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From Monte Carlo to BayesTheory: The Role of Uncertainty inPetrophysics.

Simon Stromberg



Subsurface Characterisation

The main goal of the oil-industry geoscience professional is

to characterise the complete range of possibleconfigurations of the subsurface for a given set of dataand related analogues.

This characterisation of the subsurface should lead to acomprehensive description of uncertainty that leads to thecomplete disclosure of the financial risk of further

exploration, appraisal or development of a potentialhydrocarbon resource.



Uncertainty and Risk

How to Measure Anything: Finding the Value of Intangibles in Business and The Failure of Risk Management: Why It's Broken and How to Fix It by Doug Hubbard:

Uncertainty: The lack of complete certainty, that is, the existence of more than one possibility. The "true" outcome/state/result/value is notknown.

Measurement of uncertainty: A set of probabilities assigned to a set of possibilities. Example: "There is a 60% chance this market will doublein five years"

Risk: A state of uncertainty where some of the possibilities involve a

loss, catastrophe, or other undesirable outcome.

Measurement of risk: A set of possibilities each with quantifiedprobabilities and quantified losses. Example: "There is a 40% chancethe proposed oil well will be dry with a loss of $12 million in

exploratory drilling costs".



Subsurface Realisation Tables



Petrophysical workflows

Calculate

Volume of

Clay

Calculate Clay

Corrected

Porosity

Calculate Clay

Corrected

Saturation

Apply cut-offs for

net sand,

reservoir and payand averages

Volume of Clay

Effective and Total

Porosity

Effective and Total

Porosity

Average Vcl, Por, Phi

And HPVOL



Petrophysical Base CaseInterpretation



Methods of Uncertainty Analysis

• Parameter sensitivity analysis

• Partial derivative analysis

• Monte Carlo Simulation

• Bayesian Analysis for Diagnostic Reliability



Sensitivity Analysis of InputParameters – Single Parameter

• For example:• Volume of clay cut-off

for net reservoir

• If VCL <= 0.3 and

• If PHIE >= 0.1 then

• The interval is flaggedas net reservoir



Sensitivity Analysis of InputParameters – Single Parameter



Sensitivity Analysis of InputParameters – Single Parameters



Partial Derivative Analysis

• In mathematics, a partial derivative of a function of several variables isits derivative with respect to one of those variables, with the others

held constant (as opposed to the total derivative, in which all variablesare allowed to vary). Partial derivatives are used in vector calculusand differential geometry.

• The partial derivative of a function f with respect to the variable x isvariously denoted by

• The partial-derivative symbol is ∂. The notation was introduced byAdrien-Marie Legendre and gained general acceptance after itsreintroduction by Carl Gustav Jacob Jacobi.



Partial Derivative Analysis – Example

• Volume of a cone is:

• The partial derivative of the volume with respect to theradius is

• Which describes the rate at which the volume changeswith change in radius if the height is kept constant.



Partial Derivative Analysis for Waxman-Smits Equation for Saturation

( ) ( )

( ) ( ) ( )

wt

A

Rw A

Sw

Qv B Rw Rt

n

m

n

F Qv B RwSwnSw E

E

Sw F Rw

n

Sw

E

F Rw

m

Sw

A E

F Rw

A

Sw

E

F RwmSw

Rw E

Sw

Rw

Sw

Rt E

Rw

Rt

Sw

nn

Swm

m

Sw A

A

SwSw Rw

Rw

Sw Rt

Rt

SwSw

m

Sw

⋅⋅

⋅

⋅

⋅⋅

+⋅

−

−

−

=⋅⋅+⋅−+=

⋅⋅−=∂∂⋅⋅−=

∂∂

⋅⋅=

∂∂

⋅

⋅⋅−=

∂

∂

⋅=

∂

∂

⋅−=

∂

∂

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂=

=

2

2

222222

1

;1

ln;ln;

;;

1

φ

φ

φ

φ φ

δ δ δ δφ φ

δ δ δ



Partial Derivative Analysis for A CompleteDeterministic Petrophysical Analysis

( ) ( )

( ) ( ) ( )

Swt

A

Rw A

Sw

Qv B Rw Rt

n

m

n

F Qv B RwSwnSw E

E

Sw F Rw

n

Sw

E

F Rw

m

Sw

A E

F Rw

A

Sw

E

F RwmSw

Rw E

Sw

Rw

Sw

Rt E

Rw

Rt

Sw

nn

Swm

m

Sw A

A

SwSw Rw

Rw

Sw Rt

Rt

SwSw

m

Sw

⋅⋅

⋅

⋅

⋅⋅+⋅

−

−

−

=⋅⋅+⋅−+=

⋅⋅−=

∂

∂⋅⋅−=

∂

∂

⋅

⋅=

∂

∂

⋅

⋅⋅−=

∂

∂

⋅=

∂

∂

⋅−=

∂

∂

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂=

=

2

2

222222

1

;1

ln;

ln;

;;

1

φ

φ

φ

φ φ

δ δ δ δφ φ

δ δ δ

( ) ( ) ( );

1;;

22

222

fl mab fl ma

bma

fl fl ma

fl b

ma

bb

fl fl

mama

fl ma

bma

ρ ρ ρ

φ

ρ ρ

ρ ρ

ρ

φ

ρ ρ

ρ ρ

ρ

φ

δρ ρ

φ δρ

ρ

φ δρ

ρ

φ δφ

ρ ρ

ρ ρ φ

−

−=

∂

∂

−

−=

∂

∂

−

−=

∂

∂

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂=

−

−=

( )

( )

φ

φ

φ

δ δφ φ

δ δ

φ

⋅−=∂

∂

⋅⋅+−⋅=

∂

∂

−⋅=∂∂

⋅

∂

∂+

⋅

∂

∂+

⋅

∂

∂=

−⋅⋅=

g

n

Sw

Foil

E

F RwmSw

g

n Foil

Sw g n

Foil

SwSw

Foil Foil g n

g n

Foil Foil

Sw g

n Foil

1

1/

//

1

222

222 geol sys stat total δ δ δ δ ++= samplesof number ndev std n

stat === ;1; σ σ

δ



Partial Derivative – Input Data

Zonal Averages

well zone gross net n2g por Sw

1 HSGHL 45.6 30.8 0.676 0.264 0.364

2 HSGHL 19.9 11.9 0.595 0.243 0.4784 HSGHL 44.8 22.2 0.497 0.256 0.347

5 HSGHL 31.3 11.7 0.376 0.266 0.3196 HSGHL 5.7 0.0 0.000 0.000 1.000

7 HSGHL 40.0 12.2 0.305 0.239 0.2928 HSGHL 46.8 16.2 0.345 0.270 0.393

Sumavs



Partial Derivative – Input Uncertainty

Field: Zone:

parameter value 1 std dev part error % error parameter value 1 std dev part error % error count 6 count 6

w average 0.258 0.013 w average 0.636 0.065

δδδδstat 0.005 2.0% δδδδstat 0.027 4.2%

rhob 2.224 0.010 -0.006 -2.3% Rt 10.723 1.072 -0.026 -4.1%

rhoma 2.650 0.010 0.004 1.7% Rw 0.300 0.060 0.021 3.4%

rhofl 1.000 0.020 0.003 1.2% por 0.258 0.010 -0.017 -2.7%

A 1.000 0.001 0.000 0.0%

m 1.800 0.150 0.052 8.2%n 2.000 0.200 0.052 8.2%

B 7.000 1.400 -0.030 -4.7%

Qv 0.245 0.025 -0.015 -2.4%

δδδδsys 0.008 3.2% δδδδsys 0.089 14.1%

δδδδgeol 0.000 0.0% δδδδgeol 0.000 0.0%

average 0.258 δδδδtotal 0.010 3.8% average 0.636 δδδδtotal 0.093 14.7%

count 7 count 6

w average 0.449 0.222 w average 0.074 0.022

δδδδstat 0.084 18.7% n/g 0.449 0.131 0.021 29.1%

por 0.258 0.010 0.005 6.5%

Sh 0.636 0.093 -0.011 -14.7%

δδδδgeol 0.100 22.3% δδδδsys 0.024 33.2%

average 0.449 δδδδtotal 0.131 29.1% average 0.074 δδδδtotal 0.024 33.2%

Foil=n/g*por*Shnet/gross

Porosity Hydrocarbon Saturation



Partial Derivative Analysis Results

porosity Hydrocarbon Saturation

Uncertainty (Normal) Distribution Curves

net/gross Foil

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.10 0.20 0.30 0.40

porosity

d i s t r i b u t i o n

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

avg=0.258, std=0.010, std=3.8% µ−3σ=0.229, µ+3σ=0.287

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.20 0.40 0.60 0.80 1.00

Sh


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

avg=0.636, std=0.093, std=14.7% µ−3σ=0.356, µ+3σ=0.916

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.70.8

0.9

1.0

0.00 0.05 0.10 0.15 0.20 0.25 0.30Foil


0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.70.8

0.9

1.0

0.00 0.20 0.40 0.60 0.80 1.00net/gross


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

−



Partial derivative Analysis - Saturation

Saturation Uncertainty (dSw) vs. Saturation for various Porosity Classes

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sw

S w U

n c e r t a i n t y ( d S w )

por=0.1

por=0.1625

por=0.225por=0.2875

por=0.35



Monte-Carlo Simulation

• Multiple repeated calculation of all deterministic equations

• All input parameters can besampled from a ‘distribution’ of expected range in parameters

• Co-dependency can behonoured

• All output can be analysed

• Relative contribution to error can

Be analysed

Requires a computer

Calculate

Volume of

Clay

Calculate Clay

Corrected

Porosity

Calculate Clay

Corrected

Saturation

Apply cut-offs for

net sand,

reservoir and payand averages



Monte-Carlo Simulation

Reference Case Uncertainty Definition

MONTE CARLO RESULTS HISTOGRAMSTest Well 1

Number of Simulation : 188PhiSoH Pay Zone : All

Mean : 24.84 Std Dev : 6.775

0

10

20

30

40

50

-4.77 52.5

PhiSoH Res Zone : AllMean : 30.69 Std Dev : 9.804

0

10

20

30

-5.62 61.8

PhiH Pay Zone : AllMean : 41.49 Std Dev : 14.45

0

10

2030

40

50

60

-11.9 131

PhiH Res Zone : AllMean : 128.5 Std Dev : 28.29

0

10

20

30

40

50

-15.9 175

Av Phi Pay Zone : AllMean : 0.2042 Std Dev : 0.02894

0

10

20

30

40

50

60

70

-0.0262 0.289

Av Phi Res Zone : AllMean : 0.1945 Std Dev : 0.02663

01020304050

607080

-0.0231 0.254

Phi Cut Res/PayMean : -0.00038 Std Dev : 0.02399

0

10

-0.0687 0.0723

ResultsMONTE CARLO TORNADO PLOT

Test Well 1

Error Analysis for : PhiSoH Reservoir

Rho GDSD Son Clean1

SP CleanNeu Clay

SD Den Clean1Neu CleanPhiT Clay

SD Son Clean2SD Den Clean2

DTLNRho Dry C lay

SP ClaySD Den Clay

Sw Cut Res/PaySD Son ClayRho mud filt

Rmf MSFL

Rxo ClayRes Clean

Gr ClayRes Clay

ND Den ClayND Den Clean1ND Neu Clean1ND Den Clean2

ND Neu ClayHc Den

ND Neu Clean2SGR

Res ClayPhi Cut Res/Pay

LLDNeu Wet ClayRho Wet ClayTNPH

Gr Cleann exponent

a factorRHOB

Rwm exponent

Vcl Cut Res/Pay

0.03 2.65 0.033 55 310 0 100.05 0 0.050.03 2.65 0.030.02 0 0.020.05 0 0.053 104 30.03 2.05 0.032 20.1 2.775 0.110 0 100.05 0 0.050.2 0.7 0.25 0 50.02 1.01594 - 1.01829 0.0220% 0.136 20%0.005R 0.005R 20% 0.47 - 3.08 20%20% 72.46 20%10 114 - 139 1020% 1.14 - 2.36 20%0.05 2.538 0.050.03 2.65 0.030.02 -0.04 0.020.03 2. 048 - 2. 05 0.030.05 0.461 0.050.2 0.476 - 0.8 0.20.02 0.217 - 0.3 0.025 520% 1.42 - 3.2 20%0.05 0.1 0.050.005R 0.005R 0.05 0.461 - 0.588 0.050.05 2.538 - 2.625 0.055% 5%

10 13.88 100.2 2 0.20.1 1 0.10.02 0.0220% 0.0895 20%0.2 2 0.20.3 0.3 0.3

ShiftLow

Initial Values

ShiftHigh

31.228-0.385 62.842PhiSoH Reserv oir Zone : All

Sensitivity



Models and Equations



ResultsMONTE CARLO RESULTS HISTOGRAMS

Test Well 1Number of Simulation : 408PhiSoH Pay Zone : All

Mean : 24.86 Std Dev : 6.198

0

20

40

60

80

100

-5.49 60.4

HPVOL (ft)



SensitivityMONTE CARLO TORNADO PLOT

Test Well 1Error Analysis for : PhiSoH Pay

DTLN

Rho mud filt

Rho GD

Rho Dry C lay

PhiT Clay

Res Clean

MSFL

Rxo Clay

ND Den C lay

Res Clay

ND Den C lean1

Rmf

Gr Clay

ND Neu C lean1

ND Den C lean2

ND Neu C lay

ND Neu C lean2

Phi C ut Res/Pay

Res Clay

TNPH

LLD

Hc Den

SGR

Neu Wet Clay

Rho Wet Clay

Gr C lean

a factor

n exponent

RHOB

Rw

m exponent

Sw Cut Res/Pay

Vcl Cut Res/Pay

2 2

0.02 1.01594 - 1.01829 0.02

0.03 2.65 0.03

0.1 2.775 0.1

0.05 0 0.05

20% 72.46 20%

0.005R 0.005R

20% 0.47 - 3.08 20%

0.05 2.538 0.05

20% 1.14 - 2.36 20%

0.03 2.65 0.03

20% 0.136 20%

10 114 - 139 10

0.02 -0.04 0.02

0.03 2.048 - 2.05 0.03

0.05 0.461 0.05

0.02 0.217 - 0.3 0.02

0.05 0.1 0.05

20% 1.42 - 3.2 20%

5% 5%

0.005R 0.005R

0.2 0.476 - 0.8 0.2

5 5

0.05 0.461 - 0.588 0.05

0.05 2.538 - 2.625 0.05

10 13.88 10

0.1 1 0.1

0.2 2 0.2

0.02 0.02

20% 0.0895 20%

0.2 2 0.2

0.2 0.7 0.2

0.3 0.3 0.3

ShiftLow

Initial Values

ShiftHigh

9.9920.023 19.962PhiSoH Pay Zone : 3



Bayesian Analysis For Reliability

Bayes allows modified probabilities to be calculated basedon

1. The expected rate of occurrence in nature

2. A diagnostic test that is less than 100% reliable

For example• There is a 1:10,000 (0.0001%) occurrence of a rare

disease in the population

• There is a single test of the disease that is 99.99%

accurate• A patient is tested positive for that disease

• What is the likelihood that the patient tested positive

actually has the disease?



Bayesian Theory

• Bayes’ theory is the statistical method to revise probability based on aassessment from new information. This is Bayesian analysis.

• To set up the problem:

• Consider mutually collective and collectively exhaustive outcome(E1, E2…….En)

• A is the outcome of an information event, or a symptom related to E.

• If A is perfect information, Bayes theorem is NOT needed.

)(

)(

)()(

)()()(

1

A P

E A P

E P E A P

E P E A P A E P i

N

j

j j

ii

i

•==

∑=



Bayes Theory

Input Number

Calculation

Probability of (state of nature)

occurring P(A) Input 0.10

Probablity of state of nature not

occuring P(nA) 1-P(A) 0.90Probability of True Positive Test

(that B will be detected if A

exists) P(B|A) Input 0.90

Probability of False Positive Test

(That B will be detected if A does

not exist) P(B|nA) Input 0.20

False Negative Test P(nB|A) 1-P(B|A) 0.10

Probability of a true negativetest. P(nB|nA) 1-P(B|nA) 0.80

Total probability of detecting A

(whether it present or not) P(B) P(B|nA)*P(nA)+(P(B|A)*P(A) 0.27

Total probability of not detecting

A (whether it present or not) P(nB) 1-P(B) 0.73

Probability that A is present

given that it was detected P(A|B) P(B|A)*P(A)/P(B) 0.33

Probability that A is present

given that it was not detected

(probability of a a false negative) P(A|nB) P(nB|A)*P(A)/P(nB) 0.01

Probability that A is NOT present

given it was detected P(nA|B) 1-P(A|B) 0.67

Probability that A is NOT present

given it was NOT detected P(nA|nB) 1-P(A|nB) 0.99

INPUTS

OUTPUTS



Using AVO to De-risk Explorationand The Impact of Diagnosis Reliability

• AVO Anomaly

• Information• Our geophysicist has evaluated AVO anomalies and has

assessed that:

• There is a 10% chance of geological success

• There is a 90% chance of seeing an AVO if there is a discovery• There is a 20% chance of seeing a false anomaly if there is no

discovery

• Question

• If we have a 10% COS based on the geological interpretationand an anomaly is observed, what is the revised COS if anAVO is observed

• What is the added value of AVO



AVO

Geologocal COS

1 Discovery

1.1 Anomaly 9%90%

1.2 No Anomaly 1%10%

10%

2 No Discovery

2.1 Anomaly 18%20%


90%



AVO

Discovery No Discovery

Anomaly 0.09 0.18No Anomaly 0.01 0.72

Geologocal COS

1 Discovery

1.1 Anomaly 9%90%

1.2 No Anomaly 1%

10%

10%

2 No Discovery

2.1 Anomaly 18%20%


90%



AVO

• The probability that an anomaly is a discovery is in Bayestheorem

Discovery No DiscoveryAnomaly 0.09 0.18 0.27

No Anomaly 0.01 0.72 0.73

0.1 0.9

33.0

27.0

09.0=

)()""()()""(

)()""(""(

D P D A P D P D A P

D P D A P A D P

+=

A i C iti l P it i



Assessing Critical Porosity inShallow Hole Sections

• Synopsis of SPE paper in press

• New technique for deriving porosity from sonic logs inoversize boreholes

• Sonic porosity used to determine of porosity is at criticalporosity

• Critical porosity 42 to 45 p.u.• If lower than critical porosity rock is load bearing

• Near or at critical porosity the rocks may have insufficientstrength to contain the forces of shutting in the well

• Sonic porosity (using the new technique) is used todetermine if rock is below critical porosity and used as partof the justification for NOT running a conductor

C St d A i C iti l P it i



Case Study: Assessing Critical Porosity inShallow Hole Sections

• LWD acoustic compression slowness below the drive pipe

• LWD sonic device optimised for large bore holes

• Data showed conclusive evidence of absence of hazards and

were immediately accepted as waiver for running the diverter

and conductor string of casing

• DTCO used to asses if rock consolidated (below critical

porosity)• Gives the resultant DTCO and porosity interpretation

• Raymer Hunt Gardner model with shale correction

• No mention of the reliability of the interpretation• Processed DTCO accuracy

• Interpretation model uncertainty

A i C iti l P it i




• To obtain P-Wave Velocity in a very slow formation:

• Excite and measure the (leaky P wave) in the low frequencyrange

• When the formation shear-wave falls below the boreholefluid velocity, a portion of the compression wave converts to

shear and radiates into the formation• The attenuation due to radiation loss causes dispersion of

the waves travelling along the bore-hole

• Only at low frequency range 2kHz does the semblance (STC

processing) does the result approach P-wave slowness.

• The tool described can acquire low frequency acousticwaves

Assessing Critical Porosity in




• Question• What is the potential impact of tool accuracy and model

uncertainty on the interpretation

• Should this be considered as part of the risk management

for making the decision on running the conductor

• What is the Efficacy of the method and measurement for

preventing the unnecessary running of diverter and

conductor string of casing

Baseline Interpretation




• DT matrix = 55.5 usec/ft

• DT Fluid = 189 usec/ft

• DT wet clay = 160 usec/ft

• (uncompacted sediment)

• VCL from GR using linear method

BigSonicScale : 1 : 500

DEPTH (999.89FT - 1500.37FT) 09/01/2010 15:16DB : Test (-1)

1

GR (GAPI)0. 150.

2

DEPTH(FT)

3

DTCO (usec/ft)300. 100.

4

PHIT (Dec)0.5 0.

1100

1200

1300

1400

11

GR (GAPI)0. 150.

2

DEPTH(FT)

3


4

PHIT (Dec)0.5 0.

Rock above CP



Rock above CP

BigSonicScale : 1 : 500

DEPTH(999.89FT- 1500.37FT) 10/09/201010:47DB: Test (1)

1

GR (GAPI)0. 150.

2

DEPTH(FT)

3


4

PHIT(Dec)0.5 0.

ResFlag()0. 10.

1

1100

1200

1300

1400

15001

GR (GAPI)0. 150.

2

DEPTH(FT)

3


4

PHIT(Dec)0.5 0.

ResFlag()0. 10.

Phi Cut ResCutoff SensitivityData

Wells: BigSonic

Net Reservoir - All Zones Phi Cut ResCutoff

0.40.30.20.1

300

250

200

150

100

50

P10P50P90

Uncertainty Analysis (Monte Carlo)




Distribution Default + -

DTCO (usec/ft) Gaussian log 5 5

GR clean Gaussian 17 10 10

GR Clay Gaussian 84 10 10

DT wet clay (usec/ft) Gaussian 159 5 5

DT Matrix (usec/ft) Gaussian 55.5 5 5

DT Water (usec/ft) Gaussian 189 5 5Cutoff for critical

Porosity (v/v) Square 0.42 0.03 0.01





• Footage where porosity exceeds critical porosity

• Gross Section =• P10 = 0ft

• P50 = 23 ft

• P90 = 62.5 ft

MONTE CARLO RESULTS HISTOGRAMSBigSonic

Number of Simulation : 2000Net Res Zone : All

Mean : 29.01 Std Dev : 25.93

0

50

100

150

200

250

300

-10 110

MONTE CARLO TORNADO PLOTBigSonic

Error Analysis for : Net Reservoir

Sonic Wet Clay

DTCO

Sonic matrix

Sonic water

Gr Clay

Phi Cut Res/Pay

Gr C lean

5 159 5

5 5

5 55.5 5

5 189 5

10 84 10

0.03 0.42 0.01

10 17 10

ShiftLow

Initial Values

ShiftHigh

12.207-57.575 81.988Net Reserv oir Zone : 1

Results of Monte-Carlo



Results of Monte Carlo

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

Net Rock > CP

P r o b a

b i l i t y

Tornado Plot



Tornado Plot

MONTE CARLO TORNADO PLOTBigSonic

Error Analysis for : Net Reservoir

Sonic Wet Clay

DTCO

Sonic matrix

Sonic water

Gr Clay

Phi C ut Res/Pay

Gr C lean

5 159 5

5 5

5 55.5 5

5 189 5

10 84 10

0.03 0.42 0.01

10 17 10

ShiftLow

Initial Values

ShiftHigh

12.207-57.575 81.988

Net Reservoir Zone : 1

Conclusions from Monte Carlo



Conclusions from Monte Carlo

• The model processed DTCO and model uncertainty leads

to significant doubt that the rock is below critical porosity• The most important considerations are:

• The clay volume and clay correction

• The actual porosity value for critical porosity (0.41 to 0.45)

• Tool Accuracy is not a concern (+/- 5 usec/ft)

• A good question to ask is what is the tool accuracy given the

new processing technique and challenges of running sonicin big bore-holes?

Bayesian Analysis of Reliability




• Lets assume that the regional data shows that the 90% of

all top hole sections will be below CP and stable• Based on Monte-Carlo it is judged that the interpretationwill be 80% reliable

Well Bore Stability

1 Below CP

1.1 Sonic Log Shows Below CP 72%£0.00

80%£0.00

1.2 Sonic Log Shows AboveCP18%

£0.00

20%

£0.00

£0.0090%

£0.00

2 Above CP

2.1 Sonic Log Shows Above CP8%

£0.00

80%

£0.00

2.2 Sonic Log Shows Below CP2%

£0.00

20%

£0.00

£0.0010%£0.00

£0.00

Bayesian Inversion of Tree



y

Critical Porosity

1 Log Shows "BCP"

1.1 BCP (0.72/0.74) 72%97%

1.2 ACP (0.02/0.74) 2%3%

74%

2 LOG Shows "ACP"

2.1 BCP (0.18/0.26) 18%69%

2.2 ACP (0.08/0.26) 8%31%

26%

% Wells below CP regionally 0.9

Reliability of the log 0.8

"BCP" "ACP"

BCP 0.72 0.18

ACP 0.02 0.08

0.74 0.26

If Log shows BCP 0.97

If Log Shows ACP 0.31




y y yJoint Probability Table

• If the log shows that the section is below critical porositythen we can be 97% certain that it is below criticalporosity



"BCP" "ACP"

BCP 0.72 0.18

ACP 0.02 0.08

0.74 0.26

If Log shows BCP 0.97

If Log Shows ACP 0.31

Bayesian Analysis 2



y y

• What if only 60% of the wells in the area are below criticalporosity. What is the value of the sonic?

• If the sonic log shows below CP there is an 86% chance it

is really below CP



"BCP" "ACP"BCP 0.48 0.12

ACP 0.08 0.32

0.56 0.44

If Log shows BCP 0.86If Log Shows ACP 0.73

Reliability of Diagnosis Chart



y g

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.5 0.6 0.7 0.8 0.9 1

Log Diagnostic Reliability

P r o

b a b i l i t y t h a t I n t e r v a l i s B e l o w

C P

0.2

0.4

0.60.8

1

Regional Data

% of well above

Below Cp

Conclusions From Reliability Analysis



• If the Sonic is 100% reliable as a diagnosis of the wellbeing above or below CP then the log data then we can besure the interval is above/below CP

• If the Sonic log is not 100% reliable then we need to takeinto account

• Regional data trends• Reliability of the Sonic log

• If we believe that the sonic log is less than 100%

diagnostic then there is always a risk that the well will beabove CP, even if the log data shows otherwise.

Conclusions



• The most important task of the geoscientist is to make statementsabout

• Range of possible subsurface outcomes based on:• Uncertainty

• Diagnostic reliability of data

• There are several ways to analyse the range of outcomes based onuncertain input parameters

• Single parameter sensitivity

• Partial derivative analysis

• Monte-Carlo simulation

• Bayes’ analysis for diagnostic reliability

• The results of uncertainty and reliability analysis can be counter intuitive

fro mba yes to monte carlo

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