PE4051_Set3_2010

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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

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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

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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

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“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

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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


p– Ranking of investment opportunities– Reserve estimates– Failure/reliability analysis– Geostatistics

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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


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 =


(1/6) (1/6) = 1/36

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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


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

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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



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

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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

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“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


The mean represents the “long-run average” resulting from repeated trials.

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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

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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.

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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


Lognormal - Logarithm of variable follows Normal distribution

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Graphical Representation of “Roll of a Die” Probability

DistributionPr

obab

ility

1/6

1 2 3 4 5 6


Some Types of Probability Distributions

Uniform Normal

Triangular Lognormal

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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( )

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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

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P50 = median

mode = 38P10

mean

P90

Graphing Cumulative Probability

“Special” graph paper is availablefor normal and lognormal probabilitydistributions that yields a straight line

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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.


Normal Normal×

Lognormal

=

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Many Things in Nature Followa Lognormal Distribution

Central Limit TheoremSum of any independent random variables = Normal Distribution

1,000Samples

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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


Can estimate the mean using the approximation Swanson’s Rule – μ ≅ 0. 3 (10% point) + 0. 4 (50% point) +

0. 3 (90% point)

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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.

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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!

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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

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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

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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

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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 )

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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 )

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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

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( 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

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Using Excel for Monte Carlo Simulation

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1000

100Area

Thickness

10

RecoveryFactor

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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

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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

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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

PE4051_Set3_2010

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