1Uncertainty Analysis in ReservoirCharacterization and Management:How Much Should We Know AboutWhat We Don’t Know?

Y. Zee MaSchlumberger, Greenwood Village, Colorado, U.S.A.

ABSTRACT

A reservoir is the result of geologic processes and is not randomly generated. However, manyuncertainties exist in reservoir characterization because of subsurface complexity and limiteddata. Uncertainties can be mitigated by gaining more information and/or using better scienceand technology. How much uncertainties should be mitigated depends on the needs of de-cision analysis for reservoir management and the cost of information. Uncertainty analysisshould be conducted for investigational analyses and for decision analysis under uncertaintyand risk. To know what needs to be known and what can be known should be the main focalpoints of uncertainty analysis in reservoir characterization and management.

INTRODUCTION

No uncertainty exists in a reservoir, only uncertainty inour understanding and description of it. A reservoir isnot random; it was deposited geologically and evolvedinto a unique hydrocarbon-bearing entity. Matheron(1989, p. 4) once remarked ‘‘Randomness is in noway auniquely defined or even definable property of the phe-nomenon itself. It is only a characteristic of the model.’’If a reservoir is not random, why should we use a prob-abilistic method to build a reservoir model? If a reser-voir is deterministic in nature, why should we analyzeuncertainties in reservoir characterization?

Before answering these questions, it is useful to brief-ly review the history of uncertainty analysis and twodoctrines regarding the course of nature in other scien-

tific fields. Many consider that, historically, probabilityas a measure of uncertainty was first introduced bymonks at the Port-Royal monastery in Paris (Arnauldand Nicole, 1662), although this may not be universallyagreed on. The uncertainty notion in their exemplaryargument, that fear of harm should be proportional tothe probability of an event inspired the development ofthe probability theory for uncertainty analysis, whichin the beginningwas appliedmostly in games of chance.Uncertainty analysis was largely ignored in the sci-entific world until the advent of quantum mechanics,which revolutionized physics in the 20th century. Al-though nearly all physical sciences were traditionallyconsidered deterministic,many phenomena in quantummechanics could not be explained byNewtonian deter-ministic laws (Popper, 1979; Hilgevoord and Uffink,

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Ma, Y. Zee, 2011, Uncertainty analysis in reservoir characterization and

management: How much should we know about what we don’t know?, in Y. Z.

Ma and P. R. La Pointe, eds., Uncertainty analysis and reservoir modeling:

AAPG Memoir 96, p. 1 – 15.

Copyright n2011 by The American Association of Petroleum Geologists.

DOI:10.1306/13301404M963458

2006). As a result, indeterminism and uncertainty anal-ysis (initially commonly referred to as uncertainty prin-ciple) were proposed by several prominent physicistsin the late 1920s. The debate centered on whether theworld was indeterministic or whether quantum me-chanics simply provided an incomplete description ofa fully deterministic world. This is now referred to asthe ‘‘EPR paradox’’ (Honderich, 2005, p. 237), after theinitials of Einstein, Podolsky, and Rosen, who related‘‘randomness’’ and uncertainty in quantum mechan-ics to one’s ignorance or inability to fully understandor describe some properties of reality (Einstein et al.,1935). This is very similar to reservoir descriptions. Isa reservoir random? Or is that our description cannotbe complete while the reservoir itself is deterministicin the underlying course of nature?

Reservoirs are not random, but are causal in na-ture, yet indeterministic because of our inability to for-mulate a complete and precise description and char-acterization. In other words, although a reservoir wasgeologically deposited, and thus causal, it cannot bedeterministically defined or completely determined be-cause of subsurface complexity and limited data. How-ever, despite the fact that a reservoir or, more generally,a geologic phenomenon is not random, it is legitimatetomodel it using a stochastic function (commonlywith a‘‘random’’ component) because of the indeterminacyaspect. In fact, probabilistic approaches provide meth-ods of choice for modeling subsurface heterogeneities,including geologic facies (Ma, 2009b) and petrophysicalproperties (Journel and Alabert, 1990). Furthermore, re-gardless of the methodology used, uncertainties willalways exist in reservoir characterization, and proba-bilistic methods provide viable tools for uncertaintyanalysis and quantification (Deutsch and Journel, 1992).

Although the uncertainty principle may have beensomewhat accepted in reservoir characterization, itspracticality has been rather limited. Some consider tra-ditional frequency probability theory impractical be-cause of infrequent data (Gillies, 2000; Ma, 2009b).Others think that the Bayes belief network cannot befully believed because it is too subjective (Popper, 1959).Still others feel that spatial statistics, of which geosta-tistics is a major branch, is too specialized and too dif-ficult to be popularized within the main geosciencecommunity. In other words, three roadblocks limit themore extensive use of probability theory for reservoiruncertainty analysis: (1) unbelieved Bayes belief net-work (e.g., Popper, ‘‘I do not believe in belief’’, 1979,p. 25, although this is not on geoscience application),(2) infrequent frequency probability, and (3) too spe-cialized spatial statistics.

From a practical point of view, many examples offailed projects have occurred because of the failure to

study subsurface heterogeneity and the lack of un-certainty analysis for reserve estimates and risk mit-igation in the reservoir management and decisionanalysis. A successful drilling technology project some-times becomes a great failure economically for thesereasons.

The two goals of uncertainty analysis are to quantifythe reservoir uncertainty and to reduce it. This is criticalbecause optimal reservoir management, including pro-duction forecasting and optimal depletion, requiresknowledge of the reservoir characterization uncertain-ties for business decision analysis. Three questions needto be answered in reservoir uncertainty analysis.

� ‘‘How much do we know?’’ This means that weneed togather asmuchknowledge as possible aboutthe reservoir, and analyze and interpret all availabledata.

� ‘‘How much can we know?’’ Not only should wegather as much data as we can, but we should alsothoroughly explore the data to extract the maxi-mum information. Thus, we would conduct ex-ploratory data analysis using geologic, geophysical,and petrophysical principles, as well as geostatis-tical methods. Moreover, we ought to weigh thepotential benefits of acquiring additional data togain sufficient information and knowledge tomake critical business decisions. In other words,we must weigh the value of information (VOI) ver-sus the cost of information (COI).

� ‘‘Howmuch should we try to know?’’ Like the EPRparadox in quantum mechanics, subsurface com-plexity and limited data prevent us from achievinga total understanding of a reservoir or completelydescribe every detail of subsurface heterogeneities.Therefore, our main emphasis in reservoir charac-terization should be to define relevant objectivesthat impact the business decision and to find re-alistic solutions accordingly. In other words, wemust define what we really need to know or howmuch we should try to know to make the neces-sary business decisions.

PREDICTION, MODELING, ANDUNCERTAINTY ANALYSIS

Prediction and Modeling

Many have aptly expressed the importance of predic-tion in science. Take, for example, Poincare’s well-known saying, ‘‘It is far better to foresee even withoutcertainty than not to foresee at all’’ (Poincare, 1913,

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p. 129). In reservoir characterization, prediction is notonly necessary, but also ubiquitous. Because of sub-surface complexity and limited data, however, ourpredictions of reservoir properties inevitably involveuncertainties.

Geologic and reservoir models are commonly builtto help field development planning or simply to en-hance our geologic understanding of a reservoir. Mod-eling is a form of prediction using one or more quanti-tative methods, that involve uncertainty. It is highlychallenging to analytically define three-dimensional(3-D) reservoir properties because complex subsurfaceheterogeneities occur at many different scales. We canmodel them, however, using geologic and petrophys-ical principles and probabilistic methods. For a modelto represent reality as accurately as possible, the mod-eling method should integrate all the available data in acoherent way. Because no unique solution can be foundwith the lack of data, a good modeling method shouldenable uncertainty analysis and quantification.

In reservoir characterization, geostatistical methodshave been used for both modeling and uncertainty anal-ysis (Deutsch and Journel, 1992; Ma et al., 2008). Advan-tages of geostatistical methods, as compared with tra-ditional mapping techniques, include the capabilitiesof better modeling subsurface heterogeneities and un-certainty analysis of geologic and petrophysical vari-ables, which will be further discussed in a followingsection, Inference Uncertainty.

Uncertainty Analysis and Modeling

To analyze uncertainty in reservoir characterization andmodeling, it is convenient to put it under the frame-work of a scientific process. That is, uncertainty in thereservoir characterization and modeling is caused bythe uncertainty in the input data and uncertainty in theinference, as shown in Figure 1. Uncertainty analysis,therefore, should include studies of data inaccuraciesand inference uncertainty from data to estimation of

reservoir properties. These will allow amore accuratequantification of the uncertainty in themodeling result,which then can be used for reservoir management de-cision analyses.

An integrated reservoir modeling workflow typi-cally includes data analysis, interpretations, selectionof an appropriate method that integrates the availabledata from various disciplines, and definitions of themodeling parameters. All these tasks involve uncer-tainties. In other words, uncertainty can be an inaccu-racy in data or indeterminacy in reservoir parameters.

Measurement Uncertainty

Uncertainty in data is mainly caused by uncertaintiesin measurements and secondarily by data handling.Several organizations, including the International Stan-dardization Organization (ISO), the Bureau Interna-tional des Poids et Mesures (BIPM, 2008; 2009), and theNational Institute of Standardization and Technology(Taylor and Kuyatt, 1994), have proposed guidelinesfor reporting measurement uncertainties. Tradition-ally, the most common approach was to offer the bestestimate that includes the expression of systematic andrandom errors in measurements. More recently, ISOand BIPM have changed their guidelines to recom-mend an approach involving the best estimate withassociated uncertainties. The rationale they offeredfor this approach was that ‘‘no measurement is exact’’and ‘‘it is not possible to state how well the essentiallyunique true value of the measurand is known, butonly how well it is believed to be known’’ (BIPM, 2009,p. 2–3).

Uncertainty should be defined so as to characterizethe dispersion of a measurement. Probability distribu-tions are typically used to characterize such measure-ment uncertainties. The most commonly used proba-bility function for stochastic uncertainty characterizationis the Gaussian density, in which variance or standarddeviation characterizes the dispersion of the measure-ment values. However, a nonaleatoric uncertainty com-monly, but not always, requires a non-Gaussian den-sity function.

Inference Uncertainty

Inference uncertainty is much more complex than mea-surement uncertainty and is more difficult to deal with.In reservoir characterization, inference uncertainties in-clude conceptual, interpretational, and methodologicaluncertainties, to name a few. Because of the high ex-pense of drilling, geoscientists commonly have tomake assumptions and create conceptual deposition-al models, even with limited data in the early stage of

Figure 1. Illustration of the relationships among the inputdata, reservoir characterization and modeling process, andoutput result. Uncertainty in the result is caused by uncertaintyin the input data and/or uncertainty in the inference.

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exploration. Therefore, conceptual depositionalmodelstypically contain significant uncertainties. In addition,geologic and seismic interpretations are commonly lim-ited by the quantity and quality of data and are proneto cognitive biases, especially a confirmation bias (Ma,2010). As such, interpreted results also contain signif-icant uncertainties. Examples of uncertainty analysisin well-log and petrophysical interpretations can befound in the chapter by Moore et al. (2011).

The conceptual and interpretational uncertaintiespreviously discussed can propagate into reservoir mod-eling. Moreover, geologic, petrophysical, and statisticalparameters used in modeling have uncertainties. Forexample, few data are available in early field develop-ment, and uncertainty is typically high in the net-to-gross (NTG) ratio (Journel and Bitanov, 2004). Similar-ly, uncertainties in facies proportion and other geologicand petrophysical parameters can be quite significant(Haas and Formery, 2002; Friedmann et al., 2003). Fur-thermore, choice of the modeling method (e.g., be-tween probabilistic and analytical approaches or be-tweendifferent probabilisticmethods) can be ambiguous(Falivene et al., 2006), and specifications of petrophys-ical parameters in reservoir modeling and uncertaintyanalysis are commonly ambiguous as well.

Geostatistics has been used to model geoscience phe-nomena in the last half century (Journel and Huijbregts,1978; Chiles and Delfiner, 1999; Hu and Le Ravalec-Dupin, 2004). Geostatistical techniques enable integra-tions of various data types andmodeling of subsurfaceheterogeneities in depositional facies (Ma et al., 2009)and petrophysical properties (Goovaerts, 2006); how-ever, both kriging and stochastic simulation have sig-nificant limitations in assessing local uncertainties. Forexample, multiple realizations in a traditional stochas-tic simulation are equally probable, and their inferen-tial differences lie only in the random seeds (althoughnew information can change this, which is discussedbelow). It is generally difficult to quantify spatial un-certainties of a reservoirmodel using geostatistics alone.The selection of a model among many realizations fordynamic simulations can be highly challenging as well.Several methods, including gradual deformation, opti-mization (such as probabilistic perturbation and simu-lated annealing), and the Kalman filter (Deutsch andJournel, 1992; Hu et al., 2001; Caers, 2007; Zafari andReynolds, 2007), have been proposed to help select orpostprocess geostatistical reservoir models, with vari-ous degrees of success.

It is, nonetheless, more straightforward to assess un-certainties of some ‘‘composite’’ reservoir variables,such as pore and hydrocarbon volumetrics, using sto-chastic simulations. In fact, applications to volumetricassessments have been quite extensively discussed

(Berteig et al., 1988; Ma et al., 2008), and more exam-ples can be found in several chapters in this volume.

Uncertainty analysis in reservoir characterizationalso includes transfer of the geologic uncertainty in areservoir model to reservoir performance forecasting(Ballin et al., 1993; Vanegas et al., 2011), a stochastichistory match that deals with both geologic and engi-neering uncertainties (Amudo et al., 2008), and uncer-tainty reduction and quantification using model updat-ing with production data assimilation (e.g., Devegowdaand Gao, 2011). To date, several methods have beenproposed to dealwith dynamic uncertainties or transferof the static uncertainty to dynamic uncertainty, in-cluding experimental design (Friedmann et al., 2003),gradual deformation (Hu andLe Ravalec-Dupin, 2004),proxy models (Vanegas et al., 2008), Kalman filters(Zafari and Reynolds, 2007), and distance-based ap-proaches (e.g., Caers and Scheidt, 2011).

Parameter Uncertainty Quantification

Similar to the uncertainties in measurements, uncer-tainties of reservoir parameters are typically charac-terized by a probability distribution. Uncertainty quan-tifications of these parameters in early field developmentbased on various data sources and geologic scenarioshave been discussed by many (Haas and Formery, 2002;Journel and Bitanov, 2004; Caumon et al., 2004). Param-eter uncertainty in experimental design is commonlycategorized into two or three levels, such as low, medi-um, and high, instead of using a probability distribution.

Remarks

Uncertainties in data and inference (e.g., reservoir mod-eling) naturally lead to uncertainties in the results ofreservoir characterization. As previously mentioned,some ‘‘composite’’ reservoir parameters, such as hy-drocarbon pore volume, are expressed by several othergeologic and petrophysical variables, and their globaluncertainties can be quantified by a probability dis-tribution constructed by building multiple scenariomodels and multiple realizations of each contributingproperty.

Probability distributions of uncertain quantities inreservoir characterization are generally empirical, al-though some general mathematical laws may causethem to be nearly normal or lognormal. Addition ofmultiple random variables tends to produce a normaldistribution as a consequence of the central-limit the-orem (Papoulis, 1965, p. 266), and multiplication ofmany variables tends to yield a lognormal distribution(Aitchison and Brown, 1957). The resultant probabi-listic distribution, however, will be highly influenced

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by the distributions of the input variables and theircorrelations, especially when data are limited. For ex-ample, the histogram of the oil-pore volume is highlyinfluenced by the histograms of porosity and oil sat-uration (So), as shown in Figure 2.

THE VALUE OF INFORMATION

Because uncertainty can be considered as an under-determination problem, more data could naturallyreduce the uncertainty. This is the ‘‘value of informa-tion.’’ The VOI is discussed mostly under the frame-work of decision analysis in the oil and gas industry(Cunningham and Begg, 2008; Bratvold et al., 2009;Bailey et al., 2011). The focus here is on the VOI inreducing uncertainties in reservoir characterizationand management. The VOI is most obvious in a totallyrandom system, such as a lottery. The Powerball anal-ogy (Ma and Gomez, 2009) can be used to illustrate theVOI. Powerball currently has two drums of numbers:the first is composed of 59 numbers and the second iscomposed of 39. Five numbers are drawn from the firstdrum, and one (the Powerball) is drawn from thesecond. The probability of winning the grand prize foreach ticket (a combination of six numbers) is approx-imately one in 195 million (Powerball, 2010), such as

ð59� 5Þ! 5!59!

� 1

39¼ 1

195; 249; 054

Assuming one could play after the first two draw-ings, the winning probability would increase to one inslightly more than one million, such as

ð57� 3Þ! 3!57!

� 1

39¼ 1

1; 141; 140

In other words, knowing two numbers would yielda 17,100% increase in the winning odds (i.e., 195.25/1.14 ffi 171). The VOI in reducing the uncertainty inthis case is self-evident.

Similarly, using more data could significantly re-duce uncertainty in reservoir characterization. An ex-ample of VOI by comparing depositional facies clas-sifications using two well logs versus one well log isfound in chapter 6 in this volume. Pitfalls exist, how-ever, because a reservoir is not random or data qualitymay be questionable. More data sometimes causemore ‘‘apparent uncertainties.’’ This complication canbe caused by subsurface heterogeneity or data hetero-geneity or quality. Before examining examples of in-ference bias related to subsurface heterogeneities, var-ious types of data in reservoir characterization arebriefly reviewed.

Because reservoir characterization is a multidisci-plinary process, it involves different types of data,including hard data and soft data. Typically, hard dataprovide direct information, whereas soft data are con-sidered to be indirect information. For example, coredata commonly provide direct measurements of rockformations, whereas seismic data commonly provideindirect information of rock formations. Some welllogs may be used as direct information, whereas othersare rather indirect information (Moore et al., 2011).Although interpretations are not primary data, some-times they can be used as soft data. For example, anearly task in the reservoir characterization is geologicinterpretation, the results of which are commonly usedfor reservoir modeling.

Hard Data and the Value of Information

In petroleum geostatistics, hard data are commonlysynonymous with reliable information. A simple syn-thetic example shown in Figure 3 demonstrates theVOI using NTG ratios. Assume that only three harddata points at the well locations in Figure 3A are ini-tially available. It would be difficult to know if the areacontains significant hydrocarbon resources because con-siderable uncertainty exists regarding the global NTGand depositional model. By adding four wells, whichshow high NTG ratios (Figure 3B), it would be reason-able to delineate a channel complex in the area, assuming

Figure 2. Histograms of(A) porosity, (B) oil satu-ration, and (C) oil porevolume.

Uncertainty Analysis in Reservoir Characterization and Management 5

that the delineation is consistent with the regional ge-ology. Thus, the four additional data points would pro-vide valuable information that improves our under-standing of the prospect.

Soft Data and the Value of Information

Typically, reservoir characterizations lack sufficient di-rect measurements. Thus, reservoir models built onlywith hard data tend to be too unconstrained and lack-ing in geologic realism (Massonnat, 1999). However,geologic conceptual models tend to capture regionalcharacteristics but inadequately represent smaller scalesubsurface heterogeneities. An integrated approach thatcombines both geologic knowledge and hard data canhelp more accurately model the reservoir and reducethe uncertainty.

Many geostatistical methods enable integration ofsoft data, provided that geologic knowledge is numer-ically represented in trend maps or trend curves. In ad-dition, a methodology based on the hierarchy of sub-surface heterogeneities can improve the integration ofa variety of geologic information, including use of strat-igraphic reference classes and spatial propensity anal-ysis (Ma et al., 2009). For example, facies data can beintegratedwith a depositional conceptual model to cre-ate facies probabilities, which then can be used to con-strain a geostatistical model to be geologically morerealistic (Ma, 2009b; Yu et al., 2011).

Value of Information and Sampling Bias

In theory, as more data become available, our knowl-edge of the reservoir should improve. A sampling bias,however, can complicate the VOI. Consider the exam-ple illustrated in Figure 3. The average NTG based onthe first three wells was 23%. After drilling the fouradditional wells, the average NTG from all the sevenwells was apparently 57%. However, the delineatedchannel complex, which represents about three-fifths(60%) of the area, had five data points. The overbankarea, which represents two-fifths (40%) of the area, had

only two data points. Thus, a sampling bias exists, whichshould be accounted for. The global NTG estimatedusing the polygonal tessellation or propensity zoning–based declustering (Ma, 2009b) is about 48%. The 57%global NTG from the average of the raw data repre-sents nearly a 19% overestimation (i.e., [0.57 � 0.48]/0.48 ffi 19%).

Sampling bias is generally unavoidable in explora-tion and production. Commonly, simply for economicreasons, it is intentional. No one wants to spend tensof millions of dollars to drill a dry hole just for collect-ing data. Operators drill wells in what is thought to bethe best parts of the reservoir or play, and data ob-tained from these wells may not represent either aver-age or the full range of reservoir properties. Whereasavoiding sampling bias is difficult in general, the maintask should be learning how to incorporate it in busi-ness and reservoir modeling decision making. Aimingto drill in the ‘‘sweet spots’’ in light of subsurface het-erogeneities makes predictions from data to a reser-voir model particularly treacherous. In some cases,declustering can be performed to mitigate samplingbias. In other cases, inference needs to be made thataccounts for the sampling bias (Ma, 2009a).

Value of Information versus Cost of Information

The VOI lies in reducing uncertainty when more in-formation is used. From a perspective of reservoir man-agement, we must also consider the COI. The differ-ence between the VOI and COI is known as the ‘‘netvalue of information’’ (NVOI). In reservoir character-ization and management, the following questions thatweigh VOI and COI commonly arise:

� Should we acquire a new 3-D seismic survey?� Should we drill a few new wells to delineate the

reservoir?� Should we core this well?

To answer these questions, we need to estimate theVOI versus the COI. Generally, if the VOI outweighs

Figure 3. Illustration of the VOI using NTGratios for reservoir delineation. (A) Onlythree data points were initially available.(B) Four additional data points havehelped delineate the channel complex(shown between the two dotted lines).

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the COI, that is, a positive NVOI, it is worth obtainingnew information. In short, more data will help moreaccurate reservoir characterization; but for reservoirmanagement, the benefit of collecting more data (i.e.,VOI) might be weighed against the cost of collectingdata. Obtaining the necessary data at the minimumcost is one important principle for reservoir manage-ment (Thakur, 1990).

Cost of Misinformation

An important element of the VOI is the quality of in-formation or data quality. If data quality is very poor, itdoes not provide useful information. In fact, it maylead to erroneous results. This is the ‘‘cost of misin-formation’’ (COM). The COM includes both the COIand the damage caused by the misinformation. Dam-age from the COM could be much greater than theCOI. In the example shown in Figure 3B, imagine thatthe actual NTG datum in the central area was 10%instead of 100%. In that case, the depositional modeland the global NTG ratio would be wrong. This likelywould lead to a misunderstanding of the reservoir anda poor decision for reservoir management.

Clearly, the COM includes the cost of errors simplybecause errors represent misinformation. Jablonowskiet al. (2008) discussed how to assess the cost of errorsin reserve estimates and other production and eco-nomic variables.

VARIABILITY, UNCERTAINTY, RISK, ANDDECISION ANALYSIS

Although the definitions of variability, uncertainty, andrisk are provided in the glossary of this volume, it isuseful to discuss them here in more detail. In particular,exploring the relationships among these concepts canilluminate our understanding of their differences andapplicabilities in reservoir characterization and reser-voir management.

Variability and Uncertainty

In reservoir characterization, ‘‘variability’’ refers to themagnitude of change of a geologic process, petrophys-ical property, or any reservoir variable. Variability istotally objective as it is simply a property of a phenom-enon. Generally, subsurface variability is high be-cause multiple scales of heterogeneities exist, such asin structure, stratigraphy, depositional facies, petro-physical properties, and fluids. As such, characteriza-tion of subsurface variability is rather complex. Geo-

statistical methods are commonly used to model thespatial variability of facies and petrophysical variables(Deutsch and Journel, 1992; Ma, 2009b; Ma et al., 2011).Note that early geostatistical approaches mostly usedkriging methods because of their local estimation ac-curacy. Conditional stochastic simulation was pro-posed (Journel and Huijbregts, 1978; Dubrule, 1989) tobetter model the spatial variability of geoscience phe-nomena. Although various kriging methods are stilllargely used in themining application, stochastic simu-lation has beenmore commonly used in petroleum geo-statistics, especially to describe and characterize the con-nectivity of rock properties (Journel and Alabert, 1990).

Uncertainty, however, is simply the result of ournot knowing enough about a variable. Uncertainty mayrefer to inaccuracy in data or indeterminacy or indef-initeness of a variable. Uncertainty differs from vari-ability in that even when a parameter has no variability(i.e., a constant), uncertainty may still exist simply be-cause we have no information about it. However, un-certainty is commonly highly correlated with vari-ability because high variability tends to cause moreunknowns, consequently more uncertainties. An ex-ample of future oil price, which is discussed below, willfurther illustrate this relationship.

Error and Uncertainty

It is important to distinguish uncertainty from error.Whereas an error is the difference between an indi-vidual result and the true value of a quantity, uncer-tainty of a quantity takes the form of a range as a resultof the unknown factors. Nevertheless, an intrinsic re-lationship exists between error and uncertainty (Ma,2010). Larger uncertainties in the input data are morelikely to cause errors in the reservoir characterizationresult and business decision. Conversely, errors in geo-logic, geophysical, petrophysical, or engineering datawill cause more uncertainties in reservoir characteri-zation and modeling.

Uncertainty and Risk

In common language, ‘‘risk’’ may simply refer to thepossibilities that undesirable events may occur. Forexample, a risk exists that a well might be a dry hole,or, smoking increases the chance of lung cancer. In thissense, risk and uncertainty have somewhat similarconnotations. More theoretically, however, ‘‘risk’’ isdefined as the product of the probability of occur-rence of an undesirable event and the consequence ofits occurrence (see Glossary). Therefore, risk has twocomponents: an uncertainty component and a conse-quence component. Note that although sometimes

Uncertainty Analysis in Reservoir Characterization and Management 7

risk apparently refers to variability (Markowitz, 1991,p. 188) or uncertainty, it may still implicitly connote anunspecified consequence of the uncertainty.

Because of the consequence component of the riskequation, risk has a direct impact on decision analysis.For example, a possible perdition (or any loss) causedby a bad prediction is part of risk analysis but not partof uncertainty analysis per se. In fact, it is commonlyargued that the consequence of being wrong must dom-inate the probability of being wrong in decision anal-ysis. As previously mentioned, the Port-Royal authors’historic argument that fear of harm should be propor-tional to the probability of an event (Arnauld andNicole,1662) supports the notion of uncertainty analysis, butit says nothing about the consequence component ofthe risk. As Bernstein (1998, p. 100) noted regardingthe Port-Royal philosophy, ‘‘only the pathologicallyrisk-averse make choices based on the consequenceswithout regard to the probability involved.’’ By con-trast, modern risk analysis, which was developed un-der the framework of utility theory, argues that ‘‘onlythe foolhardy makes choices based on the probabilityof an outcome without regard to its consequences’’(Bernstein, 1998, p. 100).

Risk and Reward

Risk should be discussed with reward because riskcould be reduced to zero if one does not care about thepotential benefit. It is well known that the oil explora-tion and production business is highly risky, but po-tential rewards should not be ignored. Otherwise, noone would take the risk to find oil. Pratt (1952) gavegood examples of risk taking in the early days of the oilbusiness. He described that many early explorers inthe United States took high risks to drill wells at ‘‘un-favorable’’ locations and discovered major oil fields.He summarized this phenomenon by saying (p. 2235)‘‘Since the very inception of the industry, the findingand producing of oil in the United States have beencarried on by literally thousands of independent enter-

prises; thousands of individuals, . . ., each spurred on tothe drilling of exploratory wells by the assurance thatif he made a discovery, he would reap a reward . . .’’This phenomenon explains why the United States pro-duced nearly two-thirds of all the oil that the world hadconsumed before 1952, and the discovered hydrocar-bon resource between 1920 and 1952 was nearly ninetimes greater than the previous estimate of the totalresource! By these historical examples, we don’t meanto imply that taking risks will always allow one to reapthe reward. By definition, risk implies possibility of anegative consequence.

Traditionally, evaluation of a project or proposal wasmostly based on cost-benefit analysis (CBA). This al-lows decision makers to assess the monetary value ofa proposal or project, in which all the costs and bene-fits at different times are converted into net presentvalues (NPVs) with assumed rates of return on invest-ments. When uncertainty is present, CBA should beevaluated with respect to the probability of occurrencefor each possible outcome. Evaluating expected mon-etary value (EMV) (Ikoku, 1985; Bell, 2007), or expectedreturn (Markowitz, 1991), is a basic method of calcu-lating the potential benefit under uncertainties. Thismethod includes the following steps: (1) defining pos-sible outcomes, (2) assigning the probability for eachpossible outcome, (3) evaluating profit or loss (e.g., interms of NPV) for each outcome, (4) computing theweighted average profit or loss (i.e., expected value[EV]), and (5) selecting the scenario that has the highestEMV or expected return.

Table 1 gives a simple example of two alternatives(drill or farm out) with four possible outcomes. The dril-ling EMV ($3.4 million) is greater than the farm-outEMV ($1.9 million) and, therefore, is a better alternative.

The EMV method considers not only the probabil-ities of different scenarios, but also their correspond-ing monetary costs and profits. As such, better deci-sions can be made using this method compared withmethods purely based on probabilities (Taleb, 2005,p. 98–101). Nevertheless, the EMVmethod has several

Table 1. Illustration of the EMV method for evaluating two alternative choices.

Drill Farm Out

Possible Outcome Probability Net Present Value Expected Value Net Present Value Expected Value

Dry hole 0.30 �$1,000,000 �$300,000 0 05 million bbl 0.35 $2,000,000 $700,000 $2,000,000 $700,00010 million bbl 0.25 $4,800,000 $1,200,000 $2,400,000 $600,00025 million bbl 0.10 $18,000,000 $1,800,000 $6,000,000 $600,000Expected monetaryvalue (EMV)

$3,400,000 $1,900,000

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pitfalls, including difficulties involved in choosing pos-sible outcomes and in accurately assigning correspond-ing probabilities of occurrences. In many cases, theseparameters cannot be adequately defined.Anothermoreimportant shortcoming of the EMV lies in no consid-eration of the consequence of the risk. In fact, the soleobjective of maximizing expected monetary return hasbeen found inadequate since the 1800s because manyrisk-taking strategies based on probability could not beexplained, notably the St. Petersburg paradox (Savage,1972, p. 93; Bernstein, 1998, p. 106–108). An investmentmay be good for a particular investor or company, but itmay not be good for another investor or company be-cause of their different financial and organizational sit-uations. Hence, the consequences differ. Some mayprefer more conservative investments with lower butsurer returns, whereas others may prefer more aggres-sive investments with potentially higher returns.

The utility theory is capable of considering the dif-ferent preferences of investors and companies (Ikoku,1985; Markowitz, 1991; Bell, 2007). Although a detaileddiscussion on utility theory is beyond the scope of thischapter, Figure 4 illustrates two simple utility func-tions. The straight dotted line represents impartialitytoward profit or loss. Impartiality here does not implystoicism. Instead, it means that the magnitude of thepain caused by the loss of money is the same as themagnitude of pleasure caused by the gain of an equiv-alent amount of money. In many real cases, the sadness(negative utility) caused by a loss affects people moresignificantly than happiness (positive utility) that a gaincreates. The solid curve in Figure 4 represents such arisk-averse preference. Notice the gradual increase

in utility for gains and the steeper drop in utility forlosses. In this risk profile, an investor (or a company)must gain about $3.5 million (utility = 5) to achieve amagnitude of happiness equal to the pain of a loss ofjust $0.5 million (utility = �5). Kahneman and Tversky(1979) derived a risk preference curve (also called val-ue function) that is concave for gain and convex forloss with steeper drops. Many consider this value func-tion representative of most people’s risk profile. Morerisk profiles can be found in Ikoku (1985, p. 203–209)and Markowitz (1991, p. 286–294).

The common saying ‘‘Am I paid for the risk I’mtaking?’’ nicely characterizes the principle of balanc-ing the risk and reward. Consider the following meth-od of achieving such a balance. Figure 5 illustrates thewell-known risk-reward relationship from the modernportfolio theory (Markowitz, 1952, 1991), which helpedMarkowitz win the 1990 Nobel Prize in economics. Ingeneral, a certain relationship, albeit a cloud relation-ship (i.e., large spread in the crossplot), exists betweenthe risk taken and the potential profit from the invest-ments. Whereas risk-averse investors should chooselow-risk investments, risk takers may want to choosehigher risk investments for potentially greater returns.Because of the loose cloud relationship between therisk and reward, regardless of the risk profile, it is wiseto choose investments close to the ‘‘efficient frontier,’’which represents the highest return for a given risktaken. An even better choice would be tomaximize thepotential reward while minimizing the risk. This prin-ciple can be applied to reservoir optimization (Baileyet al., 2011) and exploration and production project se-lection and management (Adams et al., 2001). Note that‘‘risk’’ in this context refers to the undesired variability.

Figure 4. Illustration of two utility functions for risk preferences.The horizontal axis represents loss or profit (M stands formilliondollars), and the vertical axis represents the utility or value(utility is a generalization from the old pleasure-pain index, asit has a broader connotation). The dotted line represents arisk-neutral stance toward profit or loss. The bold curverepresents a risk-averse preference.

Figure 5. Illustration of the relationship between risk andpotential reward or loss.

Uncertainty Analysis in Reservoir Characterization and Management 9

When the consequence component is also considered,the risk is nicely described by the common saying, ‘‘Isthe probability that I end upwith a ‘goose egg’ too highfor me to handle the consequence?’’

Decision Analysis Under Uncertainty or Risk

Pascal’s famous ‘‘wager’’ in his book, Pensees (Pascal,1670; Hajek, 2008), may have been one of the earliestand most widely known contributions to decision anal-ysis under uncertainty (Miles, 2007). Modern decisionanalysis, however, has developedmostly since the endof World War II, incorporating utility theory and riskpreferences into making decisions under uncertaintyand risk.

In general, it is wise to reduce uncertainty to a rea-sonable level before making a decision, advice aptlycaptured in the saying, ‘‘Nothing can be concludeduntil the uncertainty is mitigated.’’ As previouslynoted, the VOI is one way of mitigating uncertainties.Improving the technologies, methods, and processes ofreservoir characterization can also mitigate uncertain-ties for reservoir management. Even after mitigatinguncertainty, however, we must still make decisionsunder uncertainty, albeit less than before. Therefore,our ability to make decisions under uncertainty is crit-ical both in business and in life. Here, the risk equation,and more specifically, the consequence component ofthe risk equation, is a factor. Figure 5 shows that as riskincreases in a project or investment, uncertainty in thereturn increases, whereas the probability of a higherreturn improves. The question is: Can we afford theloss? If not, then reducing the risk will be in order,although this will also reduce potential returns on theinvestment.

Table 2 illustrates a simple example of a decisionanalysis under uncertainty and risk in which two choicesexist. The drilling EMV ($2,500,000–$500,000 = $2.0 mil-lion) is greater than the farm-out EMV ($1.25 million).Therefore, based on the EMV, which considers the un-certainties of these two outcomes, we should drill. How-ever, if we are very risk averse, having the risk prefer-

ence of the solid curve in Figure 4, we could calculatethe utilities of the drilling and farm out as follows:

Utility of drilling ¼ 0:5 ð�5Þ þ 0:5 ð4Þ ¼ �0:25

Utility of farm out ¼ 0:5 ð0Þ þ 0:5 ð2Þ ¼ 1:00

Because the utility of farming out is greater than thatof drilling, we should farm out the property. This ex-ample shows how EMV enables us to make decisionsunder uncertainty,whereas the utility theory enables usto make decisions under risk, provided that the riskprofile is defined. Obviously, a simple utility functionwas used in this example. In practice, the definition ofthe utility function based on the company’s particularsituations may be quite challenging, and utility alsochanges with time (Grayson, 1962).

Other methods of decision analysis under uncer-tainty have been proposed. For example, material un-certainties (e.g., rig or seismic equipment uncertainty)can be integrated into a business decision through anappropriate valuation of future information using de-cision trees or Monte Carlo simulation (Prange et al.,2006; Bailey et al., 2011). For cases of extremely highrisk, we may need to make special considerations indecision analysis (Taleb, 2007).

Example: Oil Price Variability, Uncertainty, and Risk

Crude oil price (COP) is volatile and its impact on theexploration and production business is extraordinarilylarge. Figure 6 shows the weekly COP from January1997 to August 2009. Overall, the COP shows a veryhigh variability, but it varies more in some periodsthan others, notably during the last 2 to 3 yr. BetweenAugust 2006 and August 2009, the COP average was$73.08, with a standard deviation of 24.84. By contrast,the COP average from January 1997 to December 1999was $15.75, with a standard deviation of 4.21. Obvi-ously,COPvariability during the last 3 yr hasbeenmuchgreater. As previously mentioned, because variabilityis simply a property of a phenomenon, it is totally

Table 2. Illustration of the utility method for evaluating two alternative choices.

Drill Farm Out

Possible Outcome ProbabilityNet Present

ValueExpected Monetary

ValueNet Present

ValueExpected Monetary

Value

Dry hole 0.50 �$1,000,000 �$500,000 0 010 million bbl 0.50 $5,000,000 $2,500,000 $2,500,000 $1,250,000

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objective.However, uncertainty is a result of unknowns.No uncertainty is seen in the COP shown in Figure 6simply because it is a collection of historic data. None-theless, the high variability of COP over time wouldimply a high degree of unpredictability or high uncer-tainty in the future COP.

Figure 7A shows a plot of monthly COP data fromJanuary 2000 to September 2002 and the predicted COPfor October 2002 to December 2003 by the U.S. EnergyInformation Administration (EIA). Because of the rel-atively low variability in the COP between January 2000and September 2002, the uncertainty range of the pre-diction was quite small. Figure 7B shows the actualweekly COP data. It fell within the EIA (U.S. EIA, 2009)predicted uncertainty range, which had a 95% confi-dence interval. Nevertheless, variability in the actualCOP was greater than in the prediction.

It is possible for the COP to go outside of the 95%confidence uncertainty range. In the fall of 2007, forexample, no one predicted that the COP would reach$140 per barrel in 2008. When it actually reached thatprice in the summer of 2008, no one predicted itwould drop below $40 per barrel by the end of 2008.Some even suggested that the $50 COP was history,and that it would never happen again. Needless to say,it did. The high variability in the COP during the last2 to 3 yr caused the EIA and other institutions to in-crease the uncertainty range of their COP predictions.Figure 8 shows the EIA (U.S. EIA, 2009) prediction ofthe yearly average COP from 2009 to 2030. The largeuncertainty range in the prediction reflects mainly theimpact of the high COP volatility during the last 3 yr.Note also that the range increases as a function of time,

reflecting greater uncertainty as the prediction movesaway from the existing COP data.

Although uncertainty in the COPmay be essentiallythe same for all the exploration and production com-panies, each company’s risk relative to it may be verydifferent because the consequences of dramatic changesin the COP vary from one company to another. A dra-matic plunge in the COP during a short period, as wewitnessed during late 2008 to early 2009, caused severebudget constraints in some companies. Others experi-enced less impact because of their asset base, diversi-fication, and/or hedging.

CONCLUDING REMARKS

Although a reservoir is the result of geologic processesand is not random (when we say that it is random, wesay it heuristically), it is legitimate to use probabilisticapproaches in reservoir characterization and model-ing. Probabilistic methods should be used in combina-tion with established geologic, petrophysical, geophys-ical, and reservoir engineering principles in reservoircharacterization.

Generally, many types of uncertainties exist in res-ervoir characterization. Data uncertainties, if not miti-gated, can propagate into subsequent reservoir charac-terization processes and cause greater uncertaintiesor errors in the results. Improving measurement toolscan mitigate measurement uncertainties. Inferential un-certainties can be mitigated by acquiring more data,and/or using better science and technology.

Figure 6. Weekly crude oil price (COP)from all countries from January 3, 1997, toAugust 14, 2009 (from U.S. EIA, 2009).

Uncertainty Analysis in Reservoir Characterization and Management 11

Figure 7. (A) Monthly crude oil price (COP)projection by the U.S. EIA (2009) for theperiod from October 2002 to December2003. Data from January 2000 is shownfor reference. The band between the twodotted lines represents the uncertaintyrange with 95% confidence. (B) Actualweekly COP during the same time span(from U.S. EIA, 2009).

Figure 8. Yearly average crude oil price(COP) from 1980 to early 2009 (historicdata) and projections (solid curve) to2030 with an uncertainty range (bandbetween the two dashed curves) (from U.S.EIA, 2009). Note that the yearly averageCOP has less variability than monthly,weekly, or daily averages. For example, theaverage COP in 2008 was about $100,whereas the weekly average COP rangedfrom $36 (in the week of December 26) to$137 (in the week of July 4).

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One of the main reasons to analyze and quantifyuncertainty is to enhance decision analysis. In general,business decisions are made under uncertainty be-cause uncertainty may be mitigated, but cannot becompletely eliminated. How much we attempt to mit-igate uncertainty should depend on the needs of de-cision analysis and the cost of information. Uncertaintyanalysis should not be for its own sake, but insteadshould support investigational analyses, decision anal-ysis under uncertainty and risk management. With lim-ited data, it is impossible to describe subsurface het-erogeneities at all levels of detail. But it is possible todescribe them in relevant details. To know what needsto be known and to know what can be known (e.g., bybalancing the complexity of the problem, availability ofinformation, cost of acquiring new information, andtimeline, etc.) should be the main focus of uncertaintyanalysis in reservoir characterization and management.

ACKNOWLEDGMENTS

The author thanks Schlumberger Ltd. for permissionto publish this work and the reviewers for the usefulsuggestions.

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