chapter 4 sp 2014 economic uncertainties part 2

8/12/2019 Chapter 4 Sp 2014 Economic Uncertainties Part 2

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PE 3003/6413

PE 3003/6413 Spring 2014

Chapter 4

Part 2

Econom ic Uncertaint ies


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Outline

1. Range Analysis

2. Probability

3. Cumulative Probability and Density Functions

4. Expected Value

5. Decision Three Analysis6. Distribution Functions

7. Monte Carlo Simulation


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Distribution Functions

Density functions are commonly used in engineering

to describe physical systems

Density distribution of a loading in a beam

For any point x along the beam , the density c an bedescr ibed b y a funct ion (in grams /cm )

The total loading between points a and b is

determ ined as the integral of the density fu nc t ion

f rom a to b


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Distribution Functions

Common Density Functions

Uniform

Triangular Distribution

Normal Distribution

Log-Normal Distribution


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Uniform Distribution

A continuous random variable X with p robabi li ty densi ty

funct ion


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Let the continuous random variable X denote the porosity

measured in core samples. Assume that the range of X is [0,

20 %], and assume that the probability density function of Xis

f(X)=0.05, 0X20

1. What is the probability that a measurement of porosity isbetween 5 and 10 %?

2. What is the mean and variance of the core samples?

E(X)= 10% and Var(X)=33.33 %


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Random Number

A set of numbers that have nothing to do with the other

numbers in the sequence

Random numbers have no defined sequence or formulation.

Thus, for any n random numbers, each appears with equalprobability

All computer languages and spread sheet has a random

function that return a random number uniformly distributed

between 0 and 1

a b


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VBA Random number initialization

Const Seed As Double = 1000

' Random numbers Seed set

Dim RndVal As Double

Randomize (Seed)RndVal = Rnd(-Seed)

' After this command random numbers can be generated as

follows

RndVal = Rnd()

We will use this in Monte Carlo section


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Random number Generation between aand b

VBA

Function RandAB(a As Double, b As Double) As Double

RandAB = (b - a) * Rnd() + a

End Function

Excel

=rand()*(b-a)+a


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Strengths

Easy to create

Easy to understand

Simple implementation and calculation

Weakness Almost always inappropriate, except as random number

generation

Use cautiously with low iterations (Monte Carlo)


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a bc

f(x)

x


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Cumulative Distribution

bx

bxccbab

xb

bxaacab

ax

xax

PxF

1

1

0

)(

2

2

1. a is the minimum outcome of the random

variable.

2. cis the mode (most likely) of the random

variable.3. bis the maximum outcome of the random

variable.

4. F(x) is the cumulative probability of all

possible outcomes of the random variable

between aandx.


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Random numbers generation

generate a random number (P) uniformly distributed between

0 to 1 (refer to uniform distribution random numbers)

random number is calculated using the inverse of the

triangular distribution as follows:

See VBA template on WebCT

1

11

0

0

Pb

Pab

accbabPb

ab

acP

aacabP

Pa

x


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Strengths

Easy to create

Easy to understand

Simple implementation

Weakness Likeliestoften confuse with mean or P50 and very difficult to

defend

End points are sometimes difficult to estimate

Poor approximation of Log-Normal


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Normal Distribution

The most widely used model for the distribution of a

random variable

Whenever a random experiment is replicated, the random variable

that equals the average (or total) result over the replicates tends

to have a normal distribution as the number of replicates becomes

large.


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Normal Distribution

Probability density function

Mean and Variance


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Normal Distribution

Let the continuous random variable X denote the porosity

measured in core samples. Assume that X is normally

distributed with 10% mean and 4% variance

1. What is the probability that a measurement of porosity

exceeds 13%?

Excel => =1-NORM.DIST(13,10,SQRT(4),TRUE) = 0.06681


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Normal Distribution

Probabilities associated with a normal distribution


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Normal Distribution




random number is calculated using the inverse of the normal

distribution as follows:

Excel

=NORM.INV(RAND(), Mean,Sigma) where p=rand()


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Normal Distribution

Strengths

Easy to create

Easy to understand

Simple implementation

Weakness Not really true nature at the ends


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Variables in a system sometimes follow an exponential

relationship as x=exp(w)

If the exponent is a random variable, say W, X=exp(W)is a random

variable

An important special case occurs when W has a normal

distribution. In that case, the distribution of X is called alognormal distribution.


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Probability density function

Mean and Variance


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random number is calculated using the inverse of the normal

distribution as follows:

Excel

=LOGNORM.INV(rand(),,)


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Strengths

Easy to create

Easy to understand

Logical starting points for many inputs

Extremely common in nature Weakness

Not really true nature at the ends


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Binomial Distribution

Bernoulli Trial or Experiment:

Flip a coin 10 times. Let X=number of heads obtained.

A worn machine tool produces 1% defective parts. Let X

=number of defective parts in the next 25 parts

produced.

The random variable in each case is a count of the

number of trials that meet a specified criterion

Assumed that the trials that constitute the random

experiment are independent

probability of a success in each trial is constant


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A random experiment consists of n Bernoulli trialssuch that

1. The trials are independent

2. Each trial results in only two possible outcomes,

labeled as success and failure3. The probability of a success in each trial, denoted as p,

remains constant

The random variable Xthat equals the number of

success has a binomial random variable with

parameters n(number of trials or experiments) andsuccess probability p


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We are drilling 5 wildcats in a new basin where thechance of a discovery (wildcat success ratio) is 0.15

on each well. Assuming each well is a Bernoulli trial,

what is the probability of only one discovery in the

fives wells drilled?

n=5

p=0.15

S=1

P(S=1)=0.3915


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Outline

1. Range Analysis

2. Probability

3. Cumulative Probability and Density Functions

4. Expected Value

5. Decision Three Analysis

6. Distribution Functions

7. Monte Carlo Simulation


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Gas Metering Problem

wq Oq

Gq GP GT


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The gas flow rate is measured a certain pressure andtemperature

Gas flow rate need to be reported at standard

conditions (Deterministic Model)

)67.459(0283.0)(

G

GGSCG

Tz

Pqq

qG(SC): gas flow rate at standard conditions [MSCF/D]qG: gas flow rate at in-situ conditions [MCF/D]

PG: gas pressure [psia]

TG: gas temperature [F]

z: gas compressibility factor [-]. For simplicity z=0.9


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For a given time

][19.203

)67.459120(9.00283.0

30010)( MSCF/D

SCGq

qG: 10 [MCF/D]

PG: 300 [psia]

TG: 120 [F]z: 0.9 (for simplicity)


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Unfortunately qG, PGand TGhas associateduncertainties

0

0.1

0.2

0.3

0.4

0.5

0 10 20

f(qG

)

qG[MCFD/D]

0

0.02

0.04

0.06

0.080.1

0.12

0.14

280 300 320

f(PG

)

PG[psia]

0

0.05

0.1

0.150.2

0.25

100 120 140

f(TG

)

TG[F]

qGis normally distributed

with mean of 10MCFD

and 1 MCFD of standard

deviation

PGis normally

distributed with mean of

300 psia and 3 psia of

standard deviation

TGis normally

distributed with

mean of 120 F and

2 F of standard

deviation


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Uncertainties need to be propagated to get theuncertainty in qG,

)67.459(0283.0)(

G

GGSCG

Tz

Pqq

0

0.1

0.2

0.3

0.4

0.5

0 10 20

f(qG

)

qG[MCFD/D]

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

280 300 320

f(PG

)

PG[psia]

0

0.05

0.1

0.15

0.2

0.25

100 120 140

f(TG

)

TG[F]

0

0.1

0.2

0.3

0.4

0.5

0 10 20

f(qG

)

qG[MCFD/D]

Monte Carlo is used to

propagates this uncertainties

Though the Deterministic

Model


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Assume qG, PGand TGare not correlated

Assume

qGis no rmal ly dis tr ibuted w ith mean of 10MCFD and 1

MCFD of s tandard deviat ion

PGis normally dis tr ibuted with mean of 300 psia and 3 psia

of standard deviat ion

TGis norm all y d is tr ib u ted w ith mean of 120 F and 2 F o f

standard deviat ion


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Follow the steps below:

1. Calculate the average qG(SC) using the average values for qG,PGand TG

2. Generate a random number following qGdistribution

3. Generate a random number following PGdistribution4. Generate a random number following TGdistribution

5. Calculate qG(SC)using the deterministic model

6. Repeat from steps 2 to 5 until maximum number of iterations is

reached or the average value of all calculated qG(SC)is close

enough to the qG(SC)

calculated in step 1


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192

194

196

198

200

202

204

0 200 400 600 800 1000 1200

Average

Iterations

Convergence Plot

Simulation Deterministic Model


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Histogram

Min 141.0443

Max 271.6897

N of classes 10

Step 13.06453288

Left Right Interval Freq Freq [frac]

141.04 154.10 [141.04-154.1] 6 0.006006006

154.10 167.16 (154.1-167.16) 22 0.022022022

167.16 180.22 (167.16-180.22) 102 0.102102102

180.22 193.28 (180.22-193.28) 185 0.185185185

193.28 206.34 (193.28-206.34) 269 0.269269269206.34 219.40 (206.34-219.4) 222 0.222222222

219.40 232.46 (219.4-232.46) 128 0.128128128

232.46 245.52 (232.46-245.52) 52 0.052052052

245.52 258.58 (245.52-258.58) 11 0.011011011

258.58 271.64 (258.58-271.64) 2 0.002002002

Total 999 1


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Histogram

0

0.05

0.1

0.15

0.2

0.25

0.3

Histogram


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Average and Standard Deviation and Normal

Approximation

Average 203.28

St Deviation 20.38

Pessimistic Mean Optimistic162.53 203.28 244.03


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0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

140 160 180 200 220 240 260

F(qG(SC)

qG(SC) [MSCFD]


Cumulative Freq DistributionPercentile qG(SC)

2.50% 161.89

5.00% 169.61

10.00% 176.58

15.00% 182.73

20.00% 187.41

25.00% 190.97

30.00% 193.8035.00% 196.52

40.00% 200.47

45.00% 202.97

50.00% 205.85

55.00% 208.28

60.00% 210.93

65.00% 213.48

70.00% 216.58

75.00% 219.57

80.00% 223.08

85.00% 227.71

90.00% 232.09

95.00% 239.09

97.50% 247.22


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Monte Carlo Process

1. Define all the variables

Identify all the measure variables of interest (i.e. NPV, oil price,

etc.) or all variables that affect the evaluation

2. Define the variables relationships in the deterministic

projection model Equations or other numerical calculations

3. Sort the input variables into two groups

Variables that we known with certain (represented by single

point value)

Significant unknowns will be represent by random variables(exact value can not be specified)

4. Define random variable distributions

Usually, the best available expert is assigned to judge each

random variable


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Monte Carlo Process

5. Perform the respective simulation trials1. Calculate the deterministic average using the average values all

random variables into the deterministic equation

2. Generate a random number following the distribution of each

random variable

3. Calculate the outcome using the deterministic model4. Repeat from steps 2 to 5 until maximum number of iterations is

reached or the average value of all calculated values in step 3 is

close enough to the calculated in step 1

6. Calculate

Plot the histogram Expected and standard deviation value of the outcomes

Plot the cumulative frequency distribution

P90

P50

P10


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P90, P50 and P10

Other disciplines

Mao-Jones, J. (2012). Decision & Risk Analysis. Merrick & Company Technical Paper.


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P90, P50 and P10

Other disciplines

Holm, S. Suleymanov, V. Vanvik, N.

(2011). Shtokman flow assurance

challenges a

Systematic approach to analyzeuncertaintiesPart 2.15thMultiphase

Production Technology Conference,

Canes (2012)


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P90, P50 and P10

Other disciplines

Holm, S. Suleymanov, V. Vanvik, N.

(2011). Shtokman flow assurance

challenges a

Systematic approach to analyzeuncertaintiesPart 2.15thMultiphase

Production Technology Conference,

Canes (2012)


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P90, P50 and P10

Other disciplines

P10, P50 and P90 are defined as the percentile of the

outcome

This classical definition will apply if you are dealingwith

1. Facility engineers or any other engineer who is

designing an equipment

2. Flow assurance engineers


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P90, P50 and P10

Petroleum Resources Management System (PRMS)

(Society of Petroleum Engineers 2007)

f

Reserves [MMBBL or BSCF]

1P

10

th

2P

50th

3P

90th

Proved Provable Possible


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P90, P50 and P10



Low Est imate:

Conservative estimate of the quantity that will

actually be recovered from the accumulation bya project

There should be at least 90% probability (P90)

that the quantities actually recovered will equal

or exceed the low estimate

This is the definition that will be used in this

course owing to we are working directely with

the reserves


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P90, P50 and P10

Reserves [MMBBL or BSCF]1P

10th

Should be at least a 90%

probability that the quantities

actually recovered will equal or

exceed the estimate

P90

Low Estimate

This is the definition that will be used in this course


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P90, P50 and P10

Reserves [MMBBL or BSCF]

2P

P50

there should be at least a 50%

probability that the actual

quantities recovered will equal or

exceed the 2P estimate


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P90, P50 and P10



High Estimate:

Optimistic estimate of the quantity that will

actually be recovered from an accumulation by aproject

There should be at least a 10% probability (P10)

that the quantities actually recovered will equal

or exceed the high estimate


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P90, P50 and P10

Reserves [MMBBL or BSCF]3P

90th

there should be at least

a 10% probability that

the actual quantities

recovered will equal

or exceed the 3P

estimate.

P10This is the definition that will be used in this course


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P90 P50

P90, P50 and P10

Summary: we will use P10= 10% sure that the value (i.e. Reserve or NPV) is

higher than P10

P90= 90% sure that the value (i.e. Reserve or NPV) is

higher than P90

P10

NPV[MM$]


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Example 1

The production of a new well can be predicted using decline curve analysisand historical data from nearby reservoirs. The hyperbolic decline curve is

given by:

bi iDibqq1

0 1

Where

q= Oil flow rate [BPD]

q0= Initial oil flow rate (time zero) [BPD]Di= Initial Nominal Decline [Fraction / year]

b= Hyperbolic Exponent factor (some authors use the term n)

i = integervalue of the time [year].


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Example 1

The net revenue of the well in a particular yearican be estimated by:

DTOATvenueReNet iii 1)(

1. A(i) is the gross revenue in year igiven by

Wc

WcWaterCosticePrGasGORicePrOildaysqA ii

1365

Wcis the water cut

2. O(i)is the operational cost in yearI

3. D= is the depreciation in year i.

4

CapexD

where Capexis the initial investment and 4 years is the taxable life of the

well; Tis the income tax rate


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Example 1

Finally, the net present value of this well can be calculated as follows:

Capex

MROR

venueReNetesentValuePrNet

ii

i

1

)(

1

where MRORis the minimum rate of return. The well will be closed when the Net

Revenue becomes negative (NetRevenue(i) < 0). Thus the life of the well corresponds

to the year before the net revenue becomes less than zero. In other words the

summation will be carried out until Net Revenue becomes negative

Using this mathematical formulation, We created a user define function in excel that

can return the NPV for a well (Deterministic Model)


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Example 1

Lets consider the following random variables:1. Initial flow rate q0is normally distributed with mean =500

and standard deviation of 30 BPD

2. Oil price is uniform distributed between 85 and 120 $/BPD

3. Gas price in uniform distributed between 2 and 6 $/Mscf

The certain variables are:

GOR 0.3 [Mscf/BBL]

WaterCost 0.13 [$/BBL]

Wc 0.5

O(i) 301,000.00 [$]T 0.29

MROR 0.13

Di 0.7 [1/year]

b 1.5

Capex 2,000,000 [$]


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Example 1

Run a Monte Carlo simulation assuming that the random variablesare independent and not correlated


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Example 2

The recoverable oil using the volumetric approach in a givenprospect is given

rf

Bo

ShA,Nr W

17587

whereAis the drainage area [acres], his the net pay [ft], is porosity [-], Swis

the water saturation [-],Bois the formation volume factor [bbl/stb] and rf

is the recovery factor [-]. Nris in stb

Example 2 template contains a user define function that calculatesNrand

can be used to setup a Monte Carlo Simulation.


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Example 2

Consider the following random variables1. is porosity [-] is normally distributed with mean 0.14 and

standard deviation of 0.02

2. Swis the water saturation [-]. Uniform distributed between 0.2

and 0.4.

3. his the net pay [ft]. Normally distributed mean 15 and standarddeviation of 1.5 feet's

4. Ais the drainage area [acres]. Lognormal distributed with =4.35 and

=0.246 (mean 80 and std of 20 Acres)

5. rf is the recovery factor [-]. Normally distributed with mean of

0.34 and standard deviation of 0.05.6. Bois the formation volume. Uniform distributed between 1.15

and 1.25

chapter 4 sp 2014 economic uncertainties part 2

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