a physics-based data-driven model for history matching ...3 © 2017 chevron u.s.a. inc. | all rights...

© 2017 Chevron U.S.A. Inc. | All rights reserved.

A Physics-Based Data-Driven Model for History Matching,

Prediction and Characterization of Unconventional Reservoirs*

Yanbin Zhang

*This work has been submitted to SPEJ and under review for publication

2


Motivation

3


Reservoir Characterization with Play-Doh

4



Reservoir

5



Wellbore

Reservoir

6



Wellbore

Reservoir

7



Wellbore

1

2

3

4 56

Reservoir

8



Wellbore

1

2

3

4 56

1

2

3

4

5

6

Reservoir

9



Wellbore

1

2

3

4 56

1

2

3

4

5

6

𝑥

𝑦

10


Reservoir Characterization with Play-Doh – Fractures

fracture

1 12 2

3 3

3 3 2 21 1

4

1 1

43 3 2 2

1 1

11

𝑥

𝑦

4

3

3

2

2

1

1

11


Diffusive Diagnostic Function (DDF)

𝑃𝑉1 𝑃𝑉2 𝑃𝑉3 𝑃𝑉4𝑤𝑒𝑙𝑙

𝑃𝑉5

𝑃𝑉1𝑃𝑉2

𝑃𝑉3𝑃𝑉4

𝑃𝑉5

𝑤𝑒𝑙𝑙

𝑇0 𝑇1 𝑇2 𝑇3 𝑇4

3D

1D

Set of contours of the

pressure solution in 3D

Each “ring” is

represented as a

single cell

d𝑠

d𝑙

ȁΣ 𝑃

ቚd𝑃𝑉Σ= ර

Σ

𝜙 d𝑙 ⋅ d𝑠ቚ𝑇Σ= ර

Σ

𝑘d𝑠

d𝑙

ቚd𝜉Σ≝

ȁd𝑃𝑉 Σ

ȁ𝑇 Σ=

Σׯ 𝜙 d𝑙 ⋅ d𝑠

Σׯ 𝑘d𝑠d𝑙

=𝜙d𝑙 Σ

𝑘d𝑙 Σ

ቚ𝜎Σ≝ ቚd𝑃𝑉

Σቚ⋅ 𝑇Σ= ර

Σ

𝜙 d𝑙 ⋅ d𝑠 රΣ

𝑘d𝑠

d𝑙= ቚ𝑆

Σ𝜙d𝑙 Σ

𝑘

d𝑙Σ

ȁ𝜉 Σ = Σ0Σd𝜉 where Σ0 is the completion sand face

A contour surface of the

pressure solution in 3D

ȁΣ 𝑃+d𝑃

ቚ𝑄Σ= ර

Σ

𝑣d𝑠 = රΣ

𝑘d𝑃

d𝑙d𝑠 = ቚ𝑇

Σd𝑃

From Darcy’s Law

ቐ𝑃𝑉 = 𝜎Δ𝜉

𝑇 =𝜎

Δ𝜉

𝜎(𝜉)

In physical space, we need two functions 𝑃𝑉 𝑥 and 𝑇(𝑥)

In dimensionless 𝜉 space, we only need one function 𝜎 𝜉which is called the DDF

12


The 1D Simulation Model with DDF

1

2

3

4

5

6

𝑥

𝑦

𝜉1 𝜉2 𝜉3 𝜉𝑁

𝜎

𝜉𝑁−1

𝜎0

𝜎1

𝜎2

𝜎3𝜎𝑁−1

Well

𝑃𝑤𝑓

1-Dimension

N Grid blocks

𝜉

𝑤𝑒𝑙𝑙

𝜉4

𝜎4

PV𝑖 =𝜎𝑖−1 + 𝜎𝑖

2(𝜉𝑖 − 𝜉𝑖−1)

𝑇𝑖 =𝜎𝑖

(𝜉𝑖+1 − 𝜉𝑖−1)/2

𝐽 =𝜎0𝜉1/2

unit: ft/md1/2

unit: ft2md1/2

We are doing the same, old, regular reservoir simulation

except that we replace the 3D grid with DDF

✓ Complex fluid and rock model

✓ Changing well constraints

✓ Capillary pressure

✓ Adsorption

✓ Coupled with wellbore flow modeling and surface network

Caveat

Remember we reduce 3D reservoir into a 1D model and that is

an approximation

However, as we will show later, it is quite a good approximation

in many cases

13


All this is good, but...

1

2

3

4 56

43 3 2 2

1 1

How do I know if I should “cut” my reservoir this way or that way?

1

2

3

4

5

6

𝑥

𝑦

𝑥

𝑦

4

3

3

2

2

1

1

14



1

2

3

4 56

43 3 2 2

1 1


1

2

3

4

5

6

𝑥

𝑦

𝑥

𝑦

4

3

3

2

2

1

1

The bad news: we don’t know in

general

15



1

2

3

4 56

43 3 2 2

1 1


1

2

3

4

5

6

𝑥

𝑦

𝑥

𝑦

4

3

3

2

2

1

1

𝜉

𝜎

DDFs

𝑡

𝑞

Production

Datahistory

matching

forward

modeling

The bad news: we don’t know in

general

The good news:

(1) We can guess and we’ll make a lot

of guesses

(2) We can adjust our guesses by

history matching

History Matching using ESMDA

Ensemble Smoother with Multiple Data Assimilation

16


We don’t do wild guesses; we guess based on

DDF Characteristics

radial flow

𝑟𝑤𝑟

𝜉

𝜎

slope:2𝜋𝑘ℎinfinite acting

boundary

𝑟𝑤′ = 𝑟𝑤𝑒

−𝑠

𝜎0 = 2𝜋𝑟𝑤′ ℎ 𝑘𝜙

0𝜉 = 𝑟 𝜙/𝑘

linear flow

𝑥𝑓

𝑑

𝜉

infinite acting

boundary

𝜎0 = 4𝑥𝑓ℎ 𝑘𝜙

0𝜉 = 𝑑 𝜙/𝑘

𝜎

𝑥𝑓

𝐿𝑤

𝑁𝑓

𝑑𝑓

𝜉

boundary

𝜎0~4𝑥𝑓𝑁𝑓ℎ 𝑘𝜙

0

𝜉𝑆𝑅𝑉~𝑑𝑓2

𝜙/𝑘

𝜎

slope~𝛼𝑘ℎSRVp

𝜎~(4𝑥𝑓 + 2𝐿𝑤 + 4𝑑𝑓)ℎ 𝑘𝜙

Take out your Play-Doh and construct this DDF!

17


DDF Characteristics - Summary

Characteristics of

the DDFDiagnostic Properties

Approximate

EquationsComments

𝜎 levelFlow area and

Reservoir quality𝜎~𝐴 𝑘𝜙

Fractures cause sharp increase in 𝜎 level.

Interferences or boundaries cause drop in 𝜎 level. 𝜎

level keeps constant for linear flow

Slope of linearly

increasing 𝜎 𝜉

Reservoir

permeability

Δ𝜎

Δ𝜉~𝛼𝑘ℎ

𝛼 depends on flow pattern. 𝛼 = 2𝜋 for radial flow.

Generally, 𝛼 > 2𝜋 for irregular flow pattern.

Area under the DDF

curvePore volume 𝐴𝑟𝑒𝑎 = 𝑉𝑝

A dramatic drop in 𝜎 level signifies boundary effect or

interferences. For unconventional reservoirs, SRV may

be identified in this way.

𝜉 at which 𝜎

behavior changesDistance 𝜉~𝑑

𝜙

𝑘

The estimation of distance is difficult to be precise

because the transition of 𝜎 behavior is usually not

clear-cut and may span a wide range.

18


Synthetic Example: Vertical Well

𝑟𝑒 = 500ft

ℎ = 100ft

𝑟𝑤 = 0.3ft

𝑘 = 0.001md 𝜙 = 0.05md

3D Cartesian grid 101 × 101 × 1DX = DY = 9.9 ftDZ = 100 ft

Black oil fluid model (𝑃𝑏 = 800 psia)

Initial reservoir pressure 𝑃𝑖 = 5000 psiaInitial water saturation 𝑆𝑤𝑖 = 𝑆𝑤𝑖𝑟 = 0.15Well producing at constant BHP = 1000 psia

(a) (b)

(c) (d)

𝝃 (ft/md1/2

) Time (days)

𝝃 (ft/md1/2

) Time (days)

Before HM

After HM

19


Synthetic Example: Multiple Fractures

10 Infinite conductivity fractures

Other parameters are the same as previous slide

(a) (b)

(c) (d)

Time (days)

Time (days)

𝝃 (ft/md1/2

)

𝝃 (ft/md1/2

)

Before HM

After HM

20


Applications

History matching and forecasting

for a Gas Well• Physics Based

✓ Complex fluid, multi-phase flow

• Data Driven

✓ Extremely fast history match

• Reservoir Characterization

✓ Total fracture area and SRV

• Integrated Workflow

✓ Coupled with surface network

✓ Optimization / uncertainty

analysis

Oil Well Examples

21


Summary

1. Physics-Based:

DDF provides a general 1D

simulation framework to

approximate 3D reservoir

2. Data-Driven:

DDF is probabilistically

conditioned to production data

DDFMachine learning and big data

First-principle-based computation

Future Research

?

22


Acknowledgement

• ETC/RPP

– Jincong He

– Jiang Xie

– Xian-huan Wen

• ETC/RPS

– Robert Fitzmorris

– Shusei Tanaka

• ETC/PEWP

– Jorge Acuna

• ETC/TRU

– Reza Banki

• MCBU

– Baosheng Liang

– Hannah Luk

• AMBU

– James Wing

– Richin Chhajlani

Please reach out to me for any

questions or to connect with me.

You may contact me at:

[email protected]

mailto:[email protected]


Backup Slides

24


History Matching using ESMDA with DDFEnsemble Smoother with Multiple Data Assimilation Diffusive Diagnostic Function

𝜉

𝜎

DDFs

𝑡

𝑞

Production

Datahistory

matching

forward

modeling

𝚫𝒎 = 𝐂𝑀𝐷 𝐂𝐷𝐷 + 𝛼𝑖𝐂𝐷−1(෩𝒅𝑜𝑏𝑠 − 𝒅𝑝𝑟𝑒𝑑)

෩𝒅𝑜𝑏𝑠 = 𝒅𝑜𝑏𝑠 + 𝛼𝑖𝐂𝐷1/2

𝑧𝑑 where 𝑧𝑑~𝑁(0, 𝐼𝑁𝑑)

𝒎𝑖+1 = 𝒎𝑖 + 𝚫𝒎

perturbed observations

model updatedata mismatch

Workflow

1. Come up with initial ensemble of DDFs

2. Perform forward modeling to obtain data

prediction

a) predictions way off, go back to 1

b) if predictions follow the trend and cover

the range of observed data, go to 3

3. Randomize the model to avoid ensemble

collapse

4. For the 𝑖th (out of 𝑛) iteration, use 𝛼𝑖 = 1/𝑛in the equation to update the ensemble of

models considering the mismatch of all

data points simultaneously

5. Model regularization by smoothing the DDF

curves and eliminate negative values (set

to 0)

6. Go to 4 for next iteration

25


Synthetic Example: Single Fracture

Infinite Conductivity vs. Finite Conductivity

Time (days) 𝝃 (ft/md1/2

)

𝝃 (ft/md1/2

) Time (days)

Infinite conductivity fracture 𝑘𝑓 = 1 × 106 md

Finite conductivity fracture 𝑘𝑓 = 1 md

Infinite or finite conductivity single fracture

Other parameters are the same as previous slide

𝜎 starts at a much larger value

𝜎0 = 4𝑥𝑓ℎ 𝑘𝜙 = 353 ft2md1/2

𝜎 starts at a smaller value, but

increases rapidly near 𝜉 = 0

26


Synthetic Example: Multiple Fractures – DDF Diagnostics

(a) (b)

𝝃 (ft/md1/2

) 𝝃 (ft/md1/2

)

𝜉𝑖~140 ft/md1/2

𝜎2~1000 ft2⋅md

1/2

𝜎1~3500 ft2⋅md

1/2

Slope ~1 ft⋅md

𝑥𝑓

𝐿𝑤

𝑁𝑓

𝑑𝑓

𝜉

boundary


0


𝜙/𝑘

𝜎


𝜎2~(4𝑥𝑓 + 2𝐿𝑤 + 4𝑑𝑓)ℎ 𝑘𝜙

Lw = 450 ft

Nf = 10df = 50 ft xf ~ 120 ft

𝜙 = 0.05𝜎1

𝜎2~ 3.5

𝜎1~ 3500 ft2md1/2

k ~ 0.001 md

SRVp ~ 6x105 ft3

2𝑥𝑓𝐿𝑤ℎ𝜙 = 6.25x105 ft3

27


Field Example: Marcellus Gas Well

Well length (𝐿𝑤) 3202 ft

Number of hydraulic fractures (𝑁𝑓) 12

Reservoir thickness (ℎ) 150 ft

Reservoir porosity (𝜙) 0.065

Initial reservoir pressure (𝑃𝑖) 5008.8 psia

Reservoir temperature (𝑇) 160 F

Connate water saturation (𝑆𝑤𝑐) 0.39

Rock compressibility (𝑐𝑓) 3×10-6 psi-1

Gas specific gravity (𝛾) 0.570.0

1.0

2.0

3.0

4.0

5.0

6.0

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 500 1000 1500 2000

Ga

s p

rod

uc

tio

n R

ate

(M

MS

CF

/D)

Bo

tto

mh

ole

Pre

ss

ure

(p

sia

)

Time (days)

1E+05

1E+06

1E+07

100 1000 10000 100000

No

rma

lize

d G

as

Po

ten

tia

l (p

si2

/cp

/MS

CF

D)

Material Balance Time (hr)

Integral of Normalized Gas Potential

Bourdet Derivative

predictionHM

Not half slope

Log-Log PlotSquare Root Time Plot

A straight line can be used to fit the data

even though it is not linear flow

28


Field Example: HM and prediction with DDF

(a) (b)

(c) (d)

Time (days)

Time (days)

𝝃 (ft/md1/2

)

𝝃 (ft/md1/2

)

𝝃 (ft/md1/2

)

Before HM

After HM

29


Field Example: DDF Diagnostics

𝝃 (ft/md1/2

)

𝑥𝑓

𝐿𝑤

𝑁𝑓

𝑑𝑓

𝜉

boundary


0


𝜙/𝑘

𝜎


𝜎2~(4𝑥𝑓 + 2𝐿𝑤 + 4𝑑𝑓)ℎ 𝑘𝜙

Characteristic of finite

conductivity fractures

SRVp~2.6 × 107 ft3

𝐴𝑡𝑜𝑡𝑎𝑙 𝑘𝜙~1 × 104 ft2md1/2

𝜙 = 0.065𝐴𝑡𝑜𝑡𝑎𝑙 𝑘~4 × 104 ft2md1/2

Total fracture sandface area

30


Field Example: Probabilistic Nature of the DDF Method


)


)

HM

HM

P50 SRVp decreases

SRVp uncertainty

range decreases

a physics-based data-driven model for history matching ...3 © 2017 chevron u.s.a. inc. | all rights...

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