Distributed by permission of Hart's E & P and James...

Distributed by permission ofHart's E & P and James Murtha

2 Risk Analysis

Jim Murtha, a registered petroleumengineer, presents seminars and trainingcourses and advises clients in buildingprobabilistic models in risk analysis anddecision making. He was elected toDistinguished Membership in SPE in1999, received the 1998 SPE Award inEconomics and Evaluation, and was1996-97 SPE Distinguished Lecturer inRisk and Decision Analysis. Since 1992,more than 2,500 professionals have takenhis classes. He has published Decisions

Involving Uncertainty - An @RISKTutorial for the Petroleum Industry. In25 years of academic experience, hechaired a math department, taught petro-leum engineering, served as academicdean, and co-authored two texts inmathematics and statistics. Jim has aPh.D. in mathematics from the Uni-versity of Wisconsin, an MS in petroleumand natural gas engineering from PennState and a BS in mathematics fromMarietta College. ◆

Risk Analysis:Table of Contents

When I was a struggling assistant professor ofmathematics, I yearned for more ideas, for wewere expected to write technical papers andsuggest wonderful projects to graduate students.Now I have no students and no one is countingmy publications. But, the ideas have beencoming. Indeed, I find myself, like anyone whoteaches classes to professionals, constantlystumbling on notions worth exploring.

The articles herein were generated during afew years and written mostly in about 6months. A couple of related papers found theirway into SPE meetings this year.

I thank the hundreds of people who listenedand challenged and suggested during classes.

I owe a lot to Susan Peterson, John Trahanand Red White, friends with whom I argue andbounce ideas around from time to time.

Most of all, these articles benefited by the carefulreading of one person,Wilton Adams,who has oftenassisted Susan and me in risk analysis classes.During the past year, he has been especially helpfulin reviewing every word of the papers I wrote forSPE and for this publication.Among his talents area well tuned ear and high standards for clarity.I wish to thank him for his generosity.

He also plays a mean keyboard, sings agood song and is a collaborator in a certainperiodic culinary activity.

You should be so lucky. ◆

Acknowledgements

▲

A Guide To Risk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 3Central Limit Theorem – Polls and Holes . . . . . . . . . . . 5Estimating Pay Thickness

From Seismic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Bayes’ Theorem – Pitfalls . . . . . . . . . . . . . . . . . . . . . . . . 12Decision Trees vs. Monte Carlo Simulation . . . . . . . . 14When Does Correlation Matter? . . . . . . . . . . . . . . . . . 20Beware of Risked Reserves . . . . . . . . . . . . . . . . . . . . 24Decisioneering Company Profile . . . . . . . . . . . . . . . . 26Landmark Company Profile . . . . . . . . . . . . . . . . . . . 28Palisade Company Profile . . . . . . . . . . . . . . . . . . . . . . 30

Table of Contents

▲

Biography

▲

14 Risk Analysis

Risk Analysis:Invoking Tools

DDDecision trees and Monte Carlo simulation are thetwo principal tools of risk analysis. Sometimes usersapply one tool when the other would be more help-ful. Sometimes it makes sense to invoke both tools.After a brief review of their objectives, methods andoutputs, we illustrate both proper and improperapplications of these well-tested procedures.

Decision trees:Objectives, methodology, resultsDecision trees select between competing alternativechoices, finding the choice – and subsequent choicesalong a path – which either maximizes value orminimizes cost.

The output of a tree is a combination of theoptimal path and the expected value of that path.Thus, decision trees yield one number.The rules ofsolving the tree force acceptance of the path with thehighest expected value, regardless of its uncertainty.

In theory, preference functions can be used inplace of monetary values, but in practice, usersseldom go to this level.

Decision trees allow limited sensitivity analysis,in which the user perturbs either some of theassigned values or some of the assignedprobabilities, while monitoring the overall treevalue. Typically, the user varies one or twoparameters simultaneously and illustrates the resultswith graphs in the plane (how the tree valuechanges when one assigned value is varied) or inspace (how the tree value varies when two assigned

values are varied together).The traditional tornadochart also is used to show how each perturbedvariable affects the tree value when all other valuesare held fixed. This chart takes its name from theshape it assumes when the influences of theperturbed variables are stacked as lines or bars, withthe largest on top.

Trees, along with their cousins influence diagrams,are particularly popular for framing problems andreaching consensus. For small to moderate sizeproblems, the picture of the tree is an effective meansof communication.

One of the most important problem typessolvable by trees is assessing the value of informa-tion. In this case, one possible choice is to buyadditional information (seismic interpretation, welltest, logs, pilot floods). Solving the tree with andwithout this added-information branch and takingthe difference between the two expected valuesyields the value added by the information. If theinformation can be bought for less than its imputedvalue, it is a good deal.

Monte Carlo simulation:Objectives, methodology, resultsMonte Carlo models focus on one or moreobjective functions or outputs. Favorite outputsinclude reserves, total cost, total time and netpresent value (NPV). Their respective inputsinclude areal extent, net pay and porosity; lineitem costs; activity times; and production

Decision Trees vs.

Monte CarloSimulation

Trees, along with their cousins influence diagrams,are particularly popular for framing problems andreaching consensus.

Risk Analysis 15


forecasts, price forecasts, capital forecasts andoperating expense forecasts.

A Monte Carlo (MC) simulation is the processof creating a few thousand realizations of the modelby simultaneously sampling values from the inputdistributions.The results of such an MC simulationtypically include three items: a distribution for eachdesignated output, a sensitivity chart listing the keyvariables ranked by their correlation with a targetedoutput, and various graphs and statistical summariesfeaturing the outputs.

Unlike decision trees, MC simulations do notexplicitly recommend a course of action or make adecision. Sometimes, however, when there arecompeting alternatives, an overlay chart is used todisplay the corresponding cumulative distributions,in order to compare their levels of uncertainty andtheir various percentiles.

Example: A tree that can be converted to simulation modelSometimes a decision tree can be reconstructed as aMonte Carlo simulation. This is especially true oftrees with one root-decision node. The simulationwould present the result as a distribution, whereasthe tree solution would only give the mean value ofthe distribution.Take, for example, the tree in Figure1, where the object is to estimate reserves. Note thepattern of a sequence of chance nodes withoutinterspersed choices. This is a giveaway forconversion. Incidentally, a + sign signifies where anode and its subsequent branches and nodes havebeen collapsed. In this case, imagine copying thenode at “big area” and pasting it at the “med area”and “small area” nodes.We shall convert this tree toa simulation and compare the nature and extent ofinformation that these two models provide us.

In fact, the tree of Figure 1 is not a standarddecision tree. Instead of the convention of adding thevalues as one moves out along a path, this treemultiplies them together to get barrels.The output issimply the expected value obtained by considering thepossible combinations of area, pay and recovery

obtained by following some path, multiplying each ofthem by the corresponding probability, which isobtained by taking the product of the branchprobabilities, and summing these weighted values.

In the tree, each parameter is discretized to onlythree representative values, signifying small,medium,and large.Thus, area can take on the values 1,300,2,000 or 2,700 acres.Of course in reality, area wouldbe a continuous variable, taking on any value inbetween these three numbers. In fact, most peoplewould argue 1,300 is not an absolute minimum, and2,700 is not an absolute maximum.They might bemore like P10 and P90 or P5 and P95 estimates.Wecan think of a small area being is some range, sayfrom 1,000 to 1,500 with 1,300 being a suitablerepresentative of that class. Similarly, 2,700 mightrepresent the class from 2,500 to 3,000 acres. Eachof these subranges of the entire range carries its ownprobability of occurrence.

For simplicity, we have made all the triples ofvalues symmetric (for example, 1,300, 2,000 and2,700 are equally spaced), but they could beanything. For instance, area could have the values1,300, 2,000 and 3,500 for small, medium and large.Likewise, we have assigned equal weights to thesmall and large representatives, again for simplicityand ease of comparison.

We have made another simplification:we assumeall possible combinations are realizable. Sometimes,the large value of area would be paired with three

P0 P5 Mode P95 P100

Area 976.3 1,300 2,000 2,700 3,023.7

Pay 10.75 20 40 60 69.25

Recovery 76.9 100 150 200 223.1

Table 1. Defining parameters for inputs (Area, Pay,Recovery) to Monte Carlo model

Failure

Success

small Area

1.0%%0

tree#1

99.0%0

med Area

25.0%1300

50.0%2000 +

+

big Area

small Pay

25.0%2700

med Pay

25.0% 20

50.0% 40 +

+

big Pay

small Recov

25.0% 60

med Recov

25.0% 100

50.0% 150

big Recov25.0% 200

Figure1. A decision tree begging to become a Monte Carlo model.

16 Risk Analysis


relatively large values of pay, in thebelief these two parameters aredependent.This is simple to accom-modate in a tree.

Building the corresponding Monte Carlo simulation modelConverting to an appropriateMonte Carlo model requiresfinding a suitable distribution foreach of the three inputs – area, payand recovery. In light of thediscussion above, we took thesmall values to be P5 and the bigvalues to be P95.We also selected

triangular distributions is each case (which is howwe obtained our P0 and P100 values). Theresulting distributions are shown in Table 1.

From the tree analysis,we can calculate the expectedvalue as well as finding the two extreme values (smallestand largest) and their respective probabilities:

Expected value 12MMSTBMinimum value 2.6MMSTB, P(min) = 1/64Maximum value 32.4 MMSTB, P(max) = 1/64What we cannot determine from the tree analysis is

how likely the reserves would exceed 5 MMSTB, howlikely they would be between 5 MMSTB and 15MMSTB, how likely they would be less than 12MMSTB, and so on.

The histogram from the Monte Carlo analysis isshown in Figure 2, and its corresponding percentilesin Table 2.Thus,while the mean value coincides withthe mean from the tree analysis (in part because of thesymmetry of the inputs and lack of correlation), welearn much more about the range of possibilities:

• 90% of the values lie between 4.9 and 20.9MMSTB;

• there is about a 56% chance of finding less than12 MMSTB (it is close to the median);

• there is only about a 20% chance of exceeding16 MMSTB; and

Modifying the tree to account fordependency among inputsConsider the modified tree branches shown inFigure 3, where the user believes larger areascorrespond to larger net pays. Without going intodetail, the corresponding Monte Carlo simulationwould handle this relationship by specifying a numberbetween -1 and +1 to indicate the degree ofcorrelation between the inputs. If the correlation were

between 0 and +1, for example, then during thesimulation, realizations sampling larger values of areawould tend to be matched with larger values of pay.This technique is routine, simple to implement andcan use historical data when it is available to estimatethe correlation coefficients.The resulting simulation inthis case would feature more extreme values (bothlarge and small) resulting in a larger standard deviationand wider 90% confidence interval than theuncorrelated case. Somewhat surprisingly, the meanvalue will also increase, but not as much as the range.

Introducing correlation and examining its effectis a standard exercise in Monte Carlo classes. Ifanything, the simulation handles these pairedrelationships more easily than the tree, where thedependency is more hard-wired.

Example: Comparing alternative mud systemsSometimes a problem can be described with a tree andenhanced with a simulation.A case in point would bea drilling cost estimate where there is a choice of mudsystems. Consider the following problem.

For 4 years, you have been drilling in a fieldwhere there is a danger of differential sticking. Ofthe 20 wells drilled to date, six of them encounteredstuck pipe. In five of those wells, the drillers fixed theproblem, spending from 4 to 18 days of rig timewith an average of 9 days. In one well, theyeventually had to sidetrack at a marginal cost U.S.$700,000 for materials and 12 days of rig timebeyond the attempt to free the stuck pipe, whichhad been 18 days.Average time to drill the wells isapproaching 45 days.The rig rate is $60,000/day.

One option for the next well is to use oil basedmud, which would greatly reduce the chance ofstuck pipe (to 1/10) and the time to fix it (to 6 days),but cost $70,000/day,

Ordinarily, one would look to the decision treeto compare these two alternatives. First, we calculatethe expected value of drilling with the conventionalwater based mud system for the three cases: noproblem, stuck but fixed and sidetrack.

No problem cost: (45 days) * ($60,000) = $2,700,000Stuck and fixed cost: (45 + 9 days)*($60,000) = $3,240,000Sidetrack cost: (45 + 30)*($60,000)

+ $700,000 = $5,200,000The respective probabilities we assign are 14/20,

5/20 and 1/20.Next, we estimate the costs and probabilities for

the oil based mud system.

Percentage MMSTB5% 4.9

10% 6.015% 6.820% 7.525% 8.230% 8.935% 9.540% 10.145% 10.850% 11.455% 11.960% 12.665% 13.470% 14.275% 15.080% 16.085% 17.390% 19.095% 21.2

Table 2. Percentiles for output (reserves, MMSTB)from simulation

Distribution for Reserves/Sample/F6Distribution for Reserves/Sample/F6

0.1000.0900.0800.0700.0600.0500.0400.0300.0200.0100.000

0 7 14 21 28 35

5% 5%4.85 20.88

Meanv=11.96266

Figure 2. Histogram from simulation

Risk Analysis 17

No problem cost: (45 days)*($70,000) = $ 3,150,000Cost when stuck pipe: (45+6)*($70,000) = $3,570,000The respective probabilities are 9/10 and 1/10.The resulting tree is shown in Figure 3, indicat-

ing the correct decision would be to use the waterbased mud with an expected value of $2,960,000rather than the oil-based alternative with anexpected value of $ 3,171,000.

The Monte Carlo approach The Monte Carlo model shown in Table 3 capturesthe same information as the tree model, but uses itdifferently. Specifically, the simulation model allowsthe user to estimate ranges for the various activitiesand costs. For comparison purposes, the historicalminimum and maximum values are treated as P5 andP95 to build input distributions, with the option forthe user to set these percentages. Most peopleestimating ranges of this sort, especially costestimators, tend toward right-skewed distributions.

P(StuckWater) 30.0%P(SideTrack) 5% LowP 5P(StuckOil) 5% HighP 95

Sample Plow Mode PhighDayRateWater $ 60 60 60 60DayRateOil $ 70 70 70 70DaysWater_NoStuck 45.0 40.5 45 49.5DaysOil_NoStuck 45.0 40.5 45 49.5DaysSideTrack 12.0 10.8 12 13.2StuckWater? 0 1 = stuckSideTrack? 0 1=sidetrackStuckOil? 0 1 = stuckDayWater_Stuck 10.4 4 8 18DaysOil_Stuck 7.2 3 6 12ExtraCost_ST 700.0CostWater $ 2,700 CostOil $ 3,150 TimeWater 45.0TimeOil 45.0

Table 3. Monte Carlo model to compare oil based and water based mud systems


18 Risk Analysis


Certainly, in thismodel, we shouldcare about morethan just the ex-pected value.For in-stance, we may wantto know the prob-ability of exceedinga certain value, suchas $4,000,000. We

may also want to question other inputs (see dis-cussion below).

Figure 5 shows the comparison with the assump-tions taken from the problem statement. Figure 6shows a variation where the oil-based mud takes only40 days on average to drill and the chance of sidetrackwith water mud is 10%, not 5%. The differencebetween the two cases is clear: as the probability ofsidetracking increases, the oil-based mud presents aless risky alternative (the density function of

outcomes is much narrower). Similar analyses can bemade easily in the Monte Carlo model.

Which model is better in this instance?Both the tree and the simulation offer informationto the user and to the larger audience of thepresentations that can be prepared from them.Thetree emphasizes the two contrasting choices andidentifies the extreme outcomes and their prob-abilities of occurrence. It is both simple and effec-tive.Yet, the tree does not explicitly acknowledge theunderlying uncertainty in each of the contributoryestimates (such as days for problem free drilling, costof mud system,days for trying to free the stuck pipe,days for sidetracking). The user can handle theseuncertainties – one or two at a time – by using thetree framework to do a trial-and-error sensitivityanalysis, and any user would be remiss in avoidingthis. But the overall uncertainty is made moreexplicit in the Monte Carlo simulation, where the

Decision Trees Monte Carlo simulationObjectives make decisions quantify uncertaintyInputs discrete scenario distributionsSolution driven by EV run many casesOutputs choice and EV distributionsDependence limited treatment rank correlation

Table 4. Comparison between Decision trees and Monte Carlosimulation

Risk Analysis 19


very nature of the model begs the user to specify therange of uncertainty.

On balance, if I had to pick one model, I wouldpick the simulation, in part because I have had farbetter results analyzing uncertainty and presentingthe details to management when I use simulation.Butwhatever your preference in tools, for this problemthe combination of both a tree and the simulationseems to be the most useful.

Which model is better in general?When is Monte Carlo more appropriate? Ingeneral, Monte Carlo seems to have more variedapplications. Any (deterministic) model you canbuild in a spreadsheet can be converted to aMonte Carlo model by replacing some of theinput values with probability distributions. Thus,in addition to the popular resource and reservevolumetric product models, people build MonteCarlo models for costs of drilling and facilities,time for projects (up to a point, when projectschedule software is more applicable), productionforecasts, and all sorts of cashflows, includingthose with fiscal terms. Decision trees must comedown to something that compares alternativechoices under uncertainty.

Combining simulation and treesIt is now possible to build a decision tree wheresome or all of the branches from chance nodescontain probability distributions rather than

values. Then one can run aMonte Carlo simulation,where each iteration is anew solution of the tree.This technology is relativelynew and relatively untested.Although it holds promise,this combination requiresthe user to be expert at bothdecision trees and simula-tion. Some people raisequestions of logic when wesolve a tree populated withvalues that are not meane s t ima t e s , bu t r a t he rextreme possibilities of theinputs. In time, whentechnical papers appear,many of these issues will be resolved.

SummaryTable 2 summarizes some ofthe features of the twomethods, illustrating theirprinciple differences. Don’tbe surprised to find someone using a tree solutionto a problem you elect to solve with simulation orvice versa. Do try to use common sense and do asmuch sensitivity analysis as you can regardless ofyour choice. ◆

E22: Meanv=2937.594

E23: Meanv=3148.974

Distribution for CostWater/Sample/E22

1.6001.4001.2001.0000.8000.6000.4000.2000.000

2 3 4 5

Valu

es in

10

-3vv

Values in Thousands

Figure 5. Comparing oil and water based mud systems

Distribution for CostWater/Sample/E22

1.6001.4001.2001.0000.8000.6000.4000.2000.000

2 3 4 5

E22: Meanv=2833.63

E23: Meanv=2826.767

Values in Thousands

Valu

es in

10

-3vv

Figure 6. New cost comparison when oil based mudresults in faster drilling.

Failure

Success

small Area

1.0%%0

tree#1

99.0%0

med Area

25.0%1300

50.0%2000

25.0%2700

small Pay

med Pay

25.0% 40

50.0% 60 +

+

big Pay25.0% 90 +

small Pay

med Pay

25.0% 30

50.0% 40 +

+

big Pay25.0% 80 +

big Area

small Pay

med Pay

25.0% 15

50.0% 30 +

+

big Pay25.0% 60 +

Figure 3. Tree incorporating dependence between area and pay

Water Mud

Oil Mud

FALSE0

tree#1

TRUE0

Stuck/Fix

No Problem

5.0%3570

95.0%3150

Chance3171

Decision2960

Stuck/Fix

No Problem

25.0%3240

70.0%2700

Chance2960

Stuck/Fix 5.0%5200

Figure 4. Tree to compare oil based vs. water based mud systems

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