Geostatistics In Reservoir Charactorization_a Review

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 variable l A 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 Function Frequency of Occurrence

Mean

Variance

zi Geostatistical Reservoir Characterization

Histograms and Cumulative Distribution Function 250

100.00% 90.00% 80.00% 70.00%

150

60.00% 50.00%

Frequency Cumulative %

100

40.00% 30.00%

50

20.00% 10.00%

0

1000.

464.2

215.4

100.0

46.42

21.54

4.642 10.00

2.154

1.000

0.464

0.215

0.100

0.046

0.022

0.010

0.005

0.002

.00%

0.001

Frequency

200

Permeability Range, md

Geostatistical Reservoir Characterization

Producing Cumulative Distribution Function from the Data • Sort the data in increasing order

X 1 ≤ X 2 ≤ X 3 ≤ ... ≤ X N • Assign a probability pi to the event ( X • pi =(i-1/2)/N

≤ Xi )

• Plot Xi versus pi Geostatistical Reservoir Characterization

PROBABILITY PLOT Calculated Permeability Data Set

Estimated Permeability Data Set 109 Data Points, Mean = 0.36 Median = 0.10 10

1

0.1

0.01

0.001 1

2

5

10 15 20

30

40 50 60 70 80 85 Probability , % Less Than

90

95

98

Geostatistical Reservoir Characterization

99

Statistics Review Univariate Statistics: Basics Expected value = Mean = Arithmetic Average N E( z) = m =

∑

1 pi z i ≅ mˆ = N

∑z

i

i =1

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

VAR( z ) = σ 2z = E ([ zi − m])2 = E ( zi2 ) − m2 1 2 ˆ ≅σz = ( N − 1)

N

∑

( z i − mˆ ) 2

i =1

Geostatistical Reservoir Characterization

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

σˆ CV = m

Geostatistical Reservoir Characterization

Cv for Different Rock Types 23

Carbonate (mixed pore type) 22 S.(4) North Sea Rotliegendes Fm (6) 21 Crevasse splay sst (5) 20 Shallow marine rippled micaceous sst 19 Fluvial lateral accretion sst (5) 18 Distributary/tidal channel Etive ssts 17 Beach/stacked tidal Etive Fm. 16 Heterolitthic channel fill 15 Shallow marine HCS 14 Shallow marine high contrast lamination 13 Shallow marine Lochaline Sst (3) 12 Shallow marine Rannoch Fm 11 Aeolian interdune (1) 10 Shallow marine SCS 9 Large scale cross-bed channel (5) 8 Mixed aeolian wind ripple/grainflow (1) 7 Fluvial trough-cross beds (5) 6 Fluvial trough-cross beds (2) 5 Shallow marine low contrast lamination 4 Aeolian grainflow (1) 3 Aeolian wind ripple (1) Homogeneous core plugs 2 1 Synthetic core plugs

Very heterogeneous

Heterogeneous

Homogeneous

0

0

1

2

3

4

Cv Geostatistical Reservoir Characterization

Q-Q / P-P Plots

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

a straight line implies that the two distributions have the same shape.

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

Q-Q plot of permeability vs. porosity 1.E+02

Permeability

1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 0.001

0.010

0.100

1.000

Porosity

Geostatistical Reservoir Characterization

Data Transformation Why 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 Transformation l Many geostatistical techniques require the data to be transformed to a Gaussian or normal distribution:

Geostatistical Reservoir Characterization

Standard Normal Distribution z = (w-µ)/σ Cum. Normal pdf Normal

-3

-2

-1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.6827 0.9545 0.9973 0

1

2

Geostatistical Reservoir Characterization

3

Exercises l Univariate analysis of well log data l Distribution Characteristics l Heterogeneity 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 Review Covariance COV ( A, B ) = E ([ Ai − mA ][ Bi − mB ]) = E ( Ai Bi ) − mA mB

1 ∃ ≅ CAB = N

N

∑ ( a b ) − m∃ m∃ i i

A

B

i =1

>0 if A, B are positively correlated CAB = 0 if A, B are independent < 0 if A, B are negatively correlated Geostatistical Reservoir Characterization

Statistics Review Variogram 2γ ( A, B ) = E ([ A − B]2 ) 1 ≅ 2γ∃ = N

γÙ0 γÙ∝

N

∑ (a

i

− bi )

2

i =1

A is increasingly similar to B A is increasingly dissimilar to B Geostatistical Reservoir Characterization

Spatial Variation Assume: Variation in a property between two points depends only on vector distance, not on location. Model Variability: Variogram γ ( h ) = Covariance

1 2N h

 1 c( h ) =   Nh

Nh

2 [ ζ ( x ) − ζ ( x + h )] ∑ i i i =1

 2 ζ ( x ) ζ ( x + h ) − m  ∑ i i  i =1 Nh

Geostatistical Reservoir Characterization

Modeling Spatial Variation l zi =z(xi) is some property at location xi l Interpret zi as a random variable with a probability distribution and the set of zi to define a random function z. l Assume the variability between z(xi) and z(xi+h) depends only on vector h, not on location xi*.

Geostatistical Reservoir Characterization

Modeling Spatial Variation l Use variogram and/or covariance to model variability 1 2γ (h) = 2γˆ ( h) = Nh  1 COV ( h) = cˆ(h) =   Nh 

Nh

∑

[ z ( xi ) − z ( xi + h)]2

i =1

Nh

∑ i =1

 z ( xi ) z ( xi + h)  − mˆ z2  

Geostatistical Reservoir Characterization

Data Sources l Lots of wells in subject reservoir l Lots of wells in similar reservoir l Outcrops l Secondary and soft data (seismic, interval constraints, expert judgement) Geostatistical Reservoir Characterization

Porosity Log 11600

11500

Depth, ft

11400

11300

11200

11100

11000 0

0.1

0.2 0.3 Porosity, fraction

0.4

Depth Porosity 11060 0.083 11060.5 0.074 11061 0.062 11061.5 0.058 11062 0.061 11062.5 0.066 11063 0.07 11063.5 0.073 11064 0.078 11064.5 0.079 11065 0.075 11065.5 0.072 11066 0.072 11066.5 0.074 11067 0.075 11067.5 0.077 11068 0.098 11068.5 0.129 11069 0.151 11069.5 0.157

Geostatistical Reservoir Characterization

Variogram Calculation φ (u) 0.083 0.074 0.062 0.058 0.061 0.066 0.07 0.073 0.078 0.079 0.075 0.072 0.072 0.074 0.075 0.077 0.098 0.129 0.151

φ (u+h) 0.074 0.062 0.058 0.061 0.066 0.07 0.073 0.078 0.079 0.075 0.072 0.072 0.074 0.075 0.077 0.098 0.129 0.151 0.157

φ (u) 0.083 0.074 0.062 0.058 0.061 0.066 0.07 0.073 0.078 0.079 0.075 0.072 0.072 0.074 0.075 0.077 0.098 0.129

φ (u+h) 0.062 0.058 0.061 0.066 0.07 0.073 0.078 0.079 0.075 0.072 0.072 0.074 0.075 0.077 0.098 0.129 0.151 0.157

Geostatistical Reservoir Characterization

Variogram Calculation 0.35

0.35

0.3

0.3

R 2 = 0.8761

R 2 = 0.9812 0.25

0.25

0.2

0.2

0.15

0.15 0.1

0.1

Lag=1.5

Lag=0.5 0.05

0.05

0

0 0

0.1

0.2

0.3

0.4

0.35

0

0.1

0.2

0.3

0.4

0.35 R 2 = 0.7653

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

R2 = 0.352

Lag = 10 Lag=2.5 0.05

0.05 0

0 0

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0.4

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

Geostatistical Reservoir Characterization

Variogram Definition Sill - No correlation 1

Variogram

Increasing variability

1.2

0.8 Range 0.6

0.4

Model Fit Experimental

Nugget Effect

0.2

0 0

5

10

15

20

25 30 Distance

35

40

45

50

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 Terminology l 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 Characteristics γ

γ

h

h

Low Spatial Correlation

γ

High Spatial Correlation α1 α2 α3

h Anisotropic

All geological inference is buried in the variogram. Geostatistical Reservoir Characterization

Variograms Modeling 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 l Assuming second order stationarity, the following relationship applies.

γ (h) = var(z)− cov(h) ⇒ cov(h) = cov(0)−γ (h) l These are important relationships to be used during kriging using variograms. Geostatistical Reservoir Characterization

Variogram Interpretation Geometric Anisotropy Same shape and sill but different ranges

Geostatistical Reservoir Characterization

Variogram Interpretation Cyclicity 1

3

1

Sill 4

2 Distance

φ Depth

1

2

3

4

Geostatistical Reservoir Characterization

Variogram Interpretation Cyclicity

Geostatistical Reservoir Characterization

Variogram Interpretation Zonal Anisotropy Both sill and range vary in different directions 1

Variability ‘between wells’ ‘Within well’ variability Positive correlation over large distance

Well 1

Well 2

Well 3

Geostatistical Reservoir Characterization

Variogram Interpretation Zonal Anisotropy

Geostatistical Reservoir Characterization

Variogram Interpretation Trend Negative correlation

1 Positive correlation

Depth

Distance Trend » non stationarity the mean is not constant

φ 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 50

45 Regression: y = -1.5807x + 51.611 40

35

Depth

30

25

20

15

10

5

0 0

5

10

15

20

25

30

Porosity

Geostatistical Reservoir Characterization

Vertical Well Profile and Variogram after Removal of the Vertical Trend 50

45

40

35

Depth

30

25

20

15

10

5

0 -8

-6

-4

-2

0

2

4

6

8

Re siduals

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 l Vertical variogram calculations l Areal variogram calculations l Variogram modeling l Inference of spatial variation/correlations

Geostatistical Reservoir Characterization