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History Matching Of A Realistic Stochastic Sub-Seismic Fault Model
Marius Verscheure (IFP)Jean-Paul Chilès (Mines Paris)André Fourno (IFP)
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Introduction
Forward model
Inverse modeling
Example
Conclusions
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Construction of fractured reservoir models
Why ? Predict reservoir behaviour Enhance production
How ? Integration of : static data (geostatistics, reservoir structure, etc...) dynamic data (well tests, production history)
Dynamic data integration = history matching
Static data
Dynamic data
Model
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Multiscale fracturing
Large scale : Seismic faults Seismic image Deterministic modeling
Medium scale : Sub-seismic faults Invisible on seismic and well logs Large objects Stochastic modeling
Small scale : joints Well logs Stochastic modeling
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Sub seismic faults : issues
Strong influence on hydrodynamic behaviors High permeabilities (or low : sealing) Large objects
Difficult to characterize Invisible on seismics
Problems with modeling How do we model what we cannot see?
Problems with history matching No efficient methods yet
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Objective : History Matching of Subseismic faults
Fluid flow simulation
Optimization
Modify fault positions to reduce the objective function
Statistical coherency must be preserved
Upscaling Simulated data Field data
Objective function
Field observations
Stochastic fault generator
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Introduction
Forward model
Inverse modeling
Example
Conclusions
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Fractal Modeling Definition
Fractal object that is similar whatever the observation scale
Fractal Dimension 1 < Df <2 Df1 : clustered
Df2 : uniformly distributed
Arguments in favor Observation of real fractured networks
Analysis of seismic faults Length distribution Number and shape of faults Spatial distribution Fractal Dimension
Model Extrapolation of seismic faults to smaller observation scales
Df=1.65
Df=1.58
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Modeling fault length distribution
Power Law : n(l) = Al-a
Determination of parameters Analysis of seismic faults determination of a, lmin, lmax
and number of faults to generate
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Modeling spatial distribution
Multiplicative cascade algorithmInitiated with 4 weights
Subdivision of each blockRandom permutation of each Pi in
each blockMultiplication with upper scale
0
1
5.0
4321
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i
D
P
PPPP
PPPP f
Density map characterized by fractal dimension Df
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Modeling spatial distribution
Fault centers drawn from constrained Poisson point process
Centers are fractal with dimension Df
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Modeling spatial distribution : Conditioning to seismic faults
Algorithm initiated with a low resolution density map
Initial resolution depends on fractal properties
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Flow simulation model : Analytical discretization
Equivalent flow properties computed for each cell
Dual porosity model Porosity : fault volume / cell Equivalent permeability tensor
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Introduction
Forward model
Inverse modeling
Example
Conclusions
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Gradual deformations
Combination of two gaussian realizations : y0 and y1
y0 and y1: same mean, same variance y(t) new realization (same mean, same variance)
Gradually deformable gaussian numbers
y(t) = y0cos(tπ) + y1sin(tπ)
New realization Realization 0 Realization 1
t : gradual deformation parameter
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Gradual deformation of a Poisson point process
Non stationarity : density map
Each point drawn independantly
Y(t) U(t) X(t)gaussian numbers uniform numbers coordinates
inverse CDFSequential algorithm
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Gradual deformation of the multifractal map
P1(t) P2(t) P3(t) P4(t) so that:
variation of t continuous deformation of the cascade process (patented)
0)(
1)()()()(
5.0)()()()(
4321
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tP
tPtPtPtP
tPtPtPtP
i
D f
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Reduction of the objective function
Fault migration continuous Hydrodynamic response continuous Objective function continuous
Optimization with gradient based algorithms (Gauss-Newton, SQPAL, etc...)
Initial Model
Objective function
History matched Model
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Introduction
Forward model
Inverse modeling
Example
Conclusions
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Example : Synthetic model 1060 subseismic faults Generated with commercial software 60 years of water flooding 121 x 150 x 7 3D simulation grid 4 producers , 11 Injectors
Reference fault network
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Example : Initial model
-Seismic fault network used to calculate initial low resolution density map (a)
-multiplicative cascade algorithm (b)
-final density map used to draw a population of fault centers (c)
-Initial fault model (d)
(a) (b) (c)
(d)
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Initial hydrodynamic response
Response different from reference model History matching necessary
Initial water cut levels for P1, P2, P3 P4. Field production data (red), simulated values (yellow)
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History Matching
Reservoir divided by zone Objective function evolution
Gauss Newton optimizer Good convergence speed few objective function evaluations
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History Matching
History matched water cut levels for P1, P2, P3 P4. Field production data (red), simulated values (yellow)
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Conclusions
Forward model Realistic Well suited to petroleum problems Gradually deformable
Inverse modeling Model consistency is unchanged Few parameters to optimize Gradient based optimization possible (fewer fluid flow
simulation runs) Good geological model given the restrictions brought
by the gradual deformation method Efficient optimization
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Future work
Account for sealing faults
Include diffuse fracturing history matching
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Some more slides...
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Example : Reference model
450 subseismic faults Df=1.65 matrix = 50 mD faults = 3.5E5 mD,
e=0.1m 10 years of water flooding
P1
P2
P3
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Example : Initial model
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Example : Matched Model
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Objective function evolution
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Generation of Poisson points
Points generated using density map
Calculate Discrete Cumulated Distribution Function from density map in X direction, then in Y direction
Generate uniform numbers on the CDF axis
Inversion of these number yields non stationnary Poisson point coordinates
Gradual deformation is done on the uniform numbers
1 – Generate Points
2 – Propagate Faults
3 – Merge Families
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Interpolation
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Determination of fractal dimension
Two point correlation function
Compute the distance between each couple of fault centers
If points are fractalfD
d rrNN
rC )(1
)(22
N: Number of points, Nd(r): number of pairs of points with distance d < r
Df : fractal dimension
small faults are invisible deviation
Modeling objectiveAdd the missing faults without modifying the fractal dimension
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3D faults
2D faults are extended and clipped on a corner point grid
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Deformation of the network
Gradual deformation Density map Poisson point process Length distribution
Pros Fractal dimension and length distribution unchanged One parameter to control all fault positions Local deformations (to match specific well)