History matching under geological controlThe probability perturbation method

Jef Caers

Department of Petroleum Engineering

Stanford University, Stanford, California, USA

Stanford Center for Reservoir Forecasting

Motivation

• The goal of history matching is not just to match history

Prediction power of models cannot be verified Geological realism enhances prediction

• Interaction between geological model and flow model

Not all geology matters for flow Iterative process between static/dynamic, not sequential

HM itself is not difficult

P1

P2

P6

P5

P4

P3

I1I2

I3

Initial Model

ProposedMatched Model

P1

P2

P6

P5

P4

P3

I1I2

I3

P1

P2

P6

P5P4

P3

I1I2

I3

Eclipse (SimOpt)Matched Model

Regions (SimOpt) 500

300

0

100

Log Scale

md

Geological scenario

A prior geological scenario defines what remainsconstant during history matching

prior geological scenario =

set of decisions about the styleof geological structures/features or aboutthe parameterizations of these structures/features

prior geological scenario =

set of decisions about the styleof geological structures/features or aboutthe parameterizations of these structures/features

• permeability/porosity variogram• Boolean model with shape distributions• Training image model and seismic derived facies probabilities• Training image with unknown Net-to-Gross

Example geological scenario

Quantify geological scenario

Prior geological scenario defines conditional probabilities

P(A|B) A = “channel sand occurs”B = known “conditioning” data

Key idea: Perturb the probability P(A|B) such that* a history match is achieved* the geological scenario remains unchanged

Key idea: Perturb the probability P(A|B) such that* a history match is achieved* the geological scenario remains unchanged

Probability perturbation

Perturb the conditional probabilities P(A|B) using another conditional probability that depends on the production data D

P(A|D) = (1-rD) i(o)(u)+ rD P(A)prior information on A

Binary case: A=“channel occurs”, then i(o)(u)=1

Combine P(A|D) and P(A|B) into P(A|B,D)

rD=0 P(A|B,D) = i(o)(u) : No perturbation

rD=1 P(A|B,D) = P(A|B) : equiprobable realization isgenerated if random seed is changed

Example

Reference model

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

training image

East

No

rth

0.0 150.0000.0

150.000

facies 0

facies 1

Geological scenario 1. Two hard data 2. Training imageI

P

Assume permeability of each facies known (1500 vs 50mD)

Creating perturbations

Seed= 76845

Seed= 36367

Perturbations preservegeology

Perturbation are betweentwo equiprobable models

Optimize on rD

i(o)(u)

P(A|D) = (1-rD) i(o)(u)+ rDP(A)

Graphical representation

high

low

rD=0

rD=1

Initial model

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 1, r_D=0.21

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 3, r_D=0.52

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 5, r_D=0.50

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 7, r_D=0.31

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 9, r_D=0.24

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

seed=76845

initial model

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

r_D = 0.05

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

r_D = 0.1

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

r_D = 0.2

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

r_D = 0.5

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

r_D = 1

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

seed=36367

Initial model

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 1, r_D=0.21

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 3, r_D=0.52

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 5, r_D=0.50

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 7, r_D=0.31

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Iteration 9, r_D=0.24

East

No

rth

0.0 50.0000.0

50.000

facies 0

facies 1

Space ofAll realizations

Mismatch between simulated and field data

Basic Algorithm

Generate initialguess realization

Outer Loop

Change random seed Choose value for rD

Define P(A|D)

Generate a new realization

and run flowsimulation

Convergedto best rD ?

Inner Loop

YES NO

History match ?

Done

YES

NO

Result

Outer iteration 4

General method

Training image

Reference Match

flow

pressure

Multi-category

Junrae KimHistory matching on N/G

History matching:

Most critical : Finding a good geological model

PP : searches within a fixed geological model

Junrae:

Critical parameters such as Net-to-Gross needto be part of the history matching process

Todd HoffmanRegional probability perturbation

PP: one parameter creates same perturbation everywhere

Todd:

Create regionsAttach a parameter rD to each region

Regional perturbation method (RPP)

Challenges: no artifact discontinuities at region borderSolve a multiple-parameter problem

P(A|D) = (1-rD) i(o)(u)+ rDP(A)

P(A|D) = (1-rDk) i(o)(u)+ rDkP(A)

Satomi SuzukiHierarchical history matching

1) Perturb Facies by Prob. Perturbation

Perm Fixed

Change Random #

History Matched ?

END

YES

NO

2) Perturb Perm within

Facies

Facies Fixed

Inanc TureyenJoint fine and coarse scale HM

Static model : fine scaleFlow simulations : coarse scale

Traditional approach

Initial fine-scale realization

Downscaled realization

Non-uniformcoarsened realization

Initial coarsened realization

History matched coarsened realization

HistoryMatching

Joint optimization

GridOptimization

Mismatch Between fine and coarse minimized

HistoryMatching

Mismatch on coarse scale usedfor fine scale perturbations

Result: History matched non-uniform gridded model Fine-scale model also History matched

Joe VoelkerApplication to Ghawar Field

dep

th

BBL/day/ft

Super-K= extreme flowBut not caused by extreme K

Driving mechanism = combinedFacies and fracture model

Super-K determinedBy HM flow meter data