An Effective Reservoir Management by

Streamline-based Simulation, History

Matching and Optimization

Shusei Tanaka

May, 2014

• Development of a general purpose streamline-based reservoir simulator: Inclusion of diffusive flux via Orthogonal Projection Illustration by black oil model Extension to a multicomponent system

• Application to Brugge benchmark case: Streamline-based simulation Streamline-based BHP/WCT data integration Flow diagnostics for streamline-based NPV optimization

• Conclusion

Outline

2/50

Streamline Technology: Overview

3/50

• Key concept of Streamline: Fast IMPES-based reservoir simulation

History matching(HM) by calibration of travel time

Improves sweep efficiency by streamline information

Pressure field Streamlines Connection map

Problem Statement:

SL-based Reservoir Management

4/50

• Challenges for mature field, multiple well… Quick forecasting

HM for individual well

Improve NPV by reallocating well rate

• Streamline is efficient, but can we apply all the time? What if flow is not convective dominant?

How about prior to breakthrough for HM?

Can we improve NPV?

Mature field with multiple

wells

Development of a General Purpose Streamline-

based Simulator

Motivation

6/50

Solve 1D Convection EquationsCalculate Diffusive Flux on Grid

Compute Pressure & Velocity Field

• Streamline simulation is difficult to apply if…

System of equation is highly nonlinear (ex. Gas injection)

Capillary and gravity effects are dominant

Error by Operator-Split

Error by IMPES

; 0

w

w ut

S

Why Split the Equation?

• Water velocity does not follow total velocity with capillary (and gravity)

7/50

tuwu

Streamline

cowowtww pkFuFu

• Split equation by physical mechanisms

Convective

Transport

Capillary

Diffusion 0

cowow

w pkFt

S

0

w

w ut

S 0

wt

w Fut

S

cowowtww pkFuFu

Saturation Transport

Equation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

Wa

ter

Sa

tura

tio

n

Normalized Distance

Correct Solution

Convection Flow

Too much diffusion

with large time step

(SPE 163640)

Operator Splitting

Capillary after

convection

8/50

0~

wtw

w Fuut

S

cowowtwwwtw pkFuFFFuu

~

~

• Split equation by physical mechanisms

• Anti-diffusive corrections

Computationally expensive:Function of (P, T, composition,

Initial state) for each grid, time step

0

w

w ut

S 0

wt

w Fut

S

Splitting with anti-

diffusive flux

Convection Eq.

Corrected Operator Splitting

Anti-diffusive concave envelope

9/50

0

w

w ut

S

wtww uufu

Parallel component,

calculate along

streamline

Anti-diffusive correction

not needed

Orthogonal Projection

• Split equation into parallel and transverse flux terms

twf u

wu

tuwu

Streamline

10/50

twf u

wu

0

w

w ut

S 0

tw

w uft

S

0

w

w ut

S

tuwu

Orthogonal Projection

Parallel to Ut

(Solve along streamline)

Transverse to Ut

(Solve on grids)

Streamline

• Split equation into parallel and transverse flux terms

wtww uufu

Parallel component,

calculate along streamline

Anti-diffusive correction not

needed

11/50

1.Compute pressure & velocity field

Include capillary effects

2.Trace streamlines

Solve 1D convection equations

Include capillarity and gravity

3.Map back saturation to grid

Calculate corrector term

Predictor-Corrector Workflow

Iterative IMPES

Orthogonal Projection

12/50

• Pressure equation(IMPES)

• Transport equation (along SL)

Orthogonal Projection:

Application to Multicomponent System

0

owgj

j

owgj

j

owgj

jj

owgj

jjr Qupuct

pScc

i

sl

ii fmt

cfgDpFyu

kyFfSym

sl

ii

owgj owgj jogwmm

m

jmjmcmjjij

t

jijj

sl

ij

ogwj

jiji

1

,2 , ,

Δ

• Transport equation (on Grid, corrector)

ogwj jmogwm

m

jmcjmmjjijtti DgpFykuuI

t

m

,

ˆˆ

Pc,Gravity along streamline

Transverse Pc,Gravity on grid

0

owgj

ijijjjijjjij qyuySyt

• Governing equation

13/50

Illustrative Example

100 mD

5 mD

• Water injection 0.2PVI, then CO2 0.2PVI

• Single time step for each injection period

• Observe capillarity by parallel/transverse to Ut

tw uf

wu

14/50

Water Saturation and Capillary Flux

Distributions

• Capillarity traps water at center by J-Function

• Capillarity flows back water towards injector during gas injection period

Sw after water injection

Arrow: water capillary flux

Sw after gas injection

Arrow: water capillary flux

1)(

kSwJpcow

15/50

Water Capillary Flux:

Parallel and Transverse to Total Velocity

Total capillary fluxCapillary flux transverse

to total velocity

Capillary flux along

total velocity

• Most of the capillary effects can be included along the streamlines

cow

t

tt

t

ow

t

w pu

uuI

u

ku

2

Along streamline On grids

16/50

cow

t

t

t

ow pu

uk

2

Water Saturation Distribution

Commercial, FD Operator Splitting

(no correction)Orthogonal Projection

• OP can take large time step without anti-diffusive correction

17/50

Injection :: CO2

10 rb/D – 1000 [Days]

Production :: BHP

(1900 psi)

2D Cross-Section CO2 Flooding Model

Pc, Convection

Pc, Gravity

Simulation model:• 7 HC component + Water• Rel-Perm by Corey• Water-wet Capillarity

Initial & Boundary Condition• 2000+ psi , 212F˚• Constant production BHP, constant CO2 injection at 10 rb/D• 1000 days

18/50

CO2 Mole Fraction Distribution:

Along Streamline

Including Pc & GravityConvection only

19/50

Production Mole Fraction of CO2

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 200 400 600 800 1000

Pro

du

ctio

n M

ole

Fra

ctio

n(C

O2

)

Time [Days]

Streamline

Commercial Simulator

Number of time step:

Commercial FD = 56

Streamline = 21

20/50

CO2 Mole Fraction Distribution:

Final Distribution

Orthogonal Projection

(After corrector term)Commercial FD

(E300)

21/50

0

50

100

150

200

250

300

350

400

450

500

2D Areal 2D Cross-Section 2D Cross-SectionHetero

Goldsmith Field

E300 FIM

Streamline

Previous case

Comparisons of Number of

Time Step

Nu

mb

er

of

Tim

e S

tep

Tested simulation cases

in the paper

10×

2×4×

3×

22/50

0

50

100

150

200

250

300

350

400

450

500

2D Areal 2D Cross-Section 2D Cross-SectionHetero

Goldsmith Field

E300 FIM

Streamline

Previous case

Comparisons of Number of

Time Step

Nu

mb

er

of

Tim

e S

tep

Tested simulation cases

in the paper

10×

2×4×

3×

23/50

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 180 360 540 720 900 1080

Pro

du

ctio

n M

ole

Fra

ctio

n (

CO

2)

Time [Days]

Streamline

Commercial Simulator0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0E+00 2.5E+04 5.0E+04 7.5E+04 1.0E+05

Pro

du

ctio

n M

ole

Fra

ctio

n (

CO

2)

Time [Days]

Streamline

Commercial Simulator

Conclusions

24/50

• Developed a new SL-based simulation method to incorporate capillarity and gravity and applied to CO2 injection cases

• Computational advantages:• Minimizes the saturation correction term

• Can take large time steps without anti-diffusive corrections

• Demonstrated by synthetic and field case:• Iterative IMPES approach handles nonlinearity

• Larger time stepping obtained compared with commercial FD simulator

Application to Brugge Benchmark:

- Streamline-Simulation

- History Matching

- NPV Optimization

Brugge Benchmark Example

26/50

• Benchmark model for HM, optimization problem• 20 producers, 10 injectors in complex geometry• Conduct 40 years of waterflood, 1000 stb/d per wel

Oil saturation and well location

Initial So Net gross ratio

PorosityRock table ID

ECLIPSE vs. Streamline Simulation:

Water-Cut (4 producers)

27/50

Circle : ECLIPSE

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 7500 15000

Pro

du

ctio

n W

ate

r C

ut

Time [Days]

BR-P-18 ECL

BR-P-8

BR-P-12

BR-P-1

BR-P-18 SL

BR-P-8

BR-P-12

BR-P-1

Line: Streamline

- ECLIPSE without NNC option

Comparisons of Oil Saturation

Distribution

28/50

Initial oil saturation

After 20 years

Streamline Commercial (ECL)

Presented at student paper contest 2013

Application to Brugge Benchmark:

- Streamline-Simulation

- History Matching

- NPV Optimization

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 500 1000 1500 2000

Wat

er

Cu

t

Time [Days]

Streamline-based Inverse Modeling

30/50

min 𝛿𝐝𝑤𝑐𝑡 − 𝐒𝑤𝑐𝑡𝛿𝐤

𝛿𝐝

1. Run reservoir simulation by given model

2. Trace Streamlines and calculate parameter sensitivity

3. Update parameters to satisfy:

Observation

Prediction

Motivation and Objective

31/50

Streamlines0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 500 1000 1500 2000

Wat

er C

ut

Time [Days]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000

CO

2M

ole

Fra

ctio

n

Time [Days]

WCT

• What can we tell prior to breakthrough? Pressure data can be used while not considered previously

• Study objective New approach to calculate pressure sensitivity along SL

Simultaneous inversion of pressure and water-cut data

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 50 100 150 200

Bo

tto

m H

ole

Pre

ssu

re

Time [Days]

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 50 100 150 200

Bo

tto

m H

ole

Pre

ssu

re

Time [Days]

BHP

Observation

Initial

ik

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 500 1000 1500 2000

Wat

er

Cu

t

Time [Days]

𝛿𝒕𝑤𝑐𝑡

Production WCT

Parameter Sensitivity Along Streamline

32/50

• TOF( ): Travel time of neutral tracer along streamlines

, ,

, ,x y z

Inlet

dsx y z

u

𝜕𝑡

𝜕𝑘𝑖= −

𝜕𝑆

𝜕𝜏

𝜕𝜏

𝜕𝑘𝑖∙𝜕𝑆

𝜕𝑡

−1

=1

𝑓′(𝑆)

∆𝜏𝑖𝑘𝑖

• Water-cut travel time sensitivity:

injectorProducer

[He et. al,2003]

𝜕𝑝𝑏ℎ𝑝𝜕𝑘𝑖

=𝜕∆𝑝𝑖𝜕𝑘𝑖

≈∆𝑝𝑖𝑘𝑖

• Bottom hole pressure sensitivity: [new]

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 50 100 150 200

Bo

tto

m H

ole

Pre

ssu

re

Time [Days]Production BHP

𝛿𝒑𝑏ℎ𝑝

𝜕𝑝𝑏ℎ𝑝𝜕𝑘𝑖

≈𝜏𝑖𝜏

𝜕∆𝑝𝑖𝜕𝑘𝑖

≈𝜏𝑖𝜏

∆𝑝𝑖𝑘𝑖

Rate-Rate constraint

Rate-BHP constraint

Sensitivity Results: 1D CPG

(3phase Gas Injection)

-20.0

-15.0

-10.0

-5.0

0.0

0.0 0.5 1.0

Pre

ssu

re S

en

siti

vity

, wrt

k

Normalized Distance

Analytical (Stremaline)

Adjoint Method

0.0

5.0

10.0

15.0

20.0

0.0 0.5 1.0

Pre

ssu

re S

en

siti

vity

, wrt

k

Normalized Distance

Analytical (Stremaline)

Adjoint Method

33/50

Inj: Gas Rate

Prd: Rate

Producer BHP sensitivity to k

Injector BHP sensitivity to k

Sensitivity Results: 2D Areal

34/50

Inj

P1

P2P3

P4

Injector BHP sensitivity by k

P1 BHP sensitivity of by k

Permeability field(Wells by rate constraint)

Adjoint Proposed

Inversion of Permeability by LSQR

35/50

• Run simulation and get following parameter

• Solve LSQR Matrix :

• Advantages:• Find pressure/WCT sensitivity during SL simulation• Localized (high resolution) changes in permeability

min 𝛿𝐝𝑤𝑐𝑡 − 𝐒𝑤𝑐𝑡𝛿𝐤 + 𝛿𝐝𝑏ℎ𝑝 − 𝐒𝑏ℎ𝑝𝛿𝐤 + 𝛽1 𝐈𝛿𝐤 + 𝛽2 𝐋𝛿𝐤

𝐒𝑤𝑐𝑡𝐒𝑏ℎ𝑝𝛽1𝐈𝛽2𝐋

∆𝐤 =

𝛿𝐝𝑤𝑐𝑡𝛿𝐝𝑏ℎ𝑝00

Water-Cut Pressure - Smoothness- Consistency with

static model

Scaled by stdev

History Matching of Brugge Field

• Use simulation result of Real.77 as observed data• Use Real.1 as initial model• Assume 3 years of data is available

Reference model Initial model

36/50

Available Observation Data

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

144.0 644.0 1144.0

Pro

du

ctio

n W

ate

r C

ut

[-]

Time [Days]

BR-P-11

BR-P-12

BR-P-15

BR-P-18

BR-P-11

BR-P-12

BR-P-15

BR-P-18

• Only 4 producers have water breakthrough• Pressure data is available for 30 wells

Water cut:

InitialObserved

37/50

Reference kx Initial kx

Change of kx, WCT Change of kx, WCT&BHP

High perm at middle layer

Change of Permeability

38/50

Reduction of Data Mismatch

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100

No

rmal

ize

d A

bso

lute

Err

or.

Pre

ssu

re

Number of Iteration

0.0

0.3

0.6

0.9

1.2

1.5

0 20 40 60 80 100

No

rmal

ize

d A

bso

lute

Err

or,

Wat

er

Cu

t

Number of Iteration

Pressure RMSE error WCT RMSE error

Individual well

Mean

39/50

• Have developed a new SL-based method to integrate pressure data into prior geologic models

• Same advantages as prior streamline work:• Analytic calculation of streamline sensitivities• Requires only a single flow simulation per iteration

• Can be applied to field pressure/rate data prior to water breakthrough

• Can be integrate pressure with water-cut or GOR simultaneously, for black-oil and compositional simulation

Conclusion

40/50

Presented at student paper contest 2014

Application to Brugge Benchmark:

- Streamline-Simulation

- History Matching

- NPV Optimization

Overview

42/50

• Problem: Determining optimal injection/production rates to

maximize NPV• Solution:

Developed a new streamline and NPV-based rate allocation method

• Advantages: Visualize efficiency of injector and producer Extensible to any secondary recovery process with

commercial simulator

- Improve oil production rate

- Works only after breakthrough

SL-based Flow Rate Allocation

Optimization: Previous Study

43/50

• Use of Well Allocation Factors (WAFs): [Thiele et. al, 2003]

Well Allocation Factor map [SPE84080]

[SPE113628]

- WAFs: offset oil production of well-pair

• Equalize arrival time of injection fluid: [Al-Hutali et. al, 2009]

Norm Wt. - 0

Aft

er

2 y

ears

Aft

er

5 y

ears

Aft

er

10 y

ears

Base

Base Improved

Norm Wt. - 0

Aft

er

2 y

ears

Aft

er

5 y

ears

Aft

er

10 y

ears

Base

- Control well rate to have equivalent

‘breakthrough’ time

- Increase well rate of high WAFs

Decrease

Increase

Decrease

Decrease

Decrease

Increase

- Improves sweep efficiency- Works only before breakthrough

• Fast

• Not robust

• Does not optimize NPV

Proposed Optimization Method:

Overall Workflow

44/50

2. Trace Streamlines and Find connection map

3. Calculate NPV diagnostic plot

4. Reallocate well rate

via efficiency

1. Run simulation model

I1 I2 I3

I6

I5

I7 I8

NPV-based Efficiency of Streamline

P1 P2

P3 P4 P5

P6 P7

Hydrocarbon value, along SL

NPV along SL, integrate over reservoir life time

𝑣𝑠𝑙

= 𝑞𝑠𝑙

𝑛𝑜𝑑𝑒

𝑆𝑜𝑏𝑜𝑅𝑜 ∆𝜏

𝑟𝑠𝑙 = 𝑞𝑠𝑙

𝑛𝑜𝑑𝑒

𝑆𝑜𝑏𝑜𝑅𝑜 + 𝑆𝑤𝑏𝑤𝑅𝑤 ∆𝜏 ∙ 1 + 𝑑 −∆𝜏/365∉

𝑝𝑟𝑑

𝑛𝑜𝑑𝑒

∆𝜏 > 𝑡𝑟𝑠𝑚

• Hydrocarbon value and NPV along streamline

Pore volume × Saturation × FVF × Price

Discount rate Reservoir life

I4

45/50

NPV-based Flow Diagnostics

I1 I2 I3

I6

I5

I7 I8

P1 P2

P3 P4 P5

P6 P7

𝑒𝑝𝑎𝑖𝑟 =σ𝑠𝑙 𝑟𝑠𝑙σ𝑠𝑙 𝑣𝑠𝑙 Total value

NPV

5-connection from Inj-4

Total value (Normalized)

NP

V (N

orm

aliz

ed)

𝑰𝟒𝐆𝐨𝐨𝐝

𝑷𝟒

𝑰𝟒𝐏𝐨𝐨𝐫

𝑷𝟕

NPV-based diagnostic plot

I4

46/50

NP

V (N

orm

aliz

ed)

Streamline-based Rate Allocation:

A New Approach

47/50

𝑞𝑛𝑒𝑤 = 𝑞𝑜𝑙𝑑𝑒𝑝𝑎𝑖𝑟ҧ𝑒𝑓𝑖𝑒𝑙𝑑

ത𝐞𝐟𝐢𝐞𝐥𝐝

decrease rate

Increase rate

Before update After update

Total value (Normalized)

NP

V (N

orm

aliz

ed)

Total value (Normalized)

Streamline-based Rate Allocation:

A New Approach

48/50

𝑞𝑛𝑒𝑤 = 𝑞𝑜𝑙𝑑𝑒𝑝𝑎𝑖𝑟ҧ𝑒𝑓𝑖𝑒𝑙𝑑

ത𝐞𝐟𝐢𝐞𝐥𝐝

decrease rate

Increase rate

Before update After update

• Advantages:• Dynamically visualize efficiency of the injector and producer

• Able to propose ‘better’ well rate during SL-simulation

Oil Saturation and Well Location

• Constraints:- Field water injection qt <= 20,000 bbl/d- Well flow rate qti <= 6000 bbl/d- Producer BHP > 100 psi, Injector BHP < 6000 psi

• Simulation Model:- Synthetic water flooding - 20 producers, 10 injectors- 20 years of simulation- Relative oil, water price = 1, -0.2 $/bbl

Brugge Benchmark Application

• Compare developed model with 3 approaches: • Uniform injection (Uniform), Well allocation factors

(WAFs), Equalize Arrival Time (EqArrive), Developed model (SLNPV)

49/50

0.00

0.04

0.08

0.12

0.16

0.20

0 1200 2400 3600 4800 6000 7200

Re

cove

ry F

acto

r [-

]

Time [Days]

SLNPVEqArriveWAFsUniform

0.E+00

5.E+06

1.E+07

2.E+07

2.E+07

3.E+07

3.E+07

4.E+07

0 1200 2400 3600 4800 6000 7200

Net

Pre

sen

t V

alu

e [

$]

Time [Days]

NPVEqArriveWAFsUniform

Recovery Factor Net Present Value

Recovery Factor and NPV

Injection Rate Production Rate

Updated Well Rate by SLNPV

0

1000

2000

3000

4000

5000

6000

7000

0 1200 2400 3600 4800 6000 7200

Pro

du

ctio

n R

ate

[b

bl/

day

]

Time [Days]

BR-P-1 BR-P-2BR-P-3 BR-P-4BR-P-5 BR-P-6BR-P-7 BR-P-8BR-P-9 BR-P-10BR-P-11 BR-P-12BR-P-13 BR-P-14BR-P-15 BR-P-16BR-P-17 BR-P-18BR-P-19 BR-P-20

0

1000

2000

3000

4000

5000

6000

7000

0 1200 2400 3600 4800 6000 7200

Inje

ctio

n R

ate

[b

bl/

day

]

Time [Days]

BR-I-1 BR-I-2BR-I-3 BR-I-4BR-I-5 BR-I-6BR-I-7 BR-I-8BR-I-9 BR-I-10

50/50

Streamlines by Sw

SLN

PV

Un

ifo

rm In

ject

ion

Streamlines by Injector

Example of SLs: After 10 Years

Not sweep aquifer region

Sweep aquifer region

Increased Inj-Prd

connection

51/50

MCERI

• Have developed a new SL-based rate allocation method to improve recovery considering NPV

• Proposed a new diagnostic plot to visualize the relative value and efficiency of a well in the asset

• Results in greater NPV compared to prior streamline-based rate allocation methods

• Can be applied to IOR/EOR simulation study with any commercial simulator, with low computational cost

Conclusions

54