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1
PETE 4051 Reserve Evaluation and Reservoir Management
Dr. R. G. Hughes
Financial AccountingProbabilistic Reserves Assessment
L O U I S I A N A S T A T E U N I V E R S I T YA N D A G R I C U L T U R A L A N D M E C H A N I C A L C O L L E G E
Craft and Hawkins Department of Petroleum Engineering
Professional Ethicsand
Reserves Reporting
2
Financial Accounting(Grossly Oversimplified)
Net Income = Equivalent Barrels* × Income per Barrel
* Gas is usually converted to equivalent barrelsat a ratio of 6 MCF / STBO.
Income per barrel = Sales price per barrel
How is Income per Barrel Determined?
p p p– operating cost per barrel
– investment (or capital) cost per barrel
K DD&A t l l t d bKnown as DD&A rate – calculated byinvestment ÷ proved reserves
Depreciation, depletion & amortization
3
The Potential Conflict of Interest when Reporting Reserves
Increasing proved reserves increases reported net incomeincomeReserve estimation is an interpretive process leading to almost qualitative valuesStock analysts rely on reported income for buy/sell recommendationsA trend toward awarding stock options and g p“performance based” compensation results in reserve estimates affecting the bottom line on managements paycheck!
Proved Reserves“Reasonable certainty”
♦ 90% probability that amount recovered will equal or exceed the estimated value
Economic under existing conditionsEconomic under existing conditions♦ Economic conditions♦ Operating conditions♦ Government regulations
Supported by actual tests or productionFluid contacts are base upon the lowest known unless engineering data indicates otherwiseConforms to existing well spacing and regulationsImproved recovery only considered if– successful pilot– response to an installed program– proven in the same or an analogous reservoir
4
“Figures don’t lie,
but liarsSource: unknown
but liars figure…..”
ProbabilityProbability is a number between 0 and
Won’t Happen
Will Happen
1 which expresses the likelihood of an event.
0 0.25 0.50 0.75 1
After Capen, Rose & Clapp
5
Objective and Subjective Probabilities Examples
Objective“Th i 50% h f h d i t ”– “There is a 50% chance of heads on a coin toss”
– “There is a chance of 1/36 of throwing a pair of sixes with two dice”
Subjective “Th i 2% h h S i ill i h S– “There is a 2% chance the Saints will win the Super Bowl”
– “There is a 25% chance that this prospect contains hydrocarbons”
ProbabilitiesIn our business, chances or probabilities are largely subjective, reflecting an individual’s betting odds that a given event will occur.betting odds that a given event will occur.
Common uses of statistics in the oil and gas business– Competitive bidding (lease sales, acquisitions)– Prospect economics
After Capen, Rose & Clapp
p– Ranking of investment opportunities– Reserve estimates– Failure/reliability analysis– Geostatistics
6
Addition RuleIf two events are mutually exclusive, the
probability of one or the other occurring isprobability of one or the other occurring is the sum of their individual probabilities.
Example:The probability of a two or a five on the
After Capen, Rose & Clapp
p yroll of a die = 1/6 + 1/6 = 1/3
Multiplication RuleIf two events are independent, the probability of both occurring is the product of their individualboth occurring is the product of their individual probabilities.
Example: The probability of 2 fives on the roll of two dice =
After Capen, Rose & Clapp
(1/6) (1/6) = 1/36
7
Probability DistributionsA Probability Distribution is a collection of
Experiment Possible Outcomes Probability(number of dots)
events and their associated probabilities covering every possible outcome of an
experiment or course of action:
Roll of 1 1/6single die. 2 1/6
3 1/64 1/65 1/66 1/6
After Capen, Rose & Clapp
Probability DistributionsPossible Outcome Probability Cumulative
ProbabilityR ll f T di 2 1/36 1/36
3 2/36 3/364 3/36 6/365 4/36 10/366 5/36 15/367 6/36 21/36
Roll of Two dice
8 5/36 26/369 4/36 30/36
10 3/36 33/3611 2/36 35/3612 1/36 36/36
8
Probability DistributionsPossible Outcome Probability Cumulative
ProbabilityR ll f T di 2 1/36 1/36
3 2/36 3/364 3/36 6/365 4/36 10/366 5/36 15/367 6/36 21/36
Roll of Two dice
8 5/36 26/369 4/36 30/36
10 3/36 33/3611 2/36 35/3612 1/36 36/36Histogram or PDF
Probability DistributionsPossible Outcome Probability Cumulative
ProbabilityR ll f T di 2 1/36 1/36
3 2/36 3/364 3/36 6/365 4/36 10/366 5/36 15/367 6/36 21/36
Roll of Two dice
8 5/36 26/369 4/36 30/36
10 3/36 33/3611 2/36 35/3612 1/36 36/36CDF
9
Possible Outcomes Probability(Reserves Found)
Prospect Reserve Distribution
(Reserves Found)0-10 MMB 0.1510-20 MMB 0.4020-30 MMB 0.2530-40 MMB 0.1540-50 MMB 0.05
lity
ty
After Capen, Rose & Clapp0-10 10-20 20-30 30-40 40-50
Prob
abil
Den
sit
Reserves Found Probability Reserves Found Cummulative(MMB) (MMB) Probability
0-10 0.15 10 0.15
Prospect Cumulative Probability Distribution
0 0 0. 5 0 0. 510-20 0.40 20 0.5520-30 0.25 30 0.8030-40 0.15 40 0.9540-50 0.05 50 1.00
1.0
After Capen, Rose & Clapp0 10 20 30 40 50
10
“Statistics don’t lie,
but statisticiansbut statisticians do…..”
Source: unknown
Expected Value or Mean The Expected Value (or Mean) of a distribution i th i ht d f ll ibl t
Outcome Probability Product1 x 0.167 0.1672 x 0.167 0.3343 x 0.167 0.5014 x 0.167 0.668
is the weighted average of all possible outcomes, each weighted by its probability:
4 x 0.167 0.6685 x 0.167 0.8356 x 0.167 1.002
3.5
After Capen, Rose & Clapp
The mean represents the “long-run average” resulting from repeated trials.
11
Statistical Parameters :
Mean = μ ≈ ∑ y / n (average)Mean = μ ≈ ∑ y / n (average)
Variance = σ 2 = ∑ ( y - μ ) 2 (n – 1)
Units are distance squared!
Standard Deviation= σ 2 = σ = ∑ (y-μ) 2√ √ n - 1
How good is this guess?
Could sum up all of the distances to seehow we did. But some distances arepositive and some are negative.
Distance frommean = yi - μ
Instead we sum the square of thedistances. This is called the “variance”.σ 2 = ∑ (yi - μ) 2 n
12
Other Statistical Parameters :
Mode – if a probability distribution p yis randomly sampled one time, the value with the most likely probability of being sampled is k th dknown as the mode.
Suppose we had a very special die that hadinstead of the normal 1-6 dots on the six sides,
it had 2 sides with 2 dots, and 3-6 dotson the remaining four sideson the remaining four sides.
The mean value that could be expected fromThe mean value that could be expected fromrolling the die a large number of times is 3.667.
But if the die is rolled only once, the mostlikely outcome is to role the mode, or 2.
13
Other Statistical Parameters :
• Median – the value that corresponds to a cumulative probability of 50%, also referred to as the “P50”.
• There is an equal likelihood of and individual sample being either above or below the median.
Probability DistributionsMany different theoretical distributions
Normal-familiar bell-shaped curve
Lognormal Logarithm of variable follows
After Capen, Rose & Clapp
Lognormal - Logarithm of variable follows Normal distribution
14
Graphical Representation of “Roll of a Die” Probability
DistributionPr
obab
ility
1/6
1 2 3 4 5 6
After Capen, Rose & Clapp
Some Types of Probability Distributions
Uniform Normal
Triangular Lognormal
15
The Normal Distribution
Mathematical equation knownas the probability density.
f N (x) =e - ( x - μ ) / 2 σ2 2
σ √ 2 π
μ (mean) = 0
σ 2 (variance) = 1
σ 2 (variance) = 4
Note:mean =mode =median !median !
What Does Cumulative Probability Look LikeFor the Normal Distribution?
μ (mean) = 0
σ 2 (variance) = 1
σ 2 (variance) = 4( )
16
The Lognormal DistributionMathematical equation knownas the probability density.
f ln (y) =e - ( ln y - μ ) / 2 σ2 2
y σ √ 2 π
The Lognormal Distribution
P90
Can estimate the mean using the approximation Swanson’s Rule
μ ≅ 0. 3 × (10% point)
+ 0. 4 × (50% point)
+ 0 3 × (90% point)
P50 = median
+ 0. 3 × (90% point)
μ ≅ (0. 3 × 28) + ( 0. 4 × 100)
+ (0. 3 × 355) = 106.5P10
17
P50 = median
mode = 38P10
mean
P90
Graphing Cumulative Probability
“Special” graph paper is availablefor normal and lognormal probabilitydistributions that yields a straight line
18
19
20
Statistically,this duck is
dead!
But that doesn’t
putmeat on
the table.
Why Bother with Lognormal?Variables which are the product of several variables...
R A S ti thi k N t tReserves = Area × Section thickness × Net to Gross Ratio × Porosity × Oil Saturation ×Percent Recoverable…tend to follow the Lognormal.
After Capen, Rose & Clapp
Normal Normal×
Lognormal
=
21
Many Things in Nature Followa Lognormal Distribution
Central Limit TheoremSum of any independent random variables = Normal Distribution
1,000Samples
22
Central Limit Theorem (continued)Sum of any independent random variables = Normal DistributionProduct of any independent random variables = Lognormal
1,000Samples
Plotting Data on Lognormal Graph Paper
When number of data points is small, plot each point instead of grouping.
Sort data in ascending order and number from 1 to n.
For each point calculate the cumulative probability.Probability = Point number / (n + 1)
Can estimate the mean using the approximation
After Capen, Rose & Clapp
Can estimate the mean using the approximation Swanson’s Rule – μ ≅ 0. 3 (10% point) + 0. 4 (50% point) +
0. 3 (90% point)
23
24
Monte Carlo SimulationA method of determining the probability distribution of an event.
The e ent is a f nction of ariables that can themsel esThe event is a function of variables that can themselves be expressed as individual probability distributions.
Each probability distribution is sampled randomly to calculate a value of the event.
The random sampling and the calculation of the value of the event is repeated many times Each calculation isof the event is repeated many times. Each calculation is know as a “pass”.
The numerous passes define the probability distribution of the event.
25
Monte Carlo SimulationExample:
Prospect Reserves = Area × Net Thickness × Recovery Factor
Acres Feet STBO/Acre-ft
Area1
Net Thickness1
Recovery Factor1
R d
0 0 0
RandomGuess
Repeat the calculation many times!
26
Monte Carlo Simulation – Beans problem
What parameters might we use to compute a “better” estimate of the number of beans in the jar?– Jar volume → Jar height; jar diameter– Bean volume → Bean shape approximately an
ellipsoid so …♦ Length of long axis (length of bean); g g ( g )♦ Length of “width” axis (diameter or radius of
bean); ♦ Length of “height axis (assume the same as the
width axis)♦ Void space amount
27
Monte Carlo Simulation – Bean problem
Jar volume:2
Ellipsoid
hdV2
Jar 2⎟⎠⎞
⎜⎝⎛= π
244 ⎞⎛⎞⎛ dlEllipsoid volume Bean 223
434
⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛== bdlabcV ππ
Monte Carlo Simulation – Beans problem
N b f bNumber of beans:
( ) Jar
BeanJar
1VolumeBean )1(
VVn
VnVφ
φ−
=
==−
BeanV
28
Monte Carlo SimulationUsing Excel
Dr. R. G. Hughes
First – Some Useful Excel Functions
L O U I S I A N A S T A T E U N I V E R S I T YA N D A G R I C U L T U R A L A N D M E C H A N I C A L C O L L E G E
Craft and Hawkins Department of Petroleum Engineering
29
30
31
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A Closer Look At the TriangularProbability Distribution
a b c
h
A 1 = ( b - a ) h / 2 A 2 = ( c - b ) h / 2
Remember, A1 + A2 = 1 h = 2 / ( c - a )
33
A Closer Look At the TriangularProbability Distribution (cont.)
a b c
h
A 1 = ( b - a ) h / 2, also equals the probability at the mode!
Pmode = ( b - a ) / ( c - a )Substituting h yields
What is the cumulative probability of theoccurrence a value “x” between a and c?
a b c
hh’
x
If b P ( ) h’ / 2If x < b, P x = ( x - a ) h’ / 2
From similar triangles we know: h’x - a
hb - a
=
P x = ( x - a ) 2
( c - a ) ( b - a ) x = a + √ Px ( c - a ) ( b - a )
34
What is the cumulative probability of theoccurrence a value “x” between a and c?
a b c
hh’
x
If b P 1 ( ) h’ / 2If x > b, P x = 1 - ( c - x ) h’ / 2
From similar triangles we know: h’c - x
hc - b
=
P x = 1 -( c - x ) 2
( c - a ) ( c - b ) x = c - √ ( 1 - Px ) ( c - a ) ( c - b )
Example TriangularProbability Distribution
0.08
0.10
0.12
0.14
0.00
0.02
0.04
0.06
10 15 20 25 30
Porosity
35
( c - x ) 2
Example TriangularCumulative Probability Distribution
P x = ( x - a ) 2
( c - a ) ( b - a )
P x = 1 -( )
( c - a ) ( c - b )
Function in SpreadsheetExample Analysis.XLSp y
36
Using Excel for Monte Carlo Simulation
37
1000
100Area
Thickness
10
RecoveryFactor
38
Graphical Monte Carlo Simulation
Can be used to combine lognormal probability distributionsCan be used to combine lognormal probability distributions.
When combining two variables, multiplying two P 90%gives P 92.7%, multiplying two P 10% gives P 7. 3%
When combining three variables, multiplying three P 90% givesP98 7%, multiplying two P 10% gives P 1 3%P98.7%, multiplying two P 10% gives P 1. 3%
1000
P =
100,000,000
Prospect mean (Swanson’s Rule) =0.3×2.25 + 0.4×6.06 + 0.3×16 = 7.9 MMSTBO 98.7
100
P90 = 16,000,000
10,000,000
10 1,000,000
P10 = 2,250,000
1.3
39
Discovery Probability
Usually involves estimating probability of several factors thatmust exist for a commercial HC accumulation to be present:
Example:
Reservoir quality rockStructureHC SourceReservoir seal
Discovery probability is the product of each of theindividual factor probabilities (multiplication rule):
Pdiscovery = P rock × P structure × P source × P seal
Prospect Economics
Dry HoleProbability = 1 - Pdiscovery
Drill Well
DiscoveryProbability = Pdiscovery
Value of Prospect =
Pdiscovery × (Value of mean reserves less completed well cost)
less ( 1 - Pdiscovery ) × Dry hole cost
40
Proved Reserves“Reasonable certainty”
♦ 90% probability that amount recovered will equal or exceed the estimated value
Economic under existing conditionsEconomic under existing conditions♦ Economic conditions♦ Operating conditions♦ Government regulations
Supported by actual tests or productionFluid contacts are base upon the lowest known unless engineering data indicates otherwiseConforms to existing well spacing and regulationsImproved recovery only considered if– successful pilot– response to an installed program– proven in the same or an analogous reservoir