Post on 25-Nov-2015
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Lecture 6
Spatial Statistics
Spring 2014
PETE-322GEOSTATISTICS
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Sample with Preferential Clustering
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
DECLUSTERING
Detect presence of clusters by preparing acumulative distribution of distance to nearestneighbor.
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Declustering Method 2
Decompose the clustering.
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Declustering
Preferential sampling shows as poor overlapping between the two histograms.
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Declustered Sample
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 16
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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.
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Bivariate Statistics
Data pairs might be and k at each well 1, k 1 2, k 2 3, k 3
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Spatial Statistics
Data pairs might be at pairs of well 1, 2 1, 3 1, 4 2, 3
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 25
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Trivial Example
0.5 1.0 1.5
0 0.5 1.0 1.5
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 35
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 36
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 37
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
The Variogram Map
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Analytical Semivariogram Models
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 40
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)
Spatial Analysis and Modeling
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Analytical Forms for Semivariograms
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
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
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Spatial Analysis and Modeling
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Spatial Analysis and Modeling
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Spatial Analysis and Modeling
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Spatial Analysis and Modeling
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Typical Variogram Shapes
Structure, Isopach
Perm, Por
Trend
(Poor sampling seismic striping?)
Guaranteed Positive Definite Matrix (add a small nugget club)
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships
Nested Model for Variogram
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 49
Spatial Analysis and Modeling
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 50
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 51
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 52
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 53
Spatial Analysis and ModelingAnisotropic, nested model Structure 1 (Red)
Structure 2 (Green)
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 54
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 55
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
Spatial Analysis and Modeling
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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 56
1014
8
6
12
22
4
Porosity %
Spatial Analysis and Modeling
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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PETE-322 GEOSTATISTICS Modeling Spatial Relationships 57
1014
8
6
12
22
4
Porosity %
Spatial Analysis and Modeling
n
hiin
ih
XX2
)(1
2
)(
)(
hTest Question
What is for a lag of 5h?