Log Normal DistributionSebuah distribusi log normal adalah berhubungan erat dengan distribusi normal. Jika log dari variabel acak yang terdistribusi normal, variabel yang diasumsikan log terdistribusi normal.

Transformation of Log-Normal to Normal Distribution

Probability Plot of Sample Permeability

22

2ln 1

2

ln 2

Standar Deviation

• Sebuah survei dari reservoir minyak dan gas dalam yang diberikan suatu cekungan menunjukkan bahwa mean aritmetika dari semua cadangan adalah 200.000 BBLS dan deviasi standar 400.000. Jika distribusi dari cadangan adalah log-normal, menghitung probabilitas yang belum dibor prospek akan menghasilkan kurang dari 400.000 BBLS.

Solution

22

2

2

2

ln 1

400,000ln 1200,000

1.609

2ln 2

1.609ln 200,000 211.4

In (400,000) = 12.90

ln 12.9 11.41.609

1.18251.1825 0.88 from Table 2.2

xz

F

…………………….……………………..……………………2.30

Dykstra Parson Coefficient

k kVk

E. Interference and Estimation of Parameters

Objective of any statistical analysis is to make interferences about population based on the information contained in a sample. In addition to making interfrence, we would also like to know the uncertainties associated with the estimation

3. Type of Estimation

In interfering about population parameters, two types of estimates are possible. An interval estimate allows an estimation of probability that a random variable within population will fall within certain region.

4. Desirable Properties of Estimation

a. Unbiasedness

E ……………….……………………..……………………2.31

b. Minimum Variance

Fig. 2.11: Unbiasedness Criterion

Fig. 2.12: Variance of Estimators

E. Bivariate Distribution

1. Linear Regression

1 2

,x

C X Y

……….……………………..……………………2.32

0 1E Y E X .……………………..……………………2.33

Example 2.7Using the data from Example 2.4, calculate the slope and the intercept for a linear regression between log of permeability and the porosity.

SolutionC(X,Y) = 0.8875

The varianve of X is also calculated as,

SX = 2.3291, therefore, SX2 = 5.4247

11

1

1 29.49 26.79 28.74 27.65 27.69922.69 23.3 23.81 25.54

26.19

n

iE X x x

n

1 3.063 2.725 3.025 2.915 3.00692.037 2.140 2.22 2.559

2.632

E Y y

1 2

,

0.8875 0.16365.4247

x

C X Y

0 1

2.632 0.1636 26.191.5584

E Y E X

Fig. 2.13: Log. Of Permeability Vs. Porosity Plot

Fig. 2.14: Inappropriate Use of Linear Regresion

2. Spatial Relationship

1

21 1

1,

1

n

i i i iin n

i ii i

C V X V X h V X V X hn

V X V X hn

……2.33

Example 2.8

The following well bore porosity data are collected from a sandstone reservoir. The data were collected at every one foot interval

Solution

A plot of covariance versus a lag distance is shown in Figure 2.15. The plot indicates that the intuitive relationship between the covariance and the lag distance has been borne out.

Fig. 2.15: Covariance versus Lag Distance

2

11 8.25 9.0 9.0 6.25 ... 8.25 9.0 9.0 9.25261 8.25 9.0 ... 8.25 9.0 9.0 6.25 ... 9.0 9.2526

1.126

C x x x x

Fig. 2.16: Scatter Plots of Porosity Data

2

1

1 n

ii

h dn

…………..…………………………………………2.35

2

11

12

n

ii

h V X V X hn

…………………………2.36

Example 2.9Using the data in Example 2.8, estimate the variogram as a function of lag distance

SolutionAs evident from Figure 2.16, the variogram has to be zero at h equal to zero, and it should increase as the distance increases.

This type of behaviour is exactly opposite of covariance.

2 2 2 211 8.25 9.0 9.0 6.25 ... 8.25 9.0 9.0 9.252 260.385x

Fig. 2.17: Variogram and Covariance as a Function of Distance