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