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PPT – PETROLEUM ENGINEERING 689 Special Topics in Unconventional Resource Reserves Lecture 6 Probabilistic PowerPoint presentation | free to view - id: 121483-YzJiO

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PETROLEUM ENGINEERING 689Special Topics

inUnconventional Resource ReservesLecture

6Probabilistic Reserves SimulationTexas AM

University - Spring 2007

Probabilistic Reserve Estimates

- Part 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

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

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

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

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

Monte Carlo Simulation Application

- Calculate cdfs for each independent variable

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)

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)

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)

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

Monte Carlo Simulation Application

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

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)

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)

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

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

Correlations Example Calculation

Correlations Example Calculation

- From figure, we can determine P10, P50, P90 and

calculate skew

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

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?

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

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

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

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

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

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

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

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

Expected Value of Reserves

Expected Value of Reserves

- From figure

Expected Value of Reserves

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

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

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

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

SPEE Risk Adjustment Factors

SPEE Risk Adjustment Factors

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

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

Probabilistic Reserve Estimates

- Part 3

Simulation

- Application of Simulation to Probabilistic

Reserve Estimates Part 1

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

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

Steps in Simulation Modeling Reserves

Steps in Simulation Modeling NPV Reserves

Steps in Simulation Modeling

- Access input variables
- Perform preliminary sensitivity analysis of input

variables - Assess probability distributions of input

variables

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

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

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)

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

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)

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

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

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

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

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

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

Example random number table

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

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

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

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)

Manual Monte Carlo Sampling Graphical

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)

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

Manual Monte Carlo Sampling Analytical

- Sampling from triangular distributions,

T(XL,XM,XH)

Manual Monte Carlo Sampling Analytical

- Sampling from uniform distribution U(xmin,xmax)

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

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

Latin Hypercube Sampling Need

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

Latin Hypercube Sampling Methodology

MCS, LHS Compared

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

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

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

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

Example Simulation MC Sampling

Example Simulation LHC Sampling

Example Simulation LHS Results

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

Simulation

- Application of Simulation to Probabilistic

Reserve Estimates Part 1