Veldhoven Aug 2005 Jansen

8/2/2019 Veldhoven Aug 2005 Jansen

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DISC Summer School, Veldhoven, August 2005

1

Smart wellsmodeling, estimation and control of oil and gas reservoirs

Jan Dirk Jansen

TU DelftCiTG Geotechnology

Shell International E&PExploratory Research


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2

Smart wells, Smart Fields, Closed-LoopReservoir Management, .A growing research theme

Jorn van Doren2,1 Okko Bosgra2 Roald Brouwer5

Talal Esmaiel1 Jan Dirk Jansen1,5 Sippe Douma5

Renato Markovinovi1 Arnold Heemink3 Hans Kraaijevanger5

Joris Rommelse3,1 Paul van den Hof2Maarten Zandvliet2,1 Cor van Kruijsdijk1 Geir Naevdal6

Justyna Przybysz-Jarnut4,1

1) TUD CiTG - Geotechnology - Petroleum Engineering

2) TUD Delft Institute for Measurement and Control

3) TUD EWI Applied Mathematical Analysis

4) TUD Applied Physics

5) Shell International E&P - Exploratory Research

6) RF Rogaland Research - Reservoir engineering

Cooperation: University of Bergen, MIT, Stanford University, TNO (VALUE, ISAPP)

Sponsoring: Shell, TNO, DELPHI, KISR

www.dietzlab.tudelft.nl


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3

Oil reservoirs

oil trapped in porous rock below an impermeable cap rock

oil

anticline

fault

oil

water

water


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Oil production mechanisms

Primary recovery expansion of rock and fluids,decreasing reservoir pressure

(depletion drive, compaction drive, 5-40% recovery) Secondary recovery injection of water or gas to

maintain reservoir pressure and displace oil actively

(water flooding, gas flooding, 10-60% recovery) Tertiary recovery injection of steam or chemicals

(polymers, surfactants) to change the in-situ physicalproperties (viscosity, surface tension, wettability)

(steam flooding, polymer flooding, 20-80% recovery)


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Reservoir models

(104 106 grid blocks)


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... maar in de porien van gesteenten !..

Geological heterogeneity

50 m

20 m

0.1 mm

Pore scale

Reservoir simulator gridblock scale


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Governing equations

isothermal single-phase flow

Conservation of mass:

Darcys law (replaces conservation of momentum):

Equation of state:

Unknows:p, v, (and known algebraic functions ofp)

( )( )

0qt

+ =

v

( ) p g d

= K

v

0

1

T

cp

=


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Resulting differential equation

isothermal single-phase flow

Full form (nonlinear inp):

Linearized form (small compressibility), no gravity:

Parabolic (diffusion) equation One state variable:p

( ) 0p

p g d c qp t

+ + =

K

2 0p

p c qt

+ =

K


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Governing equations

isothermal two-phase (oil-water) flow

Conservation of mass:

Darcys law :

Equations of state:

Unknows:p, v,, S

( )( )

0S

qt

+ =

v

( )rk

p g d

= K

v

0

1

T

cp

=

{ },o w


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Water saturation Sw, -

Relativepermeability

kr

,-

krow

krw

Relative permeabilities (oil-water, imbibition)


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Resulting differential equations (1)

isothermal two-phase (oil-water) flow

Full form (nonlinear inp and S):

Closure equations:

pc (Sw) is the capillary pressure

( ) 0r

k S p

p g d S c qp t

+ + =

K

( )

1o w

o w c w

S S

p p p S

+ = =


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isothermal two-phase (oil-water) flow p-S form (nonlinear inpo and Sw):

( )

( )

( ) ( )

0 ,

1 0

rw cw o w w

w w

o ww w w r w w

roo o o

o

o w

o w o r o o

k p p S g d

S

p SS c c q

t t

k p g d

p S

S c c qt t

+

+ + =

+

+ =

K

K

Parabolic system?


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isothermal two-phase (oil-water) flow 1-D, K = k,g= 0,pc = 0 , co = cw = cr = 0, qo = qw = 0:

0w w wtw

f S S v

S x t

+ =

ro rw

t w o

o w

kk kk

= + = + ( )1t w w w o r c S c S c c= + +

w ww

w o w o

vfv v

= =+ +

( )t o w o w pv v vx

= + = +

2

2 0t tp p

cx t

+ = parabolic

hyperbolic


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Reservoir simulation

3-phase (gas, oil, water) or multi-component

PDEs discretized in time and space FD/FV

Cornerpoint grids or unstructured grids Fully implicit (Newton iterations) or ImPES

Large variation in parameter values: 10-15 < k< 10-11 m2

Typical model size: 104 106 blocks, 50 500 time steps

Typical code size: 106 lines (well models, PVT analysis)

Primarily used in design phase: field (re-)development

Traditional research focus on upscaling, discretizationmethods, gridding, history matching


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Capital intensive well: 1-100*106 $

field: 0.1-10*109 $

Uncertain geology oil price

limited amount of data

Stretched in time scales production operations: day weeks field development years

reservoir management: 10s of years

Slow in response production: months

reservoir drainage: years

Discipline orientedgeology, geophysics,

reservoir engineering,

production, drilling

Remote deserts

swamps

offshore Speeding up!

horizontal drilling

multi-laterals

time lapse seismics smart wells .

Oil industry - characteristics


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Lower margins, higher complexity of developments

easy oil has been found

reducing cycle times

Lack of human resources, skills erosion

the big crew change

Increasing knowledge- and data intensity proliferation of cheap sensors, data transmitters:

pressure/temperature/flow sensors, time-lapse seismics, passiveseismics, fiber glass cables ,

proliferation of computing power

Oil industry - trends


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Smart Fields


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E&P activity domains

production

operations

days years decadestime

space

OU

asset

field

well

reservoir

management

field dev. planning

portfolio

management

business planning

historic data

& forecasts

objectives

& constraints

objectives

& constraints

historic data

& forecasts

production

operations

reservoir

management

field dev. planning

portfolio

management

business planning

historic data

& forecasts

objectives

& constraints

objectives

& constraints

historic data

& forecasts


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Closed-loop reservoir management

Controllable

input

Identificationand updating

Low-order model

with or w/o physics

Optimization

High-order model

up/down

scaling

Geology, seismics,well logs, well tests,

fluid properties, etc.

Noise OutputInput NoiseSystem

(reservoir, wells

& facilities)

Control

algorithmsSensors


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Smart Fields project characteristics

Key elements:

Optimization under physical constraints and

geological uncertainties Data assimilation aimed at continuous updating

of system models

Up-scaling and down-scaling between

hierarchical system models Inspiration:

Measurement and control theory

Meteorology and oceanography


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Virtual asset model

Controllable

input

Identification

and updating

Low-order model

with or w/o physics

Optimization

High-order model

up/down

scaling

Geology, seismics,

well logs, well tests,



(reservoir, wells

& facilities)

Control

algorithmsSensors


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Waterflooding with smart wells

Fixed configuration Pressure and bulk rate measurements


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Smart well

Network Splitter Isolation Unit (SIU)

SCSSV

Gas Lift Device

Wet Disconnect Unit

Zonal Isolation Packer

ICV with Sensors


ICV with Sensors


ICV with Sensors

Production Packer

SCSSV Control Line

Flat Pack with Single Hydraulic and Single Electrical Line

Dual Flat Packs each containing a Single Hydraulic and Single Electrical Line


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Virtual asset

45 x 45 grid blocks

45 inj. & prod. segments

po, Swat segments known

oil in streak: 15%

1 PV injected, qinj= qprod

oil price ro = 80 $/m3

water costs rw= 20 $/m3

discount rate b = 0%-13.5

-13

-12.5

-12

-11.5

-11

permeability field

10 20 30 40

5

10

15

20

25

30

35

40

45

10log(k) [m2]


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Results: conventional production

0 100 200 300 400 500 600 7000

200

400

600

800

r

ates[m3/d]

cum time [d]

water, oil and liquid production rates (m3/d) as function of time

0 100 200 300 400 500 600 7000

1

2

3

4

5x 10

5

cum.production[m3]

cum time [d]

cumulative water, oil and liquid production (m3) as function of time

ref wat

ref liqref oilopt watopt watopt oil

0

0.1

0.2

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0.5

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25

30

35

40

45 0

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5 10 15 20 25 30 35 40 45

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35

40

45 0

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5 10 15 20 25 30 35 40 45

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25

30

35

40

45 0

0.1

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1

5 10 15 20 25 30 35 40 45

5

10

15

20

25

30

35

40

45

Equal pressures in all segments, at injection and production well


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Step 1: open-loop control

Controllable

input


Reservoir

Model

Optimization

High-order model

up/down

scaling




(reservoir, wells

& facilities)

Control

algorithmsSensors


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Optimization techniques

Global versus local

Gradient-based versus gradient-free

Constrained versus non-constrained

Classical versus non-classical (genetic algorithms,simulated annealing)

We use optimal control theory local, gradient-based Has been proposed for history matching (Chen et al.

1974, Chavent et al. 1975, Li, Reynolds and Oliver 2003 )and for flooding optimization (Ramirez 1987, Asheim

1988, Virnovski 1991, Zakirov et al. 1996, Sudaryanto andYortsos, 1998)


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Constrained optimization

( )

( ) ( ) ( )

2 2

2 2

, subject to

2 2

2 0 2

2 02 2

0

J x y x y k

J x y x y k

J J J

J x yx y

x x y y x y k

x x

k y y x y

x y k k

= + + == + + +

= + + = + + + + +

+ = =

+ = = = =

+ = =

x

y

Top view


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System theory - notation

System equations:

Initial conditions:

LTI system:

LTI output:

x = state (pressures and saturations),u = input (well rates, BHPs, valve settings),

k= discrete time, = system parameters (permeabilities, porosities, etc.)

( ) ( ) ( ) ( ), 1 , ,k k k k + = g x x u 0

( ) 0 =x x

( ) ( ) ( )1k k k+ = +x Ax Bu

( ) ( )k k=y Cx

( ) ( ) ( ) ( )1 , ,k k k k + = x f x u


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Optimization problem

System equations:

Initial conditions:

Objective function:

Constraints:

Problem statement: max Ju

( ) ( ) ( ), 1 ,k k k+ = g x x u 0

( ) 0 =x x

( ) ( ) ( )1 1

0 0

,K K

k k

k k

J J k J k k

= =

= = u x

( ) ( ),k k c x u 0


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Objective function

Simple Net Present Value (NPV) Ktime steps

Nprdproducers

( )( )( ) ( )( )

( )

( )( )

, , , ,

11

prodNw w j w j o o j w j

k t kj

r q S k r q S k J k t k

b =

+ =

+

( )1

0

K

k

k

J J k

=

= x


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( )( ) ( )( ) ( )( ) ( )

1K

i i

ki i

J J k k J u

u k u

=

= +

x

x

Sensitivities

( ) ( )1

0

,K

k

k

J J k k

=

= x u

Wanted: change Jias a result of perturbation ui(k) at k=

cannot be

determineddirectly


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Optimal control (1a)

Modified objective function:

First variation:

( ) ( ) ( ) ( ) ( ) ( ){ }

( ) ( ) ( ) ( ) ( )

1

0

1 1

0 0

, 1 , 1 ,

, 1 , , 1

KT

k

k

K K

k k

J J k k k k k k

k k k k k

=

= =

= + + +

= + + =

u h g x x u

x x u L L

( )

( )( )

( )

( )( )

( )

( )( )

( )( )

( )

1 1 1

0 0 0

1

0

11

11

K K K

k k k

K

k

k k k J k k k

k k k

k kk

= = =

=

= + + +

+

+ + +

x x ux x u

L L L

L


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Optimal control (1b)

After shifting indices and splitting summations:

( )

( )( )

( )

( )

( )

( )( )

( )( )

( )( )( )

( )( )

( )( )

1

1

1 1

0 0

0 10

0

11 .

1

K

k

K K

k k

k kJ k

k k

K k k K k k

K k k

=

= =

= + +

+ + + + +

x xx x x

x u x u

L L L

L L L

(initial condition)( )0 =x 0 (system equations)( ) ( )1k k + = = g 0L

( )( )

( )( )

1k kk k

+ =

0x x

L Land that

Furthermore, require that

( )( )

1K

K =

0

x

L


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Optimal control theory, summary

Gradient based optimization technique

Objective function: NPV or ultimate recovery

Controls: injection/production rates (rate-controlled) or

valve openings (pressure- controlled) Gradients of objective function with respect to controls

obtained from adjoint or co-state equation

Results in dynamic control strategy, i.e. controlschange over time

Computational effort independent of number ofcontrols


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Simulate with initial control strategy (forward in time)

Calculate objective function J Simulate adjoint equation (backward in time)

Use derivatives to choose new control strategy,e.g. with steepest descent or quasi-Newton method

Repeat cycle until optimum is reached

Typically 5-10 cycles required

Optimization procedure


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Rate constrained operation

Enough energy to maintain flow rates for any numberand combination of active wells

Total injection/production rate independent of number

of active wells


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Conventional (equal pressure in all segments, no control)

0

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30

35

40

45

0

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5 10 15 20 25 30 35 40 45

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25

30

35

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5 10 15 20 25 30 35 40 45

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25

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5 10 15 20 25 30 35 40 45

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25

30

35

40

45

Best possible (identical field rate, no pressure constraints)

Results: rate-constrained (1)

l i d (2)


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NPV+60%

Production

+ 41% cum oil- 45% cum water

0 100 200 300 400 500 600 7000

200

400

600

800

rates[m3/d]

cum time [d]


0 100 200 300 400 500 600 7000

1

2

3

4

5x 10

5

c

um.production[m3]

cum time [d]


ref watref liqref oilopt watopt watopt oil

R lt t t i d (3)


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Injection and production rates as function of time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x 10-6

cum time [yr]

wellnumber(45wellsintotal)

inj. rates (m3/d)vs. time for all wells

5

10

15

20

25

30

35

40

45 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x 10-6

cum time [yr]

wellnumber(45wellsintotal)

prod. rates (m3/d)vs. time for all wells

5

10

15

20

25

30

35

40

45



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Pressure-constrained operation

Limited energy available

Total injection/production rate dependent on number

of active wells

R l i d


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0 100 200 300 400 500 600 7000

200

400

600

800

rate

s[m3]

cum time [d]


0 100 200 300 400 500 600 7000

1

2

3

4

5x 10

5

c

um.production[m3]

cum time [d]


ref watref liqref oilopt watopt liqopt oil

Improvement in NPV

+53%

Production+16% cum oil

-77% cum water

Injection-32% cum water

Results: pressure-constrained

O ti l tti (1)


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cum time [yr]

wellnumber

n. va ve se ngvs. time for all wells

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cum time [yr]

wellnumber

pro . va ve se ngvs. time for all wells

5

10

15

20

25

30

35

40

45

Optimum valve-settings (1)

100 200 300 400 500 600 700 800 9000

0.2

0.4

0.6

0.8

1

valve-setting

optimum valve-position for injector segment 12 as function of time step

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1time step (n)

inje

ctsegm1

2

optimum valve-position for injector segment 12 as function of time step

100 200 300 400 500 600 700 800 900

12

12

12

O ti l tti (2)


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cum time [yr]

wellnumber

.vs. time for all wells

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cum time [yr]

wellnumber


5

10

15

20

25

30

35

40

450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cum time [yr]

wellnumber


5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cum time [yr]

wellnumber


5

10

15

20

25

30

35

40

45

Optim m al e settings (3)


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sw at 2 days sw at 12 days sw at 129 days sw at 199 days

sw at 272 days sw at 386 days sw at 603 days

Zone A

Zone B

Zone C

5

10

15

20

25

30

35

40

45

injectorsproducers

10 20 30 4010 20 30 40

5

10

15

20

25

30

35

40

45

injectorsproducers

Streak I

Streak P

Zone A

Zone B

Zone C

Optimum valve settings (3)

Optimum valve settings (4)


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cum time [yr]

wellnumber


5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cum time [yr]

wellnumber


5

10

15

20

25

30

35

40

45

3 valvesin injector

4 valves inproducer

Example dynamic optimisation (1)


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2 4 6 8 10 12

x 10-12

distance [m]

distance[m

]

100 200 300 400 500 600 700 800 900 1000 1100 1200

100

200

300

injectorsproducers

P.A. 1 P.A. 2 P.A. 3I.A. 1 I.A. 2


Top view line-drive water injection

producers

injectors



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conventional

distance m

distance[m]

100 200 300 400 500 600 700 800 900 1000 1100 1200

100

200

300

P.A. 1 P.A. 2 P.A. 3I.A. 1 I.A. 2


Conclusions


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ConclusionsWater flood optimization

Improvement in NPV for all cases considered

Degree of improvement depends on

Operating conditions Pressure: mostly water production reduction

Rate: acceleration of production, increased oilrecovery, reduced water production

Heterogeneity type Restrictions on water production

Restricted number of segments per well may often besufficient

Local method: improvements are lower bounds

Step 2: closed-loop control


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Step 2: closed-loop control

Controllable

input


ReservoirModel

Optimization

High-order model

up/down

scaling

Geology, seismics,

well logs, well tests,fluid properties, etc.


(reservoir, wells

& facilities)

Control

algorithmsSensors

Data assimilation (automatic history


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Data assimilation (automatic history

matching) Variational methods - history match problem defined asminimization problem:

Ensemble Kalman filtering meteorology, oceanography,groundwater flow

Reservoir-specific methods

Streamlines: Datta-Gupta (Texas A&M), Thiele (Streamsim)

Probability perturbation: Caers (Stanford)

( ) ( ) ( ) ( ){ }1

1

T

m y m

i J i i i i

== y y R y y( ) ( ) ( ) ( ) }1

T

u ui i i i

+ R

Kalman filtering


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Kalman filtering

Ordinary Kalman filtering for linear systems 1960s Determines weighted average of model results and measured

data

Weighting based on uncertainty in model and data

Ensemble Kalman filtering: model uncertainty propagatedthrough simulation of large number of realizations

Can be used to update state variables (pressures, saturations)and model parameters (permeabilities, porosities)

Originally developed in meteorology 1990s

First applied in reservoir engineering by Geir Naevdal(Rogaland Forskning)

Also being investigated by Evensen (Norsk Hydro),Oliver (U of Oklahoma), Reynolds (Tulsa U)

Estimation


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Estimation

Consider two estimates and of random variablex,with

What is the best estimate?

Try: To be unbiased:

Therefore:Variance:

Minimum occurs when:

Optimal estimate:

1x 2x

( ) ( ) ( )2

11 2 2

2

0 ,

0xE x E x E x R

= = =

1 1 2 2

ux a x a x= + 1 2 1a a+ =

( )2 1 2 2 1u x a x a x= +( ) ( )2 22

2 1 2 21ux a a = + ( )2 2 22 2 1 2a K = = +

( )1 2 1 ux x K x x= +

Ordinary Kalman filtering


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Ordinary Kalman filtering

Linear system:

1. Start from xu(i) and Rx,u(i), set x(i) = xu(i) and propagate the

system model: x(i+1) = Ax(i) + Bu(i) + v(i)

2. Set Rx(i) = Rx,u(i) and propagate the error covariance:

Rx

(i+1) = A(i)Rx

(i)AT(i) + Rv

(i)

3. Kalman gain:

4. Assimilate: xu(i+1) = x(i+1) + K(i+1) [ym(i+1) - y(i+1)]

5. Update the error covariance: Rx,u(i+1) = [I-K(i+1)C]Rx(i+1) .

( ) ( ) ( ) ( ) ( )1 , , vk k k k N + = + + =x Ax Bu v v 0 R ( ) ( ) ( ) ( ), , yk k k N = + =y Cx w w 0 R

( ) ( ) ( ) ( )1

1 1 1 1T T x x y

i i i i

+ = + + + + K R C CR C R

Ensemble Kalman filtering


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se b e a a te g

Nonlinear system:

Propagate all ensemble members and compute

Compute the error covariance from the ensemble:

where

For reservoir engineering, use extended state:

( ) ( ) ( ) ( ) ( )1 , , , , vk k k k N + = = x f x u v v 0 R ( ) ( ) ( ) ( ), , yk k k N = + =y Cx w w 0 R

( ) ( ) ( )1

1 1 11

T

xi i i

J+ = + +

R X X

( ) ( ) ( ) ( ) ( )11 1 1 , , 1 1Ji i i i i + = + + + + X x x x x

1

1 J

j

jJ =

= x x

=

xx

Closed-loop optimization procedure


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1. Start from initial ensemble of reservoir model estimates

2. Determine most likely model

3. Determine optimal control u (over lifetime of reservoir)4. Simulate true reservoir behaviour (over measurement

interval). Generate synthetic measurements y. Add noise.

5. Update all ensemble members i.e. estimate x and 6. Go back to 2.

Note: step 2 may be skipped if multiple models are optimized.

p p p

Virtual asset model


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Virtual asset model

Controllable

input

Update allmodels

Noise OutputInput NoiseSystem(reservoir, wells

& facilities)

Select optimal

inputSensors

Ensemble of

reservoir models

Optimize

updated

models

Simulate overlifetime

Simulate over

measurementinterval

Closed-loop optimization example 1


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p p pPermeability estimates with Kalman filtering

-12.6

-12.4

-12.2

-12

2.3148 days

20 40

10

20

30

40-12.5

-12

-11.5

4.6296 days

20 40

10

20

30

40

-12.5

-12

-11.5

-11

6.9444 days

20 40

10

20

30

40-13.5

-13

-12.5

-12

-11.5

9.2593 days

20 40

10

20

30

40-13

-12

-11

11.5741 days

20 40

10

20

30

40

-13

-12

-1123.1481 days

20 40

1020

30

40-13

-12.5

-12

-11.5

46.2963 days

20 40

1020

30

40-13

-12.5

-12

-11.5

-11

69.4444 days

20 40

1020

30

40

-13

-12.5

-12

-11.5

-11

92.5926 days

20 40

10

2030

40-13

-12.5

-12

-11.5

-11

115.7407 days

20 40

10

2030

40

-13

-12

-11

0 days

20 40

10

20

30

40

Brouwer, Naevdal et al.,SPE 90149, 2004



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Saturations after 116 days

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10 20 30 40

5

10

15

20

25

30

35

40

45 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

10 20 30 40

5

10

15

20

25

30

35

40

45

Estimate fromKalman filter

Real fromVirtual Asset



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Final saturations

0.2 0.4 0.6 0.8

conventional

distance[m]

distance [m]

100 200 300 400

100

200

300

400

0.2 0.4 0.6 0.8

known

reservoir

distance [m]

100 200 300 400

100

200

300

400

0.2 0.4 0.6 0.8

unknown

reservoir

distance [m]

100 200 300 400

100

200

300

400



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62

p p pvertical wells (4 injectors, 9 producers)

Measurements

0.2 bar accuracy

Down-hole pressuremeasurements for producers

Surface pressuremeasurements for injectors

Total flow rates per well

Control at surface Unknowns to be estimated from

production data:

Pressure, saturation, andpermeability distribution

-13.5

-13

-12.5

-12

-11.5

distance[m]

distance [m]100 200 300 400

100

200

300

400

injectorsproducers

True permeability field

Closed-loop optimization example 2bili i i h l fil i


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63

Permeability estimates with Kalman filtering

-13.5

-13

-12.5

-12

-11.59 days

-13.5

-13

-12.5

-12

-11.519 days

-13.5

-13

-12.5

-12

-11.5

28 days

-13.5

-13

-12.5

-12

-11.5

37 days

-13.5

-13

-12.5

-12

-11.5

46 days

-13.5

-13

-12.5-12

-11.593 days

-13.5

-13

-12.5-12

-11.5185 days

-13.5

-13

-12.5-12

-11.5278 days

-13.5

-13

-12.5

-12

-11.5370 days

-13.5

-13

-12.5

-12

-11.5463 days

-13.5

-13

-12.5

-12

-11.50 days



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64

p p pSaturation distributions

Conventional water flood

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1465 days

10 20 30 40

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

12201 days

10 20 30 40

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

13796 days

10 20 30 40

5

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1463 days

10 20 30 40

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

12205 days

10 20 30 40

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

13796 days

10 20 30 40

5

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1465 days

10 20 30 40

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

12201 days

10 20 30 40

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

13796 days

10 20 30 40

5

10

15

20

25

30

35

40

45

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1463 days

10 20 30 40

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

12205 days

10 20 30 40

5

10

15

20

25

30

35

40

45 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

13796 days

10 20 30 40

5

10

15

20

25

30

35

40

45

463 days 2205 days 3796 days

Closed-loop optimized water flood

oil

water



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65

110 x 110

grid blocks

10 inj.segments (x)

10 prod.segments (o)

1100 m

1100 m

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

p p pVirtual asset

Closed-loop optimization example 3Results


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66

esu ts

Saturation forconventionalwaterflooding.

Saturation for

optimizedwaterfloodingwith updatedpermeability field.

Estimatedpermeabilityfield.[20 7000 mD]

Injection rates.The total

injection rate isconstant and2600 m3/day

46 days

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

10

116 days

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

10

463 days

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

10

750 days

1

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

10

0 days

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

Closed-loop optimization example 3Permeability estimates with Kalman filtering


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67

Permeability estimates with Kalman filtering

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

Real permeability field

Virtual Asset

Estimated average

permeability field

Model reduction


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68

Why are such simple models

working so well?

Model reduction


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69

What ? High order: 103-106 state variables

Low order: 101-102 state variables

Why ?

Reduce computational burden of optimization Regularize data assimilation problem

Adjust model size to what you can observe and control

How?

Proper Orthogonal Decomposition (Karhunen Love decomposition,Principal Component Analysis), Heijn et al. SPEJ June 2004

www.win.tue.nl/macsi-net

Example POD image compression (1)


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50 100 150 200

50

100

150

200

-13.5

-13

-12.5

0 50 100 150 200 2500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

50 100 150 200

50

100

150

200

-13.5

-13

-12.5

Original image: 62500 pixels

200 samples of 20 x 20

95 % energy level retained

147 base functions

Example POD image compression (2)average and first 8 basis functions


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average and first 8 basis functions

5 10 15 20

5

10

15

20

-12.8

-12.75

-12.7

5 10 15 20

5

10

15

20

-0.05

0

0.05

5 10 15 20

5

10

15

20 -0.05

0

0.05

5 10 15 20

5

10

15

20 -0.02

0

0.02

0.04

0.06

0.08

5 10 15 20

5

10

15

20 -0.1

-0.05

0

0.05

5 10 15 20

5

10

15

20

-0.05

0

0.05

0.1

5 10 15 20

5

10

15

20 -0.1

-0.05

0

0.05

0.1

5 10 15 20

5

10

15

20 -0.1

-0.05

0

0.05

0.1

5 10 15 20

5

10

15

20 -0.1

-0.05

0

0.05

0.1

POD the recipen


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72

Original equation:

Collect snapshots:

Decompose X with SVD:

Select ln singular values:

Transformation:

Reduced equation:

( ) ( ) ( )1 - , 2 - , ..., -= X x x x x x x

( ) ( )11 i i

== x x

2 2

1 1

l

i ii i

E

= ==

T=X

, ll + x z x z

( ) ( ) ( )( )1 ,Tl lk k k + = + z f z x u x

( ) ( ) ( ) { }1 , , , ,n

o wk k k+ = x f x u x x p S

POD computational advantage


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73

Original equation: Semi-implicit time dicretization:

Reduced equation; substitute :

( )( ) ( ) ( ) ( )( ) ( )1c cn n

t k k k t k k I A x x x B x u

+ = +

( ) ( ) ( )( ) ( ) ( ) ( ) ( ),c c ct t t t t = = +x f x u A x x B x u

( ) ( )( )( ) ( ) ( )( ) ( )

11c c

k kk k k k

t

x xA x x B x u

+ = + +

( )( ) ( ) ( ) ( )( ) ( )1c ct k k k t k k I A z x z z B z x u + + = + +

( )( ) ( ) ( ) ( )( ) ( )1T Tc clxl

t k k k t k k I A z x z z B z x u + + = + +

= +x z x

POD for optimal control Original adjoint equation:


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74

( )( )

( )( )

( )

( )

( )

( )

11

T T

n n

k k J k k k

k k k

= +

g g

x x x

( )( )

( )

( )( )

( )

( )

( )1

11

T T kT T

lxl lxl xl

k k J k k k

k k k

= +

g g

x x x

Original adjoint equation:

Reduced objective function:

Reduced adjoint equation:

( ) ( )( ) ( ) ( ) ( ) ( )( )1

01 1

, 1 1 , ,K

T Tred k

kl l

J J k k k k k k

=

= + + +

z u g z z u

Optimal control

Apply initial input u

START


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75

with POD

Calculate NPV reduced model

No

Yes

Simulate full forward model

Calculate NPV full model

Full NPVconverged?

No Yes

Calculate (truncated) transformation matrix

Substitute x =z into the Hamiltonian andrun the reduced adjoint

Produce optimized input

Simulate reduced forward model

Reduced NPV

converged?

DONE

Example POD dynamic reductionFull-order Reduced-order Difference


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Original model: 4050 states

Reduced model 41 states 99.9 % of signal energy maintained

Sw

po

Full order Reduced order Difference

Van Doren, Markovinovi, 2004

NPV optimization POD vs. full model


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0 2 4 6 8 10 12 14 16 18 20

10

11

12

13

14

15

16

17

18

19

20

21 Net present value ($) vs Number of iterations updating matrices

Number of iterations ()

Netpresentvalue(million$)

fullorderreducedorder 1reducedorder 2reducedorder 3reducedorder 4

POD - remarks


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78

Method to quantify complexity

as present in the geology

as present in the process dynamics

Limited computational advantage, so far

Reservoir dynamics lives in low-order space

Model reduction


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79

Does this mean that we dont need

geology?

Model reduction


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80

NO!

I mean yes, we do need geology!

Uncertainty requires use of many realizations:

Different scenarios + stochastic variations Our models are heavily over-parametrizedAdditional constraints can only come from geology !

Model-based closed-loop control


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Controllable

input


Optimization

up/down

scaling




& facilities)

Control

algorithmsSensors

Low-order models

with or w/o physicsHigh-order models

Closed-loop reservoir management conclusions so far


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Large scope for optimization of flooding processes(open-loop control)

Adjoint based-optimization techniques available Proper treatment of constraints still an issue

Data assimilation techniques from meteorology/oceanography promising but available for research only

Combined optimization and data assimilation (closed-loopcontrol) in early development

Low-order modelling (POD) fascinating but not yet

applicable

Closed-loop reservoir management - nextsteps ?


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83

p

More realistic models (3D, complex, well constraints)

More complex physics (WAG, polymers, fractures, steam)

More measurements (time-lapse seismics)

Optimization of multiple models

Optimization of configuration (well positions)

Multiple geological scenarios - ensemble management

Active input control to obtain information

Field experiments

Reduced-order modeling, multi-scale modeling

Value of information what, when and where to measure?

Questions?


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84

Controllable

input


Optimization

up/down

scaling




& facilities)

Control

algorithmsSensors

Low-order models

with or w/o physicsHigh-order models

Veldhoven Aug 2005 Jansen

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