Estimation of Petroleum Exploration 

Success and the Effects of Resource 

Base Exhaustion Via a 

Simulation Model 

By LAWRENCE J. DREW 

GEOLOGICAL SURVEY BULLETIN 1328 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHING TO~ 1974 
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary 

GEOLOGICAL SURVEY 

V. E. McKelvey, Director 

Library of Congress catalog-card No. 73-600342 

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 -Price 45 cents (paper cover) 

Stock Number 2401-02458 
CONTENTS 

Page 

Abstract ------------------------------------------------------- 1 

Introduction --------------------------------------------------- 2 

Acknowledgments ----------------------------------------·--- 4 

Exploratory well allocation scheme and input data arrays ---------- 4 

The hindsight model -------------------------------------------- 5 

Historical aspects ------------------------------------------ 6 

Response characteristics of the hindsight model --------------- 6 

The similarity assumption and an example -------------------- 10 

Exhaustion ---------------------------------------------------- 14 

Compound-discovery-event probabilities --------------------------- 20 

Conclusions ---------------------------------------------------- 2'4 

References cited ------------------------------------------------ 25 

ILLUSTRATIONS 

FIGURE 1. Diagram of idealized part of a digitized basin showing 

field locations, potential well sites, and barren points 

within a field -------------------------------------- 2-13. Graphs showing2. Probability curves for the discovery of specific 

quantities of petroleum --------------------- 3. Average quantity of petroleum discovered versus 

exploratory drilling intensity ----------------

4. Average incremental quantity of petroleum discovered ------------------------------------ 5. Quantity of petroleum in resource base ---------

6. Number of deposits in resource base -----------

7. Average quantity of petroleum discovered ------

8. Average number of deposits discovered ---------

9. Probability of gambler's ruin ------------------

10. Average quantity of petroleum discovered for 

three levels of exhaustion -------------------

11. Partial derivatives of the average quantity of 

petroleum discovered for three levels of exhaustion --------------------------------------- 12. Average quantity of petroleum discovered per 

well versus level of exhaustion for four levels 

of exploratory drilling ---------------------- 13. Number of exploratory wells drilled in the Powder 

River Basin, 1952-71 -----------------------

Page 

5 

9 

10 

11 

15 

15 

16 

17 

18 

18 

19 

20 

21 

III 
IV CONTENTS 

TABLES 

Page 

TABLES 1-2. Summary statistics and probabilities of discovering 

various1. Quantities of petroleum -------------------- 6 

2. Numbers of commercial deposits ------------ 7 

8-5. Estimates of the probabilities of8. Discovering any of six deposits within each 

year ------------------------------------ 12 

4. Compound discovery events ----------------- 22 

5. Discovering any of six deposits with the actual 

drilling effort expended between 1952 and 

the actual year of discovery of each deposit_ 28 
ESTIMATION OF PETROLEUM EXPLORATION 

SUCCESS AND THE EFFECTS OF RESOURCE BASE 

EXHAUSTION VIA A SIMULATION MODEL 

By LAWRENCE J. DREW 

ABSTRACT 

A random-walk model was used to study several as,pects of the process of 

exploring for petroleum deposits. The Powder River Basin, Wyo., was 

chosen as a control area for this study. 

The outcome of exploration is expressed by performance criteria; these 

include the average quantity of petroleum discovered, the average number 

of deposits discover·ed, the probability of "gambler's ruin," and the probability of attaining various levels of success. The incremental return per 

exploratory well was determined. 

Several of the more significant results predicted by the mod,el are: 

1. Increasing the number of exploratory wells per program increases, but 

at a decreasing rate, the average number of deposits and the average 

quantity of petroleum expected to be discovered. (Ten wells yield an 

average of 0.22 deposits containing an average of 13.1 million barrels 

of petroleum; 100 wells yield an average of 2.11 deposits containing an 

average of 122.8 million barrels of petroleum.) 

2. The probability of gambler's ruin (drilling consecutive unsuccessful 

wells) decreases as the number of wells per program increases. (Ten 

wells have a p·robability of gambler's ruin of 0.801; with 100 wells, the 

probability is 0.103) 

3. The expected incremental return per exploratory well decreases as the 

number of exploratory wells drilled increases. (At the 5-well intensity, 

the incremental (next) well returns 1.31 million barrels; at the 450-

well intensity, the incremental well returns 0.77 million barrels) 

A resource base exhaustion study determined the level at which different 

sizes of exploration programs will not achieve a s~pecified rate of return. 

Random-walk allocation (which excludes geologic information from influencing discovery potential) of the number of exploratory wells to be drilled 

in an area was used to estimate the probability of discovering specific deposits during various time periods. If the exploratory drilling effort expended 

in the Powder River Basin between 1952 and 1968 (2,812 wells) was allocated according to the random-walk strategy, the probability of discovering 

the Hilight deposit would be 0.999+, a virtual certainty. 

1 
2 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

INTRODUCTION 

A computer simulation model was used to study several aspects 

of exploring for petroleum deposits within the Powder River 

Basin, Wyo. The purpose of this study is to estimate the performance of the random-walk exploration model in the hypothetical exploration of a large region in which a substantial number of deposits has been discovered, such as the Powder River 

Basin. When a technique becomes available which will yield a 

reliable estimate of the total frequency distributions for the size, 

shape, reserves, and depths of the deposits contained in an area, 

a valid comparison can be made between the performance of the 

random-walk model and the success rates actually achieved in 

the basin. Such a comparison cannot be made here because an 

estimate of the total resource base of deposits contained in the 

basin was not available for this study; only the suite of deposits 

discovered with the network of exploratory wells drilled through 

the year 1970 was available. If these exploratory wells had been 

located differently, a somewhat different suite of deposits would 

almost surely have been discovered. 

The term "gambler's ruin" is used in this report to denote the 

probability of not discovering a single deposit during the execution of a given exploratory drilling program. Several additional 

terms used in this paper are defined below: 

1. Simulation modeL-A device (a computer program) which is 

used to mimic the exploration process. 

2. Random-walk allocation rule.-A rule making it equally likely 

for any point in the basin to receive a well in an exploration 

program, given that the point is not closer than a specified 

distance (lag) from the location of the last well drilled or 

that it is not within the boundary of a previously discovered 

deposit. 

3. Resource base exhaustion.-The sequential depletion of deposits from a resource base through the exploration process. 

4. Hindsight simulation modeL-A model which assumes that an 

analysis of historical data can predict the outcome of future 

exploration. A static time reference is used-that is, time is 

not a factor in itself (in that it passes) but only in that it 

refers to a point at which the full resource base is present. 

In this study, 1899, the year when the first deposit was discovered in the basin, was chosen as the time reference point. 

5. Sequential simulation modeL-A variant of the basic hindsight 

model which employs a dynamic time element-that is, at 
INTRODUCTION 3 

any given time only those deposits remaining to be discovered between that year and the year 1971 are retained in 

the resource base. 

The hindsight simulation model was used to demonstrate the 

relationships between the intensity of exploratory drilling (the 

number of wells per program) and the outcome of exploration, as 

summarized by a suite of performance criteria. The criteria include (1) the average quantity of petroleum discovered, (2) the 

average number of deposits discovered, (3) the probability of 

gambler's ruin, and (4) the probabilities of obtaining specific 

levels of success. 

The hindsight model simulated the discovery of that part of the 

entire petroleum resource base discovered in the Powder River 

Basin (approximately 27,000 mP) during the 1889-1970 time interval. During this period 154 deposits were discovered in the 

basin. This suite of deposits is conservatively estimated to contain 

1.6 billion equivalent barrels of producible petroleum; approximately 35 percent of this quantity is contained in a single deposit, 

the Salt Creek Deposit. 

In the hindsight simulation model the exploratory wells were 

allocated according to the random-walk strategy. The intensity of 

exploratory drilling was varied from a minimum level of 10 

wells to a maximum level of 500 wells. From the raw simulation 

results, tables and graphs were constructed which display the 

response characteristics of the exploration model for each of the 

performance criteria within this range of exploratory drilling. 

The effect of incremental exploratory drilling upon the average 

quantity of petroleum discovered was also examined. 

The sequential variation of the exploration model was used to 

isolate the effect of resource base exhaustion upon the exploration 

performance criteria. The same method was used to allocate the 

exploratory wells in the sequential model as was used in the hindsight model. The resource base on which exploratory drilling was 

applied in this model was the suite of 111 deposits discovered during the actual exploration of the basin between the years 1952 

and 1971. The effects of exhaustion of this resource base upon the 

exploration performance criteria were determined by removing 

each deposit discovered during each yearly time period and redrilling the basin with the same intensity as actually used. 

The study also estimated the probability of certain compound 

discovery events, the discovery of a deposit at least once during 

a sequence of time intervals. 
4 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

ACKNOWLEDGMENTS 

The input data were collected and the majority of the computer simulation runs were completed while the writer was a 

research geologist with Cities Service Oil Company, Tulsa, Okla.; 

the writer wishes to gratefully acknowledge the support he received from Cities Service. Particular acknowledgment goes to 

Joseph Huffstetler who prepared several of the basic documents 

for this study and to Phillip Fincannon who wrote the majority 

of the logic for the basic simulation model. 

EXPLORATORY WELL ALLOCATION SCHEME AND 

INPUT DATA ARRAYS 

In both model variations the random-walk method of allocation 

was used to locate exploratory wells. Using this allocation rule, 

any point within the basin is equally likely to receive a well, given 

that it is not closer than a specified distance (lag) from the location of the last well drilled or that it is not within the boundary 

of a previously discovered deposit. The magnitude of the lag was 

chosen arbitrarily to range from a maximum of 10 miles (16 km) 

for a lO-well exploration program to a minimum of 2 miles (3.2 

km) for exploration programs with 100 or more wells. 

Two types of input data arrays are required by the models. 

First, the configuration of the boundary of the basin and the 

geometry of the target deposits must be input in the form of a 

grid. This formulation allows the boundaries of the target area 

and the geometry of the target deposits to be approximated to any 

desired degree of precision by simply varying the spacing of the 

input array. The second type of data input required is estimates 

of the quantity of petroleum that can ultimately be produced from 

each target deposit. When such estimates were unavailable, cumulative production figures through February 28, 1971, were used. 

The grid characterizing deposit geometry and the configuration 

of the boundary of the basin was obtained by digitizing a base 

map with a :1;4 by :1;4-mile (0.4 by 0.4-km) grid. This density is 

equivalent to one datum point per 40 acres, which is the smallest 

of the commonly used development drilling spacings. Each grid 

point of the basin was classified as being either barren or within 

a productive petroleum deposit. Those points located within the 

boundaries of productive deposits were assigned a code number 

which was keyed to the estimated productivity of the deposit. In 

simulating the exploration of the basin, these code numbers 

served as the basis for an accounting system utilized to summarize 
WELL ALLOCATION SCHEME AND INPUT DATA ARRAYS 5 

the performance of each exploration program. An idealization of 

this digitization process is given in figure 1. The principle advantage of formulating the input data array in this manner is that 

an explicit relationship between the intensity of exploration and 

both the probabilities of discovering each target deposit and the 

values of each performance criterion were determined. 

THE HINDSIGHT MODEL 

Two central elements are common to any variation of a hindsight model. First, the spatial distribution of the targets and the 

distribution of their values are derived from records of past exploration activity in mature or exhausted areas and not from predictions based upon geologic or geophysical data of the resource 

potential remaining to be discovered within the target area. Second, to predict future performance from historical data, a similarity assumption is made. The similarity assumption nearly always 

is an assertion that the resource base discovered in the past in 

one geographic region (the control area) is similar to the resource 

base remaining- to be discovered in another geographic area (the 

target area) and henceforth the control area resource base can be 

Barren well locations 

=I 

(I .,.:;{ 

:( I"· 

' • ' ~'/"1 :V • & / I ..._"'!:......_ •. __...., •. 

Powder River Basin 

0 10 20 MILES I I I I I I 0 10 20 30 KILOMETERS 

Barren point within a field 

FIGURE !.-Idealized part of a digitized basin showing field locations, potential well sites, and barren points within a field. 
6 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

used as a basis for predicting the results of future exploration in 

the target area. For example, Allais (1957) used an assumption of 

this type to justify the use of several sets of historical resource 

data from regions within the United States and Europe as a basis 

for predicting the outcome of exploring the Algerian Sahara for 

base and precious metals. 

HISTORICAL ASPECTS 

At least three reasons can be suggested to explain the extensive 

use of the hindsight model. First, the exploration data commonly 

available to exploration analysts are historical records summarizing past exploration performance. These data, which are useful 

only for retrospectively describing exploration, are usually available for a region in the form of lists of past discoveries and yearly 

drilling intensity statistics. Second, a hindsight model alone will 

usually yield adequate estimates of future exploration performance if the principal exploration objective is to predict the expected gross return or range of return. Third, several analysts 

who have studied the exploration process have done so by adapting specific hindsight models developed in the field of military 

operations-research to search for hostile submarines, antiship 

mines, or pilots downed at sea (Engel, 1957; de Guenin, 1962; 

Griffiths and Drew, 1964; Griffiths and Singer, 1972). 

RESPONSE CHARACTERISTICS OF THE HINDSIGHT MODEL 

The response characteristics of the random-walk exploration 

model are the values, particular to this study, which are associated with the performance criteria. Table 1 shows the relationship 

TABLE 1.-Summary statistics and probabilities of discovering various quantities of petroleum, in millions of equivalent barrels 

[Gas converted to barrels at 26 cents per 1,000 ft 3] 

Summary statistics, Probability of discovering indicated quantity of Proba- minimum quantity of petroleum Num- petroleum bility of ber of discover- wells Standard ing zero Mean deviation petroleum 2 5 10 20 

10 ----------- 13.1 59.1 0.801 0.194 0.191 0.179 0.159 0.121 20 ----------- 26.0 86.7 .639 .350 .343 .324 .291 .226 

30 ----------- 38.9 102.4 .509 .484 .476 .440 .418 .345 

40 ----------- 51.2 116.0 .415 .567 .558 .536 .490 .405 

50 ----------- 63.3 128.5 .326 .665 .657 .631 .589 .504 

75 ----------- 91.8 153.7 .188 .804 .795 .769 .727 .642 

100 ----------- 122.8 176.8 .103 .890 .882 .858 .813 .739 

150 ----------- 178.1 203.1 .036 .958 .952 .939 .914 .864 

200 ----------- 228.9 226.4 .013 .982 .984 .969 .956 .904 

300 ----------- 327.3 254.9 .003 .995 .994 .991 .983 .973 

400 ----------- 405.6 265.9 .000 .999 .998 .996 .996 .993 

500 ----------- 483.0 274.1 .000 .999 .999 .999 .999 .998 
THE HINDSIGHT MODEL 7 

TABLE 1.-Summa.ry statistics and probabilities of discovering various quantities of petr.oleum, in millions of equiv·alent barrels-Continued 

Num- Probability of discovering indicated minimum quantity of petroleum berof 

wells 80 40 50 100 200 400 600 800 

10 ---------- 0.086 0.067 0.051 0.087 0.018 0.008 0.000 0.000 

20 ---------- .156 .182 .118 .054 .026 .012 .002 .000 

80 ---------- .271 .208 .177 .118 .086 .027 .008 .000 40 ---------- .820 .287 .228 .140 .051 .080 .005 .000 

50 ---------- .419 .885 .295 .175 .087 .056 .007 .000 

75 ---------- .557 .468 .420 .259 .104 .077 .017 .000 

100 ---------- ,660 .575 .588 .887 .160 .120 .048 .000 

150 ---------- .814 .764 .701 .490 .254 .171 .065 .003 

200 ---------- .870 .843 .815 .624 .858 .214 .124 .010 

800 ---------- .961 .952 .939 .800 .552 .818 .237 .OS3 400 ---------- .988 .982 .977 .908 .714 ,405 .380 .058 

500 ---------- .997 .996 .995 .958 .818 ,507 .428 .157 

between the values of the performance criteria describing the 

distribution of the quantity of petroleum discovered and the intensity of exploratory drilling. The values of the performance 

criteria which describe the distribution of the number of deposits 

discovered at each level of drilling intensity are shown in table 2, 

as is the probability of encountering a given number of unsuccesful exploratory wells in succession before the first discovery. A 

TABLE 2.-Summary statistics and probabilities of discovering various numbers of commercial deposits 

Probability of Probability of discovering indicated 

Num- Summary statistics, discover- minimum number of deposits 

berof number of deposits ing exactly 

wells zero Mean Range deposits 2 3 4 5 6 

10 0.22 0-3 0.801 0.199 0.018 0.001 0.000 0.000 0.000 

20 .43 0-4 .689 .881 .071 .011 .001 .000 .000 

30 .67 0-5 .509 .411 .146 .028 .005 .001 .000 

40 .843 0-6 .415 .575 .212 .051 .009 .002 .001 

50 1.08 0-6 .826 .674 .294 .096 .024 .005 .001 

75 1.58 0-7 .188 .812 .477 .207 .066 .019 .004 

100 2.11 0-8 .108 .897 .685 .852 .152 .055 .016 

150 3.06 0-11 .086 .964 .888 .616 .371 .183 .068 

200 3.98 0-13 .018 .987 .980 .788 .586 .866 .197 

800 5.72 0-15 .008 .997 .988 .950 .860 .711 .521 

400 7.34 1-17 .000 1.000 .998 .989 .960 .893 .785 

500 8.93 1-18 .000 1.000 .999 .998 .990 .969 .926 

Num- Probability of discovering indicated minimum number of deposits 

ber of 

wel1s 7 8 9 10 12 14 16 18 20 

10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 

20 .000 .000 .000 .000 .000 .000 .000 .000 .000 

30 .000 .000 .000 .000 .000 .000 .000 .000 .000 

40 .000 .000 .000 .000 .000 .000 .000 .000 .000 

50 .000 .000 .000 .000 .000 .000 .000 .000 .000 

75 .001 .000 .000 .000 .000 .000 .000 .000 .000 

100 .003 .001 .000 .000 .000 .000 .000 .000 .000 

150 .020 .007 .002 .001 .000 .000 .000 .000 .0()(). 

200 .091 .035 .014 .004 .001 .000 .000 .000 .000 

300 .341 .191 .100 .040 .005 .001 .000 .000 .000 

400 .627 .468 .306 .179 .044 .007 .001 .000 .000 

500 .838 .704 .552 .392 .148 .084 .006 .001 .000 
8 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

minimum of 3 million simulation cycles was used to obtain the 

results shown in these tables. 

Inspe·ction of table 1 suggests some general conclusions about 

the response characteristics of the model. First, the relationship 

between the average quantity of petroleum discovered per program and the intensity of drilling is nonlinear. Thus, doubling the 

size of the exploration program does not cause the average quantity of petroleum discovered to double. Second, as the intensity of 

exploration is doubled at higher and higher levels, the deviation 

from nonlinearity increases. This phenomenon of increased deviation from linearity of response is usually referred to as the exponential saturation effect or the effect of diminishing returns. This 

effect is a common consequence arising from the normal use of 

many search procedures actually used in exploration. 

The response characteristics of the exploration model are shown 

in figures 2 through 4. In figure 2, the probability of gambler's 

ruin and the probabilities of attaining six specific levels of discovery are plotted as functions of the intensity of exploration. 

In figure 3, the average quantity of petroleum discovered and the 

standard deviation of the distribution of the quantity of petroleum 

discovered are plotted as functions of the intensity of exploration. In figure 4, the average incremental quantity of petroleum 

discovered is plotted as a function of the intensity of drilling. 

Within the 10- to 100-well range, the probability of gambler's 

ruin, the probability of not making a single discovery, declines 

very rapidly as the intensity of exploration is increased (fig. 2) ; 

from 0.801 for 10 wells to 0.103 for 100 wells. Further increases 

in the intensity of drilling (beyond 100 wells) cause little additional decrease in the probability of gambler's ruin. Thus, the 

proba1bility of discovering one or more deposits is strongly affected 

by changes in the intensity of exploration drilling at the lower 

intensities. 

Six curves in figure 2 show how the probabilities of achieving 

or exceeding six levels of discovery (1, 10, 50, 100, 200, and 400 

million barrels) varies in response to changes in drilling intensity. 

The curve for the 1-million-barrel level is essentially the complement of the gambler's-ruin curve because less than 15 percent 

of the deposits in the resource base have estimated potentials of 

less than 1 million barrels. Therefore, when a deposit is discovered there is at least an 85 percent chance that the 1-millionbarrel level has been attained or exceeded. 

The form of the response curves changes significantly as the 

discovery goal is set at successively higher levels. At the 200-
THE HINDSIGHT MODEL 

0.6 

i 

> 1- 0.5 ~ 

iii 

<( 

Ill 

0 

a: 

ll. 

0.4 

NUMBER OF WELLS 

FIGURE 2.-Probability curves for the discovery of specific 

quantities of petroleum, in millions of barrels. 

9 
10 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

a 

w 

a: 

~ 400 

0 

u 

Cl) 

ocn 

~~ wa: 

c5 ~ 300 a: Ill 

1-u.. 

~0 

u..CI) 

0~ 

~ j 200 

~~ c(z 

::J0 

w 

C) 

c( 1 ()() a: 

w 

> c( 

1 ()() 200 300 

NUMBER OF WELLS 

FIGURE 8.-Average quantity of petroleum discovered versus exploratory drilling intensity. 

million-barrel level and beyond, the response of the probability 

curve is essentially linear over the entire 10- to 500-well range; 

additional increments of exploration drilling return nearly equal 

increments in the probability of achieving the goal. 

The average quantity of petroleum discovered per program increases as the intensity of drilling increases but at an everdecreasing rate (fig. 3) . The incremental return is seen (fig. 4) to 

decline from 1.31 million barrels per (incremental) exploration 

well to 0.77 million barrels per (incremental) exploration well 

over the 10- to 500-well range. At approximately the 250-well 

level, the rate of change of this curve is 1 million barrels per 

exploraton well. 

THE SIMILARITY ASSUMPTION AND AN EXAMPLE 

The similarity assumption is required to convert the results 

obtained from a hindsight model into a basis for predicting the 

outcome of future exploration in a target area similar to the control area. Hindsight model results as they stand, however, are 
...J 

...J 

w 

1.3 

• ;: 1.2 za: a:W 

:>a.. 

tu ~ 1.1 [[W 

...Ja: <(a: 

1- ~ 1.0 z 

WLL 

::::?!0 

~en 0.9 uz zQ 

-...J 

= 0.8 ::::?! 

0.7 

THE HINDSIGHT MODEL 

0.6 L....----.L..-___ ...__ ___ ........__ ___ ........__ __ ----J 

0 100 200 300 400 500 

NUMBER OF WELLS 

FIGURE 4.-Average incremental quantity of petroleum discovered. 

11 

completely adequate to determine what would have happened if 

actual past exploration had been carried out according to the 

tenets of the hindsight model. Extrapolation from hindsight summary to the prediction of future discovery without adequate consideration being given to the similarity assumption is a short but 

peril-laden leap. 

Often it is possible to support the use of the similarity assumption with specific geologic data. Criteria which should be examined in judging the validity of the similarity assumption for any 

specific target area are (1) the thickness of the stratigraphic section, (2) the presence of transgressive and regressive sediments, 

( 3) the source bed characteristics, and ( 4) the types of structures 

within the basin. 

In other cases, the exploration analyst may have to rely on the 

fact that no serious objection can be raised to negate the use of 

the assumption. In those cases where little explicit data exists to 

support the use of the assumption, the analyst is forced into the 

position of making the assumption knowing that subsequent evidence may be obtained which would severely compromise the 

validity of the predictions. 

Once the control and target areas are selected and the assumption of similarity is made, however, the outcome of future ex-
12 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

ploration can be predicted from the model results. The model 

results displayed in tables 1-3 and figures 2-4 can be used for 

predictive purposes if it can be assumed that the undiscovered 

deposits in the target area are similar to the deposits discovered 

in the past in the Powder River Basin and if the method of allocating exploration wells in the target area is similar to that used 

to develop the Powder River Basin model results. 

The question of how deep exploration wells should be drilled 

must be resolved by establishment of a rule. Such a rule might 

specify drilling to a target formation, say the Minnelusa (Pennsylvanian) or its equivalent. An additional cost restraint might 

produce a joint rule which might specify drilling each exploration 

well to the Minnelusa or to 13,000 feet (3,950 m) whichever is 

shallower. 

The information contained in table 1 suggests probabilistic 

statements about the relationship between the quantity of petroleum discovered and the size of the exploration program. For 

example, if only 10 wells can be allocated for exploration the risk 

of gambler's ruin is very large (80.1 percent). The chance of 

discovering a million barrels or more is only 19.4 percent. Ten 

million barrels or more will be discovered only 15.9 percent of 

the time, 50 million barrels or more only 5.1 percent of the time, 

and 100 million barrels or more only 3.7 percent of the time. On 

the average a total of 13.1 million barrels will be discovered when 

TABLE 3.-Estimates of the probabilities of discovering any of six deposits 

within each year according to random-walk allocation of actual drilling 

intensities 

Wells Deposit Year 

drilled Springen Gas Bell 

Hilight Ranch Draw Recluse Creek Kitty 

1952 38 0.122 0.010 0.012 0.015 0.037 0.042 1953 59 .183 .018 .018 .023 .059 .068 

1954 62 .191 .018 .019 .023 .062 .070 

1955 56 .175 .015 .017 .022 .051 .059 

1956 76 .231 .021 .023 .029 .078 .074 

1957 98 .285 .026 .030 .039 .098 .110 

1958 93 .272 .025 .027 .035 .091 .103 

1959 138 .378 .036 .040 .052 .133 .151 

1960 227 .542 .056 .066 .083 .210 .230 

1961 217 .524 .054 .062 .082 .203 .226 

1962 188 .480 .052 .056 .070 .182 .197 

1963 182 .466 .049 .053 .069 .171 .188 

1964 168 .444 .045 .049 .063 .163 .177 

1965 205 .511 .054 .061 .075 .192 .213 

1966 183 .471 .049 .053 .069 .174 

1967 236 .557 .060 .070 .090 .221 

1968 586 .868 .154 .162 1969 445 .785 1970 292 

1971 142 

Total ------- 3,691 
THE HINDSIGHT MODEL 18 

10 exploratory wells are allocated. At first glance this average 

may appear to be disconcertingly large when compared with the 

probabilities of reaching or exceeding specific levels of discovery. 

The apparently large size of this average relative to the probabilities of achieving various discovery goals is caused by the 

large skewness in the right tails of the distribution of petroleum 

deposits lying within the basin. For example, the largest deposit, 

the Salt Creek field, is nearly three times larger than the next 

larger deposit. As a result, the distribution of the quantity of 

petroleum discovered is highly skewed to the right when a small 

exploration program is employed because the probability of discovering the largest field, or one or more of the other large fields, 

is small (P = 0.14 for discovering the Salt Creek Field with a 

lO-well exploration program). Thus, the discovery of the Salt 

Creek Field "pushes" the frequency distribution of petroleum 

discovered far to the right. When this field is not discovered the 

aggregate quantity of petroleum discovered with a lO-well program is far smaller, usually less than a third as large. 

The skewness of the distribution of the quantity of petroleum 

discovered decreases as the intensity of exploratory drilling is 

increased because the chance of discovering the larger deposits 

approaches certainty. 

The most likely use that an exploration analyst may choose to 

make of the information displayed in table 1 is to determine the 

intensity of exploratory drilling required to achieve a specific 

goal and at the same time not to violate a specified risk level. For 

example, if the management of an exploration company sets a 

goal of discovering at least 100 million barrels of petroleum in 

the target area and at the same time is not willing to assume 

more than a 20 percent risk of not reaching that goal, then a 

300-well exploration program must be scheduled (fig. 2). It does 

not follow, however, that one out of five times the firm must face 

gambler's ruin in the exploration project. On the contrary, the 

probability of gambler's ruin is only 0.003, three chances in a 

thousand. The probabilities of achieving lesser levels of discovery 

ranges from 0.996 for the 1-million-barrel level to 0.939 for the 

50-million-barrel level. 

The expectation is that a 300-well exploration program would 

result in the discovery of 327.3 million barrels in the target area 

(table 1), that this average quantity would be contained in 5.72 

deposits (table 2), and that the number of deposits discovered 

would range from 0 to 15. The probability of discovering at least 

one deposit is then 0.997, a virtual certainty. The probability of 
14 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

discovering at least five deposits with a program of this size is 

0.711, or about seven chances in ten. The probability of discovering 10 or more deposits is only 0.040, or four chances in one 

hundred. 

The probability of encountering a specific sequence of unsuccessful exploration wells in the normal course of exploration in 

the target area prior to the first discovery can be computed by 

interpolating the results listed in the fourth column of table 2. 

The probability of drilling a series of 10 unsuccessful wells in a 

row is 0.801, a rather large risk. Approximately half of the time 

(P = 0.509) 30 consecutive unsuccessful exploration wells would 

be encountered prior to the first discovery. 

EXHAUSTION 

The effect of exhaustion of the petroleum resource base of the 

basin upon several exploration performance criteria is investigated. (The sequential model must be used to develop the results 

discussed in this and the following sections.) The 20-year time 

interval 195·2-71 was chosen for study. During this period, 111 

deposits were discovered which were conservatively estimated to 

have contained 545,727,000 equivalent barrels of petroleum. It is 

pointed out that this resource base is only a part (approximately 

40 percent) of the total petroleum resource base of the basin discovered since 1889 ; the total resource base could not be utilized 

because of restraints on the availability of computer time. Therelationship between the exhaustion of the resource base and the 

values of each performance criterion were computed by diminishing the resource base by yearly time increments (exhaustion sequence) and simulating the exploration of the basin with program.s of 10 to 500 exploratory wells at the start of each new time 

increment. The discovery sequence actually encountered during 

the exploration of the basin during the 1952-1971 period was 

chosen as the exhaustion sequence. The quantity of petroleum and 

the number of undiscovered deposits remaining in the resource 

base at the beginning of each yearly time increment are shown 

in figures 5 and 6, respectively. The results of this exhaustion 

study are again hindsight, and extrapolation of thes·e results into 

a different petroleum province must be made cautiously. 

The effect of the exhaustion of the basin's resource base upon 

three exploration performance criteria were evaluated and are 

displayed in figures 7 through 12. Figures 7 through 9 show the 

surfaces for the expected value of the quantity of petroleum discovered, expected numbers of deposits discovered, and the prob-
120 

100 

Cl) 

... 

c;; 

0 80 ~ 

w 

0 

u. 60 0 

a: 

w 

a:l 40 ~ 

::::> 

z 

EXHAUSTION 15 

1962 1964 1956 1968 1960 1962 1964 1966 1968 1970 1972 

YEAR 

FIGURE 5.-Quantity of petroleum in resource base. 

1952 1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 

YEAR 

FIGURE 6.-Number of deposits in resource base. 

ability of gambler's ruin. Figures 10 and 11, respectively, show 

the expected value of petroleum to be discovered for three intermediate exhaustion levels and the partial derivative curves for 

the expected quantity of petroleum discovered with respect to the 

size of the exploration program. 

From figures 7 through 9, the performance of various exploration programs can be determined with respect to the level of 

exhaustion. For example, an operator planning, say, a 20-well 
16 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

450 

400 

~ 350 

~ ..J 

~ 300 

u.. 

0 250 a: 

w 

~ 200 :I 

z 150 

100 

50 

PERCENTAGE OF EXHAUSTION 

90 80 70 60 50 40 30 20 1 0 I 0 

100 200 300 400 500 

QUANTITY OF PETROLEUM IN RESOURCE BASE, 0 

IN MILLIONS OF BARRELS 

FIGURE 7.-Average quantity of petroleum discovered, E(q) = 

10"1.u1Q0·u8W0·

876, in millions of barrels. 

exploration program can estimate what effect exhaustion of the 

resource base will have upon the expected results of such a program. At the start of the 20-year interval a 20-well program will 

on the average discover 0.303 deposits containing 8. 7 million 

barrels. The probability of gambler's ruin (not discovering a 

single deposit) is 0.601. When the resource :base is at the 50 percent exhaustion level, 0.21 deposits with 6.5 million barrels are 

expected to be dis·covered when 20 wells are drilled. The chance 

of ruin is 0.926 for this situation. 

For a petroleum company or group of companies undertaking 

a 200-well exploration program, it is expected that an average of 

2.49 deposits containing 66.3 million barrels would be discovered 

when the full resource base is present; the chance of gambler's 

ruin is estimated to be only 3.6 percent. If the resource base is at 

50 percent exhaustion level, a 200-well program is expected to 

lead to the discovery on the average of 1. 75 deposits containing 

an average of 49.6 million barrels; the chance of gambler's ruin 

for this case is 5.8 percent. 
450 

400 

! 350 

...J 

...J 

~ 300 

LL 

0 250 a: 

w 

a:l 

::2: 200 ::> 

z 150 

100 

50 

EXHAUSTION 

PERCENTAGE OF EXHAUSTION 

90 80 70 60 50 40 30 20 1 0 0 

1 0 20 30 40 50 60 70 

NUMBER OF DEPOSITS IN RESOURCE BASE, N 

FIGURE B.-Average number of deposits discovered, E(d) = 

l o-2. 715 No. ,ea wo. 01e. 

17 

It is noteworthy that a reduction of the resource base by half 

(reduced to the 50-percent exhaustion level) does not reduce the 

levels of the performance criteria proportionally. This phenomenon is perhaps better illustrated by constructing plots like those 

given in figures 10 and 11. In figure 10, the expected quantity of 

petroleum to be discovered is plotted against exploration intensity 

for three intermediate levels of exhaustion, 500 (8.4 percent exhaustion), 300 ( 45.0 percent exhaustion), and 100 million barrels 

(81.7 percent exhaustion). Note that, irrespective of the size of 

the exploration program, increasing the quantity of petroleum in 

the resource base by a given factor does not cause a similar increase of the expected quantity of petroleum discovered. This is 

again a result of the exponential saturation effect (the effect of 

diminishing returns). Figure 11 gives the partial derivative functions ( oE (q) jaW) for the same three intermediate exhaustion 

levels given in figure 10. The symbol E (q) denotes the expected 

quantity of petroleum to be discovered by a given program. The 

symbols Wand Q denote, respectively, the number of exploratory 
18 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

PERCENTAGE OF EXHAUSTION 

90 80 70 60 50 40 30 20 10 () 

450 

400 

~ 350 (I) 

..J 

..J 

w 300 

3: 

~ 

0 250 

IX: 

w 

~ 200 

::> 

2 

150 

100 

50 

10 20 30 40 50 60 70 80 90 100 110 

NUMBER OF DEPOSITS IN RESOURCE BASE, N 

FIGURE 9.-Probability of gambler's ruin, P(ruin) =10u02N-0

'

701W-u23 

20 

0~-------L--------L--------L--------L-------~ 

0 100 200 300 400 500 

NUMBER OF WELLS 

FIGURE 10.-Average quantity of petroleum discovered for 

three levels of exhaustion. 
EXHAUSTION 

Q = 500,000,000 bbl 

Q = 300,000,000 bbl 

Q = 100,000,000 bbl 

100 200 300 400 500 

NUMBER O.F WELLS 

FIGURE 11.-Partial derivatives of the average quantity of 

petroleum discovered for three levels of exhaustion. 

19 

wells drilled and the quantity of petroleum remaining in the 

resource base at a particular time. The variation in the rate of 

diminishing return per well is most extreme for those exploration programs with less than 50 wells. For example, when the 

resource base contains 100 million barrels, the rate declines from 

220,000 barrels per well for a lO-well program to 170,000 barrels 

per well for a 50-well program. Beyond this initial range the rate 

still diminishes, but much more slowly. At the 500-well level, the 

rate has only declined to the 125,000 level. 

The effect of exhaustion of the resources upon the quantity of 

petroleum expected to be discovered is shown in yet another manner in figure 12, where this performance criterion is expressed 

as the average quantity of petroleum discovered per well and 

plotted against the quantity of petroleum remaining in the resource base for each of four exploration programs. For example, 

assume that a discovery rate of 250,000 barrels per well must be 

attainable to attract funds for exploration. Given such a level, it 

can be determined from this graph that a 20-well exploration 

program can be carried out successfully down to the point where 

only 145 million barrels (73.4 percent) is left in the basin. If a 

100-well exploration program is planned, then at least 225 million 

barrels (58.8 percent exhaustion) must remain undiscovered in 
20 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

fa 400,000 a: 

w 

> 

0 

u 

(/) 

0 (/) 

~ ~ 300,000 

~a: ...JII: 

O< 

a::m 

tu~ ~ . u.....J 

0 ~ 200,000 

>-~ 

!:a: 

1-W 

z~ 

< 

:;::) 

d 100,000 w 

" < 

a: 

w 

> 

< 

o~--~--._--,~50---2~o-o--~25~0---3o~o---35~o---4~oo---4~~---~~o--~5~ 

QUANTITY OF PETROLEUM IN RESOURCE BASE, 

IN MILLIONS OF BARRELS 

FIGURE 12.-A verage quantity of petroleum discovered per well versus level 

of exhaustion for four levels of exploratory drilling. 

the basin to attain or surpass the desired goal. For a 200-well 

program, 280 million barrels or more ( 48.7 percent exhaustion or 

less) must be present; and 345 million barrels (36.8 percent exhaustion or less) must be left for the goal to be achieved if a 

400-well program is to be undertaken. 

A tendency therefore exists for the larger operators to decrease 

and finally cease exploration as the level of exhaustion increases 

in a region. This occurs because the larger operators cannot reasonably expect to discover, at an acceptable rate of return, the 

volume of petroleum necessary to induce them to continue to 

explore in the region. Despite the fact that the larger operators 

leave a region because of the effects of exhaustion, a smaller 

operator can still obtain an acceptable rate of return, assuming 

that the risk of not finding the remaining deposits is not too 

great. 

COMPOUND-DISCOVERY-EVENT PROBABILITIES 

The efficiency of the aggregate exploration process in discovering specific deposits is often a topic of debate among petroleum 
COMPOUND-DISCOVERY-EVENT PROBABILITIES 21 

explorationists. It is often the case that the larger deposits in a 

basin are discovered early in the exploration history of the basin. 

In other cases, a major deposit may remain undiscovered for 

decades even through significant exploration activity has been 

undertaken within a short geographic or stratigraphic distance of 

its location. As a first approximation to the solution of estimating the chance of compound discovery events, the probabilities of 

discovering six of the larger deposits actually discovered during 

the 1952-71 time span were estimated. The actual intensities of 

exploratory drilling expended during this period were used in 

these computations. The results (table 3) are again based upon 

the random-walk method of allocating exploration wells. These 

results therefore represent the worst that could be expected to 

occur because no information was used to remove those regions 

with low discovery potential. 

The number of exploratory wells actually drilled from 1952-71 

are shown in figure 13 (see also table 3). During this period the 

yearly drilling rates ranged from a minimum of 38 for the year 

1952 to a maximum of 586 for 1968. The probabilities of discovering each of the six deposits within a particular year are shown 

in table 3. Note that these probabilities vary as a direct function 

of the drilling intensity. For example, the year in which the 

probability of discovering the Hilight deposit is the largest (P= 

0.868) is 1968 when the greatest number, 586, of exploratory 

wells were drilled and smallest (P = 0.122) in 1952 when only 38 

exploratory wells were drilled. 

(I) 

..J 

~500 

3: 

>-

~ 400 lei 

0: 

g 300 0.. 

X 

w 

u.. 200 

0 

0: 

w 

~ 100 

:::> 

z 

1962 1964 1956 1958 1960 1962 1964 1966 1968 1970 1972 

YEAR 

FIGURE 13.-Number of exploratory wells drilled in the Powder River 

Basin, 1952-71. 
22 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

The probabilities of discovering each deposit in at least one 

year during the period beginning in 1952 and up to and including 

the year of discovery are shown in table 4. These computations 

were made with a computer program designed to solve the hinominal equation for the case where the probabilities of success 

vary from trial to trial. The probability of discovering the Hilight 

deposit at least once when the random-walk strategy is used to 

allocate exploration wells is 0.99998, a virtual certainty. In 

fact, the chance of discovering this deposit at least four times 

with the 3,2,57 exploratory wells drilled between 1952 and 1969 

is also nearly certain (P= 0.985). The chance of discovering the 

Bell Creek deposit, the largest deposit discovered in the basin in 

the last several decades, at least once with the random-walk strategy is estimated to be 0.903. The discovery of relatively small 

deposits, such as the Gas Draw or Springen Ranch fields, at least 

once using the random-walk strategy is estimated to occur at 

least 50 percent of the time in each case. 

Table 5 gives another sequence of probabilities which show the 

progressive increase in the probability of discovering the six deposits over the time period starting in 1952 and ending in the 

year prior to actual discovery of each deposit. The probabilities 

shown in this table were computed sequentially by re-exploring the basin according to random-walk strategy and by increasing the intensity of drilling by yearly increments. The probability 

of discovering the Bell Creek deposit when 38 exploration wells 

are drilled (1952) is 0.037. If the two years 1952 and 1953 are 

considered as a single time unit, the probability of discovery increases to 0.093; for the time unit ending in 1960 the probability 

increases to 0.590 and it increases further to 0. 763 for the time 

unit ending in 1963. When the total period from 1952 through 

1966 (the year prior to the actual discovery of the deposit) is 

considered as a single time unit, the probability increases to 

TABLE 4.-Estimates of the probabilities of compound discovery events using 

random-walk allocation of exploratory wells 

[Reserves in millions of equivalent barrels] 

Total wells Probabilities of discovering drilled Number deposit at least these between of years number of times during 

Ultimate 1952 and through time period: Year of primary year of year of Deposit discovery reserves discovery discovery 2 3 4 

Hilight ----------- 1969 55.4 3,257 18 0.99998 0.9997 0.997 0.985 

Gas Draw 1968 12.8 2,812 17 .573 .195 .043 .007 

Springen Ranch --- 1968 9.7 2,812 17 .541 .170 .035 .005 Recluse ----------- 1967 27.0 2,226 16 .580 .203 .047 .008 

Bell Creek -------- 1967 121.4 2,226 16 .903 .654 .362 .152 Kitty ------------ 1965 28.9 1,807 14 .879 .597 .300 .111 
COMPOUND-DISCOVERY -EVENT PROBABILITIES 23 

TABLE 5.-Estimates of tke probabilities of discovering any of si:» deposits 

witk tke actual drilling effort e:»pended between 1952 and tke actual year 

of discovery of eack deposit 

Wells Deposit 

Years drilled Springen Gas Bell 

Hilight Ranch Draw Recluse Creek Kitty 

1952 38 0.122 0.010 0.012 0.015 0.037 0.042 

1952-53 97 .279 .027 .030 .039 .093 .108 

1952-54 159 .420 .043 .050 .063 .146 .178 

1952-55 215 .543 .061 .063 .086 .204 .226 

1952-56 291 .632 .077 .083 .102 .259 .294 

1952-57 389 .736 .108 .115 .134 .826 .862 

1952-58 482 .804 .134 .148 .165 .389 .440 

1952-59 670 .887 .154 .180 .217 .472 .517 

1952-60 847 .942 .216 .238 .287 .590 .631 

1952-61 1,064 .975 .257 .286 .829 .664 .715 

1952-62 1,252 .990 .297 .317 .373 .713 .765 

1952-68 1,434 .990 .335 .855 .418 .768 .806 

1952-64 1,602 .996 .361 .388 .467 .809 .849 

1952-65 1,807 .998 .401 .428 .502 .855 

1952-66 1,990 .999+ .436 .458 .554 .886 

1952-67 2,226 .999+ .467 .487 

1952-68 2,812 .999+ 

0.886. Thus, when the wells are allocated according to the randomwalk .strategy, there is an 88.6 percent chanc·e of discovering the 

Bell Creek deposit with the 1,990 exploratory wells drilled in the 

time period 1952-66. 

Caution must be observed in using the probability estimates 

developed from this model study to evaluate the effectiveness of 

historical performance in discovering specific deposits; these estimates are subject to both positive and negative biases, the magnitudes of which can be only roughly estimated. For example, these 

probability estimates are conservative on the one hand because 

the target area in the simulation study is the entire basin, in 

which an exploration well can be located anywhere, even near the 

boundary of the basin where potential is low. Such areas are 

roughly estimated to comprise 150 townships, or more than 5,000 

mi 2• Exploration wells can also be located within the relatively 

unexplored central and north-central parts of the basin 

where, in 1970, 139 townships, or about 5,000 mP, were untested. 

Thus, of the basin's 27,000 mP, approximately 40 percent which 

did not physically contain any of the deposits used to make the 

probability estimates and which would not be given much consideration by the industry as having discovery potential was considered to be in the target area. The magnitude of this negative 

bias upon the estimates would, however, be less than 40 percent 

because of the effect of diminishing returns. On the other hand, 

the discovery probabilities are larger than those to be expected in 

reality because the depth to which each exploration well needs to 

be drilled to insure discovery is not explicitedly considered in the 
24 MODEL FOR PETROLEUM EXPLORATION SUCCESS 

model. A rule such as the one mentioned earlier (drill to the 

Minnelusa or 13,000 feet (3,950 m), whichever is shallower) 

would be completely effective but such a rule might take a great 

deal of exploration effort to establish. In the case of the six deposits in this sample, establishing a rule would be a somewhat 

easier task because the depths of these deposits range from 4,500 

feet (1,360 m, Bell Creek) to 9,600 feet (2,910 Il,l' Hilight). 

Other problems of biases arise from land-leasing practices and 

variations in exploration economics. 

CONCLUSIONS 

The process of exploring for petroleum deposits with a randomwalk strategy in the Powder River Basin, Wyo., was investigated 

over a range of drilling intensity. The effectiveness of this strategy on the outcome of exploration was analyzed with respect to 

a suite of performance criteria. Examples of the criteria employed are (1) the average quantity of petroleum discovered, (2) 

the average number of deposits discovered, (3) the probability of 

gambler's ruin, and ( 4) the probabilities of obtaining various 

levels of success. Before the model results can predict future exploration results in a target area, it is necessary to make an 

assumption of similarity between the resource base in the control 

area (in which the probabilities of success were generated) and 

the resource base remaining to be discovered in the target area. 

If this assumption can be justified for the target area in question, 

the following are several results predicted by the model: 

1. Increasing the number of exploratory wells per program increases, but at an ever-decreasing rate, the average number 

of deposits and the average quantity of petroleum expected 

to be discovered. (Ten wells yield an average of 0.22 deposits 

containing an average of 13.1 million barrels of petroleum; 

100 wells yield an average of 2.11 deposits containing an 

average of 122.8 million barrels of petroleum.) 

2. The probability of drilling consecutively unsuccessful wells 

(gambler's ruin) decreases as the number of wells per program increases. (Ten wells have a probability of gambler's 

ruin of 0.801; with 100 wells it is 0.103.) 

3. The expected incremental return per exploratory well decreases as the number of exploratory wells drilled increases. 

(At the 5-well intensity, the incremental (next) well returns 

1.31 million barrels; at the 450-well intensity, the incremental well returns 0.77 million barrels.) 
CONCLUSIONS 25 

The simulation model also isolated the effect of resource base 

exhaustion upon the exploration performance criteria; this effect 

wa~ found to progressively force the larger exploration interests 

out of the target area even though a large percentage of the resource base remains. For example, if the accept~ble rate of return 

is set at 250,000 barrels per well and a 200-well program is required by the firm to obtain the volume of discovery necessary to 

command investment in an area, the firm would be forced out 

when only 48.7 percent of the resource base was exhausted. In 

contrast, an operator planning to drill only 20 wells can continue 

to obtain an average rate of return of 250,000 barrels p~r exploratory well until 73.4 percent of the resource base is exhausted. 

Another result obtained from the model is the estimation of the 

prob~bility of various intensities of exploratory drilling discovering specific deposits over various time periods. For example, it is 

estimated that if the exploratory drilling effort expended in the 

Powder River Basin between 1952 and 1968 (2,812 wells) was 

allocated according to the random-walk strategy, the probability 

of discovering the Hilight deposit would be 0.999, a virtual certainty. The prob~bility of discovering this deposit with 1,252 exploratory wells drilled between 1952 and 1962 would also be 

nearly certain (P = 0.990). 

REFERENCES CITED 

Allais, M., 1957, Method of appraising economic prospects of mining exploration over large territories-Algerian Sahara case study: Management 

Sci., v. 3, no. 4, p. 285-347. 

De Guenin, Jacques, 1962, Exploration and search theory-Mathematical and 

computer applications in mining and exploration: Computer short course 

and symposium on mathematical techniques and computer applications 

in mining and exploration, 196·2, Tucson, Arizona, Univ. Arizona; College 

Mines, symposium proc., p. IL1-IL17. 

Engel, J. H., 1957, Use of clustering in mineralogical and other surveys: 

1st Inti. Conf. on Operations Research Proc., Baltimore, Operations Research Soc. America, p. 176-192. 

Griffiths, J. C., and Drew, L. J., 1964, Simulation of exploration programs 

for natural resources by models: Colorado School Mines Quart., v. 59, 

no. 4, p. 187-206. 

Griffiths, J. C., and Singer, D. A., 1972, The Engel simulator and the search 

for uranium: lOth annual council on applications of computers and 

mathematics in the mineral industries: Johannesburg, South Africa, v. 

1, p. 1-9. 

*u.s. GOVERNMENT PRINTING OFFICE: 1974-543-581/58