Importance of input error in reservoir simulations fileReservoir Simulator Accounting for input error in 4D-Var Results Summary Importance of input error in reservoir simulations Joanna

Project MotivationReservoir Simulator

Accounting for input error in 4D-VarResults

Summary

Importance of input error in reservoir

simulations

Joanna S. Pelc Remus G. HaneaArnold W. Heemink Jan-Dirk Jansen

Delft University of Technology

The Eighth International Workshop on Adjoint Model Applicationsin Dynamic Meteorology, Tannersville 18-22 May 2009

J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations



Summary

Outline

1 Project Motivation

2 Reservoir Simulator

3 Accounting for input error in 4D-Var

4 Results

5 Summary




Summary

How is oil produced?

With primary recoveryoil production is usuallyless than 30%.




Summary

How is oil produced?

Water is injected into the reservoir field to force more oil intoproduction wells




Summary

What mathematics has to do with oil?

A suitable reservoir rock must be porousand permeable




Summary

Reservoir Simulator

State vectorPressure at time 451.454 days

5 10 15 20

2

4

6

8

10

12

14

16

18

20 2.55

2.6

2.65

2.7

2.75

2.8

2.85

2.9

2.95

3

x 107

pressure

Saturation at time 451.454 days

5 10 15 20

2

4

6

8

10

12

14

16

18

20 0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

saturation

ParametersPermeability (log scale)

5 10 15 20

2

4

6

8

10

12

14

16

18

20−31

−30.5

−30

−29.5

−29

−28.5

permeability

xi = Mi(xi−1) where xi = [pi si ]T

inputs: prescribed bottom holepressures and prescribed flow rates




Summary

4D Variational Data Assimilation

Cost function in 4D-Var

J(p) =n∑

i=1

(Hi (xi) − yi)T

︸︷︷︸Distances

R−1i (Hi (xi ) − yi )︸︷︷︸

Distances

+ (p − pb)TB−1(p − pb)︸︷︷︸

Background termn∑

i=1

(Mi (xi−1) − xi)T Q−1

i (Mi (xi−1) − xi )︸︷︷︸Model error term

minimization withconstraints:

xi = Mi(xi−1) + ε

Control variables:

parameters p (permeabilities)

model error ε




Summary

Twin Experiment

The Truth

One model simulation is generated, saved and called the

Truth.

Observations

Observations are generated with a chosen measurement errorfrom the Truth.




Summary

Imperfect truth construction

In reservoir simulator perturbing grid pressure is not effective andunrealistic

White noise

0 500 1000 15002.45

2.5

2.55

2.6

2.65

2.7

2.75x 10

7

Time [days]

Grid

Pre

ssur

e at

the

wel

ls

NW − 1SW − 21NE − 421SE − 441Deterministic

Red noise

0 500 1000 15002.4

2.45

2.5

2.55

2.6

2.65

2.7

2.75x 10

7

Time [days]

Grid

Pre

ssur

e at

the

wel

ls

NW − 1SW − 21NE − 421SE − 441Deterministic

Perturbed grid pressure at the production wellsJ.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations



Summary


The Idea: perturb inputs instead of the state vector

Red noise

0 500 1000 15002.3

2.4

2.5

2.6

2.7

2.8x 10

7

Time [days]

Bot

tom

hol

e pr

essu

res

121421441Prior

Bias

0 50 100 150 200 250

2.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

2.65

x 107

Time [days]

Pre

scrib

ed b

hp

121421441Prior

Perturbed prescribed bottom hole pressures at the producers




Summary


Obtained model perturbation looks realistic

4D-Var can be applied adding the inputs to thecontrol variable

Red noise

0 500 1000 15002.52

2.54

2.56

2.58

2.6

2.62

2.64

2.66

2.68

2.7

2.72x 10

7

Time [days]

pres

sure

s in

wel

l grid

blo

cks

SW − 421

PerturbedDeterministic

Bias

0 500 1000 15002.35

2.4

2.45

2.5

2.55

2.6

2.65

2.7

2.75x 10

7

Time [days]

Grid

pre

ssur

e

SW − 421

DeterministicPerturbed

Effect of different input perturbations on the grid pressureat the producers




Summary

Accounting for input error in 4D-Var

Cost function in 4D-Var

J(p, u) =n∑

i=1

(Hi (xi) − yi)T

︸︷︷︸Distances

R−1i (Hi(xi ) − yi )︸︷︷︸

Distances

+ (p − pb)T B−1(p − pb)︸︷︷︸

Background term

minimization withconstraints:

xi = Mi(xi−1)

Control variables:

parameters p (permeabilities)

inputs u




Summary

Accounting for input bias - Results

PriorPRIOR permeability (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5

TruthTrue permeability (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5

With InputsRESULTING PERMABILITY RMSE = 0.683751

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5

Without InputsRESULTING PERMABILITY RMSE = 2.84442

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5

0 500 1000 15002.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

2.65

2.7x 10

7

Time [days]

Pre

scrib

ed b

otto

m h

ole

pres

sure

4D−Var (solid line), Truth (dashed line), Prior (dash−dotted line)

Prod: NWProd: NEProd: SWProd: SEPrior

very good permeability estimationwith accounting for input bias

very bad permeability estimationwithout accounting for input bias

very good estimation of inputs




Summary


Prior4D−Var permeability estimation with inputs

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−34

−33

−32

−31

−30

−29

−28

−27

−26


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30

−29

−28

−27

−26

With Inputs4D−Var permeability estimation with inputs

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30

−29

−28

−27

−26


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30

−29

−28

−27

−26

0 500 1000 15002.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

2.65

2.7x 10

7

Time [days]

Pre

scrib

ed b

otto

m h

ole

pres

sure



very good permeability estimationwith accounting for input bias

very bad permeability estimationwithout accounting for input bias

very good estimation of inputs




Summary


Total flow rate

0 200 400 600 800 1000 1200−10

−8

−6

−4

−2

x 10−4

Time [days]

Tot

al fl

ow r

ate

in th

e w

ells

Production well: NW

0 200 400 600 800 1000 1200

−10

−8

−6

−4

−2

x 10−4

Time [days]

Tot

al fl

ow r

ate

in th

e w

ells

Production well: NE

0 200 400 600 800 1000 1200−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

x 10−3

Time [days]

Tot

al fl

ow r

ate

in th

e w

ells

Production well: SW

0 200 400 600 800 1000 1200−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

x 10−3

Time [days]

Tot

al fl

ow r

ate

in th

e w

ells

Production well: SE

TruthPrior4D−Var With Inputs4D−Var Without InputsObs 5%




RESULTING PERMABILITY RMSE = 2.84442

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30

−29

−28

−27

−26

0 500 1000 15002.2

2.25

2.3

2.35

2.4

2.45

2.5

2.55

2.6

2.65

2.7x 10

7

Time [days]

Pre

scrib

ed b

otto

m h

ole

pres

sure






Summary


Water break trough time prediction:

Water saturation

0 500 1000 15000.2

0.4

0.6

0.8

1NW − 1

satu

ratio

ns in

wel

l grid

blo

cks

Time [days]

0 500 1000 15000.2

0.4

0.6

0.8

1SW − 21

satu

ratio

ns in

wel

l grid

blo

cks

Time [days]

0 500 1000 15000.2

0.4

0.6

0.8

1NE − 421

satu

ratio

ns in

wel

l grid

blo

cks

Time [days]

0 500 1000 15000.2

0.4

0.6

0.8

1SE − 441

satu

ratio

ns in

wel

l grid

blo

cks

Time [days]

TruthPrior4D−Var

u

4D−Var strong




Summary


PriorPRIOR permeability (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5

With InputsRESULTING PERMABILITY RMSE = 0.683751

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5

PRIOR permeability (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5

TRUE PERMABILITY (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−31.5

−31

−30.5

−30

−29.5

−29

−28.5


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−31.5

−31

−30.5

−30

−29.5

−29

−28.5


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−31.5

−31

−30.5

−30

−29.5

−29

−28.5

PRIOR permeability (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5

TRUE PERMABILITY (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30.5

−30

−29.5

−29

−28.5




Summary


Prior4D−Var permeability estimation with inputs

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−34

−33

−32

−31

−30

−29

−28

−27

−26


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30

−29

−28

−27

−26

With Inputs4D−Var permeability estimation with inputs

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30

−29

−28

−27

−26


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

−31

−30

−29

−28

−27

−26

4D−Var permeability estimation with inputs

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−34

−33

−32

−31

−30

−29

−28

−27

−26

True permeability (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20 −31.5

−31

−30.5

−30

−29.5

−29

−28.5

−28

−27.5


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20 −31.5

−31

−30.5

−30

−29.5

−29

−28.5

−28

−27.5

4D−Var permeability estimation without inputs

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20 −31.5

−31

−30.5

−30

−29.5

−29

−28.5

−28

−27.5


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−34

−33

−32

−31

−30

−29

−28

−27

−26

True permeability (log scale)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−34

−33

−32

−31

−30

−29

−28

−27

−26


2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−34

−33

−32

−31

−30

−29

−28

−27

−26

4D−Var permeability estimation without inputs

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20−34

−33

−32

−31

−30

−29

−28

−27

−26




Summary

Summary

Perturbing the state vector in reservoir simulations is noteffective neither realistic

Model inputs have a big impact on the reservoir behavior

Accounting for the input bias as control variable in 4D-Vargives very good estimations of the permeability, the outputsof the model, as well as the inputs themselves




Summary

Thank you for your attention!

Questions?


Importance of input error in reservoir simulations fileReservoir Simulator Accounting for input error in 4D-Var Results Summary Importance of input error in reservoir simulations Joanna

Documents