Uncertainty,sensitivityandrejectionin ! …cees.stanford.edu/docs/Caers-slides.pdf ·...

Jef Caers

Stanford University, USA

Uncertainty, sensitivity and rejection in predictive reservoir modeling

Quantitative modeling of geological heterogeneity Modeling uncertainty in the context of decision making Building 3D/4D models accounting for scale and accuracy of geological, geophysical and reservoir engineering data

Stanford Center for Reservoir Forecasting

SCRF overview Reservoir geology Multiple-‐point / pattern-‐based geostatistics Surface-‐based geostatistics Structural modeling Basin modeling

Reservoir geophysics Seismic reservoir characterization Rock physics 4D seismic

Reservoir Engineering Sensitivity analysis / History matching Upscaling Uncertainty, decision analysis and value of information Proxy models / model complexity

Jef Caers

Stanford University, USA

Uncertainty, sensitivity and rejection in predictive reservoir modeling

Current practice

Industry practice of

The reservoir

Life-‐tim

e

2D seismic

3D seismic

3D seismic+production

4D seismic+production

sensitivity/rejection

Life-‐tim

e

The reservoir

Why?

History matched permeability in a real field currently in production

The sensitivity argument

Any modeling of uncertainty is irrelevant/impossible without a decision or prediction goal Need for an understanding and discovery of what impacts flow processes and decision variables More than just a computational issue !

Challenge: Flow: very non-‐linear process Most important and impacting variables are discrete

The rejection argument

Karl Popper (1959): physical processes are laws that are only abstract in nature and can never be proven correct, they can only be disproven/falsified with facts or data Popper-‐Bayes

(data|m(model|d odel) ata del)) (moPPP

Application to reservoir case study

New well planned

P1

P2

P3

P4

West-‐Coast Africa (WCA) slope-‐valley system

Data courtesy of Chevron

Sensitivity

Depositional model (Training Image) Spatial uncertainty (for given depositional model) Kv/Kh ratio

Residual oil saturation Maximum water relative permeability value Water Corey exponent

What matters for prediction ?

Generalized Sensitivity Analysis (GSA) underlying principle

input parameters

A big modeling box

Geology/geophysics Stochastic

Flow

output response

A measure of sensitivity is the difference between the frequency distributions of input parameters per each class

Dim. Reduction classification

C1 C2 C3

Distance-‐based GSA

Structure Rock Fluid

stochastic

Proxy flow model complexity

Response r

Time

p

m

Dim. Reduction classification

2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

watExp

cdf

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

KvKh

cdf

pi pj

CDF

CDF

Kv/Kh Corey Water Exp

Generalized sensitivity any parameter, any response

A measure of sensitivity is the L1 norm difference between a class-‐conditional and marginal cdfs A measure of interaction sensitivity is the L1 norm difference between a conditional class-‐conditional and conditional cdfs

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

cdf

( | )i kF p c

( )iF p

pi = corey exponent

TI1 TI3 TI8 TI9 TI10 TI130

0.2

0.4

0.6

0.8

1

TIcd

f

TI|krwMax - class # 1pj = TI ; pi = corey exponent

cdf

( | , )i j kF p p c

( | )i kF p c

Application to WCA

28 wells, Response = oil production in 20 producers Two classes, kernel k-‐medoid clustering Proxy simulator : streamlines, gave same classification as Eclipse, 150.000 cells 110 reservoir models, 5 parameters (rock + fluid), total CPU = 700 min.

0 0.5 1 1.5 2 2.5

SOWCR

KvKh

TI

krwMax

watExp

SensitiveNotSensitive

Water Corey exponent

Max Water rel perm

Res. Oil Sat.

Training Image

Kv/Kh

Rejection

P odel, | ata

: to reject scenarios without

P( | , )

any HM: HM per ac

P |

ceptabl P( e scenario P |

| , )

kk

k

k

k

ScenariScenar o

Scenar

io

Sc

Sc

e

e

nar

na

k

io

o

rio

i

M D

MD

M D D

D

Geosciences are interpretative sciences Depositional model Type of fracture hierarchies Rock Physics model Fault Hierarchy

Data: geology and production

TI1: 50% TI2: 25% TI3: 25%

geological scenario

uncertainty: 3 training images

Production Data:

Water rate/well

Two modeling questions

P odel, | ata

: reject data-‐inconsistent training images

: sampling with the remaining ones

P |

P

P( | , )

P( | , )

|

k

k

k

k

k

TI

TI

I

I TIT

T

M DM

D

D D

D

M

Generate initial ensemble of 180 scoping models

TI1: 50% TI2: 25% TI3: 25%

Production data & 180 Scoping runs Water ra

te

Time/Days

Well 1 Well 2

Well 3 Well 4

Trying to falsify TIs with data represent data in lower dimensions using

multi-‐dimensional scaling

MDS: distance = difference in water rate response for all wells 9 dimensions = 99% of variance

Production data

TI1 responses TI2 responses TI3 responses

Eigencom

pone

nt 2

Eigencomponent 1

History match for each TI Regional probability perturbation

Why regional PPM? Geological realism Works for facies models Easy optimization with region parameters

Streamline geometry at final time step

Example of region geometry

History match results for all TIs CPU: Average of 24 flow simulations/model

A few history matches

Notice the absence of any region artifacts

From TI2 From TI3

Rejection sampler on TI and facies

1. Draw randomly a TI from the prior

2. Generate a single geo-‐model m with that TI

3. Run the flow model simulator to obtain a response d=g(m)

4. Accept the model using the following probability

2

RMSE( , ( ))exp

2obs gp

d m

Rejection sampler results

Prediction in newly planned well for next 1 year

2000 2100 2200 23000

200

400

600

800

1000

Time, days

Wat

er R

ate,

stb

/day

RejectionPPM

P10

P50

P90

Comparison

P(TI1|D) P(TI2|D) P(TI3|D) Runs/ model

Method 1% 38% 61% 24

Rejection Sampler 3% 33% 64% 250

Further speed-‐up by realizing that HM per scenario had little impact on reduction of uncertainty use of proxy flow models because only relative likelihood accuracy is needed

Some observations

There is no need for a history match to get a good prediction (this is case dependent)

No need to run full-‐physics on all models Sensitivity: proxies may provide accurate classification Rejection: only relative likelihood is needed

Increased importance on providing geological uncertainty through multiple scenarios

The importance of quantitative geological modeling variogram

MPS

Boolean

Process based

Surface based

The missing link Geological interpretation: attempting to understand the genesis and process of past deposition Geostatistics: attempting to model the geometries currently present with a practical application in mind

Two challenges 1. What methodology bridges this gap? 2. If so, how to bridge this gap?

?

Limitation of covariances

0.4

0.8

1.2

10 20 30 40 0

0.4

0.8

1.2

10 20 30 40 0

3

1 2

data model

Variograms EW Variograms NS

1 2 3

Training images From Boolean From high resolution seismic

From process-‐based models

Conditioning data Training image

Geostatistical model

High performance Training image

Geostatistical model

4.5 million cells, 7 seconds 1 million cells, 1 second

Honarkhah, M. and Caers, J. (2012) Math. Geosc., 44:651 672. Direct pattern-‐based simulation of non-‐stationary geostatistical models Pejman Tahmasebi et al. (2012) Comp. Geosc., 16:779 797. Multiple-‐point geostatistical modeling based on the cross-‐correlation functions Fenwick, D., Scheidt, C., and Caers, J. (2012) submitted A distance-‐based generalized sensitivity analysis for reservoir modeling Park, H., Scheidt, C. Fenwick, D. Boucher, A and Caers, J. (2012) submitted History matching and uncertainty quantification of facies models with multiple geological interpretation Scheidt, C., Renard, P and Caers, J. (2012) submitted Uncertainty Quantification in Inverse Problems: Model-‐based versus Prediction-‐Focused Aydin, O. and Caers, J. (2012) submitted Image transforms for determining fit-‐for-‐purpose complexity of geostatistical models in flow modeling PDFs available

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