-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
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 as ans\...er::! in science and engineering is, independent
"automatic penny-tossing machine" and a cumulatiye fre-
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- paper has been prepared to stimulate intereRt in the
teristics of the Monte Cado method, with particular ref-
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 a "feeling" foL' recognition of applications as they oc-
in statistics, mathematics, physics and engineering. Sam-
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

T HE Monte Carlo method may have originated

with a mathematician wishing to know ho\\o" many
gteps a drunkard would have to take to get a speci-
of design and study of a stochastic model which simu-
lates, in all essential aspects, a physical OL' mathema-
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 contLibution 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 COlllimuation 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
ing random variables.
randomness must be defined with reference to "in-
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-
ties. The "short term" and the "long run" may be as-
,~
SALIENT CHARACTERISTICS

'. Drawing from various publications (1, 2). the sali-

sumed to be two facets of probability; namely, the
subjecti,'e and objective probabilitYl respectively..
1
;
ent characteristics of the Monte Carlo method appear
to be as follows;
Att1"ibutes of Pseudo-Random.. Numbe1s
(I)-The Monte Carlo' method is associated with
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
obtained b~r standard rlata processing procedures. SOU1ce 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. ,':

paid for complexity is increased computing time and Physically, random numbers may be produced by
costs. the following: (a) flipping of a coin; (b) roll of a
(6)-A practical consideration is that the iterative dice; (c) draw of a card; (d) spinning of a wheel
calculations necessary for attaining a certain level of of chance, e.g., roulette; or (e) using a randomizing
confidence can be distributed among several com- machine, e.g. combination of electric motor - num-
puters, working simultaneously in one or more places. bered disc - instantaneous flash.
(7)-The l\ionte Carlo solutions are approximate, The simplest way of obtaining random numbers is
however, they can be up-graded commensurately with by reference to formal tables l some of which are the
the time and money allocated to the problem. result of compound u randomization. The random
numbers thus produced ma)r be visualized as being
(8)-The Monte Carlo method's purely numerical generated by two roulette wheels arranged in series
character requires careful scrutiny of all results. so that the first wheel controls the arrangement of
(9)-Solutions of the Monte Carlo method are symbols on the second wheel and the second wheel
numerical and apply onbr to the particular case stud- determines which of its positions will be recorded.
ied_ Electronic computers can generate pseudo-random
numbers internally (2). One of the simplest methods ',.
RANDOM NUMBERS Uses a binar~' digital machine to evaluate the equa-
In the application of the Monte Carlo method it is tion:
necessary to repeatedly generate random numbers. (Eg. 1)
where:
Today. there are no difficulties in the generation R n is the nth random number; Rn+l is the (n+l)th
of the random numbers required by the petroleum random number;
engineer. For reasons of completeness, however, ran- Ie is a constant multiplier - preferably an odd I=0wer
domness will be defined and sources of random num- of five;
N is usually tl:e nlnnter of binary di~its in the
bers briefly reviewed. macl:ip..e ,mrd. . t,. ...~.'-. .'.
~
~' .
~
Technology, July September, 1965, Montreal 121 ~~:~ ...
 To obtain a landom number, RII+J, it is necessary (a). Tracing flow paths of particles of displacing
only to multiply K by R,. and retain the least signifi- fluid.
cant half of the product. Starting with an odd R.., (b). Determining the outcome of exclusive "yes-no"
one will run through 2:\'-" numbers before repetition decisions regarding states (e.g., producer, dry well);
sets in. designating the amount of time spent in each of sev-
Alldiliullal method::; include the "middle-of-the- eral possible .states or sets of states (e.g., dLilling,
square" method, multiplicative and additive congru- flowing, on pump).
ential rnethod~, and numerous recursive schemes. (c). Introducing correlated or c:hained pl'OCeSHCs
(e.g" flow of fluids through porous medial.
RANDOM VARIABLES
(d I. Assigning position or mapped characteristicH
Petroleum ellgineers are familiar with cumulative through random coordinates, as well as accounting for
frequenc.}' plots or histograms of physical variables interferences in terms of occupied space and blocleed
such a~ porosity, permeabilit}r and flo\v capacity. Ran- channels (e.g., assigning permeability; "yes-no" con-
dom variables may be obtained from these and simi- cept of fluid displacement from pores).
lar cumulative fl-equenc.y or probability distributions,
F (r.P), using random numbers according to the tech- MONTE CARLO SIMULATION
nique of "random sampling":
The Monte Carlo simulation model c:an be viewed a!-l
(1 )-For a ~et of numbers CPI, ep~ . . ., ordered by an experimental device:
increasing (or decreasing) size, a cumulative fre-
quency curve is c:onstruc.ted to represent F(ep) ver- (1 )-A single run is synon)mous with an experi-
sm; cp. F(ep) .'iimply represents pel' cent of sam- ment, and the output constitutes an observation. An
ples, say porosit.y, smaller (or greater) in size than experiment. run or "his tor}" refleet~ the outcome of
the corresponding attribute ep read from the curve. an individual case le.g., the history of an individual
In pmctice, F(ep) may be represented by an equation. oil company \\jthin the frHrnework of un economic sy.';-
or preferably by a table of paired values. Less con- tern l. Branching techniques and sequential sampling
ventional forms of probability representations are may be usefuL
free-spinning dials or pie charts \:ith markers, cut- (2 I-If a sufficiently large number of obsen:ll-
out patterns and c~tegory bar plots. tions are averaged, the integrated outcome "in the
(2 )-A random number is drawn by procedures al- long run" represents the expected :o;olution. The square
ready revie\... ed and entered in the F lep) versus root of the average deviations squared will be called
plot on the cumulative probability scale F(). A~ standard deviation. It usually ~erves as a criterion for
probability cannot be greater than 100 per cent or terminating the iterations.
unity. the l'Jlndom numbers are entered as probability
A complete sequence of gteps in a simulation com-
values in the range 0 - 1.0 (e_g. a four-digit ran-
prises: (al design, model building and testing, (b)
dom number such as 5018 will be simply entered as
gathering of input data. (c) strategic (Le., design-
0.5018; FigUTC 1).
ing the experiment) and tactical planning, (i.e., de-
(3)-The ep value conesponding to the F (ep) en-
termining the amount of testing); (d l implementation
tered is the desired random variable (e.g., ep = 9.6
of the simulation, (e) analysi~ and appraisal of re-
per cent; FiYll1"e 1),
sults and (f J recommenllations.
ProbafJilify FU'/lcliun,c; Coordination between simulation and analytical
techniques, in many cases, provides for economy.
Probability functions and representations vary
greatly in origin. form, accuracy, reliability and con- VARIANCE-REDUCING PRoceDURES
fidence. This valiety reflects an intensive t.:oncerll for
simulating the l'andom impacts encountered in prat.:- Monte Carlo simulation must be repeated many
tice. [f dabl Jlos~e~'S a. proved definite mathematical times to plovide "expected !-iolutions." The accllrac:,\'
distribution, such as normal, log-normal, binomial 01' of ~olutions !mpl'Oves only a~ the square root of the
Poisson, it i~ advantageous to use a well-studied rela- number of experiments: to double the accuracy of the
tionship rather than the empirical frequency distribu- expected answer, one mmit quadnlple the numbel' of
tion de\"eloped in a particular ca:-;e. Finding mathe- trials. Therefore, it b important to find W.l)'S of
maticai relationships to represent distribution::;, how- increasing the efficiellcy of the :'lampling process.
ever, may prove to be time-consuming, bothersome or [n general, the amount of calculation can be re-
beyond the needs of the petroleum engineer. In fact. duced by using relative rather than absolute values.
the Monte Carlo method is particularly suited to ca:-;es In certain cases, the actual process may be replaced
where the frequency C:lIrve does not conform to any b.\! something le::;s erratic that will yield an accuraLe
well-known or analytically manageable distribution. answer more Quickly. A judicioLis reformulation of the
The analysis of field and laboratory data (e.g., pro~ problem can prove L1seful provided that the "qu icle"
ce~sing of exploratioll ~tatistics or c.ore anaI:',rges) to
answer i~ rebteu in a known manner to the solution
arrive at experimental distribution::. is essential to of the origi nal problem.
the development of many Monte Callo models_ A }l~
troleum engineel', however. can develop excellent sta- Typical \'aL'iance-rcoucing tec:hniqucs BI'C briefly
tistlLal and pl"Obabilistic models using data ~mpple l'eyiewed.
mented by experienc.e and personal judgment. Use of
intuitive probabilities in Monte Cado experiments Importance SalJlplit!(J
can bE:' very instructi\"c. Thb i~
a :3tandard variallce-I'cduc:ing procedure Ctlll-
Di:)i:ributions L1~ually serve to determine the per ~bting of drawing samples from a distribution,
cellt frequency of items greater in size than a :iP~ F' (r.P),
other than that suggested by the problem,
cific value of the attriLute," but may ahw be put to F(1J), and applying weighing factors to correct the
::;omewhat uncoll\entional use:3-, such a::;: final ~ulution for haYing used a distOlteri or bia:'ll'd

122 The Journal of Canodian Petroleum

:-
distribution.. F"(</ must be selected iu such a way SCRIPT, CSL, SIMPAC, SOL, GPSS). Convenient
as to' .reduce the sampling effort substantially. More- instructions are provided in the latter for organiza-
ave.!" a single distorte.d distribution "may be
used si- tion of input'statistics, generation of _random num-
multaneously to represent two or more random bers, sampling from distributions, simulating _experi-
processes. - Multi-stage sampling is related to impor- ment time and testing to determine the "last run. 1l
tance sampling. Monte Carlo simulation, howevel', requires careful
planning, and competence is needed if computers are
R1.U3S1.'an Roulette and Splitting to be used profitably.
In this case, "uninteresting" samples are "killed" The running time is dependent on the nature. of
b}r a supplementary game- of. chance and "interesting"
the particular problem, implementation of variance-
samples ate split into refined samples by a process reducing procedures, initial guess, and accurac:~r re-
of independent branching to more than make up for qUired. The 'N-....." convergence law implies that if an
the samples eliminated. additional significant figure is desired in a l\ionte
Carlo calculation, the machine time must be increased
Use of Expected Values by a factor of 100. Round-off and truncation errors
have little effect on the accuracy of Monte Carlo solu-
In cases where there is ~ mixture of analytic and tions. An accuracy of about 10 - 20 per cent is usual-
probabilistic aspects of a problem, one may calculate ly attainable without excessive labour_ lVlost experi-
analytically that part which is easy and "Monte Carlo" ments may be processed in 30 - 45 minutes on a large
that which is difficult. machine, although a few applications may reqUire in
excess of 2 - 4 hours of computer time.
C01'relation and Regression
The narne "swindle" was adopted to describe the ApPLICATION AREAS
use of correlated variables in Monte Carlo calcula-
tions.. Hammersley and Morton (3) described "nega- The application of the Monte Carlo method should
tively correlated" variables in the Buffon experiment. be given consideration in cases where:
They used two needles rigidly fixed in the form of a (I)-Solutions are sought for physical processes
Cross and, assuming twice as much labour as that that are either entirely or partially stochastic and the
involved in tossing and recording a single needle, dem- outcome seems to depend, in some important wa)r,
onstrated that to obtain the same level of accuracy upon probability.
One needs to expend 12.2 times less work. Similarly,
three needles will reduce the work by a factor of 44.3 (2)-Actual physical experimentation would be ex-
and four needles by 107.2. An equilateral triangle pensive, time-consuming or hazardous.
lnay be used in the same ,';.ray. The gains that can be (3)-Analytical solutions are prohibitivel)' cumber-
obtained by procedures of statistics are even greater. some or, indeed, impossible.
In other cases, two or more solutions may be involved (4)-Discrimination between important and insig-
simultaneously and 'we can simplify the calculation nificant features of a problem is desired; frequent-
by analysing -differences only (e.g., comparison of al- ly, an analytical treatment is clearly indicated follO\\.~
ternative policies)_ ing a Monte Carlo experiment_
Systematic Sampling (5)-Scattering of data and complexity of geometr)'
cannot be accommodated in any other model without
In this procedure, the amount of sampling for each
region is determined explicitly. This approach is used making defeating assumptions regarding the inherent
frequently in the fil'st stage of a multi-stage sampling variation in some important input parameters_
process. Double systematic sampling is a particular (G)-Experimental data are insufficient and must
form of the above. be "patched upJl with assumptions arising from theo-
retical and intuitive considerations.
St'ratified or Quota Sarnpling (7)-An aggregate behaviour is involved in some
This is a combination of importance sampling and implicit fOI'm in a .dynamic, uncertain environment
systematic sampling. Each region is assigned a fixed such that a macroscopic solution is indicated.
number of samples in order to minimize the variabil- (8)-A measure of effectiveness, success 01' equiva-
ity of the solution (e.g_, proportional to the product lent is to be determined for a planned operation rela-
..
.....
of the relative size of the subregion and its standard tive to its behaviour in an entirely random fashion_
deviation). Stratified sampling is equivalent to break- ~~

(9)-Probabilistic data corresponding to months of

ing the region investigated into several parts and ap-
plying standard Monte Carlo methods to each part ac.tual experience must be obtained quickh~ and cheap-
.:
~~; .... ..
~
ly.
independently. In terms of costs, stratified sampling
is preferable to importance sampling. Many "rigid" calculations can be modified or ex-
Variance-reducing techniques are well treated in tended to become standard applications of the Monte
the literature of statistics_ However, the greatest Carlo method, but the latter should, by no means, be
gains are made by exploiting specific features of the considered a truly general method_ In a few situa-
particular problem rather than by routine application tions, the Monte Carlo method is.the only feasible
one_
of general procedures.

USE OF COMPUTERS ApPLICATIONS IN MATHEMATICS

l\ionte Carlo is a computer-based method because The indiscriminate application of !'rlonte Carlo tech:
it requires a great number of iterative calculations_ niques in mathematics can lead to stepped-up numer-
Applications are relatively easy to program in either ical activity, which in turn may render the solution
conventional (e.g., FORTRAN, ALGOL, ACT) 01' spe- uneconomical. Thus, Monte Carlo is a method of last
cial simulation languages (e.g., GEMS, SPS-l, SIr.!- resort for mathematicians, and it is used primarily in

Technology, July - September, 1965, Montreal 123

multidimensional problems requiring approximate ApPLICATIONS TN STATISTICS
solutions at a "single point" rather than over the en- :Monte Carlo techniques are widely used in the sta-
ti re spd.ce. tistical design of experiments, and mayor may not
In addition to educational illustrations, such as the be connected \....ith techniques referred to as sequen-
multiplication of fractions b:~r playing cards and eval- tial, multiplex and inverse sampling. Typical applica-
uating integl'uls by thro\ving darts, s.Jme Monte Carlo tions include; con-elation of )andom valiables; study
applications in mathematics, of int.erest to the petro- of interdependent events; branching processes fol' de-
leum engineer, would include: sign of new equipment and analysis of failure pat-
(a). I\:Iatrix inversion (e,g.. solving systems of terns; quality control and scheduling of maintenance.
equations).
(b). l\oIultiple integration (e.g., oil-in-place calcula- ApPLICATIONS IN OPERATIONS RESI::ARCH
tions) . The .Monte Carlo method introduces a new dimen-
(c). Solving second-ordel' differential equations in sion of flexibility in many OR analJfses. Single-valued
h\:o independent variables (e.g., Laplace equation of input variables, such as mode, median or mean. are
fluid flow in homogeneous porou~ media). replaced by distribution:) expressing the situation in
(d)' Solving partial differential equations (e.g., un- a more meaningful manner and providing room for
::;teady-state flow equation for Darcy flow of a slight- "optimistic" and "pessimistic" limits, Monte Carlo
ly compressible liquid). techniques extend the procedures for solving the fol-
(e). Ev~'!.luating integration con'3tan~s in two~di lowing classes of problems:
mensional problems (e.g., de~ermining field gradients
(a), Linear programming: mixing problem using
at the edge of an eiectrode in a fuu r-electrode field by
random variables.
Schwartz-Christoffel transformation).
(b). Resource allocation: proration to markets; al-
(fl. Anabrsis of generation and transmission of er-
location of work to pl"oces~ing machines; ~tocl( dis-
1'01'S (e.g., measurements).
tribution.
The i\'1onte Carlo method's simplicity should not be (c). Optimization: warehouse location; shortest
o,,~rlooked. By way of an illustration, the Laplace dif- route problem and k-th best route through a networiL
ferential equation may be solved by "random walk" (d). Inventory: decision rules; inventory control
experiments. A "particle" begins a random walk from and procurement.
an interior point, and when it reaches the boundar}r (e L Transportation and merging: simulation and
the history is terminated with a score equal to the control of traffic and highway merging and conges-
boundary value, The solution at the interior point is tion; flow through networks.
the arithmetic mean of the scores obtained by a suf- (0. Queuing or waiting line theory: terminal op-
ficiently great number of ,mlks, Similarl.y. in solving eration of trucks, buses and ~hips; determination of
a partial differential equation, one computes the state peak demands.
of affairs at a certain time by stopping the random (g). Analysis of type PERT (Program Evaluation
walks following a fjxed number of steps correspond- and Re"ie\.... Technique).
ing to th~t time. (hl. Dynamic programming: maximum time - rate
of return; multif'tage and Markoff renewal processe~.
ApPLICATlONS IN PHYSICS
ApPLICATIONS IN THE OIL AND GAS INDUSTRY
The follo\"r'ing are t.ypical applications of 'Monte
Carlo techniques: .Many ~'lonte Carlo techniques developed in other
fields, especially statistics and operations reseul'ch.
(1 )-Collision, diffusion, absorption and reflection are applicable either directly or in modified form to
of neutrons; nuclear reactions: cascades. the oil and gas indu~try.
(2 )-Radiant heat-transfer through absorbing anJ Generally, weighted a"erages are llsert to eXl)l'e~1i
emitting media (e.g., combustion and shock-wave resen'oir properties. Determinidtic solutiolls require
processes in gas dynamics). cuntroversial assumptions regarding the types of av-
(3 )-Neutron transport and particle transmission erages to be used in calculations. The Monte Cal'io
through matter (e.g., shielding). method circumvents the need fOl" averaging, as il ~arn
('ell-Absorption of "gamma" radiation (e.g., well pies directlJ' from distl'ilmtions. In addition, the
scintillation counters). .i.\'1onte Carlo method treats the solutiollH with con-
tinuing cognizance of the limiting ranges of accuracy
(5'l-Transmission of a disturbance or signal
and reliability. In any case, failure to a}Jply Monte
through a material or region against impediment by
Carlo techniques does not avoid an inherent prolmbil-
random irregularities imbedded in the medium (e.g.,
ity aspect; jt merely transfers uncertainty in a dil'-
transmis!-:ion of pressure disturbances in the ptesence
guised form to some other f;tage of deci:;ion,
of lenses and barriers).
The probability that a variable has a certain v"lue
(G I-I\:Iacroscopic description of multi-phase fluid
but may val"~! between certain limitti often arise...; in
flow through deformable porous media (e.g., retention
by filters 1. oil and gas evaluations (<1). :Monte Carlo technique.,;
can be u:::ed in multival"iable correlation~ and applied
(7 )--Calculations of molecular flow rates through 011 distribution!-i instead of :-jingle-valued parameters.
~mall channels of complicated shape.
ProbablY the most valuable information made avail-
.J. "V. Gibbs (1876) introduced the stochastic meth- able by these technique." would be a Imnwh.!clge or
od in statistical mechanic~ through the hypothesis the oYer-ali variability of final results,
that the average random walk over the time is simi- j1!Ionte Carlo techniques can be u~ed ill illtegl'HtiollH
lal to the average over the phase space. On this basi!i. uver multi-dimensional spaces (e.g.. oil and gas-in-
Ihermodynamic quantities were determined by integ- place calculations). If desired, Monte Cal'lo integra-
rating over thp ensemble of phase ~pace (e,g., ga:; tion may be refined by introducing multivariate proll-
c.ompres~ibility factor), hbility fund ions (e.g.. distribution of formation water

T24 The Journal of Canadian Petroleum

 ~:

saturation as a function of porosity, permeability and- Some applications in property acquisition are as
structural position). follows:
Available historical or experimental data are very (I)-Use of sequential investment models, based in
'-.
important and should be u;ed ,~herever possible. "Per- part on uexperience" piobabilitJ.' distributions and
sonal probabilities" should not be underestimated, ,
landom waiting time between possible acquisitions, to , '

howeve:r;, when recommended by mature and experi- formulate decisions regarding the rejection of marg-
enced engineers. inal deals and preparations in expectation of attrac-
tive deals. ' ..
ApPLICATIONS IN OIL AND GAS EXPLORATION AND (2)-Optimal selection of investments involving
uncertain returns so that the total funds available for
PROPERTY ACQUISITION
investment will not likel.y be exceeded and the average k~: .. :
~;:...' ",.
Nature) using well the deck of cards in her posses- variance for the total investment will remain below l'"

sion, responds to the search for oil and gas b~r signals some preassigned level.
that are partially random. Exploration for oil a.nd (3)-Selection of investments to minimize the vari-
'.
gas therefol'e involves many uncertainties. Each ele- ance of a sum of rates of return subject to the con-
mental assumption in the exploratory search involves straint that the average rate of return exceeds some
its own degree of llncertainty_ Together. these as- preassigned level. This may require the application of
sumptions p:1rramid to a total uncertainty of critical modified dynamic programming techniques.
,
" proportions. The uncertainty is not really related to (4)-Determination not only of the most likely rate
the determination of the success ratio of a company of return for a particular investment, but also of the
with infinite resomces, which can conduct explora- ~hances of intermediate rates of return or even of
tion on such a large scale that it can rely on 'long- total loss. Investments! costs, market values and prices .~. ".
range" average results; nor is the problem simply to are expressed in terms of distributions.
decide whether to drill a well or not. The problem (5)-Sampling from Bayesian distributions and
seems instead to be that of the operator with limited analysis of individual, personal probability beliefs.
. funds ''''ho wishes to explore a particular basin and The experiences which can be gained through real-
seeks a decision from an individual and specific point istic simulations may explain the abundance of busi-
of view. This is so because it is well knmvn that op- ness games conducted by many organizations today.
erators proceeding in a prudent manner have drilled
as many as twentJ' unsuccessful prospects although
ApPLICATIONS IN RESERVOIR ENGINEERING
the industry-wide success ratio in the area was as
high as one discovery in five trials. Chalkey, Cornfield and Park (6, 10) described the
Monte Carlo tl2!chniques can be used profitably in measurement of porosity by a stochastic method as
simulations of exploratory programs to focus the at- follows; a pin is thrown on a photomicrograph of a
tention on critical "reas (5): Although not a sub- section of the porous material, and a "hit" is scored
stitute for the undertaking of ,,,'orthwhile risks, the if its point falls in a pore. If the experiment is re-
Monte Carlo method assists in the following: peated many times in a random manner, the -ratio ..
(I)-Reduction of the number of possible explora- of "hits" to "throws' approaches the value of porosity.
tion programs to a manageable size. Similarly, if a needle of length I is dropped a great
(2)-Application of statistical and personal proba- number of times on a photomicrograph enlarged n
bilities to change a decision from one of uncertainty times! and counts are made of the number of times
to one of assumed risk. the end points fall within a pore, h, as well as of the
Typical applications in exploration are: number of times the needle crosses the perimeters
of the pores, c, then the specific surface s may be
(I)-Tracing the history of an individual oil com- computed from: ' ,
pan]! by branching processes and sequential sampling
from appropriate probability functions. Individual ex- 4oi>c
5=---n (Eq. 2)
periments can be terminated following attainment of Ih
a fixed number of ,'eutures, a certain level of economic
- stabilit]r, an amount of profit, a merger or bank- This is considered to be the best method of determin-
ruptcy. This would include individual "random walks." ing the specific surface (6),
(2)-Bidding models on the basis of the particular Warren and Skiba (7) studied an idealized mis-
company's policy regarding returns and utilit,}, proba- cible displacement process in a three-dimensional he-
bility of discovery, likely competitors and their bid- terogeneous medium by means of experimental com-
ding patterns. The model would be useful in the de- putation based on a Monte Carlo model. In their study,
termination of the calculated risk toward the ac- displacement processes were described in terms of the
ceptance of a particular bid. probability of "residence time" of "mathematical"
(3)-Preparation of stochastic models of the dis- particles representing input fluid.
cover}' process, including: In recent years, the mathematics and application of
(a). Calculation of the number of prospects to drill statistical and Monte Carlo techniques have received
in a particular basin so that there will be a reason- increased attention in mineral sciences (8, 9).
able statistical probability of obtaining production According to Collins (6), if a mathematical theory
in at least one prospect. of flow through porous media is possible at all, it
(b). Dete-rmination of type of hydrocarbon upon must take the form of a statistical theory describing
discovery (I.e., gas or oil)- the macroscopic features of the flow, in the sense of
(c). Calculation of the size of oil or gas discovery, the "ergodic hypothesis" of Gibbs. Several investI-
using appropriate distributions_ gators (l0, 6) have heated this problem in terms of
Cd). Calculation of "extension" and secondarJ' re- random walks of a "drunkard '''ithout or with 'some
covery reserves by drawing relative appreciation memory;" Le., with or without autocorrelation. The
factors from distributions Ustratified" by "year randomness of a porous medium, however, has yet to
since discover}'." be successfully represented.

TcchnologYI July - Sep~ember, 1965, Montreal 125

 Warren and Price (11) studied the effect of the dis- TABLE
position of heterogeneous permeabilities on single-
phase flow for a known permeability distribution func- POTOsit)' CWlIlrlali,'e FrequCllcy Distribution
tion. They compared pl"obabilistic permeability solu-
tions fOi' radial flow between concentric cylinders with F(~j
permeabilities calculated on the basis of arillnuetic,
geometric and harmonic means, as well as of the ~O o
median and the mode. 57 0.1l0
6.9 0.239
78 0.306
85 0.389
Oil-i?/-Place Calculations 96 0.500
10.3 0.581
The determination of oil or gas in place provides llii 0.697
an ideal setting for the Monte Carlo approach. A com- 13.3 0.807
14 l 1.000
puter program was prepared to calculate the oil-in-
place and applied to the Judy Creek Beaverhill Lake
pooL The input consisted of:
A (acres) = 27.000 h ([eeU = 70 TABLE [1
Bot (fnction) = 1.<110 (SPE Standard Symbol's)
Formati(Jrt lVa/er FrequtJlf:Y Distrib,ll;ol/:
The porosity. {P, was introduced as a cumulative fre- Porosity Rallf!e: :1.0-5::
quency function and jH shown in Table I and Figure
1_ The formation water saturation, 8 . . . , was expressed F,(S" )
by five distributions, The effect of structural posi-
tion on the \\'ater saturation wa::; not considered in 20.5 o
this ca~e. Formation water saturation distribution~ Z3.5 0.006
26.5 0.021
therefore account only for the porosity \'ariation and 29.5 0.086
are shown in Tables II to VI. They are the result of 32.5 0.2\9
an "educated interpretation" of the "scattering" of ~55 0.501
data in a l/J-S .. plot. An alternative would have been 385 0.754
,11.5 0.913
to use simple dala proces!iing procedures to alTi\'e 44.5 0.974
fit statistical distributions. Stratified sampling ac- 47.0 1.000
cording to structura! position and type of porosity
was considered. but. in the intere;;;t of simplicity, was
not implemented, A :iequential sampling technique,
howevel', was employed. First, a pseudo-random num- used in calculation, seems to provide realistic results_
ber was generated and the porosity drawn from its Differences between evaluations using various "aver-
distribution. A second random number was entered age.:-" and Monte Carlo techniques can be computed
in the appl'opriate fOI-mation water distribution. de- for any desired cut-off Multi-modal frequency diH-
pending upon the porosity value sampled by the first tributiolls reflecting several sy.sterns m~,y be. treated
random number. Corresponding porosity and forma- either compoundly or .separately, Additional data may
tion water values were used to calculate the oil-in- be produced by "random walk" techniques (e.g.. a.s-
place using the exprelision: suming porosity 'pr.ltential" surfaces) or, \vhere resel'-
VOil- information is incomplete, by sampling from dis-
N ~ 7758 Ail" ( l - S".iIl", lEg. ,?j
tributions_
The results uf the Monte Carlo {~xperiment are Conventional splitting and weighing technique!'; may
shown in Table VII. The expected N, following 1,090 be used for sampling from distributions to some fixed
experiments, equals 819,763 10 6 STB. Using average amount of item~ (e.g., volume represented per Imm~
properties, i.e., = 9.3 per cent and S" = 16 per pie), These are applicable to either single- 01" multi-
cent, one obtains 812_372 lOr. STE_ The application of ple~liample lipaces. Although rejection of samples af-
this lVlonte Carlo technique, therefore, while by-pass- fects the probability functions. in pl'actice thili may
ing the need for selecting the type of averages to be not be significant.

,
" ---
~

""
/ - ~
..-----
~

~
/'
/
.~

z
0

:::O~
Ftl1ll' 0

~
B
~~
.. Ell
,,/ ;:/
~
./ w /,,0/ /

/
>
s;: ~
~c

:5 ,J /
G
V >

Vi
u
"
~, ,
" "
0
0'
" " '00
POR051TY ~ po. Ci>l11 PE"~EA8111TT ~ mil

Figul'e 1.- Pm'o8tfy CUlJIulative Fl'equ611cy Figure 2.-Penllcability CUlHulative Fretllw/1CY

Di,'itrib!ftion. DisfJi.butiolls.

126 The Journal of Cl::lnodion Petroleum

 '/

. ~~(:~<;
~c~:.'''::
;;;:._:~:':_'
TABLE III TABLE V
~:;;:.~
Formation Water FTequency Distribution;
POTOSUy Range: 5.3-7.6
Formation Walef Frequency Distribution;
Porosity Range: 9.9-12.2 ,.>
~.

S.. (%)
S. (%) F,(S.)
7.0 o
13.5 0 8.9 0.017
15.9 0.006 10.8 0.044
18.3 0.Ql8 12.7 0.189
20.7 0.041 14.6 0.439
23.1 0.083 16.5 0.669
25.5 0.284 18.4 0.806
27.9 0585 20.3 0.898
30.3 0890 22.2 0.956
32.7 0949 24.0 1000
35.5 1000

TABLE IV
VI
Formation Waler Frequency Di.'ilribwion;
TABLE

FOT11wJion Water Frequency Dislribution;

,..
Porosity Range; 7,6-9.9
Porn.'iity Range: 12.2-14.4

5..(%) F,CS,J S.(%) F,(S,,)

9.7 0 5.1 0
118 0.01I 6.8 0.027
13.9 0.031 8.6 0.086
16.0 0.060 10.3 0.214 . -::
18.1 0.145 12.1 0.457
20.2 0.371 13.8 0.629
22.3 0.684 15.6 0.810
24.4 0.912 17.3 0.888
26.5 0.974 19.1 0.936
28.4 1000 20.8 1.000

.,:~
.~
"

j
TABLE VII
OIL-IN-PLACE CALCULATION BY A MONTE CARLO TECHNIQUE

Fonnation Expected Standard

Experiment Porosity Water Oil-in-Place Oil-in-Place Deviation
n ~(%) S, (%) N(MMSTB) :ENjn(MMSTB) S(MMSTB)
10 13.4 12.9 1217.651 823.032 275.212
20 8.8 19.3 737.2,0 774.251 301.351
30 14.2 11.3 1310.395 799.850 304.773
40 14.4 13.7 1293085 808.604 325.986
50 5.6 27.6 425.127 790.657 315.583
60 5.7 19.3 479.882 812.068 316.727
70 14.2 19.7 1I88.813 799.383 328.290
80 10.3 15.4 905.860 811.405 326.595
90 8.0 22.0 645.354 800.999 322.565 f:
100 5.8 23.4 460.447 7911I7 326.405
200 10.4 17.6 892.440 833.400 325.827
300
400
11.2
13.4 I 16.8
113
968.747
1232.252
838.692
819.106
314.459
318.590
500 143 119 1312.445 826.402 317.283
600 10.4 13.7 935.m4 819.129 317.503
700 5.7 27.6 127.020 824.285 319.679
800 13.6 10.3 1264.661 823.709 319.795
900 7.5 26.4 577.540 82I.I74 318.735
-
1000 8.4 22.0 679.361 823.625 317.379
1010 8.8 15.0 778.989 823.465 316.755
1020 7.7 22.2 626.699 823.171 316.889
1030 14.3 15.5 1260.481 821.I56 317.612
1040 12.4 20.2 1027.008 820.745 316.893
1050 10.1 15.4 887.212 820.275 316.996
1060 7.9 22.1 641785 820.026 317.319
1070 9.5 16.5 004.052 818.568 316.962
1080 3.4 35.3 228.109 818.324 317.353
I090 7.6 18.3 648.883 819.763 316.792

Technology~ July - September, 1965, Montreal 127

 Productivity TABLE IX
As a second illustration of the application of the PRODUCTIVITY INDEX DAT,\
Monte Carlo method, a simple, single-phase, steady-
state radial flo,,," problem ,",'as developed. The produc-
tivity index was calculated on the aS8umption that Pool: Leafiand Vking
the absolute horizontal permeability varies in a radial P, ~
2,~A7 psig Average f1owin~ pre~surc
B" ~
1.450 durin~ productivity test!>:
direction away from the ,vellbore, but that the drain- ~o ~
0.4 ep. p". = 2183 p<:.i~.
age area could be approximated bJr concentric zones, h avg. ~
7.5 feet Avcra~e of eleven pro-
rJ ~ rJ-I, within ' ... hich the permeabilitJr, ki, is uniform. k avg. = 6.28 md ductivil\' test5 in two wcll.,;::
(core analysis; footage J ~ 0.289
The permeabilities, k j , were obtained by sampling weighted'l
from distributions of the core analJrsis data of either k clltoff 0.1 md
~
.r (using k ""16.1 ~ 0,066
the well, ~iodel 1, or the pool, Model 2. In iVIodel 1, the
assumption was made that the lateral permeabilit}r a = 160 acres per well- r,. ~
1,490 rl~cl
assumed: r. - 0.263 feet
val'iation is of the same order of magnitude as that of
the horizontal permeability for depth intervals cored ~Vdl: 12-16-10-55
in the welL Model 2 assumed that the lateral per-
meability variation is evaluated by all samples tested
[,
- 11.6 feet A\rcragc of six: productivit)r
tC!;ts:
in the pool and would therefore be more readily ap- k = 8.06 Old J 0.291
~

(core anal)rsis; footage J (using k) ~ 0,132

plicable to cases identjfied by thin payor lack of weighted)
stratification_ The "depths" of the individual rings
were assumed to be either independent of permeabil-
ity, "Geometry a":
tiO}IS.4 or ,j and 6, depending on the model. The model
permeability is calculated from:
(Eq..1)
IOIr (r.. /r.. . \
or dependent upon both permeability size and jumps,
!;-", = ('1. /l
1: r _1_ log (_r,_)
"Geometry b": 1 k; r,1

The average pool pay thieknes~ and ten rings (Le.,

where r = 10) weL'e used in all case~.
C is a proportionality factor determined from: Table X contains partial results of the productivity
index calculations.
j = r The expected permeability, k,.~I" wa~ l:alculated from:
C 2: Ri-l R j + i = r.. - r,, ((/. 61
j =
n
using R" = 1; R"'_I = 1 and where R i is the largest of :::: kill
I
(Eq. S)
n
log (!;-l + l~
lClg (kJrl + U The produeti\'ity index equal~:

An individual experiment is c.onducted by drawing r

pseudo-random numbers and obtaining permeability J (STB/D/psil = 0.00307 kIll h_ (Eq.9)
~ B II (LII log (r,,/r.1
values from appropriate distributions. The cumulative
frequency distribution for the pool, Leafland Viking, As expected from an inspection of FiguJ'e 2, the cal-
and '...ell, 12-16-40-55, are shown in Table V[II and culated permeabilit)r and produeti"it~.. indices are high-
Figure 2. Ring boundaries are calculated from Equa- er \"'hen using the permeability distribution for the
pool (i.e., Model 2). The e.ffect of geometry Hppenr~
to be negligible. Differences between core Hnalysi::l
and actual productivit)r te~t~ cannot be reconciled by
TABLE VIII
serial flow models used in the Monte Carlo calcula-
Permeahiliry Cumu{a/ir'e Freql(e1lcy Di~rrilmlioll tion and must be explained by well stimulation 01' lacle
of stabilization during tests.
Well k (md) Pool k (md) F(k)
CONCLUSIONS
0.10 0.10 0
By its ve.ry nature, the oil and gas industry is
Oll O.I~ O.ll frequently concerned with stuchastic rather than l'igid-
Iy defined environments.
0.12 0.18 0.22
The methodology of Monte Carlo techniques is now
0.16 0.26 0.3:1 well under::;tood and can, in certain cases, offer the
0.23 0.3B 0.4. alert industry definite adnlIltages. Cooperation be-
tween engineers, mathematicians. statisticians, econo-
0.37 O.5B 0.55 mists and computer personnel i::; needed ill order to
grasp the full scope of the possibilities, as well a~
0.76 O.q. 0.66
identify and implement the applications, of the Monle
2..10 1.95 0.77 ea rio method.
1LOa 7.50 O.BB Although by no means an answer to all pt"Oblem:->,
i\!Ionte Carlo techniques should be adopted as a di~
100.00 100.00 LaO tinct component in a company's repertoire of tools.

126 The Journal of Canadian Petroleum

 "

" '

TABLE X
PRODUCTIVITY INDEX CALCULATION BY A JV:ON1E CARLO TEChNIQUE
.:.-
1l1cdell ft-Icdel2

"Georneir}' a" "Geomefry b" "GeomeiTY a" "Geometry b"

Experiment k" xr j; (bbl( k""I' .j;(bbl( k,,:o:p j; (bbl( Irexi j; (bbl(

n (md) day/~si) (md) day(p,i) (md) day/psi) (md) day(psi)

1. .... " .. 0.233 0.0025 0.232 0.0025 LI50 0.0122 0,357 0.0038
2 ......... 0.322 0.0034 0.320 0.0034 0.783 0.0083 1.240 0.0131
3 ........ 0.448 0.0047 0.459 0.0049 0,603 0,0064 0.872 0.0092
4 ....... 0,462 0.0049 O,4eO 00051 0.590 0.0062 0,769 0.0081
5 ....... 0.442 0.0047 0.455 0.0048 0.513 0.0054 0.698 0.0074
6 ...... , .. 0.435 0.0046 0.440 0,0047 0,486 0.0051 0.604 0.0064
7 ......... 0.391 0.0041 0.396 0.0042 0,452 0.0048 0,564 0,0060
8 ......... 0,465 0.0049 0.469 0.0050 0,455 0,0048 0.523 0.0055
9 ...... .. . 0.435 0.0046 0.439 0.0046 0,482 0.0051 0,482 0.0051
10 ......... 0,491 0.0052 0.508 0.0054 0,448 0.0047 0,472 0.0050

20 ......... 0.380 0,0040 0.391 0,0041 0,488 0.0052 0,462 0.0049

30 ..... ". 0.342 0.0036 0.356 0,0038 0,471 0.0050 0,401 0.0042
40 ....... ' 0.353 0.0037 0.359 00038 0.501 0.0053 0.418 0.0044
50 ......... 0.345 0.0037 0.349 0.0037 0.516 0.0055 0.501 0.0053
60 ......... 0.338 0.0036 0.346 0.0037 0.544 0.0058 0.476 0.0050
70 ...... .. . 0.348 00.037 0.357 0.0038 0.525 0.0056 0,466 0.0049
80 .... ... .. 0.349 0.0037 0.357 0.0038 0.515 0.0055 0.495 0.0052
90 ..... .. ". 0.350 0.0037 0.353 0.0037 0.505 0.0053 0,475 0.0050
100. , ....... 0.336 0.0036 0.340 0.0036 0.534 0.0056 0,466 0.0049
1I0., .... '" 0.329 0.0035 0.332 0.0035 0.540 00057 O,4N 0.0050
120 ........ , 0.320 0.0034 0.323 0.0034 0.561 0.0059 0.48<1 0.0051
.' .
~.-
126 ......... 0.316 0.0033 0.319 0.0034 0.562 0.0059 0,482 0.0051
127 ........ , 0.318 0.003i 0.320 0.0034 0.559 0.0059 0.481 0.0051
128 ....... ' 0.317 0.0034 0.319 0.0034 0,559 0.0059 0,480 0.0051
129 ...... 0.315 0,0033 0.317 0.0034 0.563 0.0060 0.482 0.0051
130 ...... .. 0.314 0.0033 0.316 0.0033 0.565 0.0060 0.482 0.0051
131. ...... .. 0.316 0.0034 0.319 0.0034 0.564 0.0060 0,482 0.0051
132 ... , , .... 0.315 0.0033 0.318 0.0034 0.563 O.OOED 0,480 0.0051 ~<. --
.0 : ' , . '
133 ........ 0.315 0.0033 0.318 0.0034 0,565 O.OOEO 0.487 0.0051 .~ '; ,
134 ... " .... 0313 ' 0.0033 0.316 0.0034 0.5f.5 O.OOED 0,486 0.0051
135 ... ._-.- - 0.312 0.0033 0.315 0.0033 0.5('2 0.0059 0.483 0.0051

.;
ACKNOWLEDGMENTS (6) Collins, R. E., "Flow of Fluids Through Porous :i\rla-
terials," Reinhold Publishing Corporatioll, New York,
The author expresses his appreciation to the Oil 1961, 270 p.
and Gas Conservation Board, Calgary, for permission (7) JVal"J"en, J. E., and Skiba, F. F.J ui\faeroscopic Dis-
persion,'J Society of Petroleum Engineers Journal,

,
(I
to use its computer facility. Special appreciation is
extended to fifess,"s. G. D. Hnlbert for completion of
several Monte Carlo programs, - N _ Collins for selec-
September, 1964, pp. 215 - 230.
(8) Hazen, S. Hr' J /7-., "Statistical Analysis of Sample
Data for Estimating Ore," Bureau of i\Iines, Vrash-
' ..
";:~."." .
.::: : ....
., tion of specific applications and E. J _ Morin for re- ington, 1961, 27 p.
,1 viewing the manuscript. The opinions expressed in (9) Hewlett, R. F_, rlSimulating Mineral Deposits Using
:, the paper, hOlrl'ever, al'e entire!)' those of the author. Monte Carlo Techniques and Mathematical Models,"
i Bureau of Mines, Washington, 1964, 27 p.
(10) Scheidegger, .A. E., "The Physics of Flow Through
Porous Media," University oj To-ronto Press, 1963,
REFERENCES 313 p,
(11) Wan'en, J. E., and Pl-ice, H. 8., "Flow in Hetero-
(1) Householder, A. 5., Editor: ".i.Vlonte Carlo IVlethod," geneous Porous Media," Society oj Pet"oleu7n Engi-
p?'oc6cdings of June 29 - July 1, 1949, Sl'mposiunl, nec?,s JOU?7u,Ll, September. 1961, pp. 153 - 169.
National Bureau of Standards, Vitashington, 1951,
42 p.
(2) Meyc1, H. A_, Editor: "Symposium on Monte Carlo
Methods," John Viriley and Sons, New York, 1956, Eliador' (Doren Stoian is manager,
:3iD p. data processing, at the Oil and Gas
(3) Ham?lZcl'slcy, J. llf., and Jllo1"ton, If.. IV., "A New Conservation Board in Calgory, Alberta.
Monte Carlo Technique: Antithetic Variates," Pro- Previously, he worked as a speciol stud-
ies and reservoir engineer for the .same .. ;
ceedings of the Camb?'idge Philosophical Society,
1956, pp. 449 - 475. orgonization, as an instructor at the
Univer.sity of Alberta in Edmonton, ond
(4) Walstrom, J. E_, ';A Statistical .i.Vlethod for Evaluat- in vorious capacities in Fronce, Ger-
ing Functions Containing Indeterminate Variables many, Austria and his native country,
and its Application to Recoverable Reserves Calcu- Roumania. He holds a B.S. degree in
lations and Vilater Saturation Determinations," mechanical and petroleum engineering
Com.puten in the iIJine?'al IndU8trics, Stanford Uni- from the Technical University of Han-
versity Publications, 1964, pp. 823 - 832. over, Germany, and is active in several
(5) Kuufm.an, G. 111., UStatisticai Decision and Related engineering, computer ond doto pracess-
Techniques in Oil and Gas Exploration:' Prentice- ing societies.
Hall, Inc., Englewood Cliffs, New York, 1963, 307 p.

Technology. July -I September, 1965. Montreal 129