June 1, 2009
Model-Reduced Variational Data
Assimilation For Reservoir Model Updating
M.P. Kaleta (TUD), A.W. Heemink (TUD), J.D. Jansen (TUD/Shell), R.G. Hanea (TUD/TNO)
June 1, 2009 2
Outline
� Background
� Model-reduced variational data assimilation (MRVDA)
� Study case
� Results
� Conclusions
June 1, 2009 3
Background
Reservoir management represented as a model-based closed-loop controlled process
Data assimilation
algorithms
Reservoir
Model
Noise OutputInput NoiseSystem
(reservoir, wells
& facilities)
Optimization
algorithmsSensors
Reservoir
ModelReservoir models Reduced reservoir
model
J. D. Jansen et al. [2005]
June 1, 2009 4
Background
Goal
To find the maximum a posteriori estimate of the log permeability field by solving the minimization problem:
� represent the vectors of observed and predicted
production data at time ,
� the uncertain log permeability vector,
� the prior knowledge about log permeability,
� covariance matrix for the log permeability vector,
� the covariance matrix for data measurements errors
,obs
i iy y
θP
priorθ
θ
iP
it
( ) ( ) ( )( ) ( )( )1 1
1
1 1min ( )
2 2
DNT Tprior prior obs obs
i i i i i
i
J θ− −
=
= − − + − −∑θ
θ θ θ P θ θ y y θ P y y θ
June 1, 2009 5
MRVDA
Advantage of adjoint method� A gradient based algorithm with gradient calculated by adjoint
method which requires one adjoint solution regardless of the number of model parameters
Disadvantage of adjoint method� Implementation of the adjoint model
SolutionsIf we re-parameterize the permeability field then:
� we can calculate the gradient by perturbations: computationally expensive for large number of uncertain parameters
� we can construct the low-order approximation of the tangent linear reservoir model for which the low-order adjoint model is derived
June 1, 2009 6
MRVDA
Proper Orthogonal Decomposition (Lumley 1967) also knows as
� Karhunen- Loève (K-L) TransformLoève (1946)Karhunen (1946)
� Empirical Orthogonal Function� Principal Component Analysis (1986)
Application � Statistical tool to analyze experimental data
The POD is used to analyze the set of realizations with an aim to extractdominant features and trends (coherent structures called patterns in space)
� Reduced Order Modeling (ROM) The POD is used to provide a relevant set of basis functions with which a low-dimensional subspace is identified, then the reduced model is constructed by projection of the governing equations on that subspace
June 1, 2009 7
MRVDA
Classical adjoint approach
MRVDA
Optimization
High-order nonlinear
reservoir model
High-order linear
adjoint model
Re-parameterization
High-order nonlinear
reservoir model
Low-order linear
reservoir model
Low-order linear
adjoint model
Optimization
Re-parameterization
Vermeulen, P.T.M., Heemink, A.W. [2006]
Altaf, U.M., Inverse shallow water flow modeling
June 1, 2009 8
MRVDA
High-order nonlinear model high-order linearized model
Consider high-order nonlinear reservoir model
Tangent linear approximation of the high-order nonlinear reservoir model around can be rewritten as
where
andStep 0Re-parameterize permeability field using set of realizations
( ) 6 6
1( ) ( ), , (10 ) | | (10 )h
i i it t h O h Oθ−= ∈ =x f x θ R θ� �
( ) ( )1 1
1
1
( ), ( ),( ) ( )
( )
i i i i
i i
i
t tt t
t
− −−
−
∂ ∂= +
∂ ∂
f x θ f x θ∆x ∆x ∆θ
x θ
b= −∆θ θ θ ( ) ( ) ( )b
i i it t t= −∆x x x
( ( ), )b b
itx θ
{ }1, , Kθ=Θ θ θK
, ,ll hθθ θ θθ θ∆ = ∈θ P r r R �
June 1, 2009 9
MRVDA
High-order linearized model low-order model
Step 1
Generate number of system solutions and define snapshots as
Step 2
Apply POD on pressure and saturation snapshots separately and use it to derive projection subspace
Project the high-order linearized model
into low-order linear model
( ) ( )1 1( ) ( ), ( ),b b b
i i i i it t tθ θ− −∆ = + −x f x θ P r f x θ
( ) ( )i it t≈∆x Pr ,l l h∈r R �
1( ) ( )i i i it t θ θ−= +r N r N r
( ) ( )1 1
1
1
( ), ( ),( ) ( )
( )
i i i i
i i
i
t tt t
t
− −−
−
∂ ∂= +
∂ ∂
f x θ f x θ∆x ∆x ∆θ
x θ
June 1, 2009 10
MRVDA
Step 3
Approximate the matrices of the low-order model
Using the chain rule differentiation
Using finite difference approximation we get
( )1
1
( ),
( )
i iT l l
i
i
t
t
− ×
−
∂= ∈
∂
f x θN P P R
x
( )1( ), l li iT
i
tθθ θ ×−∂
= ∈∂
f x θN P P R
θ
( ) ( ) ( )1 1 1 1 1 11
1 1 1 1
( ) ( ), ( ) ( ), ( ) ( ),( )
( ) ( ) ( ) ( )
b b b b b b
i i i i i i i i iij
j i i j i i
t t t t t tt
t t t t
− − − − − −−
− − − −
∂ + ∂ + ∂ +∂= =
∂ ∂ ∂ ∂
f x Pr θ f x Pr θ f x Pr θxP
r x r x
( ) ( ) ( )1 1 1( ), ( ), ( ),b b b b b b
i i i i i i
j
j j j
t t tθ θ θ θ θ θθ
θ θ θ
− − −∂ + ∂ + ∂ +∂= =
∂ ∂ ∂ ∂
f x θ P r f x θ P r f x θ P rθP
r θ r r
( ) ( ) ( )1 1 1
1
( ), ( ) , ( ),
( )
b b b b b b
i i i i j i i
j
i
t t t
t
ε
ε− − −
−
∂ + −≈
∂
f x θ f x P θ f x θP
x
( ) ( ) ( )1 1 1( ), ( ), ( ),b b b b b b
i i i i j i i
j
t t tθ θθ
θ
ε
ε− − −∂ + −
≈∂
f x θ f x θ P f x θP
θ
June 1, 2009 11
MRVDARe-parameterization
High-order Reservoir Model Simulation
Low-order Model Simulation
Gradient Calculation
Reduced Objective Function Calculation
Initial Parameters
Objective Function Calculation
Converged? Done
Building of The Low-order Model
Converged?
Low-order Adjoint Model Simulation
Sup Optimal Parameters
Parameters Update
Snapshots Simulation
June 1, 2009 12
MRVDA
Computational complexity
� Preprocessing cost of generating representative snapshotsspanning a large portion of possible permeability field
� Preprocessing cost of solving eigenvalue problems
� Preprocessing cost of building low-order system matrices
� As many model runs + multiplications of Jacobians with projection matrix as many state patterns
� As many model runs + multiplications of Jacobians with projection matrix as many parameters
� The cost of solving dense low-order linear model
June 1, 2009 13
Study case
Reservoir model assumption
�3 dimensional (60x60x7 with 18553 active grid blocks)
�Two-phase (oil-water)
�No-flow boundaries at all sides
Measurements
�Bottom hole pressures from injectors each 60 days during 3 years
�Flow rates from producers each 60 days during 3 years
Gijs van Essen [2006]
Producer
Injector
June 1, 2009 14
Results
Model-Reduced VDA
29+5
-
State
patterns
~ 67
-
Nr of model simulations
22
22
Permeability
patterns
2001301
-2700
Number of
snapshots
Objective
function
Outer
loop
True log perm field Prior log perm field Estimated log perm field
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
June 1, 2009 15
Results
0 500 1000 1500200
300
400
500
Time [DAYS]
Liquid Rate [BBL/DAY]
0 500 1000 1500500
600
700
800
900
Time [DAYS]
Liquid Rate [BBL/DAY]
0 500 1000 1500150
200
250
300
350
Time [DAYS]
Liquid Rate [BBL/DAY]
0 500 1000 1500250
300
350
400
450
Time [DAYS]
Liquid Rate [BBL/DAY]
Legend
Liquid rate in the
production wells
June 1, 2009 16
Results
Classical adjoint approach
True log perm field Prior log perm field Adjoint log perm field
0 5 10 15 20 25 30100
120
140
160
180
200
220
240
260
280Values of the objective function at the successive iterations
Number of iteration
Objective function
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
June 1, 2009 17
Results
0 500 1000 1500200
300
400
500
Time [DAYS]
Liquid Rate [BBL/DAY]
0 500 1000 1500500
600
700
800
900
Time [DAYS]
Liquid Rate [BBL/DAY]
0 500 1000 1500150
200
250
300
350
Time [DAYS]
Liquid Rate [BBL/DAY]
0 500 1000 1500250
300
350
400
450
Time [DAYS]
Liquid Rate [BBL/DAY]
Legend
Liquid rate in the
production wells
June 1, 2009 18
Results
Comparison of the methods
~26*2116Adjoint approach
~67130Model-reduced approach
-270Initial (Prior)
Time in simulations
Objective function
Method
True log perm field Model-reduced log perm field Adjoint log perm field
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
20 40 60
20
40
60Layer: 1
20 40 60
20
40
60Layer: 2
20 40 60
20
40
60Layer: 3
20 40 60
20
40
60Layer: 4
20 40 60
20
40
60Layer: 5
20 40 60
20
40
60Layer: 6
20 40 60
20
40
60Layer: 7
June 1, 2009 19
Conclusions
� Model-reduced variational data assimilation does not require the implementation of the adjoint of the tangent linear model of the original reservoir model
� Model-reduced variational data assimilation gives the estimates comparable to those from data assimilation using an adjointmethod (comparable match and predictions)
� Model-reduced variational data assimilation is easy to implement and treats simulator as a black-box
� If we have the adjoint code then we do go for the classical approach
June 1, 2009 20
Questions?
June 1, 2009 21
MRVDA: Proper Orthogonal Decomposition
Suppose we have an ensemble
POD seeks for the best / optimal linear representations of the members of the set :
Optimality condition
Solution
The above optimization problem is solved by eigenvalue analysis of the correlation matrix , that is
{ }1 , , K=X x xK6(10 ),hi h O∈x R �
[ ] [ ]
1 1
1
1 1
1
ˆ ˆ ˆ, , , ,
K
K lh K h l
l l
K l K
r r
r r× ×
×
≈ = =
X X x x p p
K
K K M O M
K
h
i ∈p R
21
1 KT
i i
iK =
−∑ x PP xminh lR ×∈P
X
TR = XX
T
i i iλ=p pXX
June 1, 2009 22
MRVDA: Proper Orthogonal Decomposition
An optimal basis is given by
where is the eigenvector corresponding to i-th largest eigenvalueof the matrix
The relative importance present in each basis vector
Choose such that
where denotes the fraction of the cumulative relative importance we want to capture
[ ]1 2, , , l…*P = p p p
ip
1
j
j K
i
i
λϕ
λ=
=
∑
1 2 0l Kλ λ λ λ≥ ≥ ≥ ≥K � K�
1
l
i
i
ϕ α=
≤∑
l
α
June 1, 2009 23
MRVDA: Reduced Order Modeling
Consider a general high-order nonlinear discrete reservoir system
The aim of the reduced order modeling is to find the projection with and where to obtain the low-order system
whose trajectories evolve in -dimensional space
( )( ) ( ),i i it t=y h x θ
( )1( ) ( ), ,hi i it t −= ∈x f x θ R
6(10 )h O����
h l×∈P RT
l=P P I l h����
( )1( ) ( ),T
i i it t −=r P f Pr θ
( )( ) ( ),i i it t=y h Pr θ
( ) ( )T
i it t=r P x
High-order nonlinear
reservoir model
High-order linear
adjoint model
Optimization
Low-order nonlinear
reservoir model
Low-order linear
adjoint model