture (mm): log 𝑎 = log 0.2 ± log 2
– Expected log density (m-1): log 𝐴 = log 1 ± log 2
• Logarithm of the upscaling equation
log 𝐾 = 𝐶1 + log 𝑓 + 3 log 𝑎 + log 𝐴
log 𝜙 = 𝐶2 + log 𝑎 + log 𝐴
log 𝜎 = 𝐶3 + 2 log 𝐴
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Predicted connectivity
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Partially connected fractures
• We set fracture size to R = 5 m
• Connectivity is computed as
𝐴 − 𝐴𝑐 2
𝑓=
𝐴 𝐴 − 𝐴𝑐 + 𝛽/𝑅
• Connectivity is then a monotonically
increasing function of fracture density
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Predicted connectivity
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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 LMEnRML
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Data match summary
Number of LM-EnRML iterations
Fracture
parameters
as primary
variables
Upscaled
parameters
as primary
variables
0
1
2
3
4
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
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
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BHP, PRO-1
WCT, PRO-11
GOR, PRO-12
Initial ensemble
Traditional approach
Our approach
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Permeability
Sigma factor
True case
Initial ensemble
Final ensemble,
our approach
Final ensemble,
traditional approach
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Final ensemble,
our approach
Final ensemble,
traditional approach
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Conclusion
• Fracture upscaling creates nonlinear relations
between the upscaled parameters
• These relations may be lost during history
matching, if upscaled parameters are used as
primary variables
• The problem can be avoided by history
matching fracture parameters directly
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