Virtual-Intell igence Applicationsin Petroleum Engineering:Part 3Fuzzy Logic

Virtual-Intell igence Applicationsin Petroleum Engineering:Part 3Fuzzy LogicShahab Mohaghegh, SPE, West Virginia U.

Distinguished Author Series

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Distinguished Author Series

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Parts 1 and 2 of this series of articles presented a generaloverview of artificial neural networks and evolutionarycomputing, respectively, and their applications in the oiland gas industry.1,2 The focus of this article is fuzzy logic.The article provides overview of the subject and its poten-tial application in solving petroleum-engineering-relatedproblems. As the previous articles mentioned, the mostsuccessful applications of intelligent systems, especiallywhen solving engineering problems, have been achieved byuse of different intelligent tools in concert and as a hybridsystem. This article reviews the application of fuzzy logicfor restimulation-candidate selection in a tight-gas forma-tion in the Rocky Mountains. We chose this particularapplication because it uses fuzzy logic in a hybrid mannerintegrated with neural networks and genetic algorithms.

BackgroundThe science of today is based on Aristotles crisp logicformed more than 2,000 years ago. Aristotelian logic looksat the world in a bivalent manner, such as black and white,yes and no, and 0 and 1. The set theory developed in thelate 19th Century by German mathematician Cantor wasbased on Aristotles bivalent logic and made this logicaccessible to modern science. Subsequent superimpositionof probability theory made the bivalent logic reasonableand workable. Cantors theory defines sets as a collectionof definite, distinguishable objects. Fig. 1 is a simple exam-ple of Cantors set theory and its most common operations,such as complement, intersection, and union.

The first work on vagueness dates back to the first decade20th Century, when American philosopher Pierce noted thatvagueness is no more to be done away with in the world oflogic than friction in mechanics.3 In the early 1920s, Pol-ish mathematician and logician Lukasiewicz4 developedthree-valued logic and talked about many-valued, or multi-valued, logic. In 1937, quantum philosopher Black5 pub-lished a paper on vague sets. These scientists built the foun-dation on which fuzzy logic was later developed.

Zadeh,6 known as the father of fuzzy logic, published hislandmark paper Fuzzy Sets in 1965. He developed manykey concepts, including membership values, and provideda comprehensive framework to apply the theory to engi-neering and scientific problems. This framework includedthe classical operations for fuzzy sets, which comprise all

the mathematical tools necessary to apply the fuzzy-set the-ory to real-world problems. Zadeh was the first to use theterm fuzzy, which provoked much opposition. A tirelessspokesperson for the field, he was often harshly criticized.At a 1972 conference, Kalman stated that Fuzzification isa kind of scientific permissiveness; it tends to result insocially appealing slogans unaccompanied by the disciplineof hard scientific work.7 (Note that Kalman is a former stu-dent of Zadehs and inventor of the famous Kalman filter, amajor statistical tool in electrical engineering. The Kalmanfilter is the technology behind the Patriot missiles used inthe Gulf War. Claims have been made that it has beenproved that use of fuzzy logic can significantly increase theaccuracy of these missiles.8,9) Despite all its adversaries,fuzzy logic continued to flourish and has become a majorforce behind many advances in intelligent systems.

The word fuzzy carries a negative connotation in West-ern culture, and fuzzy logic seems to misdirect the atten-tion and to celebrate mental fog.10 On the other hand, East-ern culture embraces the concept of coexistence of contra-dictions as it appears in the yin/yang symbol (Fig. 2).While Aristotelian logic preaches A or Not-A, Buddhism isall about A and Not-A.

Many believe that the tolerance of Eastern culture forsuch ideas is the main reason behind the success of fuzzylogic in Japan. While fuzzy logic was being attacked in theU.S., Japanese industries were busy building a multibillion-dollar industry around it. Today, the Japanese hold morethan 2,000 fuzzy-related patents. They have used fuzzytechnology to build intelligent household appliances, suchas washing machines and vacuum cleaners (Matsushita andHitachi), rice cookers (Matsushita and Sanyo), air condi-tioners (Mitsubishi), and microwave ovens (Sharp, Sanyo,and Toshiba), to name a few. Matsushita used fuzzy tech-nology to develop its digital image stabilizer for camcorders.Adaptive fuzzy systems (a hybrid with neural networks) canbe found in many Japanese cars. Nissan patented a fuzzyautomatic transmission that is now very popular with manyother manufacturers, such as Mitsubishi and Honda.10

Fuzzy-Set TheoryThe human thinking/reasoning/decision-making process isnot crisp. We use vague, imprecise words to explain ourthoughts or communicate with one another. There is a con-tradiction between the imprecise, vague process of humanreasoning, thinking, and decision-making and the crisp, sci-entific reasoning of black-and-white computer algorithmsand approaches. This contradiction gave rise to the imprac-tical approach of using computers to assist humans in thedecision-making process and is the main reason that tradi-

Copyright 2000 Society of Petroleum Engineers

This is paper SPE 62415. Distinguished Author Series articles are general, descriptiverepresentations that summarize the state of the art in an area of technology by describingrecent developments for readers who are not specialists in the topics discussed. Written byindividuals recognized as experts in the area, these articles provide key references to moredefinitive work and present specific details only to illustrate the technology. Purpose: toinform the general readership of recent advances in various areas of petroleum engineering.

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tional artificial intelli-gence or conventionalrule-based systems(also known as expertsystems) have beenunsuccessful. Expertsystems, which start-ed as a technology inthe early 1950s,remained in researchlaboratories and neverbroke through to theconsumer market.

In essence, fuzzylogic provides the means to compute with words. Usingfuzzy logic, experts are no longer forced to summarizetheir knowledge to a language that machines or comput-ers can understand. What traditional expert systems failedto achieve finally became reality with the use of fuzzy-expert systems. Fuzzy logic is made up of fuzzy sets,which are a way of representing nonstatistical uncertaintyand approximate reasoning, which includes operationsused to make inferences.9

Fuzzy-set theory provides a means of representing uncer-tainty. Uncertainty usually is the result of either the randomnature of events or the imprecision and ambiguity of theinformation we have about the problem we are trying tosolve. In a random process, the outcome of an event fromamong several possibilities is strictly the result of chance.When the uncertainty is a product of randomness of events,probability theory is the proper tool to use. Observationsand measurements can be used to resolve statistical or ran-dom uncertainty. For example, once a coin is tossed, no fur-ther random or statistical uncertainty remains.

Most uncertainties, especially when dealing with com-plex systems, are the result of a lack of information. Thetype of uncertainty that is the outcome of the complexityof a system arises from imprecision, from our inability toperform adequate measurements, from a lack of knowl-edge, or from vagueness (like the fuzziness inherent in nat-ural language). Fuzzy-set theory is a marvelous tool formodeling the kind of uncertainty associated with vague-ness, imprecision, and/or a lack of information regarding aparticular element of the problem at hand.11 Fuzzy logicachieves this important task through fuzzy sets. In crispsets, an object either belongs to a set or it does not. In

fuzzy sets, everything is a matter of degrees. Therefore, anobject belongs to a set to a certain degree. For example, theprice of oil today is U.S.$24.30/bbl. Given the price of oilin the past few years, this price seems to be high. But whatis a high price for oil? A few months ago, the price of oilwas approximately U.S. $10/bbl, which everyone agrees islow. Given how much it costs to produce a barrel of oil inthe U.S., one can say that the cutoff between low and highfor oil price is U.S. $15/bbl. If we use crisp sets, U.S.$14.99/bbl is low and U.S. $15.01/bbl is high. Imagine ifthis was the criterion used by oil company executives tomake decisions. The fact is, while U.S. $15.01 is a pricethat many people (in the oil industry) would be happywith, U.S. $16/bbl is better and U.S. $20/bbl is even better.Categorizing these three prices as high can be quite mis-leading. Fig. 3 shows the fuzzy sets that fuzzy logic pro-poses for the price of oil.

The most popular (although not yet standard) form ofrepresenting fuzzy set and membership information is

A(X)=m, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1)

where the membership of X in Fuzzy Set A is m. Accord-ing to Fig. 3, when the price of oil is U.S. $20/bbl, it has amembership of 0.15 in the fuzzy set Good and a mem-bership of 0.85 in the fuzzy set High. With these valuesto represent the oil-price-membership values,

good($20)=0.15 . . . . . . . . . . . . . . . . . . . . . . . . (2a)

and high($20)=0.85. . . . . . . . . . . . . . . . . . . . . . .(2b)

Approximate Reasoning. When decisions are made onthe basis of fuzzy linguistic variables (low, good, high)with fuzzy-set operators (and, or), the process is calledapproximate reasoning. This process mimics the humanexperts reasoning process much more realistically than doconventional expert systems. For example, if the objectiveis to build a fuzzy expert system to help us make a recom-mendation on enhanced recovery operations, we can usethe oil price and the companys proven reserves to makesuch a recommendation. Using the fuzzy sets in Fig. 3 forthe oil price and the fuzzy sets in Fig. 4 for the companystotal proven reserves, we try to build a fuzzy system thatcan help us in making a recommendation on engaging inenhanced-recovery operations (Fig. 5).

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Fig. 1Operations of conventional crisp sets.

Fig. 2Yin/yang symbol.

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Approximate reasoning is implemented through fuzzyrules. A fuzzy rule for the system being explained here canhave the following form.

Rule 1. If the price of oil is high and the total provenreserves of the company is low, engaging in enhanced-recovery practices is highly recommended.

Because this fuzzy system comprises two variables andeach variable consists of three fuzzy sets, the systemincludes nine fuzzy rules, which can be set up in a matrix(Fig. 6). The abbreviations in the Fig. 6 matrix correspondto the fuzzy sets defined in Fig. 5. One can conclude fromthis example that the number of rules in a fuzzy systemincreases dramatically with the addition of new variables.Adding one more variable consisting of three fuzzy sets tothe example increases the number of rules from 9 to 27.This is known as the curse of dimensionality.

Fuzzy Inference. A complete fuzzy system includes afuzzy-inference engine. Fuzzy inference helps us buildfuzzy relations based on the fuzzy rules that have beendefined. During a fuzzy-inference process, several fuzzyrules are fired in parallel. Parallel-rule firing, unlikesequential evaluation of the rules in the conventionalexpert system, is much closer to the human reasoningprocess. Unlike the sequential process, where some infor-mation contained in the variables may be overlookedbecause of the stepwise approach, parallel firing of therules allows simultaneous consideration of all the infor-mation. Many different fuzzy-inference methods exist. Weexamine a popular method called Mamdanis inferencemethod.12 Fig. 7 illustrates this inference method graphi-cally. In this figure, a case is considered when the price ofoil is U.S. $20/bbl and the company has approximately 9million bbl of proven reserves. Oil price is represented byits membership in the Good and High fuzzy sets, whiletotal proven reserves is represented in the Low and Mod-

erate fuzzy sets. As Fig. 7 shows, this causes the firing offour rules simultaneously. According to Fig. 6 these areRules 1, 2, 4, and 5. In each rule, the fuzzy-set operationand [the intersection between the two input(antecedents) variables] is evaluated as the minimum andconsequently is mapped on the corresponding output(consequent). The result of the inference is the collectionof the different fuzzy sets of the output variable on the bot-tom of the figure.

A crisp value may be extracted from the result asmapped on the output fuzzy sets by defuzzifying the out-put. One of the most popular defuzzification procedures isto find the center of the mass of the shaded area in the out-put fuzzy sets.

Application in the Petroleum IndustryFuzzy logic has been used in several petroleum-engineer-ing-related applications. These include petrophysics,13,14reservoir characterization,15 enhanced recovery,16,17 infilldrilling,18 decision-making analysis,19 and well stimula-tion.20-22 In this section, we review an application thatincorporates fuzzy logic in a hybrid manner in concertwith neural networks and genetic algorithms.

In this example of use of intelligent systems in petrole-um engineering, neural networks, genetic algorithms, andfuzzy logic are used to select candidates for restimulationin the Frontier formation in the Green River basin.22 Asthe first step of the method, neural networks are used tobuild a representative model of the well performance inthe Frontier formation. Table 1 lists the input parameters

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Fig. 6Fuzzy rules for approximate reasoning.

Fig. 4Fuzzy sets representing total proven reserves.

Fig. 5Fuzzy sets representing the decision to engage inenhanced recovery.

Fig. 3Fuzzy sets representing oil price.

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used in the neural-network model-build-ing process.

After training, test-ing, and validation ofthe neural networks,the training data sethad a correlation coef-ficient of 0.96 and theverification data sethad a correlation coef-ficient of 0.72. As abyproduct of the neur-al-network analysisand with a methodcalled backward elim-ination, an attemptwas made to identifythe most influentialparameters in this dataset. Fig. 8 shows theresults of neural-net-work backward-elimi-nation analysis.

This figure shows allfour categories of inputdata. The most influen-tial category has thelowest R2. The figurealso shows that reser-voir quality is the mostimportant category, fol-lowed by the comple-tion and stimulationcategories, which seemto be equally impor-tant. The location-related input parame-ters seem to be the leastimportant comparedwith the others. Note

that, among all the parameters involved inthe analysis, only the last three stimulationrelat-ed parameters in Table 1 are considered tobe controllable.

The second step of the analysis involves geneticoptimization of the stimulation parameters. Thelast three input parameters in Table 1 (fluid type,total fluid volume, and total proppant amount) areused in the optimization process. With the neural-network model developed in the first step of theanalysis as the fitness function of the evolutionprocess, the algorithm searches through all possi-ble combinations of the three stimulation parame-ters and tries to find the combination that resultsin the highest 5-year cumulative production. Thisprocess is repeated for each well. The differencebetween the optimized and the actual 5-yearcumulative production is considered to be thepotentially missed production that might berecovered by restimulation. The outcome of thisprocess is called the potential 5-year cumulativeproduction and is used as one of the three inputs

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Fig. 7Mamdanis fuzzy inference.

InputCategory Parameter Comments

Location x x coordinates of the well (east-west)y y coordinates of the well (north-south)KB elevation Kelly bushing elevation

Reservoir Permeability From type-curve-matching analysisDrainage area From type-curve-matching analysisTotal gas-feet Sum (porosity+gas saturation

+net pay) (all zones) Completion Total h completed Total completed thickness (all zones)

Total number of Total number of perforation holesholes

Completion date Date of well completionNumber of zones Total number of zones completed

Fracture Fracture number A well may have up to seven fracture jobs

Fluid type Gelled oil, ungelled oil, linear gel, crosslinked gel

Fluid volume Total amount of fluid pumped in all fractures

Proppant amount Total amount of proppant pumped in all fractures

TABLE 1INPUT PARAMETERS FOR NEURAL-NETWORK ANALYSIS

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in Step three, which is the fuzzy-decision support systemthat uses approximate reasoning.

Step three is a three-input/one-output fuzzy system.The inputs include the potential 5-year cumulative pro-duction, a calculated parameter called fractures per zone(FPZ), and pressure. Engineers in the field brought theFPZ parameter to our attention. They mentioned thatsome wells had been completed in all zones (as many asseven zones can be present) but only one hydraulic frac-ture had been performed. In other words, the ratio ofnumber of treatments performed to total number of zonescompleted is an important factor. We also found thatlong-term pressure surveys had been performed on manywells in 1995. The issue with pressure surveys is thatshut-in time and depth where the pressure readings weretaken were not consistent throughout the field. Thisintroduces serious imprecision in the pressure values as acomparative value from well to well. Therefore, we sub-jected all three input parameters to fuzzy sets using low,moderate, and high fuzzy sets. Output of the fuzzy systemis the degree to which a well is a candidate for restimula-tion. The output fuzzy sets include (1) the well is a can-didate, (2) the well may be a candidate, and (3) the wellis not a candidate. The system includes 27 fuzzy rulesthat are qualified with a set of three truth functions. Fig.9 shows the 27 rules with truth qualifications for thefuzzy systems. Fig. 10 shows the truth-qualification func-

tions used for the approximate-reasoning imple-mentation in the fuzzy system. As the figureshows, each rule can be true, fairly true, orvery true.

With this three-step process, all the wellsbelonging to a particular operator in the Frontierformation were processed and a list of restimula-tion candidates identified.

Results. The intelligent-systems approach for thisapplication was modified as a result of its applica-tion to three different formations, two in theRocky Mountains and one in east Texas. Thefuzzy-decision support system was the mostrecent addition to the process. The new andimproved intelligent-systems approach thatincluded the fuzzy logic component picked WellGRB 45-12 as Candidate 20, while this well was

missed as a candidate before the addition of fuzzy logic tothe procedure. An engineer with several years of experi-ence in this field also had suggested this well as a candi-date. The fuzzy-decision support system was able to cap-ture the engineers knowledge and use it in an automaticprocess for all the wells in the study. Fig. 11 shows theresult of restimulation on Well GRB 45-12.

ConclusionsThis series of articles presented a general backgroundand some introductory information about virtual intelli-gence and three of its most popular tools (neural net-works, genetic algorithms, and fuzzy logic). Some uses ofthese technologies in the oil and gas industry were alsopresented along with application examples for each ofthe techniques. We hope that this effort invokes someinterest in this area by demonstrating the potential thatthese methods have in solving challenging andcomplex problems.

Nomenclatureh= thickness, L, ftm= membership valueR2= correlation coefficientx,y= coordinates

X= input-parameter value= membership of a fuzzy set

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Fig. 9Rules used in fuzzy-decision support system.

JPTJPT

Fig. 8Influence of parameters in the stimulation process in Fron-tier formation.

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SI Metric Conversion Factorbbl 1.589 873 E01= m3

Shahab Mohaghegh is an associate professor of petro-leum and natural gas engineering at West Virginia U. inMorgantown, West Virginia. e-mail: shahab@wvu.edu.Involved in R&D of virtual-intelligence techniques since1991, he has applied the techniques successfully to petro-leum engineering problems in many different areas,including drilling, completion, stimulation, formation eval-uation, and reservoir evaluation. Mohaghegh holds BSand MS degrees in natural gas engineering from TexasA&I U. and a PhD degree in petroleum and natural gasengineering from Pennsylvania State U. A member of theEditorial Review Committee, he served as a Review Chair-man for SPE Reservoir Engineering and Evaluation dur-ing 199799.

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Fig. 10Truth qualification for fuzzy rules.Fig. 11Gas and water production for Well GRB 45-12before and after restimulation.