geostatistics in reservoir charactorization_a review
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Geostatistical Reservoir Characterization
Geostatistics: A Review of Basic Concepts
Univariate Statistics and Variogram
Geostatistical Reservoir Characterization
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).
Geostatistical Reservoir Characterization
Distribution FunctionMean
Variance
Freq
uenc
y of
Occ
urre
nce
z i
Geostatistical Reservoir Characterization
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 %
Geostatistical Reservoir Characterization
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 ≤
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
Normal Scores Transformationl Many geostatistical techniques require the data to be
transformed to a Gaussian or normal distribution:
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
Exercises
lUnivariate analysis of well log datalDistribution CharacteristicslHeterogeneity Measures
Geostatistical Reservoir Characterization
Statistics Review
Bivariate Statistics
The Covariance and the Variogram are
related measures of the joint variation of
two random variables.
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
*.
Geostatistical Reservoir Characterization
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
−
+== ∑
=
Geostatistical Reservoir Characterization
Data Sources
lLots of wells in subject reservoir
lLots of wells in similar reservoir
lOutcrops
lSecondary and soft data (seismic, interval
constraints, expert judgement)
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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.
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
Variogram Characteristicsγ
h
γ
hLow Spatial Correlation High Spatial Correlation
All geological inference is buried in the variogram.
γ
hAnisotropic
α1
α2
α3
Geostatistical Reservoir Characterization
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.
Geostatistical Reservoir Characterization
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
−=⇒
−=
Geostatistical Reservoir Characterization
Variogram Interpretation Geometric Anisotropy
Same shape and sill but different ranges
Geostatistical Reservoir Characterization
φ
1 2 34
1
2
3
4
1D
epth
Distance
Sill
Variogram InterpretationCyclicity
Geostatistical Reservoir Characterization
Variogram InterpretationCyclicity
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
Variogram InterpretationZonal Anisotropy
Geostatistical Reservoir Characterization
1Negative
correlation
Positivecorrelation
Trend » non stationaritythe mean is not constant
Dep
th
φ
Distance
Variogram InterpretationTrend
Geostatistical Reservoir Characterization
Variogram Interpretation Vertical Trend and Horizontal
Zonal Anisotropy
Geostatistical Reservoir Characterization
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:
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
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
Geostatistical Reservoir Characterization
Exercises
lVertical variogram calculationslAreal variogram calculationslVariogram modelingl Inference of spatial
variation/correlations
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