lecture 2a rv's

8/13/2019 Lecture 2A rv's

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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

lecture 2a rv's

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