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Fundamentals and Applications of the Monte Carlo Method

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-JC-PT6S-63-C>2 fundamentals and Applications of the Monte Carlo Method By E. STOIAN* (16th A71n'/url Technical llIeeting, The Petroleum Society of C.l_ill" Calgm'y, May, 1.?65) ABSTRACT The Monte Carlo method embarked on a course of Perhaps no industry is more vitally concerned with risk its own when calculations using random numbers than the oil and gas industry, and few professional men were s}'stematical!:r considered as a di::itinct topic uf other than petroleum engineers are required to recom­ studJ' by S. Wilks (19441- mend higher investments on the basis of such uncertain Downloaded from http://onepetro.org/jcpt/article-pdf/4/03/120/2166363/petsoc-65-03-02.pdf by guest on 25 September 2021 and limited information. Tn recent }'ears, the number of The picturesque name of Monte Carlo originuted al methods dealing with risk and uncertainty has grown the Los Alamos Scientific Laboratory with von N eu­ extensh-el}' so that the classical approach, using ana­ mann and Ulam (194--1) as a code for cla~sified worl{ lytical procedures and single-yalued parameters, has un­ dergone a significant transformation. The use of stochas­ related to the simulation of neutron behaviour. tic "ariables, such as those frequentl,r encountered in the The novelty of the Monte Carlo method lies chiefly oil industry, is now economically feasible in the eyulua­ tion of an increasing number of problems by the applica~ in the unexpected approach. l\Iore specifically, the tion of Monte Carlo techniques. Ile'wne~s relates to the suggestion that many relation­ This paper defines the Monte Carlo method as a subset ships arising in non-probabilistic contexts can be of simulation techniques and a combination of sampling evaluated more easily by stochastic experiments than theory and numerical analysis_ Briefly, the basic tech­ by standard anal:rtical methods. This, in effect. is the nique of Monte Carlo simulation inyob.-es the represen­ tation of a situation in log-ical terms so that, when the von Neumann - Ulam concept. pertinent data are inserted, a mathematical solution be­ The difficulty of accepting Monte Carlu solution::! comes Jlossible. Using random numbers generated by an "automatic penny-tossing machine" and a cumulatiye fre­ as ans\...·er::! in science and engineering is, independent Quency distribution, the beha'\'iour pattern of the particu­ of legal supedicial aspects, not to be underestimated. lar case can be determined by a process of statistical ex­ Certain signg, however, are promising. Indeed, the perimentation_ In practical applications, the probabilistic Monte Carlo techniques have received considerable data expressed in one or several distributions may pertain to geological exploration, discover).' processes, oil-in~place publicity in the paRt fifteen yean.; and the method haH e\'aluations or the productivity of heterogeneous reser­ recently attained a statLl~ of preliminar.r acceptance, Yoirs. The great variety of probabilit).' models used to Extensive references exist on both general and :;pe­ date (e..[_, normal, log-normal, skewed log~normal, linear, multi-modal, discontinuous, theoretical, experimental) con­ cific topics (1, 2). The Monte Carlo method, however, firms a broad range of experimental computations and a is still unfamiliar to those direct1)r concerned with g-enuine interest in realistic representations of random potential applications. More particularly. the method impacts encountered in practice. is seldom Llsed by petroleum engineers (3, 4). This Emphasis in this paper is directed to the salient charac­ teristics of the Monte Cado method, with particular ref­ paper has been prepared to stimulate intereRt in the erence to applications in areas related to the oil and ,gas Monte Carlo method and its applications in the oil nnd industry_ Attention is focused on reservoir engin{'erin~ gas industry. The intention is to present a deal' pic­ models. Nevertheless. management facets of the oil and ture of the method and its tools in order to provide gas business are considered alan,!! with other applications in statistics, mathematics, physics and engineering. Sam­ a "feeling" foL' recognition of applications as they oc­ ple size reducing techniques and the use of digital com· cur in practice. puters are also discussed. DEFINITION INTRODUCTION The Monte Carlo method may be defined <"8 a means HE Monte Carlo method may have originated of design and study of a stochastic model which simu­ T with a mathematician wishing to know ho\\o" many lates, in all essential aspects, a physical OL' mathema­ gteps a drunkard would have to take to get a speci­ tical proces.s. Basically, the method is one of numer­ fied distance away from a tavern, assuming that ical integration. As a combination of gampling theoLT each of his steps had an equal probability of being and numerical analysis, the fi'1onte Carlo method is a cast in any of the four principal directions. This lead~ special contL·ibution to the science of computing, Brief­ to the classic concept of "random walk" which has ly, Monte Carlo is a practical method which HotVCH great problem-solving potential. problems by numerical operations on random num­ Another classical principle is associated with Buf­ bel"S, Some experts apply the name Monte Carlo only fon (1773), who observed that if a plane is ruled to cases that are best illustrated by the use of prob­ with parallel and equally spaced lines, and a needle abilistic techniques to solve nonstatistical mathema­ just long enough to reach from one line to the next is tical problems, Other experts reserve the Monte Carlo thrown at random, the probability that the needle de5ignation onlJr for calculations implementing sophis­ crosses a line is 2/.". This leads to the important dis­ ticated variance-reducing techniques. cover~r that one can evaluate a definite quantity by a Statistical sampling procedures and must llumel'ical completel~r random process. experiments of a stochastic nature (i.e., involving a seL of ordered observations) are now included in the Monte Carlo method, and it is in this context that '''''Oil u?ld Gas COlllim·uation Board, Calgary, Alta. the name "I\'ionte Carlo" will be used, 120 The Journal of Canadian Petroleum --'- -- --,..: , ' BASIC TECHNIQUES. Concept of Ran~omness The important st~p-s in ~ lV~onte Carlo calculation The idea of mathematical randomness is that "in are: the long run" such-and-such conditions will "almost .(l)-Selecting or designing a probability model by always" prevail. ·By way of illustration, in the long statistical data reduction, analogy 01' theoretic.al con­ run approximately half of the tosses of a true coin siderations. would be -heads. Statisticians associate randomness (2)-Generating random numbers and correspond­ with probability. The "intuitive" school states that randomness must be defined with reference to "in­ ing random variables. stantaneous" probability and not to what ,~'ill happen (3)-Designing and implementing variance-reduc­ "in the long run." The proponents of the "frequency" ing techniques. theory define both randomness and probability in terms of the frequency hypothesis of equal probabili­ SALIENT CHARACTERISTICS ties. The "short term" and the "long run" may be as­ ,~ sumed to be two facets of probability; namely, the '. Drawing from various publications (1, 2). the sali­ subjecti,'e and objective probabilitYl respectively.. ent characteristics of the Monte Carlo method appear 1; to be as follows; Att1"ibutes of Pseudo-Random.. Numbe1·s (I)-The Monte Carlo' method is associated with Downloaded from http://onepetro.org/jcpt/article-pdf/4/03/120/2166363/petsoc-65-03-02.pdf by guest on 25 September 2021 probability theory_ However, whereas the relation­ Most processes of generating random numbers are ships -of probability theory have been derived from cyclic. If the cycle is relatively long for a specific ap­ theoretical considerations of the phenomenon of plication) however, the sequence can be considered .chance, the Monte Carlo method uses probability to "locally random" for all practical purposes. This con­ find answers to physical problems that maJ~ or may cept is very profitable in that we can use simple not be related to probability. processes to generate pseudo-random numbers for I i (2)-The application of the Monte Carlo method of­ practical applications. To qualify for pseudo-random­ l fers a penetrating insight into the behaviour of the ness, sequences must comply with certain require­ s~ystems studied. Frequently, problems become decep­ ments. Some of these are ~ tively simple. In this sense, effective Monte Carlo (a). In any sequence, the digits used in the num­ techniques are self-liquidating. bers must be distributed with uniform density; Le_. (3)-The results of Monte Carlo computations are they must be iu roughly equal quantities. treated as estimates within certain confidence limits (b). Successive digits must be uncorrelated; i.e.• rather than true or exact values. Actually, all mean­ no digit should tend to follow any other digit. ingful physical measurements are expressed in this (c). There must be no correlation between succes­ way. In many cases where relationships in a model sive numbers_ cannot be evaluated at all because of either mathe­ Tests of randomness apply to the generating process matical or practical considerations, Monte Carlo tech­ rather than the randomness of the sequence. Common niques can be used to obtain approximations. tests are: (a) Frequency; (b) Serial; (c) Poker; and (4)-As in any other method, there is a need for (d) Gap Tests. Additionally, there are independence, adequate basic information; data for the implemen­ normality and chi-square tests. tation of the Monte Cado method, however, may be b~r obtained standard rlata processing procedures. SOU1·ce of Random Sequences (5)-The method is flexible to the extent that the intricacies of a problem. as may be reflected by either Random numbers may be obtained bi: (a) a phys­ a great number of parameters or complicated geo­ ical process, (b) "look-up" in a formal table and (c) metry, do not alter its basic character; the penalty digital computers.
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