MCERI: Model Calibration and Efficient Reservoir Imaging
Multiscale Parameterization and
Streamline-Based Dynamic Data
Integration for Production Optimization
Norne Field E-Segment
Eric Bhark
Alvaro Rey
Mohan Sharma
Dr. Akhil Datta-Gupta
MCERI
Approach to case study
• Objective
Develop optimal production strategy (2005 to 2008)
Production and seismic data integration
• Conceptual approach
Deterministic perspective
Single, history matched model (to 12/2003)
Global parameters defined
• Faults and transmissibility multipliers
• Saturation regions
– Relative perm, capillary pressure
• Large-scale permeability & porosity
heterogeneity with multipliers
Data integration
• Minimal calibration of prior
2/23
MCERI
Structured workflow
Production data integration
• Calibrate permeability heterogeneity to fluid rates (to 12/04)
• Multiscale parameterization (global to local scales)
Seismic data integration• Match (time lapse) changes in acoustic impedance by
adjusting water front movement (Sw)
• Streamline-based techniques
Production optimization strategy
• Optimize constrained well rates through forecast period
• Objective of improving sweep efficiency (fluid arrival time equality along streamlines)
3/23
MCERI
Production data integration:
Overview
• Calibrate prior permeability model
Multiscale approach of global-to-local adjustment
Update at sensitive locations and scales
• Production data
Three-phase rates
• 12/1997 to 12/2004
Producers E-3H, E-3AH, E-2H
• Heterogeneity parameterization
Reduce parameter dimension of high-resolution model
Address parameter correlation, insensitivity
4/23
MCERI
Parameterization
• Grid-connectivity-based transform (GCT)
Parameterization by linear transformation
Characterize heterogeneity as weighted linear combination of basis vectors
• GCT basis vectors
Generalization of discrete Fourier basis vectors for generic grid geometries
• Parameterization analogous to frequency-domain transformation
• Modal shapes, harmonics of the grid
5/23
Reservoir property
=
w1 w2
+ +
w3
+
w4
… +
w15
… +
w10
Calibrated
parameters
Bhark, E. W., B. Jafarpour, and A. Datta-Gupta (2011), A Generalized Grid-Connectivity-Based
Parameterization for Subsurface Flow Model Calibration, Water Resour. Res., doi:10.1029/2010WR009982
1 2 3 4 10 15
MCERI
• Parameterize layers individually
Maintain prior vertical variability, stratification
Prevent vertical smoothing
• For each layer (21 active of 22 total):
Define perm multiplier (1) field as calibrated field
Retain prior heterogeneity at full spatial detail
Calibration approach
6/23
Prior (ln md)
Multiplier
( )
m
i
iiw
1
cells
param.
n
m
MCERI
• Adaptive refinement of multiplier fields (layers)
From coarse (global) to fine (local) scale
Successive addition of higher-frequency basis vectors
=w1
Constant (zero frequency) basis vector
21 parameters total
zonation
7/23
Calibration workflow
Layer 1
multiplier
MCERI
w2
=w1
+
w5
7/23
+ …
Calibration workflow
• Adaptive refinement of multiplier fields (layers)
From coarse (global) to fine (local) scale
Successive addition of higher-frequency basis vectors
Layer 1
multiplier
MCERI
w2
=w1
+
w5
+ …
w10
7/23
+ …
Calibration workflow
• Adaptive refinement of multiplier fields (layers)
From coarse (global) to fine (local) scale
Successive addition of higher-frequency basis vectors
• Between gradient-based minimization iterates (Quasi-Newton)
– Gradient from one-sided perturbation of transform parameters
• Based on data sensitivity (gradient contribution)
Cease (layer-by-layer) upon data insensitivity to addition of detail
Layer 1
multiplier
MCERI
Calibration results (71 param)
L2 L10 L20 L21 L22
Calibrated multiplier fields:
Permeability fields (Multiplier .* Prior):
8/23
MCERI
Production data misfitWATERCUT
OIL RATE
E-3H E-2H E-3AH
E-3H E-2H E-3AH
9/23
Lower
OWC
MCERI
Structured workflow
Seismic data integration• Match (time lapse) changes in acoustic impedance by
adjusting water flood movement (Sw)
• Streamline-based techniques
10/23
MCERI
Seismic data integration:
Overview
• Seismic inversion of reflection data
Acoustic impedance at grid cell resolution
• Dr. Gibson of Texas A&M Geophysics Dept.
• 2001 – 2003 time lapse interval
• Changes in Z (dynamic changes)
• Calibration to seismic data
Sequential integration of acoustic impedance
• Objective function weighting
– Multiple sources seismic inversion uncertainty
– Limitations in PEM
Gradient-based workflow
• Calibrate inter-well permeability based on streamline-derived sensitivities
– Grid cell resolution local calibration
11/23
Difference of
averages:
2003 - 2001
MCERI
Data misfit
Model (k)Update (LSQR)
Simulation
PEM Z(Gassman)
SL-based sensitivities
Streamline-based workflow
kkkk
P
P
S
S
S
S
g
g
w
w
ZZZZ
Water front evolution
• Positive time-lapse
changes (Sw)
Sensitivity formulation
• Two-phase (water-oil)
PEM
• Consider only variation
with saturation (Kf)
So Sw Sg
Numerical differencingPrior
Model
12/23
Streamline-derived
(analytical)
wS
Z
k
wS
kLkkGZ 21seis1
MCERI
Sensitivity formulation
• Well rates Cell saturations Acoustic impedance
Cell permeability near streamlines traced from production wells
• Trace streamlines from producers
Velocity field from finite-difference simulation
• At each cell
Map Sw, k, to intersecting streamline
Compute time of flight ()
per segment:
tzyx ,,,SS ww
Transform to streamline coordinates
,SS ww t
dru
outlet
inlet
Define semi-analytical formulation for Sw at each cell
0
ww F
t
S
13/23k
S1
k
S 'w
w
tt
MCERI
Increase in acoustic impedance
• Replacement of oil by water
Decrease in acoustic impedance
• Occurs in areas initially water-saturated infer pressure effect
Results: Seismic data integration
K = 5-9
Pre-calibrated Model Observed Calibrated Model
K = 11
14/23
Difference:
2003-2001
MCERI
Production data misfit revisited
No degradation in match quality
• Confirmation that (local, inter-well) permeability
updates for seismic data integration are consistent
with calibration from production data integration
15/23
WATERCUT
E-3H E-2H E-3AH
MCERI
Structured workflow
Production optimization strategy
• Optimize constrained well rates through forecast period
• Objective of improving sweep efficiency (front arrival time equality along streamlines)
16/23
MCERI
Optimal Production Strategy:
Overview
• Review reservoir flow pattern, connectivity
• ‘Base Case’ strategy for rate optimization
From investigation of production enhancement opportunities
• Optimal rate strategy
1) Maximize sweep (RF)
• Equalizing fluid arrival time at producers
(from injectors, aquifer)
2) Maximize NPV (indirectly)
• Accelerating production
i.e., minimize arrival time
17/23
Producer
Injector
Injector
MCERI
Reservoir Flow Pattern
Aquifer outside
of E-segment
Aquifer
Tra
cin
gfr
om
Pro
du
cers
Tra
cin
g fro
m
Inje
cto
rs
18/23
Calibrated model:
End of history
at Dec. 2004
MCERI
1) Produce at last available rates
(Dec. 2004)
RF = 47.8%
2) E-3H sidetrack well in layer 10
Highest remaining oil pore volume
3) F-1H gas injection
Higher NPV than water injection
– Lower injection/production costs
Improvement pre-optimization:
RF = 48.5%
Increment of 0.7%
Incremental NPV increase: 872 MM$
Base Case Production Strategy
19/23
Economic Parameters
Discount Rate 10 %
Oil Price 75 $/BBL
Gas Price 3 $/Mscf
Water Prod/Inj Cost 6 $/BBL
Gas Inj Cost 1.2 $/Mscf
Sidetrack 65 MM$
Production Constraints
Max. Inj FBHP 450 Bar
Min. Prod FBHP 150 Bar
Max. Water Inj Rate 12000 Sm3/day
Max. Liquid Prod Rate 6000 Sm3/day
Max. Water Cut 95 %
Max. GOR 5000 Sm3/Sm3
MCERI
j
iij
q
tS
q
Rate optimization workflow
• Consider 6-month time intervals
• Trace streamlines (using velocity field)
Compute fluid arrival time at producers
• Compute obj. fn.
Penalize water, gas production
• Minimize obj. fn. using SQP
Analytical sensitivities
Single forward simulation
iwii ftt ,' 1 qq
20/23
prodN
i
ittJ
1
2'qqq
MCERI
Rate optimization workflow
21/23
j
iij
q
tS
q
• Consider 6-month time intervals
• Trace streamlines (using velocity field)
Compute fluid arrival time at producers
• Compute obj. fn.
Penalize water, gas production
• Minimize obj. fn. using SQP
Analytical sensitivities
Single forward simulation
• Progress to next time interval
prodN
i
ittJ
1
2'qqq
iwii ftt ,' 1 qq
MCERI
300
344
434
0
100
200
300
400
500
Norm Wt.-0 Norm Wt.-100 Norm Wt.-1000
Incre
menta
l N
PV
, M
M $
Case
48.88 49.19 49.24
40
45
50
55
Norm Wt.-0 Norm Wt.-100 Norm Wt.-1000
Recovery
Facto
r (b
ased o
n O
IIP
), %
Case
Production acceleration
prodprod N
i
i
N
i
i tttJ
1
2
1
2qqqq
• Rate opt. improves recovery factors
Delays gas breakthrough (and shut-in) at E-2H and E-3H-sidetrack
• Acceleration ( ) improves NPV
Disproportionate increase – pressure support from higher gas injection rate
compensates for water injection (BHP upper limits reached)
Recovery factor Incremental NPV(over base case)
22/23
(up 0.3%)
MCERI
Summary
• Production data integration
Global to local permeability calibration
• Multiscale parameterization
Minimally update (pre-calibrated) prior model
• (Sequential) Seismic data integration
Match change in acoustic impedance between 2001 and 2003
Calibrate cell permeability based-on streamlines traced from producers
• Cell saturations through water front movement
Well-captured positive changes
• Production schedule optimization
Established base scenario of E-3H-sidetrack (large remaining oil pore
volume) and F-1H gas injection (lower costs)
Improved RF and NPV by equalization and reduction of fluid travel times
23/23
MCERI: Model Calibration and Efficient Reservoir Imaging
Norne Comparative Study
Eric Bhark
Alvaro Rey
Mohan Sharma
Dr. Akhil Datta-Gupta
MCERI
Backup slides: GCT
27/X
MCERI
Grid-connectivity-based transform basis
(1) Model (or prior) independent
Can benefit from prior model information
(2) Applicable to any grid geometry (e.g., CPG, irregular unstructured,
NNCs, faults)
(3) Efficient construction for very large grids
(4) Strong, generic compression performance
(5) Geologic spatial continuity
28
Highlights of new basis
M
N
M v
v
v
u
u
u
2
12
1
2
1
=
MCERI
Concept: Develop as generalization of discrete Fourier basis
KEY: Perform Fourier transform of function u by (scalar) projection
on eigenvectors of grid Laplacian (2nd difference matrix)
Basis development
• Interior rows Second difference
Periodic operator (circulant matrix)
• Exterior rows Boundary conditions control
eigenvector behavior
29
MCERI
• Decompose L to construct basis functions (rows of )
Always symmetric, sparse
Efficient (partial) decomposition by restarted Lanczos method
Orthogonal basis functions;
• In general (non-periodic) case
Eigen(Lanczos)vectors vibrational modes of the model grid
Eigenvalues represent modal frequencies
Basis development
vΦvΦuvΦuT 1
5 10 15 20 25 30 35 40 45 50
5
10
15
20
25
30
35
40
45
50
Grid LaplacianCPG Unstructured
2-point connectivity (1/2/3-D)
30
MCERI
• Modal shape modal frequency
• Constant basis Zero frequency
• Discontinuities honored
Basis vec. 1 Basis vec. 2 Basis vec. 3 Basis vec. 4 Basis vec. 5
Corner-point Grid
(Brugge)
Basis functions: Examples
Basis vec. 9
31
MCERI
Parameterize
multiplier field
Additional
spatial
detail?
NO
Add higher-
frequency modes to
basis
YES
Calibrated Model
(1) START: Prior model
Prior spatial hydraulic
property model
Update in transform
domain
Back-transform
multiplier field to
spatial domain
Flow and transport
simulation
Mu
ltis
cale
ite
rate
Unit-multiplier field at
grid cell resolutionG
rad
ien
t-b
ased
itera
te
Streamline-,
sensitivity-based
inversion (GTTI)
Structured workflow
Data misfit
tolerance?
NO
YES
(2) Regional update (3) Local update
FINISH
32
MCERI
Leading basis functions by modal frequency
Leading basis functions by prior model compression performance1 2 3 4 5 6 7 8 9
3D CPG 1 2 3 4 5 6 7 8 9
Coefficient spectrum: scalar proj. of prior onto 500 leading basis functions
Honoring prior by basis element selection
33
Sp
ec
tra
l
co
eff
icie
nt
Basis function by modal frequency Basis function by compression
MCERI 34/X
Pressure misfit
MCERI
E-3AH Pressure
• There is an apparent constant shift
Simulated pressure is over-estimated
• Potential Solutions
Add (negative skin), completion specific
• Skin required to lower pressure 20+ bars (e.g., s = -10) results in high
rate fluctuation as drawdown becomes too large
Add WPIMULT < 1.0
• Same result as for skin
Lower Pinit
• Improves match, but
lowered to 150 bars
MCERI
E-3AH Pressure
• Early FMT match indicates
that Pinit is consistent with
prior model specs
• This is despite isolation of
EQLNUM 3 (see below)
which would permit a very
different pressure across
the NOT formation
MCERI
Backup slides:
SL-based AI integration
37/X
MCERI
Seismic inversion
• Selected components
QC/filtering of sonic, density logs
• Well acoustic impedance
– Conditioning data
Stochastic inversion (genetic algorithm)
• Solve for acoustic impedance maps at 2001, 2003
• Average of 5 realizations
Compute change at grid cell resolution
• Observation data for model calibration
• Focus on dynamic changes
• Reduce affect of static, poorly resolved parameters
Gao, K. Acoustic impedance inversion using Petrel for the Norne Oil Field,
Texas A&M Geophysics Dept.
3rd Layer 10th layer Bottom layer
Difference of
averages:
2003 - 2001
12/24
MCERI
(Qualitative) Results
Pre-calibration Calibrated
Saturation
Changes
(Cellular Grid)
Acoustic
impedance
(Seismic volume)
Sli
ce
J =
49
Slice
J =
45
Assessment of WOC in E-segment (Ile, Tofte)
Change in Z (2001 – 2003) with Sw following production & seismic integration
• Orthogonal intersection of seismic volume slice and grid slice
• Increase in calibrated WOC more consistent with observed acoustic impedance
15/X
Pre-calibration Calibrated
MCERI
• Well rates Cell saturations Acoustic Impedance
Sensitivity (Z/k) computed along streamlines traced from producers
• Trace through velocity field at grid cell resolution
Sensitivity matrix is sparse
• non-zero components correspond to cells intersected by streamlines (localization)
tzyx ,,,SS ww
Sensitivity formulation
/SS ,SS wwww tt
kS
k
S '
ww
tt
1
Semi-analytical
13/X
I J
Transform to characteristic coordinates
t,SS ww
Define semi-analytical formulation for Sw
MCERI
Time-lapse sensitivity
• Sw depends on front location & previous state of saturation
• Perturbation in Sw
41/X
1-n
wnw
nw S,
τSS
t
1-nw1-n
w
nw
nw'n
w SS
SτS
1S
t
τS
1
S
S
S
SτS
1
S
SτS
1S
0w'
0w
M-nw
2-nw
1-nw1-n
w'
1-nw
nwn
w'n
w ttt
0S
S
FS w
w
ww =τ
+t
• 2-phase incompressible
• perturbations in properties do
not affect streamline geometry
• Mapping of Sw b/w SL’s at different ‘steady-state’ intervals
MCERI
Seismic data integration
42
Layer 10
Layer 20
MCERI
• Construct sparse sensitivity matrix
Gradient-based minimization (LSQR)
• For each cell at which acoustic impedance measured
Compute sensitivity for all cells along intersecting streamline(s)
Sensitivity definition
No
bs
Cell
s w
ith
in s
eis
mic
cu
be
NparamActive model cells
xx
xxx
xx
xx
xxx
xxx
xx
xxx
xxx
x
xx
xxx
xx
x
x
xx
x
kS
1
k
S 'w
w
tt
14/X
MCERI
Prod. data misfit
44/X
Oil rate
Gas rate
MCERI
Backup slides:
Production Optimization
45/X
MCERI
Reservoir Flow Pattern
46
Aquifer
Aquifer
Based on calibrated model at end of history ( Dec-2004)
MCERI
RF NPV Increm.
(%) (MM $) (MM $)
1 Do Nothing: Production based on last available voidage rates 47.8 3998 -
2 Case 1 + Sidetrack + Water Injection: Recomplete E-3H in layer 10 horizontally 48.8 4438 440
3 Case 1 + Gas Injection: Inject gas through F-1H (at same voidage as w ater inj.) 48.0 4574 576
4 Case 1 + Sidetrack + Gas Injection 48.5 4870 872
Case Production Strategy
Base Case
20/24
Base case for optimization
• E-3H sidetrack in layer 10
Highest remianing oil pore volume
• F-1H gas injection
Shut-in of E-2H (Feb. 2008) and E-
3H-sidetrack (Feb. 2007)
Higher NPV than water injection
• Lower injection/production costs
Production Constraints
Max. Inj FBHP 450 Bar
Min. Prod FBHP 150 Bar
Max. Water Inj Rate 12000 Sm3/day
Max. Liquid Prod Rate 6000 Sm3/day
Max. Water Cut 95 %
Max. GOR 5000 Sm3/Sm3
Economic Parameters
Discount Rate 10 %
Oil Price 75 $/BBL
Gas Price 3 $/Mscf
Water Prod/Inj Cost 6 $/BBL
Gas Inj Cost 1.2 $/Mscf
Sidetrack 65 MM$
MCERI
Enhancement scenarios tested
• Sidetrack (300m)
E-3H in layers 1-3
E-3AH in layer 5, 6, 7, 8, 9, 10
• Currently in layers 1 and 2
F-3H in layer 2, 3 for injection to support E-3AH
• Currently in layer 20
• Conversion of F-3H into gas injector
Layer 20
48/X
MCERI
Analytical sensitivity
• Producer i, well (prod. or inj.) j
• When j is producer:
Assume streamlines do not shift for perturbation in well rates
• Travel time at i sensitive only to change in well rate at producer j = i
• When j is injector:
Nfls,i,j connect wells i and j
• Requires only single forward simulation
49/X
00
0
,,
1
,,
fsl
fsl
jfsl
N
l
jil
ij
N
NqNS
jifsl
ji
jiqS
j
i
ij
0