history matching by joint perturbation of facies distribution and net-to-gross

History matching by jointperturbation of facies distribution

and net-to-gross

Junrae Kim and Jef Caers

Objectives

• Constraining geological models to production data, adjusting aspects of geological model.

• Assessing value of information of production data on N/G.

• Application to a realistic 3D reservoir and well test.

Motivation :N/G should not be fixed in history matching

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N/G=0.05

N/G=0.6

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N/G=0.05

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N/G=0.6

Ref• Elliptical bodies of high permeability (1000 mD) in a lower permeability matrix (15 mD)• Constant injection rate of 700 STB/day• No boundary conditions assumed • N/G (Reference) = 0.32

Flow response of 20 realizations

Probability perturbation method

Define: P(A|D) = (1-rD) i(k)(u) + rD P(A)

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

P(A|B)P(A|D)

Model for P(A|B,D) Generate i(k+1)(u)

rD = 1 Full perturbation

rD = 0 No perturbation

Jef Caers

rD is found by solving a simple 1D optimization problem.

• Problem

=>More difficult joint optimization of rD and N/G.

• Solution

=>Making proportional to rD : Back to 1D optimization

• Why?

=>High rD, : Big change in model

G/N

G/N

Couple perturbation of N/G with perturbation of rD.

Perturbing N/G

Make linearly proportional to GN / .Dr

GN /

GN /

max/GΔN

1Dr

max/GΔN

minDr

Couple perturbation of N/G with perturbation of rD.

Increase GN /

Decrease GN /

Probability perturbation method:basic algorithm

Generate an initial guess realization

Outer Loop

Change random seed

History match?

Done

Converged to best rD?

Inner Loop

Yes

No

Yes No

Choose value for rD

Define P(A|D)

Generate a new realization

Run flow simulation

The proposed method:basic algorithm

Choose value for rD

Calculate G/NG/N

Calculate O(+)

Choose best rD and Obest from O(-) and O(+)

Generate a newrealization & flow simulation

Calculate G/NG/N

Generate a newrealization & flow simulation

Calculate O(-)

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2-D reservoir model

Conditioning data (facies)

Training image

• Reference: 50 x 50• Elliptical bodies of high permeability (1000 mD) in a lower permeability matrix (15 mD)• Constant injection rate of 700 STB/day• No boundary conditions assumed • N/G (Reference) = 0.32• Training image: 150 x 150

GNGN // 0 GNGN // 0

Inner iteration

Optimal value GNGN // 0

GNGN // 0

Initial model

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1 2 3 4 5 6 7Inner-iteration

Outer Iterations

12.0/,31.0

,11 GNr

Iteration

D 27.0/,26.0

,22 GNr

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31.0/,16.0

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D 33.0/,12.0

,44 GNr

Iteration

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Init

Iter 1

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Iter 4 (History matched)

32.0/ 0 GN

Reference Model

20 realizations

Ref=0.32Mean(H.M.)= 0.33IQR=[0.28, 0.38]

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Initial Pressure

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History match

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0.12 0.52Initial N/G :

Selected randomly from [0.12, 0.52]

0.32

20 initial realizations

20 history matched realizations

20 realizations

Ref=0.32Mean(H.M.)= 0.33IQR=[0.19, 0.45]

Initial N/G : Always 0.5

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History match

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Unbiased!!

20 initial realizations

20 history matched realizations

• Investigate the value of information provided by a well test in a realistic 3D reservoir (Stanford V).

• First: Spatial re-sampling to model the uncertainty on N/G without well test data: Journel (1993).

• Second: Apply the proposed method. 1. Use re-sampled N/G as initial guess 2. History match 3. Check final N/G from history matched realization

Quantification of uncertainty on N/G from single well test

3-D reservoir model

•Dimensions 100x130x30 cells, three layers.

•Fluvial channels (1500mD) in (50mD) permeability matrix.

•Reference True N/G = 0.39.

•A single vertical well (N/G = 0.5) available at x=52, y=65.

Quantification of uncertainty on N/G

What is the uncertainty without well test? => Spatial bootstrapping

1. Generate a geostatistical model conditioned to the single vertical well log data only; N/G realization = 0.5

2. Randomly sample 20 vertical wells from the single geostatistical model.

3. Calculate N/G from 20 resampled wells.=> N/G histogram

Quantification of uncertainty on N/G (First: without well test)

Quantification of uncertainty on N/G (Bootstrap)

True N/G = 0.4Mean = 0.46IQR = [0.33, 0.58]

Wide confidence intervalUpward bias

A spatial resamplingof 20 wells

N/G realization= 0.5

Section z=1

50 100 150 200

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True Reference Training image

Resampled N/G

Quantification of uncertainty on N/G (Conditioning to well test data)

1. Each of the 20 re-sampled N/G is used as an initial guess.

2. 20 final history match generated- perturbing facies and N/G.

3. Calculate N/G from final HM models.=> N/G Histogram

What is the uncertainty with well test?

Well test model

•A drawdown test for 100 days.

•The rate of production is fixed to 1000 STB/day.

•Boundary effect starts at 10 days.

•

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t

d(Dp)/d(lnt)Dp

Radial flow LateTransient

PSS/Closed boundary

][][][ STB/RB.B,cp,psi/. ootc 00011111028356

Quantification of uncertainty on N/G (Conditioning to well test data)

Quantification of uncertainty on N/G (Conditioning to well test data)

987

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0.001 0.01 0.1 1 10 100

days

psia

(WBH

P)

ref

HistorymatchedPressure

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days

psia (W

BHP)

ref

Initial Pressure

WBHP (initial guesses) WBHP (history matched)

Initial guesses (N/G) without well-testMean = 0.46IQR = [0.33, 0.58]

History matched (N/G) with well-testMean = 0.44IQR = [0.40, 0.46] Reduction of uncertainty on N/G is significant.

True N/G = 0.4Original well log N/G = 0.5

Quantification of uncertainty on N/G (Conditioning to well test data)

Pseudo steady state versus transient state

• Variance in all cases are smaller than the case, where no well test is available.

• The uncertainty is largest in the 3-day case.

• Assumption: Averaging volume known (PSS)

Quantification of uncertainty on N/G (Conditioning to well test data)

100 days 10 days 3 days

Conclusions

•N/G should not be fixed in a history matching process.

•The proposed method jointly parameterizes perturbation of

N/G and of facies => simple & robust.

•The method quantifies uncertainty of N/G based on

production data.

•Well test: N/G can be quantified if averaging volume is well known.

Facies anisotropy and facies distribution

History matching by joint perturbation of

Case 1: Always starts from -45°

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-90° 90°

Case 3 : Select randomly from [-90°, 90°]

Mean: 47.6[36, 58]

Mean: 39.7[30, 52]

Mean: 44.0[35, 52]

Case 2: Always starts from 0°

IQR:

A history match was achieved in all cases…

45° 75° -45°-15°

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But still with large effort

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VR

N/G

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Reservoir N/G

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The global N/G corresponding to the volume ratio

volumereservoirTotal

welltestaroundvolumelCylindricaVR

Volume ratio corresponding to the number of days for which well test is run.