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PETROLEUM ENGINEERING 689 Special Topics in Unconventional Resource Reserves Lecture 6 Probabilistic

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Explain typical effects of correlations between variables on MCS results ... SPE 23586, 'Reserves and Probabilities Synergism or Anachronism,' by Cronquist ... – PowerPoint PPT presentation

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Title: PETROLEUM ENGINEERING 689 Special Topics in Unconventional Resource Reserves Lecture 6 Probabilistic


1
PETROLEUM ENGINEERING 689Special Topics
inUnconventional Resource ReservesLecture
6Probabilistic Reserves SimulationTexas AM
University - Spring 2007
2
Probabilistic Reserve Estimates
  • Part 3

3
Learning Objectives
  • You will be able to
  • Explain how Monte Carlo Simulation is applied in
    volumetric reserve estimates
  • Explain typical effects of correlations between
    variables on MCS results
  • Summarize strengths and weaknesses of
    probability tree, parametric, and MCS methods

4
Learning Objectives
  • You will be able to
  • Calculate expected value of reserves, given cdf
    or low, high, and likely values
  • Explain general trends in SPEE risk adjustment
    factors

5
Monte Carlo Simulation
  • Procedure in brief
  • Determine relation between set of independent
    random variables and one or more dependent random
    variables
  • Sample randomly pdfs of independent variables
  • Calculate repetitively to generate pdfs of
    dependent variables

6
Monte Carlo Simulation Application
  • Procedure applied to School Prospect
  • Establish algorithm to calculate dependent
    variable
  • ULTGAS NETPAY x AREA x RECFAC
  • Establish correlations between independent
    variables
  • Assume no correlations in this example

7
Monte Carlo Simulation Application
  • Establish pdfs for each independent variable
  • Assume triangular distributions, using minimum,
    most likely, maximum values
  • Net pay distribution truncated at 20 ft

8
Monte Carlo Simulation Application
  • Calculate cdfs for each independent variable

9
Monte Carlo Simulation Application
  • Generate random number between 0.0 and 1.0 and
    determine value for NETPAY that corresponds to
    this cumulative probability
  • If random number were 0.65, NETPAY would be 136
    ft (see previous figure)

10
Monte Carlo Simulation Application
  • Generate new random number between 0.0 and 1.0
    and determine value for AREA that corresponds to
    this cumulative probability
  • If random number were 0.15, AREA would be 728
    acres (see previous figure)

11
Monte Carlo Simulation Application
  • Generate new random number between 0.0 and 1.0
    and determine value for RECFAC that corresponds
    to this cumulative probability
  • If random number were 0.95, RECFAC would be 1495
    Mscf/AF (see previous figure)

12
Monte Carlo Simulation Application
  • Compute value for ULTGAS
  • ULTGAS NETPAY x AREA x RECFAC
  • (136)x(728)x(1495)
  • 148 Bscf ULTGAS(1)
  • Repeat and compute ULTGAS(2), ULTGAS(3),
    ULTGAS(4), , ULTGAS(n) until enough values are
    available to define pdf of ULTGAS (hundreds or
    thousands of iterations)
  • Latin hypercube sampling more efficient that
    Monte Carlo sampling, when available

13
Monte Carlo Simulation Application
14
Monte Carlo Simulation Application
  • MCS expectation curve slightly less skewed than
    Parametric method results
  • MCS sampled triangular distributions, parametric
    log normal distributions, so we might reasonably
    expect these results

15
Dependencies in Simulation
  • Broad types of dependencies
  • Correlations between input variables used to
    calculate pdfs of desired results (e.g., OIP or
    reserves for specific accumulation)
  • Correlations between calculations of desired
    results (e.g., OIP or reserves for accumulations
    that are aggregated)

16
Spearman Rank Correlations
  • We can use Spearman rank correlations (SRC) to
    detect dependencies between pairs of input
    parameters
  • SRC tests for correlation between data sets based
    on relative rankings of elements in data sets
    rather than on values of elements
  • SRC ranges from -1.0 (perfect negative
    correlation) to 1.0 (perfect positive
    correlation)

17
Spearman Rank Correlations
  • SRCs can be calculated using standard
    statistical software
  • Add-on modules for EXCEL, such as _at_RiskTM and
    Crystal BallTM include capability for SRC
    calculation

18
Correlations Example Calculation
  • Cronquist examined School Prospect using MCS
    and assuming SRC of 0.5 between AREA and NETPAY
    and between RECFAC and NETPAY
  • Software automatically adjusted input parameters
    so that output parameters were sampled at rank
    comparable to that at which input parameters were
    sampled

19
Correlations Example Calculation
20
Correlations Example Calculation
  • From figure, we can determine P10, P50, P90 and
    calculate skew

21
Correlations Example Calculation
  • General observations from examples
  • Consideration of dependencies tends to increase
    spread between P10 and P90 values
  • Larger values of SRC tend to increase spread
  • P50 may be affected less than spread (as in
    specific example) by large SRC

22
Comparison of Procedures
  • Review three procedures illustrated to estimate
    reserves probabilistically
  • Probability tree
  • Parametric method
  • Simulation
  • Question what are strengths and weaknesses of
    each procedure?

23
Critique of Probability Tree
  • Advantages
  • Simple, easy to understand and to implement on
    spreadsheet
  • Provides reasonable approximation to pdf
  • Disadvantages
  • Discrete input variables cause poorly defined
    pdfs of target variables
  • Difficult to handle correlations between
    variables
  • Complexity cascades if analysis involves more
    than minimum, most likely, and maximum values

24
Critique of Parametric Method
  • Advantages
  • Easily programmed for spreadsheet
  • Provides approximate statistical distribution of
    target function
  • Disadvantages
  • Need to understand statistical basis
  • Difficulties in handling correlations between
    variables
  • Inability to input truncated distributions
  • Assumes log normal output pdfs

25
Critique of Simulation
  • Advantages
  • Most widely used procedure for probabilistic
    reserve estimates
  • Handles variety of pdfs and potential
    correlation between variables
  • Commercial software readily available
  • Widely accepted, promotes communication between
    users

26
Critique of Simulation
  • Disadvantages
  • Requires sufficient input data to characterize
    pdfs of input variables
  • Data analysis and treatment of input pdfs tend
    to be subjective
  • Requires considerable expertise to handle
    somewhat complex software
  • Appears to be black box to some observers
  • Results may not be as precise as advocates claim

27
Triangular Frequency Distributions
  • Triangular distributions (TD) often used when
    information about parameters limited
  • TDs poor representations of strongly skewed
    distributions give more weight to skewed tail
    than appropriate
  • P10, P90 calculations may be unrealistic

28
Alternative Frequency Distributions
  • Beta distributions frequent substitutes for
    triangular distributions
  • May be scaled to fit any min-max range
  • May be either symmetric or skewed (either
    positively or negatively)
  • Users should be cautious about using unbounded
    distributions to fit parameters in bounded range
  • Users should be aware that most geoscientists and
    engineers underestimate degree of uncertainty and
    skew in stochastic variables

29
Comparison and Integration of Probabilistic and
Deterministic Reserve Estimates
  • Two references may be especially helpful
  • SPE 23586, Reserves and Probabilities
    Synergism or Anachronism, by Cronquist
  • SPE 38803, An Integrated Deterministic/Probabilis
    tic Approach to Reserve Estimation An Update,
    by Nangea and Hunt

30
Expected Value of Reserves
  • Expected value, or mean, can be calculated as sum
    of probability-weighted values in set
  • In Monte Carlo simulation, mean of output pdf (of
    reserves, for example) routinely calculated by
    software
  • Note that expected value of reserves is not the
    same as proved reserves closer to proved plus
    probable (P50) reserve value

31
Expected Value of Reserves
32
Expected Value of Reserves
  • From figure

33
Expected Value of Reserves
34
Expected Value of Reserves
  • From calculation, ER 41
  • From graph, P50 39 ER close approximation
  • Conclusion Select either P50 (if cdf known) or
    value from calculations (if P1, P2, P3, Prs
    known) to determine Expected Reserve, ER
  • P50 more accurate

35
Risked Reserves
  • Historical definition deterministic reserve
    estimate multiplied by probability that estimated
    reserve will actually be recovered
  • Probabilistic calculations may require risk
    adjustment rather than risk weighting

36
SPEE Risk Adjustment Factors
  • SPEE conducts annual surveys to determine
    opinions of practicing reserve analysts on the
    probability of realizing estimates of present
    value of future net revenue of reserves
    attributed to various
  • Classifications
  • Development status
  • Producing categories

37
SPEE Risk Adjustment Factors
  • SPEEs risk adjustment factors (RAFs) implicitly
    include risks attributable to
  • Actual expenditures to drill or recomplete wells
  • Operations associated with such activities
  • Realization of estimated operating expenses,
    taxes and wellhead prices
  • Potential drainage of behind-pipe reserves
  • Geologic and engineering uncertainties in reserve
    estimates

38
SPEE Risk Adjustment Factors
39
SPEE Risk Adjustment Factors
40
Learning Objectives
  • You can now
  • Explain how Monte Carlo Simulation is applied in
    volumetric reserve estimates
  • Explain typical effects of correlations between
    variables on MCS results
  • Summarize strengths and weaknesses of
    probability tree, parametric, and MCS methods

41
Learning Objectives
  • You can now
  • Calculate expected value of reserves, given cdf
    or low, high, and likely values
  • Explain general trends in SPEE risk adjustment
    factors

42
Probabilistic Reserve Estimates
  • Part 3

43
Simulation
  • Application of Simulation to Probabilistic
    Reserve Estimates Part 1

44
Learning Objectives
  • You will be able to
  • Explain the steps in simulation modeling
  • Explain how to implement Monte Carlo sampling
    manually, given graphical CDF
  • Explain how to implement Monte Carlo sampling
    manually, given analytical inverse CDF
  • Explain the advantages of Latin Hypercube
    sampling compared to Monte Carlo sampling

45
Steps in Simulation Modeling
  • Define the problem
  • Identify possible actions, uncertainties, and
    time horizons
  • Construct the model
  • Relate input variables to output variable
  • Be sure to include all the variables that affect
    the result but not variables that dont matter

46
Steps in Simulation Modeling Reserves
47
Steps in Simulation Modeling NPV Reserves
48
Steps in Simulation Modeling
  • Access input variables
  • Perform preliminary sensitivity analysis of input
    variables
  • Assess probability distributions of input
    variables

49
Steps in Simulation Modeling Sensitivity
Analysis
  • Purpose is to identify critical variables
    (requiring probability distributions) and
    variables not requiring probability distributions
  • Basic idea is to calculate output variable by
    varying value of each input variable
  • TopRankTM, in our software set, can perform
    sensitivity analysis for us

50
Steps in Sensitivity Analysis
  • Assume smallest, largest, and most likely values
    for each variable
  • Calculate output variable with first variable at
    its smallest value and remaining variables at
    most likely values
  • Repeat, but set first variable at its largest
    value with remaining variables still at their
    most likely values
  • Repeat process for all remaining variables

51
Steps in Simulation Modeling Evaluate
Probability Distributions
  • Probability distributions must describe range of
    likely values for each parameter of interest
  • Distributions may be standard forms (e.g., normal
    or lognormal) or intuitive (e.g., triangular,
    rectangular)

52
Steps in Simulation Modeling Evaluate
Probability Distributions
  • Methods to determine probability distribution
  • Fit theoretical distribution to historical data
    from reservoir or analogous reservoir
  • From experience of analyst
  • Heuristic approach use rule of thumb believed
    to be appropriate

53
Steps in Simulation Modeling Evaluate
Probability Distributions
  • Parameters of distribution may be chosen on basis
    of studies of geological properties in area or
    analogous area
  • In absence of data, normal or lognormal
    distributions likely appropriate for
    distributions of geological properties
  • Some properties must be bounded (to avoid
    porosities and saturations outside range 0 to
    100, for example)

54
Steps in Simulation Modeling Perform
Calculations
  • Monte Carlo simulation involves repeated sampling
    from input distributions and subsequent
    calculation of set of sample values for output
    distribution
  • Process repeated over many iterations

55
Steps in Simulation Modeling Interpret Output
  • After simulation completed, analyst should
    analyze results critically and question
    assumptions about model structure and input
    variables
  • As example, analyst may want to assess
    sensitivity of output distribution to changes in
    any input distributions or to structure of model

56
Steps in Simulation Modeling Output
Sensitivity Analysis
  • In sensitivity analysis, rerun model with
    different assumptions and assessments
  • Compare outputs to those from base case
  • If changes have minor effects on results, assume
    original model adequate

57
Steps in Simulation Modeling Analyze Simulated
Alternatives
  • Simulate all reasonable alternatives
  • Generate individual distributions and statistical
    information
  • Output probability distributions then used to
    provide insights into alternatives and to guide
    choice of optimal alternative

58
Random Sampling Methods
  • To simulate process or system with random
    components, we must generate set of random
    numbers
  • Random numbers then transformed to obtain random
    variables from other distributions
  • Transformed random numbers called random variates
  • Random numbers and probability distributions form
    building blocks of simulation

59
Generating Random Numbers
  • Spinning disks, electronic randomizers, computers
    used to generate random numbers
  • Tables generated this way
  • Most books on statistics contain table of random
    numbers
  • Example is Table 6-1 in Mian, vol II, p. 323
  • Most spreadsheet and simulation software have
    built-in capability to generate random numbers,
    avoiding need for tables

60
Example random number table
61
Characteristics of Random Numbers
  • Each successive random number in sequence must
    have equal probability of taking on any one of
    the possible values
  • Each number must be statistically independent of
    other numbers in sequence
  • Numbers need to be random observations from
    uniform distribution between 0 and 1, referred to
    as U(0,1) random numbers

62
Random Numbers Between Fixed Limits
  • Uniformly distributed random numbers between
    fixed limits A and B can be generated using
    formula
  • where x is uniformly distributed between 0
    and 1

63
Monte Carlo Sampling
  • After generating random number, we then generate
    random observations from probability
    distributions assigned to each input variable in
    model
  • Computers transform random numbers into random
    variates using complex algorithms
  • We illustrate method with manual calculations

64
Manual Monte Carlo Sampling Graphical
  • We need cumulative distribution function (CDF) of
    distribution and sampling procedure for uniform
    distribution on (0,1)
  • Following figure illustrates transformation of
    random numbers 0.92353, 0.67381, and 0.27832 into
    random variates (oil price in this case)

65
Manual Monte Carlo Sampling Graphical
66
Manual Monte Carlo Sampling Analytical
  • Graphical method cumbersome for large number of
    iterations
  • Alternative Analytical method requires us to
    find equation for CDF in form
    RN f(X)
  • and solving for X as function of RN (i.e.,
    finding inverse function of CDF)

67
Manual Monte Carlo Sampling Analytical
  • Method involves integrating PDF to obtain CDF and
    then finding inverse of CDF
  • Approach practical for some simple distributions
    (e.g., uniform, triangular, exponential) but not
    for complex distributions (e.g., normal,
    lognormal)
  • Complex functions have no simple expression for
    CDF

68
Manual Monte Carlo Sampling Analytical
  • Sampling from triangular distributions,
    T(XL,XM,XH)

69
Manual Monte Carlo Sampling Analytical
  • Sampling from uniform distribution U(xmin,xmax)

70
Manual Monte Carlo Sampling Analytical
  • Sampling from exponential distribution E(x,?)
  • PDF and CDF given by
  • where
  • 1/? mean of distribution
  • Inverse function of exponential CDF

71
Latin Hypercube Sampling
  • In Monte Carlo sampling (also known as full
    distribution sampling), each random variable
    remains element of distribution leaving entire
    statistical range available for sampling in
    subsequent iterations
  • Often results in clustering of sampling in some
    parts of distribution, leaving other parts
    unsampled
  • Figure illustrates problem

72
Latin Hypercube Sampling Need
73
Latin Hypercube Sampling Methodology
  • In Latin Hypercube Sampling (LHS), cumulative
    distribution function partitioned into
    non-overlapping intervals of equal probability,
    in line with number of required iterations
  • Example if we choose 10 iterations, distribution
    might be divided into 10 parts
  • Random samples then picked from each interval

74
Latin Hypercube Sampling Methodology
75
MCS, LHS Compared
76
MCS, LHS Compared
  • LHS guarantees all probabilities represented as
    intended
  • Gives equal weight to all probabilities on CDF
  • LHS can reduce necessary number of iterations by
    30
  • Necessary means more iterations would not change
    result appreciably

77
Example Simulation Manual
  • Even though we use computers to simulate, manual
    calculation illustrates process
  • Simulate simple case with three variables
  • Oil reserves, MSTB
  • Oil price, /STB (net of OPEX)
  • Tax rate of 40

78
Example Simulation Manual
  • Oil reserves represented by uniform distribution
    of parameters U(100,250)
  • Minimum 100 MSTB
  • Maximum 250 MSTB
  • Tax rate, 40, constant
  • Oil price represented by triangular distribution
    with parameters T(10,18,26)
  • Minimum 10/STB
  • Most likely 18/STB
  • Maximum 26/STB

79
Example Simulation Manual
  • Expected value of NCF is product of expected
    values of each parameter from its distribution
  • Expected value of uniform distribution
  • Expected value of triangular distribution
  • Expected value of net cash flow

80
Example Simulation MC Sampling
81
Example Simulation LHC Sampling
82
Example Simulation LHS Results
83
Learning Objectives
  • You can now
  • Explain the steps in simulation modeling
  • Explain how to implement Monte Carlo sampling
    manually, given graphical CDF
  • Explain how to implement Monte Carlo sampling
    manually, given analytical inverse CDF
  • Explain the advantages of Latin Hypercube
    sampling compared to Monte Carlo sampling

84
Simulation
  • Application of Simulation to Probabilistic
    Reserve Estimates Part 1
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