computational challenges in reservoir modeling sanjay...

Computational Challenges in Reservoir Modeling

Sanjay Srinivasan The Pennsylvania State University

Well Data

3D view of well paths • Inspired by

an offshore development

• 4 platforms

• 2 vertical wells

• 2 deviated wells

• 8 horizontal wells

Sample well log • Well data include:

– horizontal location – true vertical depth – porosity – permeability – density – sonic log – lithology – layer indicator

• Well markers: true vertical depth when well intercepts a layer surface

Seismic impedance

Impedance

Case Study Summary • Reference reservoir is a fluvial reservoir • Porosity and permeability histogram show distinct bi-modality

Compartmentalization of reservoir into pay / non-pay – decision of stationarity

Freq

uenc

y

porosity

0.000 0.100 0.200 0.300 0.400 0.5000.0000

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

0.0700All Facies Porosity

Number of Data 1916mean 0.16

std. dev. 0.10coef. of var 0.63

maximum 0.34upper quartile 0.27

median 0.14lower quartile 0.06

minimum 0.02

Porosity across facies Number of Data

mean std. Dev.

coef. Of var

1916 0.16 0.10 0.63

porosity 0.00 0.10 0.20 0.30 0.40 0.50

Facies classification basis

• Seismic impedance in channels, mudstone and border sands :

Pay Facies: channel sands Non-pay facies: mudstone + splays + levies

Freq

uenc

y

Impedance

2500. 4500. 6500. 8500. 10500. 12500.0.000

0.100

0.200

0.300

Impedance in mudstone



Freq

uenc

y

Impedance

2500. 4500. 6500. 8500. 10500. 12500.0.000

0.100

0.200

0.300

Impedance in channel sand



Freq

uenc

y

Impedance

2500. 4500. 6500. 8500. 10500. 12500.0.000

0.100

0.200

0.300

Impedance in border sand



Impedance in mudstone

Number mean

59311 8527

Impedance in channel

Number mean

51003 7928

Impedance in border sand

Number mean

9286 8252

Impedance 12500 8500 4500

Impedance Impedance 12500 8500 4500 12500 8500 4500

Work flow • simulate multiple pay / non-pay facies models • within non-pay, simulate border facies

• simulate porosity models separately in pay and non-pay

• same for permeability • merge pay / nonpay, porosity / permeability

models using matching facies template multiple equi-probable reservoir models

multiple facies templates

Pay / Non-pay facies modeling Available Data 1) Vertical proportion curve

uncertainty in modeled surfaces

uncertain vertical proportion curve

Reference reservoir / surfaces

Well data / reference surfaces

Well data / model surfaces

Proportion of sand 0.1 0.3 0.5 0.7 0.9

Stra

tigra

phic

dep

th

0.0

0.2

0.4

0.6

0.8

1.0

Semi-Variogram

Defined as: In the spatial context, Under intrinsic stationarity – the RF Z(u) itself might not be stationary, but increments of the RF {Z(u)-Z(u+h)} are presumed stationary. Thus:

{ }2)(21 YXEXY −=γ

( ) ( ){ }2)(),(2 huuhuu +−=+ ZZEγ

( ) ( ){ } ( ) ( )}{21)(

21),( 2 huuhuuhuu +−=+−=+ ZZVarZZEγ

stationary )(hγ=

Well markers - Top surface

0. 500. 1000. 1500. 2000. 2500.

0.

1000.

2000.

3000.

200.0

300.0

400.0

Variogram Inference • Inference of the experimental

variogram requires: – Plotting h-scatterplot by

systematically varying h in particular spatial directions.

– For each scatterplot compute the corresponding moment of inertia of the scatterplot.

– Plot the moment of interia versus the lag h in specific directions

Some remarks •There must be sufficient number of pairs in each scatterplot (statistical mass) •Outliers (abnormal high or low values) can cause the moment of inertia to fluctuate wildly

γ(h)

h 0

Curve shape depends on nature

of phenomena

Facies modeling - Data • Indicator variogram - “hard” data

γ

0 . 0 0 0

0 . 1 0 0

0 . 2 0 0

0 . 3 0 0

0 . 4 0 0

γ

I n d i c a t o r v a r i o g r a m s

0 . 0 0 0

0 . 1 0 0

0 . 2 0 0

0 . 3 0 0

0 . 4 0 0

γ

0 . 0 0 0

0 . 0 5 0

0 . 1 0 0

0 . 1 5 0

0 . 2 0 0

• Seismic impedance

45o azimuth 135o azimuth

Horizontal indicator variograms

0.0 400.0 800.0 1200.0 Distance 0.0 0.4 0.8

Vertical indicator variogram

Depth

Horizontal variogram


0.0 400.0 800.0 1200.0 Distance

Facies modeling - variograms Horizontal borrowed from seismic Vertical from “hard” data

γ

0 . 0 0 0

0 . 0 5 0

0 . 1 0 0

0 . 1 5 0

0 . 2 0 0

γ

0 . 0 0 0

0 . 0 5 0

0 . 1 0 0

0 . 1 5 0

0 . 2 0 0

0 . 2 5 0


0.0 400.0 800.0 1200.0 Distance

0.0 Depth

0.4 0.8 1.2

Spatial Estimation Simple kriging (SK) of the depth at unsampled locations from marker data

nd ,..,1),( =ααx)( od x ox

)(1)(1

* ∑ ⋅+⋅

∑−=

=

nAo DmD

ααα

αα λλ xx

nCC on

,...,1,)()(1

=−=−⋅∑=

αλ αβαβ

β xxxx

SK estimate:

SK system:

Requisites

{ } AmDE =)(x average depth-data over study A covariance modeled after a spatial average over “stationary” area A

)(hC)(hAC

( )dxu ,: =Note

Indicator Paradigm

• Consider the indicator RV:

• Important property:

or better still, given n data:

Indicator basis function • Consider the expansion defined on the

basis of n indicator random variables : - Given the n indicator random variables, this

expansion is the most complete possible. - There are a total of terms in

the above expansion - There are 2 outcomes for each RV there are

outcomes possible for the function . - These outcomes define the space

Indicator Expansion

• The conditional expectation is precisely the projection of the unknown indicator event on to .

• If instead the projection function is defined on a reduced basis Lk:

Indicator Expansion

• Another way to write the expansion: basis function

Normal Equations

• The implication of the projection theorem is that:

or in terms of projections: which is a system of normal equations

Examples

- unbiasedness - reproduction of indicator

covariances

- reproduction of the 3rd order covariances

In general: requires knowledge of up to order indicator

covariances

Facies modeling - indicator approach Indicator kriging (Simple IK)

[ ]∑=

−⋅+=)(

1

* )()()(u

uuun

pIpIα

ααλ

Interpretation

{ }datapay nuProbI |)(* ∈=u

IK System

)()()( 0

)(

1α

βαββλ hhu

u

I

n

I CC =⋅∑=

Facies modeling - indicator approach Indicator kriging local conditional probability

{ }datapay nuProb |∈

• Draw ]1,0[Uniform∈rSimulation

If then else

{ }data|pay nuProbr ∈≤pay:1)( ∈= uuI nonpay:0)( ∈= uuI

• Add simulated into data set : (n) (n+1) • Visit next node

)(uI'u

Locally varying mean Requisites

• vertical proportion of pay facies in stratigraphic layer :

• Indicator variogram model,

Facies modeling - indicator approach

[ ]∑=

−⋅+=)(

1

* )()()()()(u

uuun

VV dpIdpIα

αααλ

)(dpVd

)(hIγ

Indicator Simulation - Results • Slices through the facies model

2-D Slice # 7

East

Nor

th

0.0 100.0000.0

120.000

• Variogram reproduction

2-D XZ Slice # 40

East

Nor

th

0.0 100.000500

5002-D YZ Slice # 40

East

Nor

th

0.0 120.000500

500

Nonpa

Pay

γ

0 . 0 0 0

0 . 1 0 0

0 . 2 0 0

0 . 3 0 0

XY slice #7

XZ slice #20 YZ slice #20 pay

nonpay


0.0 400.0 800.0 1200.0 2000.0 1600.0 Distance

Horizontal indicator variograms

Indicator Simulation - Results • Vertical proportion

Freq

uenc

y

0.300 0.350 0.400 0.450 0.5000.000

0.050

0.100

0.150

0.200

0.250Hist. of global proportion (before trans)


coef. of var 0.07

Freq

uenc

y

0.300 0.350 0.400 0.450 0.5000.00

0.20

0.40

0.60

0.80

Hist. of global proportion (after trans)


coef. of var 0.01

• Global pay proportion 0.20 0.80 0.40

Stra

tigra

phic

dep

th

0.0

0.2

0.4

0.6

0.8

1.0

0.60 Sand proportion

Number of Data mean

coef. Of var

50 0.38 0.07

before trans after trans

Number of Data mean

coef. Of var

50 0.35 0.01

simulated target

0.0 1.0

Indicator Simulation - Results • Perform a histogram transformation • Simulation after trans

2-D XY Slice # 7

East

Nor

th

0.0 100.0000.0

120.000

2-D XZ Slice # 40

East

Nor

th

0.0 100.000500

5002-D YZ Slice # 40

East

Nor

th

0.0 120.000500

500

Nonpa

Pay

XY slice #7


nonpay

Facies modeling - using seismic info. • Likelihood of seismic impedance |

pay∈u{ }pay)()1( ∈≤= uu sSProbsF

Similarly,

{ }nonpay)()0( ∈≤= uu sSProbsF

• If: then impedance discriminates pay / nonpay

)0()1( sFsF ≠

Freq

uenc

y

Imp

4000. 6000. 8000. 10000. 12000.0.000

0.040

0.080

0.120

0.160

Likelihood of meas. Zimp (Non-pay)Number of Data 279

mean 8575.4std. dev. 1054.6

coef. of var 0.12maximum 9950.0

upper quartile 9450.0median 8850.0

lower quartile 7850.0minimum 5750.0

Freq

uenc

y

Imp

4000. 6000. 8000. 10000. 12000.0.000

0.100

0.200

0.300

Likelihood of meas. Zimp (Pay)Number of Data 174

mean 6540.2std. dev. 707.49

coef. of var 0.11maximum 9350.0

upper quartile 6850.0median 6350.0

lower quartile 6150.0minimum 5150.0

No. of Data mean

279 8575

Likelihood of impedance in nonpay

4000 6000 8000 10000 12000 Impedance

No. of Data mean

174 6540

Likelihood of impedance in nonpay

4000 6000 8000 10000 12000 Impedance

Facies modeling - using seismic info. • Want

• Apply Bayes’ inversion:

[ ]{ }1,)(|1)( +∈= ll sssIProb uu

{ } { }{ }

{ }AProbBProb

ABProbBAProb ⋅=||

[ ]{ }1,)(|1)( +∈= ll sssIProb uu

[ ]{ }[ ]{ }

{ }1)(,)(

1)(|,)(1

1 =⋅∈

=∈=

+

+ uu

uu IProbsssProb

IsssProbll

ll

[ ]{ })(

,)()1|()1|(1

1 uu

psssProb

sFsFll

ll ⋅∈−

=+

+

Facies modeling - Bayes’ inversion

Requisites – Prior vertical proportion, – Likelihood, – Seismic impedance distribution,

)(dp)1(sF

)(sF

[ ]{ }1,)(|1)()(' +∈== ll sssIProbp uuu

used as locally varying mean after standardization

)(),('1 dpdxpN V

xd

∑ =

Implementation

layer d pay proportion

where ),( dxu=

Bayes’ inversion - Results • Slices through facies model

Facies Slice 7 after trans -T

East

Nor

th

0.0 100.0000.0

120.000

• Vertical proportion

Facies Slice 7 before trans -

East

Nor

th

0.0 100.0000.0

120.000

pay

nonpay

XY slice #7 before trans XY slice #7 after trans

0.0 0.2 0.4

Stra

tigra

phic

dep

th

0.0

0.2

0.4

0.6

0.8

1.0

0.6 Sand proportion

target simulated

0.8 1.0

Facies modeling- object based fluvsim (Deutsch and Wang (1996)) Modeling using reversible coord. transforms

channel complex

channel

Stratigraphic coord. transformation

channel complex areal transformation

areal transformation

channel

Simulation procedure in transformed space – annealing simulation of channel complex

Facies modeling - object based

∑∑ −+−+−=n

CCn

VVGGcc IIdpdpppOz

)()(||*)()(||*| αα uu

global proportion vertical proportion well data

where if simulated 1)( =uI channel∈u

– annealing simulation of channel – perturb channel until convergence – back coord. transform to obtain reservoir model

– perturb channel complex until convergence

Fluvsim - results Facies model ( well mismatch: 11.2 % ) Vertical proportion curve

Fluvsim only vertical prop. XY Slice 6

East

Nor

th

0.0 100.0000.0

120.000

Fluvsim only vertical prop. XZ Slice 20

East

Nor

th

0.0 100.000500

500Fluvsim only vertical prop. YZ Slice 20

East

Nor

th

0.0 120.000500

500

Pay

Nonpa

XY slice #7


nonpay

0.2 0.4

Stra

tigra

phic

dep

th

0.0

0.2

0.4

0.6

0.8

1.0

0.8 Sand proportion

target

simulated

0.0 0.6 1.0

Fluvsim model - using seismic info. • Average impedance vs. areal pay proportion

Cokrige areal proportion map using:

– vertically averaged proportion data – exhaustive 2-D seismic impedance – variogram of facies proportion – co-located cokriging under MMI

Impe

danc

e

Areal Proportion

Areal proportion (wells) vs. impedance

0.0 0.20 0.40 0.60 0.80 1.007000.

8000.

9000.

10000.

Number of data 19correlation -0.49

rank correlation -0.25

Impedance vs. areal proportion

Areal proportion

Impe

danc

e

No. of Data correlation

rank correlation

19 -0.49 -0.25

Fluvsim - using seismic info. • Cokriged areal proportion map

Fluvsim Slice 20 XZ with areal prop.

East

Nor

th

0.0 100.000500

500Fluvsim Slice 20 YZ with areal prop.

EastN

orth

0.0 120.000500

500

Nonpa

Pay

Cokriged areal prop. -

East

Nor

th

0.0 100.0000.0

120.000

0.3000

0.3500

0.4000

0.4500

0.5000

0.5500

0.6000

0.6500

0.7000

0.7500

0.8000

• fluvsim slices ( well mismatch - 14 %) Areal prop. on reference reservoir

East

Nor

th

0.0 100.0000.0

120.000

0.30

0.55

0.80

Cokriged areal proportion

XY slice #7


nonpay

Fluvsim - using seismic info. • fluvsim vertical proportion

Areal prop. on reference reservoir

East

Nor

th

0.0 100.0000.0

120.000Areal prop. in sim. with no areal

East

Nor

th

0.0 100.0000.0

120.000

0.3000

0.3500

0.4000

0.4500

0.5000

0.5500

0.6000

0.6500

0.7000

0.7500

0.8000

• impact of areal proportion Areal proportion - reference Areal prop. - unconditional

0.30

0.55

0.80

Sand proportion 0.0 0.2 0.4

Stra

tigra

phic

dep

th

0.0

0.2

0.4

0.6

0.8

1.0

0.6

target

simulated

0.8 1.0

1.0

Areal prop. in sim. with no areal

East

Nor

th

0.0 100.0000.0

120.000

Areal proportion - reproduction

Facies modeling - Some remarks • Object based models geologically realistic but

require numerous shape parameters

• trans post-processing of indicator simulations essential

• Integration of seismic has significant impact

Indicator basis function • Consider the expansion defined on the

basis of n indicator random variables : - Given the n indicator random variables, this

expansion is the most complete possible. - There are a total of terms in

the above expansion - There are 2 outcomes for each RV there are

outcomes possible for the function . - These outcomes define the space

Indicator Expansion – Another Interpretation In general the extended equations are for all 2n realizations of the n data. If instead, the estimate corresponding to a specific realization of the indicator RVs, say , then: Unbiasedness:

Orthogonality:

Two bases 1,D – two equations to obtain two unknowns

Single Normal Equation

Calibration of geologic information

East

Nor

th

0.0 100.0000.0

150.000

Geologic model

i(u;sk)

st(u) = { s(u+hα), α = 1, … , nt }

Set of data events = training data set

mp statistics

Variogram – two-point statistics

γ h

traditional

GrowthSim components

Matching Configuration

Data Configuration

Presenter

Presentation Notes

components

Pattern Statistics pattern observation

Presenter

Presentation Notes

configuration observation

To condition realization to hard (static) data Sample from distribution P(A | B )

where A = Simulation pattern B = Information from conditioning data & geology

Geological / Hard information Well data

Multipoint Statistics

( , )( | )( )

P AP BAP

BB

=Use Bayes’ Rule

MP Histogram

A – single point simulation event

Strebelle (2000), Caers (1999)

A – multiple point simulation event

GROWTHSIM, Arpat (2005)

Pattern Statistics optimized template ◦ goal reduce noises reduce spurious pattern configurations

CO

RR

ELAT

ION

Training Image

(100 x 100)

Template Domain

(7 x 7)

Spatial Correlation

Pattern Statistics effects of optimized template

GrowthSim algorithm

1. gather conditioning data around some node in the simulation grid

2. look for matching pattern configurations to the conditioning data in the statistics

3. apply one of the matching pattern configuration to the simulation grid

4. repeat 1-3 around the newly simulated node, until the simulation grid is filled

GrowthSim 100x100, 2-category data TI

Multiple Grid Simulation

effects of multiple-level simulation

Calibration of geologic information

East

Nor

th

0.0 100.0000.0

150.000

Geologic model

i(u;sk)

st(u) = { s(u+hα), α = 1, … , nt }

Set of data events = training data set

mp statistics

Variogram – two-point statistics

γ h

traditional An exhaustive training image required for inference of statistics

Noisy spatial covariance (variogram) inferred using sparse data, modeled using smooth functions restrictive, unrealistic

• Develop a geostatistical simulation method that does not rely on an exhaustive training model

• Perform inference and modeling of spatial connectivity functions in the spectral domain using sparse data.

Moment Generating Function

Expanding the function eωz as a Taylor series about the origin:

Cumulant generating function of Z is the Neperian logarithm of the moments generating function M:

Moments and Cumulants

Relation between the first few moments and cumulants

Moments can be calculated from the cumulants by

Three-point moment is a measure of similarity between three spatial locations. High value implies

three locations jointly have high values of the spatial attribute Z Spatial cumulant is also a

measure of spatial connectivity.

Moments and Cumulants

Polyspectra

For a stochastic process Z(t), energy is defined as

Power Spectrum

Property of Fourier transform - if an image is rotated, its Fourier transform also rotates

changes in direction of spatial continuity rotation in Fourier transform space orientation of objects can be detected from power spectrum

Power Spectrum – Detecting orientations

Orientation calculated using eigenvectors of the inertia matrix computed by thresholding the Fourier transform

Power Spectrum – Detecting orientations

Bispectrum

Bispectrum

Bispectrum

Phase of Fourier transform is a nonlinear function of frequency, extracted by biphase

Biphase Phase of Fourier spectrum

Need to find a feature identifier from bispectrum that can distinguish object shapes within the images - invariant under translation, scaling, rotation, and amplification Define the integrated bi-spectrum: Define phase of integrated bi-spectrum

Identifier 𝑃(𝑎) is invariant to translation, scaling, rotation and amplification (Nikias and Raghuveer, 1987 and others)

Integrated Bispectrum

Detecting Shapes

So far: Higher-order spectra can provide some interesting insights into spatial characteristics of reservoir models However, FFT requires an exhaustive dataset or image (such as a training image) What about when only scattered data is available?

×106

2

6

4

0

Image reconstructed using only highest 15% of Fourier coefficients

Bispectrum from sparse data

Consider the inverse transform:

used for geostatistical simulation (Jafarpour, Goyal, McLaughlin, & Freeman, 2009) assuming a DCT unique solution – no uncertainty quantification

Compressed Sensing

Estimate Fourier transform coefficients from scattered data solving a regularization problem Optimum lambda established using jack-knife

least mixed norm (LMN) [p=2, q=1]

Band-limited coefficients

Sparse reconstruction

Log(Amplitude) Reconstruction

Conditioning data

Sparse reconstruction – some results

One idea to improve quality of reconstruction - limit the search space to low frequency coefficients

How to automatically select the search space? Apply a jackknife method.

Randomly select 10% of the conditioning data (validation data) and use

the remaining data for reconstruction

An improvement

Reference Models Sparse Conditioning Data

Can we detect shapes using sparse data?

Our goal is to use the integrated bi-spectrum computed using the Fourier transform from sparse data to classify the models reflecting different features

Can we detect shapes using sparse data?

Conclusions • Performing inference and modeling of connectivity statistics in the Fourier space

provides an avenue for modeling geological realism in a computationally efficient manner

• Computation of higher order moments, cumulant and polyspectra using sparse data appears feasible

• Detection of size, orientation and shapes of features consistent with available conditioning data is an interesting aspect of this research training image selection

• While a broad idea about reservoir structures is possible using the power spectrum, the location of objects and details of the shapes are only possible by identifying phase bispectrum, higher-order spectra

computational challenges in reservoir modeling sanjay...

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