geostatistics in reservoir charactorization_a review

Geostatistical Reservoir Characterization

Geostatistics: A Review of Basic Concepts

Univariate Statistics and Variogram


Random Variable

l In the stochastic approach (as opposed to deterministic approach), we treat reservoir properties as a random variablelA random variable, z, can take a series of

outcomes or realizations ( zi, i=1, 2, 3,.....N) with a given set of probability of occurrences (pi, i =1, 2,...N).


Distribution FunctionMean

Variance

Freq

uenc

y of

Occ

urre

nce

z i


Histograms and Cumulative Distribution Function

0

50

100

150

200

2500.

001

0.00

20.

005

0.01

0

0.02

2

0.04

6

0.10

0

0.21

5

0.46

4

1.00

0

2.15

4

4.64

210

.00

21.5

446

.42

100.

021

5.4

464.

210

00.

Permeability Range, md

Freq

uenc

y

.00%10.00%

20.00%30.00%

40.00%50.00%

60.00%70.00%

80.00%90.00%

100.00%

FrequencyCumulative %


Producing Cumulative Distribution Function from the Data

• Sort the data in increasing order

• Assign a probability pi to the event • pi =(i-1/2)/N

• Plot Xi versus pi

NXXXX ≤≤≤≤ ...321

)( iXX ≤


1510521 20 30 40 50 60 70 80 85 90 95 98 990.001

0.01

0.1

1

10

Probability , % Less Than

Cal

cula

ted

Perm

eabi

lity

Dat

a Se

t PROBABILITY PLOT

Estimated Permeability Data Set 109 Data Points, Mean = 0.36 Median = 0.10


Statistics ReviewUnivariate Statistics: BasicsExpected value = Mean = Arithmetic Average

Variance = a measure of the spread of a distribution about its mean

∑∑=

=≅==N

iiii z

NmzpmzE

1

1ˆ)(

VAR z E z m E z mz i i( ) ([ ]) ( )= = − = −σ 2 2 2 2

∑=

−−

=≅N

iiz mz

N 1

22 )ˆ()1(

1σ̂


Coefficient of Variation

mCV

σ̂=

• Coefficient of Variation Cv is a dimensionless measure of spread of the distribution and is commonlyUsed to quantify permeability heterogeneity


0123456789

1011121314151617181920212223

0 1 2 3 4

Synthetic core plugsHomogeneous core plugs

Aeolian wind ripple (1)Aeolian grainflow (1)

Mixed aeolian wind ripple/grainflow (1)

Fluvial trough-cross beds (2)Shallow marine low contrast lamination

Fluvial trough-cross beds (5)

Carbonate (mixed pore type) (4)S. North Sea Rotliegendes Fm (6)

Cv

Homogeneous

Heterogeneous

Very heterogeneous

Crevasse splay sst (5)Shallow marine rippled micaceous sst

Fluvial lateral accretion sst (5)Distributary/tidal channel Etive ssts

Beach/stacked tidal Etive Fm.Heterolitthic channel fill

Shallow marine HCSShallow marine high contrast lamination

Shallow marine Lochaline Sst (3)Shallow marine Rannoch Fm

Aeolian interdune (1)Shallow marine SCS

Large scale cross-bed channel (5)

Cv for Different Rock Types


Q-Q / P-P Plots

l Compares two univariate distributionsl Q-Q plot is a plot of matching quartiles

– a straight line implies that the two distributions have the sameshape.

l P-P plot is a plot of matching cumulative probabilities – a straight line implies that the two distributions have the same

shape.l Q-Q plot has units of the data, l P-P plots are always scaled between 0 and 1


1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

0.001 0.010 0.100 1.000

Porosity

Perm

eabi

lity

Q-Q plot of permeability vs. porosity


Data TransformationWhy do we need to worry about data transformation?l Attributes, such as permeability, with highly skewed data

distributions present problems in variogram calculation; the extreme values have a significant impact on the variogram.

l One common transform is to take logarithms,y = log10 ( z )

perform all statistical analyses on the transformed data, and back transform at the end → back transform is sensitive

l Many geostatistical techniques require the data to be transformed to a Gaussian or normal distribution.The Gaussian RF model is unique in statistics for its extreme analytical simplicity and for being the limit distribution of many analytical theorems globally known as “central limit theorems”

The transform to any distribution (and back) is easily accomplished by the quantile

transform


Normal Scores Transformationl Many geostatistical techniques require the data to be

transformed to a Gaussian or normal distribution:


Standard Normal Distributionz = (w-µ)/σ

00.10.20.30.40.50.60.70.80.9

1

-3 -2 -1 0 1 2 3

Cum. Normalpdf Normal

0.68270.9545

0.9973


Exercises

lUnivariate analysis of well log datalDistribution CharacteristicslHeterogeneity Measures


Statistics Review

Bivariate Statistics

The Covariance and the Variogram are

related measures of the joint variation of

two random variables.


Statistics ReviewCovariance

>0 if A, B are positively correlatedCAB = 0 if A, B are independent

< 0 if A, B are negatively correlated

COV A B E A m B m E A B m mi A i B i i A B( , ) ([ ][ ]) ( )= − − = −

≅ = −=∑∃ ( ) ∃ ∃C

Na b m mAB i

i

N

i A B1

1


Statistics ReviewVariogram

γÙ 0 A is increasingly similar to BγÙ ∝ A is increasingly dissimilar to B

2 2γ ( , ) ([ ] )A B E A B= −

≅ = −=∑2 1

1

2∃ ( )γN

a bii

N

i


Spatial VariationAssume:Variation in a property between two pointsdepends only on vector distance, not onlocation.

Model Variability:Variogram

Covariance

γ ζ ζ( ) [ ( ) ( )]hN

x x hh

ii

N

i

h

= − +=∑1

2 1

2

c hN

x x h mh

ii

N

i

h

( ) ( ) ( )= +

−

=∑1

1

2ζ ζ


Modeling Spatial Variationl zi =z(xi) is some property at location xi

l Interpret zi as a random variable with a probability distribution and the set of zito define a random function z.lAssume the variability between z(xi)

and z(xi+h) depends only on vector h, not on location xi

*.


Modeling Spatial Variation

lUse variogram and/or covariance to model variability

2

1

)]()([1)(ˆ2)(2 hxzxzN

hh i

N

ii

h

h

+−== ∑=

γγ

2

1

ˆ)()(1)(ˆ)( zi

N

ii

h

mhxzxzN

hchCOVh

−

+== ∑

=


Data Sources

lLots of wells in subject reservoir

lLots of wells in similar reservoir

lOutcrops

lSecondary and soft data (seismic, interval

constraints, expert judgement)


Porosity Log

11000

11100

11200

11300

11400

11500

11600

0 0.1 0.2 0.3 0.4Porosity, fraction

Dep

th, f

t

Depth Porosity11060 0.083

11060.5 0.07411061 0.062

11061.5 0.05811062 0.061

11062.5 0.06611063 0.07

11063.5 0.07311064 0.078

11064.5 0.07911065 0.075

11065.5 0.07211066 0.072

11066.5 0.07411067 0.075

11067.5 0.07711068 0.098

11068.5 0.12911069 0.151

11069.5 0.157


Variogram Calculationφ(u) φ(u+h)

0.083 0.0740.074 0.0620.062 0.0580.058 0.0610.061 0.0660.066 0.070.07 0.0730.073 0.0780.078 0.0790.079 0.0750.075 0.0720.072 0.0720.072 0.0740.074 0.0750.075 0.0770.077 0.0980.098 0.1290.129 0.1510.151 0.157

φ(u) φ(u+h)0.083 0.0620.074 0.0580.062 0.0610.058 0.0660.061 0.070.066 0.0730.07 0.0780.073 0.0790.078 0.0750.079 0.0720.075 0.0720.072 0.0740.072 0.0750.074 0.0770.075 0.0980.077 0.1290.098 0.1510.129 0.157


Variogram CalculationR2 = 0.9812

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4

Lag=0.5

R2 = 0.7653

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4

Lag=2.5

R2 = 0.8761

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4

Lag=1.5

R2 = 0.352

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4

Lag = 10

l As the separation distance increases, the similarity between pairs of values decreases


Variogram Definition

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35 40 45 50Distance

Vario

gram

Model FitExperimentalNugget

Effect

Range

Sill - No correlation

Incr

easi

ng v

aria

bilit

y


Variogram Model

Variogram improves with increasing:

- Number of data pairs at each lag spacing.

- Number of lags with data.

è Lots of data required for statistically

significant variogram.


Variogram Terminologyl Sill

– the variance of the data (1.0 if the data are normal scores)– The plateau that the variogram reaches at the range

l Range– As the separation distance between pairs increases, the corresponding variogram

value will generally increase. Eventually, an increase in the separation distance no longer causes a corresponding increase in the averaged squared difference between pairs of values.The distance at which the variogram reaches this plateau is the range

l Nugget effect – natural short-range variability (microstructure) and measurement error– Although the value of the variogram for h=0 is strictly 0, several factors, such as

sampling error and short term variability, may cause sample value separated by extremely short distances to be quite dissimilar. This causes a discontinuity from the value of 0 at the origin to the value of the variogram at extremely small separation distances


Variogram Characteristicsγ

h

γ

hLow Spatial Correlation High Spatial Correlation

All geological inference is buried in the variogram.

γ

hAnisotropic

α1

α2

α3


VariogramsModeling Spatial Correlation

l The shape of the variogram model determines the spatial continuity of the random function model

l Measures must be customized for each field and each attribute (φ,Κ)l Depending on the level of diagenesis, the spatial variability may be similar within similar

depositional environments.


Variogram and Covariance

lAssuming second order stationarity, the following relationship applies.

lThese are important relationships to be used during kriging using variograms.

)()0cov()cov()cov()var()(

hhhzhγ

γ

−=⇒

−=


Variogram Interpretation Geometric Anisotropy

Same shape and sill but different ranges


φ

1 2 34

1

2

3

4

1D

epth

Distance

Sill

Variogram InterpretationCyclicity


Variogram InterpretationCyclicity


1Variability ‘between wells’

‘Within well’ variability

Positive correlation over large distanceWell 1 Well 2 Well 3

Variogram InterpretationZonal Anisotropy

Both sill and range vary in different directions


Variogram InterpretationZonal Anisotropy


1Negative

correlation

Positivecorrelation

Trend » non stationaritythe mean is not constant

Dep

th

φ

Distance

Variogram InterpretationTrend


Variogram Interpretation Vertical Trend and Horizontal

Zonal Anisotropy


Vertical Well Profile and Variogram with a Clearly Defined

Vertical Trend

y = -1.5807x + 51.611

0

5

10

15

20

25

30

35

40

45

50

0 5 10 15 20 25 30

Porosity

Dep

th

Regression:


Vertical Well Profile and Variogram after Removal of the

Vertical Trend

0

5

10

15

20

25

30

35

40

45

50

-8 -6 -4 -2 0 2 4 6 8

Residuals

Dep

th


Methodology for Variogram Interpretation and Modeling

l Compute and plot experimental variograms in what are believed to be the principal directions of continuity based on a-priori geological knowledge

l Place a horizontal line representing the theoretical sill.l Remove all trends from data.l Interpretation

– Short-scale variance: the nugget effect – Intermediate-scale variance: geometric anisotropy. – Large-scale variance:

• zonal anisotropy • hole-effect

l Modeling– Proceed to variogram modeling by selecting a model type (spherical, exponential,

gaussian…) and correlation ranges for each structure


Exercises

lVertical variogram calculationslAreal variogram calculationslVariogram modelingl Inference of spatial

variation/correlations

geostatistics in reservoir charactorization_a review

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