SEG/EAGE DISC 2003

INTRODUCTIONOutline

INTRODUCTIONOutline

• A Brief Historical Perspective

• The interaction between 3D Earth Modeling and Geostatistics

• Basic Probability and Statistics Reminders

SEG/EAGE DISC 2003

A random variable takes certain values with certain probabilities.

Example: Z = sum of two dice

RANDOM VARIABLESRANDOM VARIABLES

HISTOGRAM

0

1

2

3

4

5

6

7

1 2 3 4 5 6 7 8 9 10 11 12 13

SUM OF TWO DICE

FR

EQ

UE

NC

Y

Série1

Each value, for instance 4, is a realization

1-12

PROBABILITY DENSITY FUNCTION

SUM OF TWO DICE

FR

EQ

UE

NC

Y (

NO

T N

OR

MA

LIZ

ED

)

SEG/EAGE DISC 2003

Scale Count Minimum Maximum Mean Std. Dev. Correlation 27x27 100 13.55% 40.73% 24.42% 6.49% 0.72 9x9 900 9.43% 53.47% 24.42% 8.27% 0.90 3x3 8100 6.12% 75.58% 24.42% 9.89% 0.99 1x1 72900 4.80% 98.87% 24.42% 10.34% 1.00

THE IMPACT OF AVERAGING (2)HISTOGRAMS

THE IMPACT OF AVERAGING (2)HISTOGRAMS

1-18P. Delfiner/X. Freulon

9x9 27x271x11x1 9x9 27x27

SEG/EAGE DISC 2003

THE SUPPORT EFFECT(FRYKMAN AND DEUTSCH, 2002)

THE SUPPORT EFFECT(FRYKMAN AND DEUTSCH, 2002)

Well log

Histogram of core

Histogram of log

2-31

Impact on Cut-off

Variance is volume-dependent!

SEG/EAGE DISC 2003

NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, =5)NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, =5)

),( called also 2

)(exp

2

1)(

2

2

mNmx

xf

1-26

CONFIDENCE INTERVAL:

95% of values fall between m-2 and m+2

Porosity Uncertainty:

15 35

95%

SEG/EAGE DISC 2003

INTRODUCTIONLessons LearnedINTRODUCTION

Lessons Learned

• Geostatistics role in geosciences still evolving

• Geostatistics more and more closely integrated with earth modeling

• Probability and statistics help quantify degree of knowledge

• Support effect : decrease of variance as volume of support increases

• Confidence interval closely related to mean and standard deviation for normal distribution

• The correlation coefficient quantifies linear relationships

• Trend surface analysis is a useful model, but too simple

SEG/EAGE DISC 2003

NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, =5)NORMAL (OR GAUSSIAN) DISTRIBUTION (m=25, =5)

),( called also 2

)(exp

2

1)(

2

2

mNmx

xf

1-26

CONFIDENCE INTERVAL:

95% of values fall between m-2 and m+2

Porosity Uncertainty:

15 35

95%

SEG/EAGE DISC 2003

THE COVARIANCE AND THE VARIOGRAMOutline

THE COVARIANCE AND THE VARIOGRAMOutline

• Stationarity

• How geostatistics sees the world. The model.

• How to calculate a variogram

• A gallery of variogram models

• Examples

SEG/EAGE DISC 2003

STATIONARITY OF THE MEANSTATIONARITY OF THE MEAN

2-3

NonstationaryStationary

SEG/EAGE DISC 2003

A spatial phenomenon can be modeled using 2 terms:

• a low-frequency trend

• a residual

Constant trend: stationary variable Quadratic trend + stationary residual

STATIONARITY OF THE VARIANCE (1)STATIONARITY OF THE VARIANCE (1)

2-1P. Delfiner/X. Freulon

SEG/EAGE DISC 2003

The residual should have a constant variance

A variable with

• constant trend and

• residual with varying variance

A variable with

• quadratic trend and

• residual with varying variance

STATIONARITY OF THE VARIANCE (2)STATIONARITY OF THE VARIANCE (2)

2-2P. Delfiner/X. Freulon

SEG/EAGE DISC 2003

WHAT TO DO WHEN NOT ENOUGH DATA ARE AVAILABLE?

WHAT TO DO WHEN NOT ENOUGH DATA ARE AVAILABLE?

Vertical WellsVertical variograms

Variance gives sill of horizontal variograms

A priori geological knowledge

Behavior at origin and nugget effect

2-39

Seismic data Horizontal anisotropy ratios and ranges

Horizontal Wells Horizontal variograms

SEG/EAGE DISC 2003

THE COVARIANCE AND THE VARIOGRAMLessons Learned

THE COVARIANCE AND THE VARIOGRAMLessons Learned

• The model: low frequency trend + higher frequency residual +noise

• Variogram model more general than stationary covariances

• Meaning of the various parameters of the variogram model

• Relationship between fractals and geostatistics, covariance and spectral density

SEG/EAGE DISC 2003

KRIGING AND COKRIGINGOutline

KRIGING AND COKRIGINGOutline

• What is kriging

• How noise is handled by kriging. Error Cokriging

• Factorial Kriging for removing acquisition footprints

• Combining seismic and well information

– External Drift

– Collocated Cokriging

• Kriging versus other interpolating functions

SEG/EAGE DISC 2003

NUGGET EFFECT VS NYQUIST FREQUENCYNUGGET EFFECT VS NYQUIST FREQUENCY

(h)

0

Minimum detectable variogram range = d

hd

Minimum detectable wavelength = 2d

Maximum detectable spatial frequency = 1/(2d)

Distance between data=d

x x x x

x x x x

x x x x

d

SEG/EAGE DISC 2003

THE FACTORIAL KRIGING MODELMARINE EXAMPLE: HORIZON-CONSISTENT VSTACK (3)

THE FACTORIAL KRIGING MODELMARINE EXAMPLE: HORIZON-CONSISTENT VSTACK (3)

3-39J.L. Piazza and L. Sandjivy

m

(m/s

)2

in-line effect(4) Spherical (D1) 300 m (D2)

450 (m/s)2

geophysicist effect (3) Spherical 1600 m (D1)

100 m (D2)100 (m/s)2

m

(m/s

)2

Geological signal(1) Spherical 7500 m, 1000 (m/s)2

(2) Spherical 1600 m, 300 (m/s)2

Final modelFinal model

m

(1) Linear 1000 (m/s)2

(2) Spherical 300 (m/s)2

(3) Spherical 100 (m/s)2

(4) Spherical 450 (m/s)2

(5) Nugget 400 (m/s)2

artefacts

SEG/EAGE DISC 2003

INTRODUCING EXTERNAL DRIFT AND COLLOCATED COKRIGING

INTRODUCING EXTERNAL DRIFT AND COLLOCATED COKRIGING

The situation

• Scattered well data giving exact measurements of one parameter (depth, average velocity, porosity, thickness of a lithology…)

• 2D or 3D seismic data giving information about the variations of this parameter away from the wells (time, stacking velocity, inverted impedance, seismic attribute…)

The problem

• How to combine well and seismic information properly, in such a way that the parameter measured at the well is interpolated away from the well using the seismic information?

SEG/EAGE DISC 2003 V. Bigault de Cazanove

THE EXTERNAL-DRIFT MODEL THE EXTERNAL-DRIFT MODEL

Two variables Z(x) and S(x)

S(x) assumed to be known at each location x

S(x) defines the shape of Z(x)

)()()( 10 xxx RSaaZ

3-56

Deterministic external-drift

Randomresidual

SEG/EAGE DISC 2003

j

jj

i

iicok ZZZ )()()( 2101 xxx

COKRIGINGCOKRIGING

Two variables Z1(x) and Z2(x) (such as porosity & acoustic impedance)

Use of Z1 and Z2 data to get a better interpolation of Z1

3-67

Porosity estimation by

cokriging

Porosity data at wells

Acoustic impedance data

from seismic

SEG/EAGE DISC 2003

j

jj

i

iicok ZZZ )()()( 2101 xxx

COLLOCATED COKRIGINGCOLLOCATED COKRIGING

COKRIGING

COLLOCATED COKRIGING

)()()( 02101 xxx ZZZ

i

iiccok

Complicated system of equations

Requires variograms of Z1, Z2, cross-variograms of Z1 and Z2

SEG/EAGE DISC 2003

COLLOCATED COKRIGING (JEFFERY ET AL., 1996)

COLLOCATED COKRIGING (JEFFERY ET AL., 1996)

3-70

Just the variance of residual gravity is used, not the

whole variogram!

WELL CONTROL DEPTHING VELOCITY

ISOTROPIC VARIOGRAM

CORRELATION 0.76

RESIDUAL GRAVITYISOTROPIC VARIOGRAM

Cross-validation shows 25 % improvement

(Mean absolute error from 22 to 15.5 m/s)

SEG/EAGE DISC 2003

EXTERNAL DRIFT OR COLLOCATED COKRIGING?EXTERNAL DRIFT OR COLLOCATED COKRIGING?

Collocated CokrigingExternal Drift

Model Seismic is low frequency termCorrelation coeff betweenseismic & primary variable

InputSeismic map and wellsVariogram of residuals

Seismic map and wellsCorrelation coefficient

Variogram of primary variableVariance of seismic data

PropertiesInteraction between variogram

model and correlation coeff

Applications

Equal to linear transform of seismic beyond variogram range

Construction of structural model Interpolation of petrophysical parameters

SEG/EAGE DISC 2003

KRIGING AND COKRIGINGLessons Learned

KRIGING AND COKRIGINGLessons Learned

• Kriging a weighted average of surrounding data points

• Nugget effect can be interpreted as variance of random errors

• Factorial kriging can handle multiscale variogram models

• Two techniques are preferred for combining seismic and wells:

- External Drift

- Collocated Cokriging

• Kriging surface expression similar to that generated by splines

SEG/EAGE DISC 2003

CONDITIONAL SIMULATIONOutline

CONDITIONAL SIMULATIONOutline

• Monte-Carlo simulation reminders

• Conditional simulation versus kriging

• How are conditional simulations realisations produced?

• Multivariate conditional simulations

• Conditional simulation of lithotypes

• Constraining conditional simulations of lithotypes by seismic

• Generalized multi-scale geostatistical reservoir models

SEG/EAGE DISC 2003

+

=

THE THREE PROSPECTSTHE THREE PROSPECTS

+

4-7

m1=75 1=15 m2=100 2=25 m3=200 3=40

m=375

Independence assumption:conclusion obtained by Monte-Carlo

simulation (or by properly combining variances)

=50

Full dependence assumption: conclusion obtained by

simply adding min and max of prospects

=80

SEG/EAGE DISC 2003

DEPENDENCE OR INDEPENDENCE?DEPENDENCE OR INDEPENDENCE?

1. Independence: Variances are added:

2. Full Dependence: Confidence Intervals (or standard deviations in the gaussian case) are added

23

22

21

2

321

SEG/EAGE DISC 2003

A KRIGING EXAMPLE IN 3D (LAMY ET AL., 1998b) A KRIGING EXAMPLE IN 3D (LAMY ET AL., 1998b)

4 9AI

km.g / s.cm3 N

4-10

Why should the reservoir be smooth

precisely away from the data

points?

Total UK Geoscience Research Centre

SEG/EAGE DISC 2003

KRIGING OR CONDITIONAL SIMULATION?KRIGING OR CONDITIONAL SIMULATION?

KrigingConditional simulation

Output Multiple realizations. One “deterministic” model.

PropertiesHonors wells,honors histogram, variogram,spectral density.

Honors wells, minimizes error variance.

ImageNoisy, especially if variogram model is noisy.

Smooth, especially if variogram model is noisy.

Data points

Image has same variability everywhere. Data location cannot be guessed from image.

Tendency to come back to trend away from data. Data location can be spotted.

4-16

UseHeterogeneity Modeling,Uncertainty quantification

Mapping

SEG/EAGE DISC 2003

CONDITIONAL SIMULATIONLESSONS LEARNED

CONDITIONAL SIMULATIONLESSONS LEARNED

• Conditional simulation generates representative heterogeneity models. Kriging does not.

• SGS and SIS most flexible simulation algorithms.

• Multivariate conditional simulation techniques can be used to account for correlations between various realizations.

• Bayesian-like techniques most suitable for constraining lithotype models by seismic data.

• Geostatistical conditional simulation provides toolkit for generating lithotype and petrophysical models at all scales.

SEG/EAGE DISC 2003

GEOSTATISTICAL INVERSIONOutline

GEOSTATISTICAL INVERSIONOutline

• What is geostatistical inversion

• Examples of geostatistical inversion

• Using geostatistical inversion results to predict other petrophysical parameters and lithotypes

SEG/EAGE DISC 2003

GEOSTATISTICAL INVERSIONLessons Learned

GEOSTATISTICAL INVERSIONLessons Learned

• Geostatistical Inversion generates acoustic impedance models at higher frequency than the seismic data.

• Non-uniqueness quantified through multiple realizations.

• Geostatistical inversion still a tedious exercise, in terms of processing time and processing of multi-realizations.

• Emerging applications for predicting petrophysical parameters and lithotypes from acoustic impedance realizations.

SEG/EAGE DISC 2003

QUANTIFYING UNCERTAINTIESOutline

QUANTIFYING UNCERTAINTIESOutline

• Why should we quantify uncertainties

• Structural uncertainties. How to quantify them?

• Combining all uncertainties affecting the 3D earth model

• Multirealization vs scenario-based approaches

• Demystifying uncertainty quantification approaches

SEG/EAGE DISC 2003

EARTH MODELLING AND QUANTIFICATION OF RESERVOIR UNCERTAINTIES

EARTH MODELLING AND QUANTIFICATION OF RESERVOIR UNCERTAINTIES

Geometry

Static properties

Dynamic properties

6-4

Impact on GRV!

Impact on OIP!

Impact on Reserves!

SEG/EAGE DISC 2003

QUANTIFICATION OF STRUCTURAL UNCERTAINTIESTHE APPROACH

QUANTIFICATION OF STRUCTURAL UNCERTAINTIESTHE APPROACH

• Estimation of uncertaintiesEstimation of uncertainties

• Identify uncertainties in the interpretation workflow,Identify uncertainties in the interpretation workflow,

• Quantify their magnitudeQuantify their magnitude (Confidence interval)(Confidence interval)

Interp

rete

r ’s in

pu

tG

eo

statis

tican

 ’s in

pu

t

• Measure of their impact on the results (GRV,OIP...)Measure of their impact on the results (GRV,OIP...)

• Geostatistical Simulation Geostatistical Simulation

• Statistical AnalysisStatistical Analysis

SEG/EAGE DISC 2003

NORTH SEA STRUCTURAL UNCERTAINTY QUANTIFICATION CASE STUDY (ABRAHAMSEN ET AL., 2000)

NORTH SEA STRUCTURAL UNCERTAINTY QUANTIFICATION CASE STUDY (ABRAHAMSEN ET AL., 2000)

GRV (Mm3)

p

df

Base case = 652 Mm3

SEG/EAGE DISC 2003

QUANTIFYING UNCERTAINTIESLessons Learned

QUANTIFYING UNCERTAINTIESLessons Learned

• Geostatistical techniques can be used to quantify the combined impact of uncertainties affecting the earth model.

• Uncertainty-quantification nothing more than translating input uncertainties into output uncertainties. Input is always subjective.

SEG/EAGE DISC 2003

• Generation of 3D heterogeneity models

• Integration of seismic data in reservoir models

• Uncertainty quantification

3 AREAS WHERE GEOSTATISTICS IS CRUCIAL 3 AREAS WHERE GEOSTATISTICS IS CRUCIAL

7-2

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www.ualberta.ca/~cdeutsch/

ekofisk.stanford.edu/SCRFweb/index.html

www.math.ntnu.no/~omre

www.cg.ensmp.fr

www.tucrs.utulsa.edu/joint_industry_project.htm

www.ai-geostats.org

WEBSITES ABOUT PETROLEUM GEOSTATISTICSWEBSITES ABOUT PETROLEUM GEOSTATISTICS

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AAPG Computer Applications in Geology, No. 3, Stochastic Modeling and Geostatistics, J.M. Yarus and R.L. Chambers eds

Chilès, J.P., and Delfiner, P., 1999, Geostatistics. Modeling Spatial Uncertainty, Wiley Series in Probability and Statistics, Wiley & Sons, 695p.

Deutsch, C.V., and Journel, A.G., 1992, GSLIB, Geostatistical Software Library and User’s Guide, New York, Oxford University Press, 340p.

Doyen, P.M., 1988, Porosity from Seismic Data: A Geostatistical Approach, Geophysics, Vol. 53, No. 10, p. 1263-1275.

Isaaks, E.H., and Srivastava, R.M., 1989, Applied Geostatistics, New York, Oxford University Press, 561p.

Lia, O., Omre, H., Tjelmeland, H., Holden, L., and Egeland, T., 1997, Uncertainties in Reservoir Production Forecasts, AAPG Bulletin, Vol. 81, No. 5, May 1997, p. 775-802.

Thore, P., Shtuka, A., Lecour, M., Ait-Ettajer, T., and Cognot, R., 2002, Structural Uncertainties: Determination, Management, and Applications,

Geophysics, Vol. 67, No. 3, May-June 2002, p. 840-852.

BOOKS AND PAPERS TO READBOOKS AND PAPERS TO READ

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