lec6_sgems and spacial statistics2014

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PETE-322 GEOSTATISTICS Modeling Spatial Relationships

Lecture 6

Spatial Statistics

Spring 2014

PETE-322GEOSTATISTICS

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PREFERENTIAL SAMPLINGBias in sampling denotes preference in taking the measurements.Although estimation and simulation methods are robust to clustered preferential sampling, parameters that need to be inferred from the same sample prior to estimation or simulation can be seriously distorted, especially for small samples.The solution to preferential sampling is preparation of a compensated sample to eliminate the clustering, for which there are several methods.Declustering is important for the inference of global parameters, such as any of those associated with the histogram.

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Declustering Spatial Data

Any bias in sampling that results in taking relatively more samples from a particular region in space can result in data clustering.

What is a Data Cluster?

Declustering techniques attempt to remove sampling bias.

Without declustering, clustered data are usually given unrealistically more weights in statistical analysis

- clustered data dominate the calculation of statistical measures.- data variability under-estimated

Analysis of clustered data without accounting for the clustering can result in biased predictions.

Why Decluster?

A major reason for sampling bias in spatial data is the interest in characterization of (and production from) high pay regions.

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A-8

Need for Declustering

Rocks: 28% porosity, 100-6000md, 25% Sw

Fluids: 34o API, 800-1100 GOR, asphaltene prone

A-8

NWFX

Appraisal wellM Sand ProducerM Sand Water Injector

A-9A-10

A-7

A-6A-3

A-2

A-5

A-1

OWC @ -13,022

OWC @ -12,890

A-4

Appraisal wellJ Sand ProducerM Sand ProducerM Sand Water Injector

A-9A-7

A-6 A-3A-2

A-5

A-1

A-4

Overbank FaciesChannel Facies

127#1 J Sand

M Sand

1999 TGS data

1999 TGS data

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Declustered StatisticsDeclustered Statistics

ii n

w 1

Declustered Naive

Variance

Mean

Ni

i

i

N

ii

Xd

w

Xw

1

1

N

iiX XN 1

1

Ni

i

N

iXdi

Xd

w

Xiw

1

1

2

2)(

N

iXiX XN 1

22 )(1

1

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Declustering Method 1

A Simple Declustering Procedure

A simple way to decluster spatial data is through cell declustering, through the following procedure:

1) Divide the reservoir into a uniform grid with L cells of size c ; the global mean for the field is

ii Ln

1

N

iii

N

ii

i

L

l

n

ii

l

L

llL XXLn

XnLL

l

111 11

** 1111

Mean for each cell

2) Count the number of data points in each cell (data in cell l = nl,; total # data = N ) and estimate the mean in each cell

L

llL mL

1

1

3) Compute a weight that is used for declustering the data by making sure that each cell has the same contribution (regardless of the number of data it contains) as follows:

ln

ii

ll Xn

1

* 1

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Declustering Example

lDeclustered Mean and Variance:

0.25

0.26

0.24

0.300.20

0.19

0.17

0.240.08

0.05

0.030.10

0.11

0.23

0.140.18

0.190.21

0.15

0.11

5 3 1

4 2 1

2 1 1

nl.25 .2 .08

.18 .13 .05

.13 .1 .03

1640.0)03...........30.26.24.25(.2011

1

N

iiXN

0057.0)1640.03(.......)1640.24(.)1640.25(.120

1

)(1

1

222

2

1

2

N

iiXN

Without declustered Mean and Variance:

1/45 1/27 1/9

1/36 1/18 1/9

1/18 1/9 1/9

i

1239.003.91......14.

361.....26.

45124.

45125.

451

1

*

N

iiiL X

0051.)1239.03(.91......)1239.14(.

361.....)1239.26(.

451)1239.24(.

451)1239.25(.

451)( 222222*

1

*2

L

N

iiiL X

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Sample with Preferential Clustering

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DECLUSTERING

Detect presence of clusters by preparing acumulative distribution of distance to nearestneighbor.

10


Declustering Method 2

Decompose the clustering.

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Declustering

Preferential sampling shows as poor overlapping between the two histograms.

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DECLUSTERING One possibility is to obtain the declustered

subset (S4) by expanding the subset without clusters (S1) by transferring a few observations (S3) from the clustered subset (S2). Transfer observations from

(S2) by decreasing distance to nearest neighbor in S4.

Stop transferring points when the distribution of distances for the original subset S1 is about the same as that for the transferred observations (S3)

S3

S 1

713


Declustered Sample

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Quiz Question

Why do we decluster data?a) To make the data look nicerb) To remove sampling biasc) To transform the data to a

normal distributiond) To interpolate data at

unknown locationse) None of the above

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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 15

Geostatistics Geostatistics is a set of statistical tools that allow us to

analyze spatially distributed data. In classical statistics, there is an underlying assumption of data independence. This is not true in the earth sciences and any science that uses data gathered from a geographical coordinate system. The vary nature of a surface requires dependency on distance and orientation.

The field of geostatistics was developed in mining industry by geological engineers. It uses both deterministic and stochastic methodologies to help us understand the behavior of spatial data. It has the unique ability to not only integrate different types of data, but also data with different scales of volume support. It is useful in exploration geology as well as reservoir characterization. It provides not only estimates of values at unsampled locations, but provides the basis for understanding the reliability and uncertainty of the estimate.

MIN= 3.1P25= 6.2P50= 8.4P75= 11.5

MAX= 19.1MEAN= 8.9STD= 3.8

STD/MEAN= 0.43

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Variography: Geologic surfaces and features

have different scales and directions of continuity.

A variogram is the metric which describes the anisotropic behavior of a regionalized variable.

Continuity:Spatial continuity involves the concept that small values of an attribute are in geographical proximity to other small values, while high values are close to other high values. Transitions are gradual.

Geostatistical Analysis

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Where it fits in the workflowSpatial Analysis & Modeling

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Random Functions

A random function is a collection of random variables, one per site of interest in the sampling space. A realization is the set of values that arises after obtaining one outcome for every distribution. Geostatistics relies heavily on random functions to model uncertainty.

10

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Spatial Statistics

We have examined bivariate statistics of two different random variables, e.g., and k

Data pairs are usually at same location Spatial statistics (geostatistics) involves bivariate

statistics of the same random variable at two different locations, e.g., (u1) and (u2)

(u1) and (u2) are two different random variables

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Bivariate Statistics

Data pairs might be and k at each well 1, k 1 2, k 2 3, k 3

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Spatial Statistics

Data pairs might be at pairs of well 1, 2 1, 3 1, 4 2, 3

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Correlation Function of Distance Wells d1 apart

1, 2 2, 7

Wells d2>d1 apart 1, 5 3, 14

Survey QuestionWhich group of wells has greater covariance?

a) Wells d1 apartb) Wells d2 apart

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Modeling Continuity, or Spatial Correlation

Assessment of covariance or its close equivalent, the semivariogram, has been the classical way in geostatistics to measure spatial correlation.

The semivariogram or the covariance are necessary in the formulation and application of most estimation and simulation methods.

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Implications of StationarityUnder Stationarity Assumption

X( ) ( ) u &E X u d E X u d

X X( ) ( ) ( ) u &E X u L X u C L L

constant in space

function of distance (given a direction), but not location

Mean

Auto-Covariance

1 1

1 1( ) ( )N N

X i ii i

x u x u dN N

N

ii

N

ii

N

iii LuxN

uxN

LuxuxN

LC111

)(1)(1)()(1)(

13

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The variogram is a model of spatial continuity that identifies and quantifies the directions and scales of continuity. That is, it identifies the orientations and grain of the underlying geological surface or body. It is ultimately used to determine the weights in the Kriging equations, the geostatistical estimation method. Variography can be calculated for any regionalized variable.

Identifies and quantifies directions and scales of spatial continuity

Used to determine the weights during interpolation or simulation

Applied to any Regionalized Variable

Varia

nce

Distance (LAG)

Spatial Analysis and Modeling

n

hiin

ih

XX2

)(1

2

)(

)(

Compute the average squared difference between pairs of measurements at different

separation intervals, known as the Lag interval.

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CovarianceIf the mean is constant and the covariance is independent of location, then always (h) =Cov(0) Cov(h),which makes it immaterial which one to use. In general, what is estimated is the semivariogram because its estimation does not require knowledge of the mean.

semivariogram

Cov(0)

Lag a

covariance

Lag a is the range and the semivariogram asymptote is called the sill, which is equal to the variance Cov(0).

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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 27Co

v (h

)

Distance

(h)

Distance

Spatial Analysis & Modeling

Relationship Between the Variogram & Covariance:

Variogram

Distance = multiples of lag intervals Y axis = variance = mean squared

difference Covariance (Actually used by algorithms)

Distance = multiples of lag intervals Y axis = Cov = Sill - Variogram

Covariance is used in the kriging equation

because it is computationally more efficient.

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Trivial Example

0.5 1.0 1.5

0 0.5 1.0 1.5

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Modeling Anisotropy

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Directions of Continuity

What is the maximum direction

of continuity? (Fabricate Variogram

Intellectually?)

Scales of Continuity

Different directions of

Continuity (scales -h).

The Variogram in Perspective

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Geostatistical WorkflowSpatial Distribution of Reservoir Properties

Geological features are not randomly distributed in a spatial context.

Reservoirs are heterogeneous and have directions of continuity because of their specific depositional, structural, and diagenetic histories.

Credit: Yarus 2006

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Two-Point Spatial ModelingTwo-Point Spatial Relationship

Variogram Estimation and Modeling

Var

iogr

am

Distance

E-W Variogram

LEW LSN

Var

iogr

am

Distance

S-N Variogram

2) Spatial Modeling

XA=?

XB

XC

Two-point Statistics

LEW

LSN

1) Data

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Variogram Estimation (Scattered Data)Data Scarcity1) Available observation points? small number2) Equally spaced observations? very few3) Equally spaced and in the same direction?

Lag and Direction ToleranceTo deal with above issues Use LL and

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Irregular Patterns

Easting

Nor

thin

g

If the observations, z (s), are not regularly spaced, they are grouped into distance classes of equal radial thickness, which customarily is set equal to twice th .

th: Lag Tolerance: angular bandwidthtb: bandthwidthtb only operates when the cone is wider than twice th

Each class contributes one value to the experimentalsemivariogram, (h ).

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North

AngleBand width

East

Lag tolerance

Lag distance (h)

Angletolerance

Points within thisarea are accepted aslying a distance hfrom origin (at adata location).

Xi

Xi+h

What is the Variogram? Search Parameters for the horizontal variogram

Because of irregular spacing of input points, search criteria must be defined to select points within distance range given by the Lag.

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N90N95N100N105

N110N115

N120N125N130

N135N140N145N150N155N160N165N170N175N180N185N190N195

N200

N205

N210N215N220N225N230N235N240N245N250N255N260N265

0 500 1000 1500 Distance (m)

0

9000000

Variogram : SEISMIC AI

Maximum direction of Continuity

Minimum direction of Continuity

Traditional Variograms and Variogram Map

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Spatial Analysis and ModelingVariogram Map Polar Plot

If you have enough well data or seismic data then you can compute the Variogram polar

plot to determine the major and minor directions of continuity and approximate

scales.These parameters are used to compute and

model the anisotropic variogram.

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The Variogram Map

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39


Analytical Semivariogram Models

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Why do I need a variogram model?

Kriging system requires knowledge of correlation function for all ranges and azimuths

Smoothes experimental statistics and introduces geological information

To estimate the weights of neighboring values

Ensures positive estimation variance (only certain mathematical functions satisfy this condition)


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41


Analytical Forms for Semivariograms

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Common Variogram Models Basic Variogram Models

Nugget EffectC0

(L)

L

000

0 LCL

L

0000

LLC

LC

0

21

23 3

LC

aLaL

aLCLLM

aS

aLCaE LLM 3exp1

223exp1 aLCaG LLM

Spherical

Exponential

Gaussian

2 wCLaP

wLLM Power C0

(L)

L

w=0.5

w=1.5

22

43



44



23

45



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High VarianceIn First Lag(Geological Phen.Less Than WellSpacing or Error?)

Minimum Number of Bins ~ 30

Number of Pairs:n*(n-1)/2

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Typical Variogram Shapes

Structure, Isopach

Perm, Por

Trend

(Poor sampling seismic striping?)

Guaranteed Positive Definite Matrix (add a small nugget club)

48


Nested Model for Variogram

25

49



Note: The Nugget acts as a low pass filter (removes short scale features) in kriging, but adds short scale uncorrelated noise when performing simulation.

Anatomy of the Variogram:

Nugget Effect

Sho

rt S

cale

Nugget Effect with Long Scale Spherical Model

Sill = Data variance

Long Scale

Nested Short and Long Scale Spherical Models

Long Scale

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Illustrates the impact of an increasing Nugget effect using a Unique (all data)Neighborhood. Regardless of the amount of nugget the data values at the wellsare always honored if the well locations reside on a grid node. The maps are theaverage porosity.

Impact of the Nugget Term

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Spatial Analysis and ModelingOmnidirectional variogram: Should be computed first

Average scale for all directions, uses all data pairs Best indicator of model type (e.g. Spherical, Exponential, Cubic, Gaussian)

Best indicator of a Nugget (discontinuity at the origin)

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Spatial Analysis and ModelingModel Types

Exponential: least smooth

Cubic

Spherical

Gaussian: most smooth

All models have the same effective spatial scale: 3100-m

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53


Spatial Analysis and ModelingAnisotropic, nested model Structure 1 (Red)

Structure 2 (Green)

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Spatial Analysis and ModelingNested Ellipse

Simulation Result

Kriging Result

Depending upon whether you will perform kriging or simulation, the

images provide a visual image of the result of applying a variogram model.

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Variograms - Special Considerations: Behavior of the variogram is poor with few pairs The data spacing and geobodies size can give a false

impression of a Nugget effect Outliers adversely affect the variogram The first few lags are most important for modeling (weights

are largest) Avoid complex nested structures The variogram should relate to a geological model The variogram polar plot contains a great deal of

information, but requires a considerable amount of data


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1014

8

6

12

22

4

Porosity %


n

hiin

ih

XX2

)(1

2

)(

)(

h

Survey QuestionHow many data pairs have a lag of 4h?

a) 3b) 4c) 5d) 6e) 7

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1014

8

6

12

22

4

Porosity %


n

hiin

ih

XX2

)(1

2

)(

)(

hTest Question

What is for a lag of 5h?

lec6_sgems and spacial statistics2014

Documents

cell data

data clustering

spatial statisticsspring

number of data points

cell declustering

sampling bias

j sandm sand1999 tgs

declustering techniques