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Geostatistics for Reservoir
Characterization
Lecture 2a - What is a Random Variable and
How Do We Describe It?
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Random Variablea random variable is
1. Some characteristic which is unpredictable
2. Has a value associated with it
Examples
Lithology: 1 = sandstone, 2 = shale
Permeability: 247md core plug
Fracture spacing: 12.3cm
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Types of Random Variables
1. Discrete
1. Nominal (names) e.g., lithologies, rock types
2. Ordinal (size) e.g., hardness, sinuosity
2. Continuous
1. Interval (arbitrary zero) e.g., position, GR, SP
2. Ratio (fixed zero) e.g., mass, length, volume
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Random Variables and Information Continuous RV's + algebra is OK
Can add/subtract interval RV's
mass + mass = mass
Ratios have to be careful
porosity + porosity porosity
Discrete RV's & algebra don't mix
Sand = 1 & shale = 2
1 + 1 = 1
Why? Continuous RV's have more info than discrete
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Working with RV's
How can we manipulate RV's?
Want some way to express the uncertainty ofvalue
Answer: use probability!
(Prob is not the only answer)
Can use fuzzy variables
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Probability Two Part Definition
An identifiable event E
Examples
E = Facies A at a particular location
E = Core plug permeability is between 100 and 300 md
A number p expressing event likelihood
Of 300m gross pay in a well, 240m is productiveE = productive or net pay
p = Prob(E) = 240/300 = 0.80
Eighteen of thirty channels lacked abandonment elementsE = channel with abandonment top eroded away
p = Prob(E) = 18/30 = 0.6
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Probability and Frequency
We interpret probability as a frequency
Prob(E) = (trials giving E)/(total number of trials)
For a coin toss, we expect from physicalarguments
Prob(H) = 0.5
Prob(T) = 0.5
Have to assume many tosses (ie experiments)
We can plot RV value versus probability
Called a probability density function (PDF)
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Coin Toss PDF Let T = 0, H = 1
Prob(0) = 0.5
Prob(1) = 0.5
Function notation Y is the value ofthe tossed coin
Y is an RV
Prob(Y=0) = 0.5
Prob(Y=1) = 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1
Probability
Value of Random Variable
Prob Density Function
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Coin Toss
Histogram
Measurement based
No. of trials known Y-axis
Counts
Frequency
Care needed
Comparing histos
Different N?
Compare this to the coin-toss PDF
if N is odd number, Histogram and PDF must be different
0
10
20
30
40
50
60
70
0 1
Frequency
Value of Random Variable
Histogram (N = 100 trials)
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PDF vs Histogram - 1
PDF does not depend on no. of trials, N PDF assumes N very large
Histogram is OK for any N
The PDF is the histogram when N gets large
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PDF vs Histogram - 2
We say the PDF shows the populationbehavior
Histogram may differ from PDF because of N small 10 coin tosses may give H = 3 and T = 7
For these tosses, Prob(H) = 3/10 = 0.3; Prob(T) = 0.7
The histogram gives the samplebehaviour Histogram also called
sample PDF
empirical PDF
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Example HistogramsN
Rose Diagram Bi-Directional
Total Number of Points = 461
Bucket Size = 10 degrees Error Size = 0 degrees
0 44
0% 20% 40% 60% 80% 100%
zone1
zone2
zone3
zone4
zone5
facies1
facies2
facies3
facies4
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Continuous Variables
For discrete RV's, we can define the event
E = Heads E = Facies #2
For continuous RV's, we have to define a range
E = 0.1 < < 0.13 E = k > 1 md
The PDF of a discrete RV is blocky
The PDF of a continuous RV is smooth
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Continuous Variable PDF
f(x)
x0
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Continuous Variable PDF - Interpretation
f(x)
x0
xxfxxYx )()Prob( 000
x0 x0+x
f(x0)
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Continuous RV PDF's
f(x0)x is the area of a box
f(x0) is the height of the box x is the width of the box
So Prob(x0
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Histograms for Continuous RV's
For samples X1, X2, , XN
Divide range Xmax - Xmin into intervals (bins) Count frequency of X's in each bin
Plot bin frequencies versus value of RV
Notes
Bins usually of equal size
Rule of thumb: bin size X = 5(Xmax - Xmin)/N
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PDF Example . . .
0
100
200
300
400
500
600
0 5 101
1 102
2 102
2 102
3 102
Frequency,number
Gamma Ray Reading, API Units
mode
Histogram Example
Coun
ts
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Getting bin size right
Bin size and start point can
affect histogram shape
Bin size too small
Looks like comb
Bin size too big
Too coarse
Changing start point will
indicate whether histo is
"stable" or not 0.0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50X
Fig. 3-9
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Another Sample PDF Example
Two modes
Two lithologies?
Preferential sampling?
Histo doesn't say why
Just shows what's
happening
Up to us to investigate
0
10
20
30
40
50
60
2.
5 5
7.
510
12
.5
15
17
.5
20
22
.5
25
27
.5
30
Frequency
Porosi ty, %
Well A-04 Core Plug Porosity Distr ibution
2 modes
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Figure 3.10
3700 m
Sandstone
Mudstone
Coal
Carbonate concretions
KEY
LOWERBRENT
Gamma ray
m m
m
m
m
TYPICAL NORTH SEA
LOWER BRENT SEQUENCE
0
50
90
0 2000 4000 6000PERMEABILITY (mD)
0
10
20
-2 -1 0 1 2 3 4LOG (PERMEABILITY)
0
25
0 10 20 30POROSITY (%)
Core Plug Data Histograms
Two modes
Multimodal
3900 m
FREQU
ENCY(%)
m Mica
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Figure 3.11
m m
m
m
Por (%)Perm (D)
RANN
OCH
0 5 10 150 10 20 30
ETIVE
Porosity (%) Log10(Perm.)
-2 0 2 40 10 20 30
-2 0 2 40 10 20 30
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Some Properties of PDFs . . . .
1. f(x) > 0 for all x
2. Area under f(x) = 1
3. Show how many of one value vs another
4. Do not show relationships eg versus depth
0
5
10
0 50 100
Depth
Permeability
01234
5
20 40 60 80 100
Frequency
Bin
Histogram
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Summary Points . . .
Random variable types
Discrete Continuous
Use probs to express likelihood of occurrence
Interpret prob as frequency Plot of value vs prob is PDF
If we use samples, sample PDF is histogram
PDF's can
Reveal multiple lithologies
Hide spatial relationships