Simulation

1
Simulation

• Application of Simulation to Probabilistic
  Reserve Estimates Part 2

2
Learning Objectives

• You will be able to
• Identify correlations between input variables in
  simulation
• Simulate correlations manually
• Simulate reserves estimates using Excel
• Simulate reserves estimates using _at_RISK

3
Identifying Correlations Between Variables

• When input variables are correlated and we assume
  them to be independent, our simulation results
  will be incorrect
• We must identify correlations and take them into
  account
• Scatter plots help us identify correlations

4
Total Linear Positive Dependence
5
Total Nonlinear Negative Dependence
6
Diffuse Positive Dependence
7
Uncorrelated Variables
8
Quantifying Dependence Correlation Coefficient

• Correlation coefficient, r, reveals degree of
  correlation between variables
• where
• sX standard deviation of Xs
• sY standard deviation of Ys
• average (mean) of Xs
• average (mean) of Ys

9
Interpretation of Correlation Coefficients

• Values of r range between -1 and 1
• Value near 1 means strong positive correlation
• Value near -1 means strong negative correlation
• Value near 0 means little or no correlation
• Linear regression analysis quantifies
  relationship between dependent variable Y and
  independent variable X, Y a bX

10
Using Excel to Determine Correlation Coefficient

• Use CORREL function
• CORREL(X-range,Y-range)
• OR
• Use Tools Data Analysis Regression
• Provides table of correlations and other
  parameters

11
Simulating Total Dependence

• When correlation coefficient r near 1, we assign
  same random number for X and Y in simulation
• When correlation coefficient near -1, use
  original random number, x, to generate value for
  independent variable and 1.0 x to generate
  value for dependent variable
• Sample independent variable, X, first
• Value of X influences value of dependent variable
  Y by restricting its range
• Alternative Use relationship Y a bX

12
Simulating Diffuse Dependence

• Data in example figure shows downward sloping
  trend with r -0.5344
• Similar situations arise frequently in practice

13
Simulating Diffuse Dependence Methodology

• Prepare cross plot of random variables X and Y
• Draw box around data points so majority is
  bounded, maximum and minimum limits defined
• Identify type of variation of Y within box as
  function of X random, clustered midway,
  clustered at upper or lower boundary

14
Simulating Diffuse Dependence Methodology

• Generate normalized Y distribution
• For each X, unique distribution of Y between
  Ymin and Ymax exists, and is conveniently
  represented in terms of normalized Y variable
• For given iteration, value of X selected
  randomly, and value of Y from normalized
  distribution corresponding to value of X selected

15
Simulating Diffuse Dependence Methodology

• Develop cumulative probability distribution for
  independent variable X and normalized Y from
  previous step
• Generate random numbers and sample distributions
• Generate two random numbers, RN1 and RN2
• Use RN1 to sample X distribution
• Use RN2 to sample YNORM distribution
• Values of X1 and YNORM1 result

16
Simulating Diffuse Dependence Methodology

• Obtain Ymin and Ymax corresponding to X1 from
  figure or, better, from fitting equations
• Calculate Y1 from YNORM1, Ymax, and Ymin
• Repeat for each iteration at fixed value of X1
• Select X2, randomly and repeat entire process
• Process illustrated in Example 6-1, Mian, pp. 342
  - 347

17
Simulation Using Excel

• Generating random numbers
• Use formula RAND() in any cell
• Properties
• Whenever function is used, numbers between 0 and
  1 have same chance of occurring numbers will be
  uniformly distributed
• Numbers probabilistically independent when one
  random number generated, we obtain no information
  about subsequent random numbers

18
Simulation Using Excel

• Inverse of probability distributions in Excel
  (built-in functions)
• BETAINV() returns inverse of cumulative beta
  function distribution
• CHIINV() returns inverse of one-tailed
  probability of chi-squared distribution
• FINV() returns inverse of F probability
  distribution
• GAMMAINV() returns inverse of gamma
  probability distribution

19
Simulation Using Excel

• Inverse of probability distributions in Excel
  (built-in functions)
• LOGINV() returns inverse of lognormal
  probability distribution
• NORMINV() returns inverse of normal cumulative
  probability distribution
• NORMSINV() returns inverse of standard normal
  cumulative probability distribution
• TINV() returns inverse of students
  t-distribution

20
Simulation Using Excel

• Excels built-in inverse probability distribution
  functions all have probability in argument
• Example NORMINV(probability,µ,s)
• We can replace probability with random number in
  simulation
• Example NORMINV(RAND(),µ,s) generates random
  variate of normal distribution

21
Simulation Using Excel

• Excel has other built-in functions that generate
  pdf and cdf but not inverse
• BINOMDIST() binomial distribution
• EXPONDIST() exponential distribution
• HYPEGEOMDIST() hypergeometric
• POISSON() Poisson distribution
• WEIBULL() Weibull distribution

22
Simulation Using Excel

• We can determine inverses of CDFs of Excels
  functions with no built-in inverse by using
  VLOOKUP function
• Table 6-6, page 350 of Mian illustrates use of
  VLOOKUP function

23
Using VLOOKUP Function
24
Example Simulation with Excel Ex. 6-2, Mian

• Objective calculate volumetric oil reserves
• Porosity normally distributed, mean 0.14,
  standard deviation 0.02
• Water saturation triangular, min, most likely,
  max values 0.2, 0.3, 0.44
• Formation thickness normally distributed, mean
  15 ft, std. dev. 1.5 ft

25
Example Simulation with Excel Ex. 6-2, Mian

• Objective calculate volumetric oil reserves
• Drainage area normally distributed, mean 77
  acres, std. dev. 63 acres (careful can cause
  negative areas in simulation)
• Recovery factor normally distributed, mean
  0.34, std. dev. 0.05
• Oil FVF uniform distribution, parameters 1.15
  and 1.5

26
Example Simulation with Excel Ex. 6-2, Mian
27
Steps in Setting Up Spreadsheet for Ex. 6-2

• Enter inputs for probability distributions in
  cells E4 to G9
• Cell B13 NORMIN(RAND(),E4,F4)
• Cell C13 IF(J13lt((F5-E5)/(G5-E5)),E5SQRT((
  F5-E5)J13),G5-SQRT((G5-F5)(G5-E5)(1-J13)
  ))
• Formula refers to Eqs. 6.1,6.2. Cell refers to
  random numbers in cell J13. Enter RAND() in cell
  J13 and copy down to cell J550.

28
Steps in Setting Up Spreadsheet for Ex. 6-2

• Cell D13 NORMINV(RAND(),E6,F6)
• Cell E13 NORMINV(RAND(),E7,F7)
• Cell F13 NORMINV(RAND(),E8,F8)
• Cell G13 RAND()(F9-E9)E9 (Eq. 6.3)
• Cell H13 ((7758B13(1-C13)D13E13)/G13)F13
  volumetric reserve equation

29
Steps in Setting Up Spreadsheet for Ex. 6-2

• Copy cells B13H13 to cells B550 to H550,
  providing 538 iterations for simulation
• Cell D10 AVERAGE(H13H550) calculates average
  reserve for 538 iterations

30
Using _at_RISK

• _at_RISK is software to analyze business and
  technical sitautions with risk exposure
• Functions as add-in to MS Excel
• Uses Monte Carlo simulation for risk analysis
• Application illustrated with reserves simulation
  model

31
Application of _at_RISK to Reserves Simulation

• Porosity normal distribution f(14,2)
• Sw triangular distribution Sw(20,30,44)
• h normal distribution h(15,1.5)
• Ad lognormal distribution Ad(77,63)
• FR normal distribution FR(34,5)
• Bo uniform distribution Bo(1.15,1.5)

32
Examples Using _at_RISK from Mian

• See Mian, pages 355-366 for details on using
  _at_RISK for this reserves simulation example
• See Mian, Example 6-3, pages 366-369 for details
  on using _at_RISK for NPV example
• See Mian, pages 370-373, for details on modeling
  dependency in _at_RISK
• See Mian, pages 373-375, for information on
  combining _at_RISK and PRECISIONTREE

33
Accomplishments

• You are or will be able to
• Identify correlations between input variables in
  simulation
• Simulate correlations manually
• Simulate reserves estimates using Excel
• Simulate reserves estimates using _at_RISK

34
Simulation

• Application of Simulation to Probabilistic
  Reserve Estimates Part 2

35
Expected Value and Decision Trees

• Value of Additional Information

36
Learning Objectives

• You will be able to
• Calculate the value of perfect information
• Calculate the value of imperfect information

37
Value of Information

• New or additional information can reduce or
  remove uncertainty
• Reduced uncertainty should increase payoff and
  reduce variance
• Additional information costs money
• Examples
• Seismic survey
• Laboratory analysis
• Services of consultant
• Market survey before launching new project

38
Questions to be Answered Before Buying Additional
Information

• Is the additional information worth the cost?
• If several potential sources of information
  exist, which one if preferred?

39
Expected Value of Perfect Information

• Expected value of perfect information (EVPI) is
  expected payoff with perfect information (EPPI)
  minus expected payoff under uncertainty
• EVPI is amount we can spend on acquiring perfect
  information
• EVPI gives upper-bound for imperfect information,
  since perfect information is rarely available

40
Expected Value of Perfect Information

• Best payoff (from perfect information) found by
  first determining maximum payoff of each event,
  then multiplying each maximum by probability of
  event
• EVPI then calculated as difference between best
  payoff and most likely payoff
• Process illustrated by example

41
Example Expected Value of Perfect Information

• For decision problem discussed earlier (leasing
  60 acres to join drilling unit and determining
  whether to drill, farm out, or back in)
• Geologists believe additional seismic data will
  significantly reduce uncertainty can tell us
  dry hole or producer, but not size of reserve
• We want to determine maximum amount we can pay
  for additional seismic

42
Example Expected Value of Perfect Information
43
Example Expected Value of Perfect Information

• Choose maximum value in each row of given data to
  represent NPV of perfect information
• Since information perfect, dry hole risk has
  vanished

44
Example Expected Value of Perfect Information
45
Example Expected Value of Perfect Information

• Multiply NPV values assuming perfect information
  by probabilities to obtain at components of
  expected value
• Add components of expected value to determine
  expected payoff of perfect information EPPI
• Subtract EMV under uncertainty (which was 25.375
  M for back-in option) from EPPI to determine EVPI

46
Example Expected Value of Perfect Information
47
Example Expected Value of Perfect Information

• EVPI EPPI EMV
• 33.387 M 25.375 M
• 8.012 M
• We can afford to pay no more than
• 8.012 M for seismic

48
Expected Value of Imperfect Information

• Imperfect information changes degree and nature
  of uncertainty without eliminating it
• Example from seismic
• Perfect information would be 100 reliable
• Actual expectations might be 90 probability that
  seismic will indicate structure when structure is
  present, and 10 probability that seismic will
  indicate structure when structure is not present

49
Expected Value of Imperfect Information

• Expected value of imperfect information (EVII) is
  expected payoff with imperfect information minus
  expected payoff under uncertainty
• Expected net gain (ENG) is expected value of
  information (perfect or imperfect) less cost of
  obtaining information

50
Expected Value of Imperfect Information

• Bayesian methodology used to revise prior
  probabilities and determine new posterior
  probabilities, calculated using new information
  available through experiments or tests
• Substitute posterior probabilities in place of
  prior probabilities in outcome state
• Expected payoff thus calculated taking into
  account posterior probabilities in place of prior
  probabilities

51
Implementing Bayesian Analysis

• Determine course of action that would be chosen
  using only prior probabilities and record EMV of
  this course of action
• Identify possible insights new information can
  provide
• Assign probabilities to new information
  (conditional probabilities)

52
Implementing Bayesian Analysis

• Calculate joint probabilities (product of prior
  probabilities and conditional probabilities)
• Calculate marginal probabilities (sum of
  appropriate joint probabilities)
• Calculate posterior probabilities (joint
  probabilities divided by marginal probabilities)
• Replace initial (prior) probabilities by revised
  (posterior) probabilities and calculate revised
  (less uncertain) EMV of project

53
Example Value of Imperfect Information

• We have discovered oil in an offshore prospect
• Studies indicate reserves in 5-25 MM STB range,
  with probabilities in table
• Two options
• Design facilities based on information available
• Drill delineation wells to improve probability
  and reservoir size estimates

54
NPV of Each Field Size and Facility, MM
55
Questions to Answer

• Determine most economical field size without
  further information, using EMV
• Calculate expected value of perfect information
  using EMV and EOL. Based on EVPI, determine
  maximum amount we can pay to acquire additional
  information

56
Questions to Answer

• Calculate expected value of imperfect information
  if we decide to drill delineation wells costing
  15MM before we decide on size of facilities
• Geologists beliefs about delineation wells
• Probability 90 that we will identify large
  reservoir if thats what is actually there
• Probability 60 that we will identify medium
  reservoir if thats what is actually there
• Probability 30 that we will identify small
  reservoir if thats what is actually there

57
EMV Using Available Information
58
Expected Value of Perfect Information (EVPI)
EVPP 0.3x4500.45x2100.25x60 244.5MM
EVPI 244.5 - 219.5 25MM
59
Expected Opportunity Loss (EOL)
Confirms selection of size C facility EVPI
25MM same as calculated by EMV method
60
Expected Value of Imperfect Information (EVII)

• To calculate EVII, consider two alternatives
• Install platform without acquiring additional
  information (calculations above)
• Drill delineation wells and decide on platform
  size based on information they provide

61
Decision Tree for Option to Drill Delineation
Wells
62
Partial Tree for Result of Delineation Wells
63
Assessment of Probabilities of Different Field
Sizes from Partial Tree

• Note joint probabilities of favorable outcomes
  when delineation wells are drilled are
• 0.3x0.9 0.27 large field
• 0.45x0.6 0.27 medium field
• 0.25x0.3 0.075 small field
• Total probability of favorable outcome 0.27
  0.27 0.075 0.615 (and probability of
  unfavorable outcome is 1 0.615 0.385)

64
Rearrangement of Tree (Inversion)

• Posterior probabilities and EMVs shown in tables
  and on inverted tree
• Process demonstrates application of Bayes rule
  (see Mian, vol. 2, pp. 94-99)

65
Application of Bayes Rule

• P(Ai/B), posterior probabilities, represent
  probabilities
• that reservoirs will be small, medium, or large
  (Ai),
• given results of delineation drilling (B)
• P(B/Ai) represent probabilities that delineation
• drilling result (B) will be favorable or
  unfavorable,
• given probabilities (Ai) that reservoirs are
  small, medium
• or large
• P(Ai), prior probabilities, represent original
  probabilities
• that reservoirs will be small, medium, or large

66
Calculation of NPV, Delineation Wells Favorable
Select size C if delineation well results
favorable
67
Calculation of NPV, Delineation Wells Unfavorable
Select size B if delineation results unfavorable
68
Value of Imperfect Information

• Table indicates we should select size C facility
  if delineation well results favorable (EMV
  273.9MM)
• Table indicates we should select size B facility
  if delineation well results unfavorable (EMV
  141.36MM)

69
Inverted Tree for Result of Delineation Wells
70
Delineation Well Decision

• Expected payoff with imperfect information, EPII,
  if we drill delineation wells
• EPII 0.615x273.90 0.385x141.36 222.87MM
• Expected value of imperfect information, EVII
  222.87 - 219.50 3.37MM
• We should pay no more than 3.37MM to drill
  delineation wells, which means we cannot support
  the proposed 15MM drilling budget
• Since EVPI is 25MM, value of information from
  delineation wells, 3.37MM, considerably less
  than value of perfectly reliable results

71
Learning Objectives

• You can now
• Calculate the value of perfect information
• Calculate the value of imperfect information

72
Expected Value and Decision Trees

• Value of Additional Information