# PETROLEUM ENGINEERING 689 Special Topics in Unconventional Resource Reserves Lecture 6 Probabilistic - PowerPoint PPT Presentation

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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
• 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
• Simple, easy to understand and to implement on
• Provides reasonable approximation to pdf
• 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
• Provides approximate statistical distribution of
target function
• 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
• Most widely used procedure for probabilistic
reserve estimates
• Handles variety of pdfs and potential
correlation between variables
• Widely accepted, promotes communication between
users

26
Critique of Simulation
• 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
• 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

36
• 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
• 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,
• Potential drainage of behind-pipe reserves
• Geologic and engineering uncertainties in reserve
estimates

38
39
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

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