GEOLOGICAL SURVEY CIRCULAR 724 

In Search of a Statistical Probability Model 

for Petroleum-Resource Assessment 

In Search of a Statistical Probability Mo':lel 

for Petroleum-Resource Assessme.nt 

By Bernardo F. Grossling 

GEOLOGICAL SURVEY 

A critique of the probabilistic significance 

of certain concepts and methods used in 

petroleum-resource assessment. To that end, 

a probabilistic model is sketched 

1975 

C I R C U l A R 724 
United States Department of the Interior 

STAN,LEY K. HATHAWAY, Secretary 

Geological Survey 

V. E. McKelvey, Director 

Library of Congress catalog-card No. 75-600065 

free on application fo the U.S. Geological Survey, National Center, Reston, Va. 22092 
CONTEN'DS 

Page 

Definition of symbols ----------------------------------------------------- III 

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

Introduction ------------------------------------------------------------- 1 

Concepts and premises --------------------------------------------------- 4 

Fundamental concepts of resources and reserves ------------------------- 4 

Assessment of the petroleum-resource base ------------------------------ 6 

Objective and bias in resource assessment -------------------------------- 9 

Problems with historical projection models ---------------------------------- 11 

Linear "drilling input-petroleum output" model -------------------------- 11 

Drilling finding rate ------------------------------------------------- 14 

Economic role of the finding rate --------------------------------------- 15 

Note on the logistic method of projection -------------------------------- 15 

Note on random search ----------------------------------------------- 16 

Sketch of a probabilistic model of petroleum-resource assessment -------------- 17 

References cited -------------------------------------------------------- 18 

ILLUSTRATION 

Page 

FIGURE 1. Diagram showing span of the various resource and reserve terms 

for oil ------------------------------------------------------ 5 

TABLES 

TABLE 1. 

2. 

Stages in petroleum-resource development ------------------------

New oil in place added per foot of exploratory drilling -------------

SYMBOLS 

Basic Resource and Reserve Quantities 

B _____________ Resource base. 

EVRD _________ Cumulative production + R2 

OIP ___________ Initial oil in place. 

p ______________ Probability of discovery of a resource element. 

pd _____________ Probability of locating a resource element. 

Pe _____________ Probability of existence of a resource element.. 

R ______________ Recoverable resource amount 

Rt _____________ Proven reserves. 

R2 _____________ Expanded proven reserves. 

R3 _____________ Undiscovered recoverable resources. III 

Page 

9 

15 
Statistical Quantities 

f(z) ___________ Probability function. 

L ___________ · __ Lower bound. 

p(x~O) ________ Probability that x~O. 

q ______________ Marginal risk. 

W ____________ Upper bound. 

x --------------Variate. 

Quantities in Production Forecasting Model 

Dn _____________ Exploratory drilling footage during yearn. 

e _____________ _ReVision to prior computations of t::,.R', t::,.R", and t::,.R"'. 

f ______________ Fraction of the current reserves to be produced each year. >.. ______________ Lead time in years. 

m _____________ Finding rate in barrels of oil or thousand cubic feet of gas per eJ~-

ploratory foot drilled. 

ILl _____________ Primary recovery factor. 

P.2 _____________ Secondary recovery factor. 

/La -------------Tertiary recovery factor. 

Pn _____________ Production in yearn. 

Rn ____________ Recoverable proven reserve at end of yearn. 

t::,.Rn ___________ Increment of recoverable proven reserve established during yearn. 

t::,.R'n __________ Revisions and extensions of earlier primary proven reserves in year n. 

!::,. R" n __________ Actual increment to secondary proven reserves in year n. 

t::,.R"'n _________ Actual increment to tertiary proven reserves in yearn. 

Quantities in Financial Model 

a ______________ Ad valorem tax. 

{3 ______________ Royalty rate. 

t::,.Pn ( r) _______ Annual production for year r of ann-project. 

Ea _____________ Expensed items. 

Eb -------------Tax credits. 

ET _____________ Cash expenditures in year r. 

Eo _____________ Initial cash expenditures. 

F ______________ Present value of discounted cash flows. 

'Y ______________ Corporation tax rate. 

mo _____________ Initial finding rate. 

7J ______________ Effective depletion as a fraction of the net revenue. 

p ______________ Unit price of oil or gas. 

Q ______________ Cumulative exploratory drilling. 

Q1 _____________ Cutoff of cumulative exploratory drilling. 

r ______________ Annual discount rate. 

S ______________ Cumulative supply of oil or gas. 

Sn _____________ Oil or gas supply in yearn. 

T1 _____________ Production decay constant. 

r --------------Time counted from initial year of n-project. 

o ______________ Depreciation for year r. 

U _____________ Unit price expectation. 

Quantities Used in Exploration Model 

ai _____________ Probability of correct recognition of trap type i. 

a1 _____________ Probability of correct recognition of trap type j. 

{B.} ___________ Set of sedimentary basins. 

t::,.S1 ___________ Average area of trap type j. 

dn1 ____________ Incremental number of exploratory holes drilled after traps of type j. 

{At} ___________ Set of well..exploredgeologic domains At. 

N1 -------------Total number of traps type j, other than i, which exist in the region. 

rt,J ____________ Ultimate recoverable resources from Vt, 1• 

S --------------Total search area. T ____________ _Number of traps discovered. 

T1,

1 ___________ Trap of type j in basin i. 

T1 _____________ Cumulative number of traps of type j discovered. 

{V1,

1} __________ Set of geologic compartments. 

. .-· v1,

1 ____________ Element of geologic compartment Vt,J • 

IV 
In Search of a Statistical Probability Model for 

Petroleum-Resource Assessment 

By Bernardo F. Grossling 

ABSTRACT 

Exploratory drilling is !till in incipient or youthful 

stages in those areas of the world where the bulk of the 

potential petroleum resources is yet to be discovered. 

Methods of assessing resources from projections based 

on historical production and reserve data are limited to 

mature areas. For most of the world's petroleum-prospective areas, a more speculative situation calls for a critical 

review of resource-assessment methodology. The language 

of mathematical statistics is required to define more 

rigorously the appraisal of petroleum resources. 

Basically, two approaches have been used to appraise 

the amounts of undiscovered mineral resources in a 

geologic province: ( 1) projection models, which use 

statistical data on the past outcome of exploration and 

development in the province; and (2) estimation models 

of the overall resources of the province, which use certain known parameters of the province together with 

the outcome of exploration and development in analogous 

provinces. These two approaches often lead to widely 

different estimates. Some of the controversy that arises 

results from a confusion of the probabilistic significance 

of the quantities yielded by each of the two approaches. 

Also, inherent limitations of analytic projection modelssuch as those using the logistic and Gompertz functions 

-have often been ignored. 

The resource-assessment problem should be recast in 

terms that provide for consideration of the probability 

of existence of the resource and of the probability of 

discovery of a deposit. Then the two above-mentioned 

models occupy the two ends of the probability range. The 

new approach accounts for (1) what can be expected 

with reasonably high certainty by mere projections of 

what has been accomplished in the past; (2) the inherent biases of decisionmakers and resource estimators; 

(8) upper bounds that can be set up as goals for exploration; and ( 4) the uncertainties in geologic conditions in a search for minerals. Actual outcomes can then 

be viewed as phenomena subject to statistical uncertainty and responsive to changes in economic and 

technologic factors. 

INTRODUCTION 

Petroleum at present provides the major 

energy for driving the economies of nations. 

The foreseen worldwide depletion of petroleum 

1 

resources in a few decades at a time of· transition to new energy technologies will impose 

hardships on many nations but will enhance 

the economic opportunities of those having a 

petroleum potential. 

The petroleum-prospective areas of the world 

consist of sedimentary basins and ge.osynclines 

that ,are not too intensely deformed te.~tonically, 

including the continental shelves down to the 

200-m depth--a total area under vr"'rious national jurisdictions of about 26.1 mill ion square 

miles. Certain oceanic areas beyond the continental shelves-the continental slop~ and continental rise-also have petroleum prospects, 

but their eventual development ma;r be some 

10-20 years away. 

Traditionally, the favorable geolog:~~ environment for petroleum has been consid~red to be 

the geosynclinal belt. Cratonic areas have been 

reluctantly accepted by many geolo~ists. The 

French first discovered the importart accumulations in Africa, actually in Algeria. in a cratonic area previously deemed inadoouat~ The 

occurrence of giant gas accumulations in1 ,the 

western Siberian platform also did some violence to some traditional views about areas in 

which petroleum should occur. Largn tracts of 

cratonic areas now believed to be c0vered by 

not-too-thick sedimentary columns ought to be 

considered another frontier in petroleum 

geology. The continental margins--'Shelf, slope, 

and rise-are yet another petroleum frontier. 

The major areas that should be considered 

for petroleum exploration appear t'"' have already been outlined in the world, yet appraisals 

of the extent of these areas are cautious and 

often underestimate what subsequert exploratory work reveals. For example, est.imates of 

the extent of the Great Artesian Basin of 
Australia have been greatly increased in the 

course of just a decade. Similarly, I expect that 

the current picture of the extent of prospective 

areas in Africa and South America is also conservative. 

The extent of prospective areas is very significant even though experience in many basins 

shows that only a small percentage of the total 

prospective area is probably actually underlain 

by commercial petroleum accumulations. 

The larger the tract of undrilled prospective 

area, the greater are the chances that thick 

sedimentary pods may occur here and there. 

Even when a few scattered pieces of evidence 

may indicate a thin sedimentary cover, prospects for generation and primary migration of 

petroleum may be enhanced by the large size 

of a prospective area-as they are in prospective areas in the interior of Africa and in the 

Amazon basin. Of the total world petroleumprospective areas, the non-OPEC, noncommunist developing countries control 48 percent; 

the developed countries, excluding the USSR, 

30.5 percent; the USSR, 13 percent; the Middle 

East, 4.6 percent; and China 3.5 percent. 

Next to be considered, after the extent of 

sedimentary areas, should be the total sedimentary volume in each region. Information on this 

factor either is not readily available or has not 

been released as yet by companies or governments that have explored certain areas. Moreover, m,any other geologic factors should be 

considered in a realistic comprehensive 3.tssessment of the world's petroleum potential. 

The examination of the distribution of petroleum occurrences throughout the world, in 

basins that have had a significant amount of 

exploratory drilling, indicates that roughly half 

the prospective basins and geosynclines do not 

yield any or much petroleum; in tl'ose areas 

having petroleum deposits, the manner of distribution, size of deposits, and geok~ic conditions that control the petroleum accumulations 

vary greatly. The situation could be described 

as a two-stake sequential decision game played 

by Nature. First, it is decided, with probability 

roughly :lf2, whether a particular basin or province will contain commercial 3.tCcumnlations or 

not. In the .second stage is decided, ·with an underlying probability distribution, the magnitude of ·the petroleum resources of tra. basin or 

geosycline. 

Before exploratory drilling, it would be difficult to ascertain which basins contain petroleum. Of the many prospective basins and geosynclines in developing countries, for example, 

roughly about half will prove disappointing and 

have minor petroleum accumulations or none at 

all. The uncertainty of the outcon1e at this 

stage is something that has to be accE:'~Jted; only 

actual exploratory work, including drilling, can 

resolve the question. 

Another ,source of surprise C'an be the size of 

the petroleum accumulations. This is especially 

important because of the large contribution to 

petroleum resources of a relative}~~ few but 

very large fields. One can only speculate, before drilling, that certain geologic conditions 

may prevail in a given basin or geosyncline 

that could lead to large accumulatio:-ts. 

Giant oil fields discovered so far F.re mainly 

concentrated in the Middle East, and giant gas 

fields, in the USSR. The number of rdants, per 

million square miles of prospective areas so far 

discovered, are : 

Oil 

Middle East ------------------------------- 45 

G(I,IJ 

3.3 

11.8 

7.9 

.6 

0 

USSR --------------------------------------

Conterminous United States -------------------

Africa and Madagascar ----------------------

Latin America ------------------------------

South and Southeast Asia, mainland ---------

6 

3.2 

3.2 

2 

0 0 

About two-thirds of all the past drilling in 

the world for petroleum took place in the 

United States. Most of the petroleum-prospective areas of the developing nations are grossly underexplored. Estimates of the t.otal number of wells drilled per square mile cf prospective area in major regions are: 

2 
Wella'J}er 

aquat'emile 

Contenninous United States --------------------------- 1.17 

USSR ---------------------------------------------- .16 

Argentina, Mexico, and Venezuela --------------------- .06 

Middle East ----------------------------------------- .01 

Latin America, except as mentioned above -------------- .01 

South and Southeast Asia and Indonesia ---------------- .01 

Africa and Madagascar ------------------------------ .003 

About the same relative values are obtained if 

either the number of exploration wells or the 

total footage drilled are considered instead of 

the total number of wells. 

The projected drilling density in the conterminous United States provides an upper bound 

for the desirable drilling density. In the early 

development of the U.S. petroleum industry, 

much unnecessary drilling took place. Even ·allowing for this factor, however, the drilling 

density would be roughly about 0.5 wells per 

square mile of prospective area. In the Middle 

East, drilling density is exceptionally low because of the giant dimensions of the fields, and 

I it cannot be considered representative of desirable drilling density elsewhere. In the USSR, 

petroleum exploration is intense, but the exploration is f·ar from passing the midpoint in 

overall development. 

For areas on the order of 1 million square 

miles or more, a desirable ultimate density of 

0.3 wells per square mile of prospective area 

seems to be a reasonable target. Then the drilling density should eventually increase by a factor of about 100 in Africa and Madagascar and 

about 30 in South and Southeast Asia and also 

in Latin America, except for Argentina, 

Mexico, and Venezuela. Even in the last three 

countries, drilling density should eventually increase by a factor of about six. 

The final exhaustion of worldwide petroleum 

resources can be foreseen to occur within a few 

decades. At the moment, the petroleum scene 

is dominated by one cartel. Looking beyond 

Region 

areas containing proven reserves, however, 

one finds a substantial amount of completely 

unexplored or insufficiently explorea lands successful development of which could wrest the 

control of the petroleum scene from the cartel. 

Most of the prospective acreage available belongs to some developing nations. Tro. more important opportunities for petroleum development in developing countries that are not already playing a major role appear to be found 

in : Mexico, Colombia, Brazil. Peru, and Argentina in Latin America and Mali, Nig·er, Egypt, 

Mauritania, Chad, Tanzania, S01nalia, and 

Mozambique in Africa. 

Vigorous exploration in many of the promising areas should result in the discovery of substantial amounts of petroleum, whicl' could ease 

the pressures for price escalation. However, 

whether the present petroleum seller's market 

will go away will depend on who C9ntrols the 

new petroleum provinces. 

The recoverable petroleum that can be said 

with certainty to have been found so far in a 

given country is the sum of the cumul·ative production plus the proven reserves. M<'reover, the 

expected value in a statistical sense from discovered fields is somewhat larger. For this purpose, I have introduced the concept of EVRD 

(Expected Value of Recoverable Discoveries, 

as defined on p. 6. 

For the major regions that may be considered, the EVRD per square mile of prospective 

area is: 

Go.e 

(108 cu ft) 

Middle East -------------------------------

Contenninous United States ------------------

Oil (bbl) 

496,000 

101,600 

610 

656 

453 

207 

USSR -------------------------------------

Western Europe ----------------------------

Africa and Madagascar ----------------------

Latin America ------------------------------

South and Southeast Asia, mainland -----------

3 

49,600 

27,000 

30,700 

24,200 

2,600 

92 

52 

60 
The amount of petroleum initially in the 

ground in a given country or region and considered to be recoverable within foreseen technological and economic limits is the Estimated 

Ultimate Recovery (EUR) of that country or 

region. 

By considering certain best explored areas, 

excluding the Middle East, as benchmarks, I 

have selected the following average figures for 

the total EUR from continental-size regions: 

100,000 to 250,000 barrels of oil and 500 million 

to 1.300 million cubic feet of gas per square 

mile of prospective area. The above figures do 

not include the eventual occurrence of Middle 

East-size accumulations. 

The above figures provide another upperbound estimate for the world's EUR, namely 

2,600 billion to 6,500 billion barrels of 

oil, and 

13,000 trillion to 34,000 trillion cubic 

feet of gas. 

In addition, some further allowance should 

be m·ade for the eventual occurrence of clusters 

of giant accumulations, as in the Middle East. 

The extended reserves of the Middle East are 

595 billion barrels of oil and 612 trillion cubic 

feet of gas. The Middle East EUR's are probably, at most, equal to a few times these 

amounts; that is, less than an order of magnitude larger. 

Several petroleum companies are known to 

have pursued the seareh for "another Middle 

East." A consideration of the broad tectonic 

framework of various Earth regions suggests 

certain likely places. Some of the places that I 

would choose are : the north slope of the USSR, 

the north slope of Canada, the Gulf of Mexico, 

the Argentine Continental Shelf, the shelf between Mozambique and Malagasy, and the 

shelf between Australia and New Guinea. Of 

these six possibilities, four are in or near developing regions, namely: the Gulf of Mexico, 

the Argentine Continental Shelf, the shelf between Mozambique and Madagascar. and the 

shelf between Australia and New Guinea. 

I propose the following upper-bound EUR's 

for Latin America : 

490 billion to 1,225 billion barrels of 

oil, and 

2,450 trillion to 6,370 trillion cubic 

feet of gas; 

4 

for Africa and Madagascar: 

470 billion to 1,200 billion l'qrrels of 

oil, and 

2,400 trillion to 6,100 trillion cubic 

feet of gas; and 

for South and Southeast Asia: 

130 billion to 325 billion l'~.rrels of 

oil, and 

650 trillion to 1,700 trillion cubic 

fMtof gas. 

Because of the very limited petroleurr development in these prospective lands, the b,dk of the 

above EUR's still remains in the gro,:tnd to be 

discovered. • 

The above remarks show that there probably 

are large potential petroleum resour~es to be 

sought in the world. For most areas, rlrilling is 

only beginning. Methods of foreca~ting resources from projections based on historical 

production and reserve data (for instance, by 

using Gompertz and logistic functi4lns) are 

limited to mature areas ·such as the conterminous United States and the Maraeaih<J basin· in 

Venezuela. For the overwhelming mt'.iority of 

the world's petroleum-prospective s.Teas, we 

have a more speculative situation w~:\ich calls 

for a critical review of · resource-aASeSSment 

methodology. 

The estimates of dimensions of pr'lSpective 

areas and of upper-bound EUR's in this "Introduction" have been taken from unpublished 

data that I submitted to the International Bank 

for Reconstruction and Development, Washington, D.C.; they provide an indicatk"'l of the 

size of the exploratory tasks ahead an ci provide 

an estimate of the maximum amounts of oil and 

gas which would be obtained if the most favorable conditions prevail. 

CONCEPTS AND PREMISEf 

FUNDAMENTAL CONCEPTS OF RESOURCES AND 

RESERVES 

The span of the various resource ard reserve 

terms for oil are shown in figure 1. The situation for gas is similar. 

First we have the initial Oil In Plar~ (OIP) , 

that is, the amount prior to any exploitation 

that is to be found in undiscovered and discovered fields. Obviously, the OIP is difficult to 

estimate. As for the undiscovered (unknown) 

fields, an estimate has to be made on the basis 
Initial Oil 

In Place 

(OIP) 

Recoverable oilEstimated Ultimate 

Recovery 

(EUR) 

Expected Value 

of Recoverable 

Discoveries 

(EVRD) 

Undiscovered 

recoverable oil 

( R 3) 

t 

Expanded proven 

reserves 

( R 2) 

FIGURE 1.-Span of the various resource and reserve terms for oil. 

5 

f 

Cumulative 

production 

Proven 

res~rves 

( R 1) 

J 
of the discovered (known) fields in relation to 

the geologic setting. This estimate would not be 

too difficult to make if the unknown fields belong to the same geologic groups as the known 

fields or if both known and unknown fields belong to a well-defined statistical population. 

Actual exploration, however, shows that often 

a certain discovery alters the concept of the 

petroleum accumulations to be found in a given 

area. Projections from an old model can thus 

become obsolete. This logic·al difficulty is hard 

to overcome. Perhaps the best we can hope for 

with respect to the OIP value is to establish 

lower bounds for it and to raise the lower 

bound whenever wider knowledge about the 

petroleum geology of a region being considered 

justifies it. 

Second comes the concept of recoverable resources; that i.s, the amount of oil that can be 

recovered, within technological and economic 

limits, from both undiscovered and discovered 

fields-often denoted as the Estimated Ultimate 

Recoyery (EUR). The relative amount of oil 

that is recoverable varies greatly and has not 

been well established on a worldwide basis. As 

a result of modern production practiceS, primary and secondary methods have become well 

integrated, and a sharp distinction between 

them is not justifiable. At the moment, 40 percent probably represents a realistic target figure for these two integrated phases of recovery. 

How much oil .still remains in the ground, even 

after the best production practices, is not really 

known. Perhaps the actual recovery can be as 

high as 80 percent. Economic limits constitute 

another factor that conditions the estimation of 

the amount of recoverable resources. These 

limits will change with time, so a forecast of 

the recoverable resources should vary with the 

time span of the forecast. 

Third comes the concept of cumulative past 

production, that is, the total oil that has been 

produced from the discovered fields. This is the 

figure that can be ascertained most readily. 

Fourth comes the concept of proven reserves 

R1. Proven reserves, designated here as R1, are 

defined as the amounts of petroleum that can 

be considered certain to be producible from explored acreage within present economic and 

technological limits. To convey a qualitative 

sense to the above definition, I woulP. say that 

6 

in known fields, the amounts of petrol~mm that 

can be obtained with certainty, say with a 

higher than 90 percent probability, ~n be estimated within +25 percent or so. The main uncertainty in this estimation process would be 

covered by the ,span of the error ( +25 percent). 

The increments of oil to be expected from 

known fields would fall off rapidly b~low a 90 

percent probability. Essentially then, we would 

have a rather narrow estimation distribution 

function for the proved reserves-the estimation of the magnitude of a quantity that is 

known to exist. 

Fifth comes the concept of expanded proven 

reserves R2, which represents the expected 

amounts, in a statistical sense, of oil from revisions and extensions of discovered fields. We 

deal here with a different kind of uncertaintythe speculation that some petroleum may exist 

beyond the known parts of fields ·and as extensions of them. For most countries, t1' e following generalization can be made : An additional 

quantity equal to the proven reserves can be obtained with probability of 80 percent, and another equal additional quantity, with probability of 50 percent. Hence, the expected value of 

the expanded proven reserves, desigiJ a ted here 

·as R2, would be (1+0.8+0.5) Rt=2.S R1. 

The recoverable petroleum that with certainty can be said to have been found so far in a 

given country is the sum of the cumulative production plus the proven reserves. The expected 

value from discovered fields is somewrat larger. 

For this purpose, I have introduced tl'~ concept 

of EVRD (Expected Value of Recove:-:"able Discoveries) , defined as 

EVRD cumulative production+ex~anded 

proven reserves R2. 

Finally, the difference between the recoverable resources and the sum ( cumulatiye production+expanded proven reserves) is the undiscovered recoverable resources R3. JT ence, an 

estimation of Ra involves an estimation of recoverable resources. 

ASSESSMENT OF THE PETROLEUM-REf"lURCE 

B,ASE 

The main issue underlying the S'tdden increase in oil prices is a wide, sudden r'-".alization 

that world petroleum resources will b~ depleted 

within a few decades. How to appraise the ex-
tent of the remaining petroleum resources is a 

source of confusion. For long-range economic 

planning, it would be useful to have rather accurate estimates of the total amount of recoverable oil in a given country. However, because 

of the high degree of unpredictability of the actual location of petroleum deposits and because 

of technological limitations in the search techniques, it is not economically viable or technically possible to discover, ahead of development, 

all the fields containing the remaining world 

petroleum resources. 

Therefore, to appraise the long-range supply 

of oil ·and gas, we need to go beyond mere projections made from present trends. First, one 

should estimate the m~agnitude of the resource 

base. regardless of economics and uncertainty 

of discovery. And then one should subdivide the 

resource base according to various intervals of 

unit costs, in increasing order starting from 

present levels. Future technological developments or changed market conditions probably 

will permit the exploitation of certain resources 

that currently are uneconomical. Second, the 

resource base should be ~subdivided according 

to certainty of occurrence. Improvement of exploration techniques and increasing knowledge 

of actual geologic conditions would allow some 

resources whose existence now is considered 

uncertain to be incorporated into the available 

supply. Such a conceptual framework for the 

long-range appraisal of resources has already 

been proposed by McKelvey (1972). It provides 

a much more meaningful basis for long-range 

forecasts than before. 

The recoverable reserves for one fully developed oil field can be estimated with relatively 

great accuracy (say within +25 percent), but 

the estimation for a new field, a petroleum district, a sedimentary basin, a nation, or the 

Earth constitutes a series of exercises that are 

increasingly difficult and that have correspondingly wider ranges of uncertainty. A few basically original estimates have been made of the 

world petroleum resources, but a vast amount 

of published data in fact amounts to "regurgitations" of someone else's data, or someone 

else's "regurgitations" of someone else's, etc. 

The language of mathematical statistics is required to define more rigorously the problem 

of appraisal of petroleum resources. 

7 

To carry the analysis a step fartr~r, one can 

pose the question of whether the amount of recoverable oil and gas of a given part of a region 

could be of a given magnitude. The answer can 

be given only in probabilistic term~s. That is, we 

are faced here with the a priori prob ~.bility density function of the recoverable amount of 

petroleum for an undrilled area. Such a prob-

.ability density function is really not known, for 

example, for many undrilled areas in developing nations, and for most of the continental 

shelves, and it can only be conjectur~. 

Conceptually, at a given time, one could first 

classify the remaining resources F-- according 

to the probability of being found as exploration 

continues. This probability p can bE: considered 

to be the product of the probability of existence 

-of a field Pe -and of the probability of actually 

locating the undiscovered field Pd· T1'" resources 

that at a given time we know with certainty to 

exist are a certain amount. One c<J llld conceptually proceed to dassify the undif~overed resources as incremental quantities .6.R corresponding to ranges of the probability p down to 

·some low value of the probability. T'1e resource 

base B could be defined as the exr~ted value 

of the resources, that is, the summation of the 

incremental resource amounts multiplied by the 

probability of finding them; that is 

B= ~ p.6.R. (1) 

for all ~R 

Upon this first dassification sche1ne, we have 

now to impose the constraints that result from 

economics. Only a fraction of the segment of 

resources within a certain probal,ility range 

can be considered to be economically recoverable and discoverable, correspondin~ to the conditions at the time when the as~essment is 

made. In the future, the economic limits may 

widen (although not necessarily so), thus permitting a larger proportion of each segment to 

be exploitable. · 

Moreover, it would seem that the a priori 

probability density function is not narrow and 

would definitely become small only beyond the 

largest conceivable size of the recoverable 

amount of petroleum. Furthermor~~, the probability density corresponding to a very small 

size of the recoverable oil, or gas, is significant 

and different from zero. Moreover, the prob-
ability density for, let us say, an amount of recoverable oil comparable on a unit ·area basis 

with a given known oil basin is significant ·and 

different from zero. As we do not know the 

shape of this probability density function for 

undrilled ·basins and as it appears to be quite 

broad, it is not proper to give only one value 

for the amount of recoverable oil or gas or to 

expect that its standard deviation is a small 

fraction of the m·agnitude of the recoverable 

amount. As a first approximation, a uniform, or 

flat, probability density function could be taken 

for the petroleum estimates of an unexplored 

area. To give one figure for the petroleum resources of an unexplored area would be a futile 

undertaking. 

Although this scheme might appear to be conceptually clear, operationally it is very difficult. 

Below a given probability, say 60 percent, the 

situation becomes highly speculative. There is 

very little basis on which to construct the actual scheme, and yet the largest expected contributions to the resource base B should come 

from resources having low probabilities of 

eventually being found. However, one could 

strive to perfect such a picture gradually, considering the past record of discovery ·as a basis 

for estimating parameters for theoretical statistical models. 

Perhaps one of the most perverse effects of 

the conceptual difficulties of petroleum-resource 

assessment is what I call the accuracy delusion. 

By that I mean the misconception that published figures for undiscovered resources have 

a somewhat narrow distribution function; that 

is, that the possible values form a Gaussian distribution about the published figure and have 

not too great a standard deviation, say 20 percent. For new undrilled tracts of territory, as 

exemplified by the continental shelves, such a 

Gaussian distribution c.annot yet be provided. 

A team of company specialists might agree 

among themselves on a "most probable" value, 

but from team to team the "most probable 

value'' will be found to vary substantially. The 

wide scatter observed in bids for offshore petroleum leases could well be attributed to these 

variations. 

A better approximation to the underlying uncertainty function of resource estimates than 

the Gaussian curve that leads to the accuracy 

8 

delusion is a modified uniform distribution 

function. It would extend, with uniform probability, from zero up to a figure somewhat 

greater than that for the richest known similar 

tract elsewhere and then would dr'l-p rapidly 

back to zero probability for great~ amounts. 

By analogy with other tracts and fP>m knowledge of adjoining areas, one might ju~tify modifying the uniform distribution on tl' o. low side 

also; that is, to drop quickly to zero probability 

for resource amounts smaller than a certain 

-amount. 

The estimated magnitude of the resource 

base, and its subdivi·sion according tc economics 

and degree of uncertainty in finding, sets the 

targets for long-range petroleum exploration. 

In this selection of targets we are not. restricted 

by the high degree of certainty r·~uired by 

short-range considerations because, in the exploration of large unknown areas, the uncerbainties diminish as the exploration proceeds. 

To apprai~ the petroleum-·resourc.~ potential 

of new tracts of territory, one should consider 

the basic scheme (table 1) that underlies petroleum-resource exploration and de"relopment. 

After the sedimentary basins have been identified, a predrilling estimate of Jl otential is 

made on the basis of factors such as the area, 

the maximum thickness of sediments, the type 

of sediments, the existence of structural traps, 

the existence of stratigraphic traps, the reconstruction of geologic history, the occurrence of 

oil and gas seeps, the adjoining petroleum provinces, and so on. 

An exploratory well is ~aimed at r. very specific target which has been identified from similar previously undrilled, unrecogniT~d targets 

on the basis of (1) the prior data obtained in a 

region plus (2) the specific geological and geophysical surveys in the particular prospect. The 

various targets that exist in the ba,qin may be 

categorized in various groups such as foreland 

anticlines, hinge-belt anticlines, platform anticlines, fault traps, reefs, pinnacle r~fs, domes 

over salt domes, regional pinchouts, shoestring 

sands, etc. Moreover, the geologic definition of 

the targets in each group could be quite specific. 

When the petroleum industry in a given basin 

is pursuing a given "play" it is in f~et running 

after targets in one of these groups. The actual 

existence of petroleum in one of th ~e targets 
---

TABLE 1.-Stages in petroleum-resource development 

Geological and geophysical exploration 

Existence of sedimentary basins 

Predrilling information : Area of the basins 

Maximum thicknesses of the sediments 

Type of sediments 

Existence of structural traps Existence of stratigraphic traps 

Reconstruction of the geologic history Oil or gas seeps 

YES NO: OU':':' 

Adjoining petroleum provinces 

Predrilling potential? (end of first phase of appraisal) Exploratory drilling campaign 

Postdrilling estimate of petroleum potential ? (end of second phase of appraisal) 

Further geological and geophysical work 

Further drilling 

Development of oil potential (subsequent phases of appraisal) 

can only be ascertained with a given statistical 

probability even after consideration of all the 

information that can be gathered before drilling. One could say, for example, that one out 

of four structures of a certain type in a certain part of a basin would contain oil. Moreover, the magnitude of the accumulation would 

be essentially determined by the group type1 the 

actual size being almost unpredictable. 

In this manner, the statistical success of drilling would be about the same almost to the very 

end of the play, except for the effect of the enhancement of knowledge because of interaction 

with previously obtained data. One could thus 

describe the outcome, in barrels of oil or thousand cubic feet of gas found per foot drilled, 

as a random sample from a normal distribution, having a certain mean and a certain variance that characterize the play. When seve~al 

plays are being pursued, the outcome would 

consist of random samplings from the various 

normal distributions corresponding to each 

group. 

In none of these models would there be a decreasing finding rate as a normal situation. The 

exploration would reach the limits of the resource with few warning signals from the finding rate, and the bottom would be hit rather 

unexpectedly. 

An analogy here may help to visualize the 

problem. Let us suppose that an experienced 

hunter with a shotgun is hunting rabbits in a 

large enclosed field. Let us further assume that 

20 rabbits are in the field and that the hunter 

requires three shots per rabbit, on the basis of 

earlier experiences in similar fields. One would 

expect that on the average he will require three 

shots per rabbit from the first one he downs 

9 

until the very last rabbit. Moreover, when a 

series of exploratory wells is drilled in a given 

region, the aim could actually impr0ve because, 

as the data from an increasing nun1ber of wells 

and exploration surveys become available, the 

geologic picture i~s clarified. 

OBJECTIVE AND BIAS IN RESOURCE ASSESSMENT 

When faced with the problem of estimating 

the magnitude of undiscovered p~~troleum resources, one needs to analyze the objectives of 

making such estimates. In situati~ns that involve uncertainty, one can rely he~.vily on the 

tools of mathematical statistics as long as the 

.g1Jatistical properties of the quantities involved 

are well defined. But such is not the case for 

most of the petroleum-prospective areas of the 

world. We are not dealing with a single statistically well defined population of petJ~oleum fields. 

As exploration enc.roaches upon the undrilled 

areas, one needs to introduce ne·w· statistical 

categories of fields, and, because of this, statistical methods have to be used with caution, and 

their limitations must be understo')d. 

Of course, situations vary from the statistically well-defined-as to size of rccumulation 

and location-to the very fuzzy. UPfortunately, 

for most of the undrilled areas, the situation is 

very fuzzy. For each petroleum pr->vince there 

is a certain histogram of petroleu'11 accumulation ranked according to .a sequence of size intervals which range from zero t<' the largest 

conceivable value. This histogram--or its conceptual limit, the probability denrity distribution----can be estimated for certain limited regions, but we have at present ver:" little to go 

on in constructing one for a sedirn entary basin 

in the interior of Africa, for exa:rrple. 
We are forced then to a blend of objective 

and subjective judgments. This situation has 

to be kept in mind because the mere fact that a 

probability density function is formulated 

mathematically does not necessarily mean that 

subjective elements have been removed. Ignoring these underlying logical difficulties, and 

also ignoring the various points of view that 

can legitimately be used in re~ource appraisal, 

has given rise to much confusion and argument. Petroleum-resource data have been misused by being taken out of context. 

The amount of recoverable resources in a 

certain geologic domain can be viewed as a 

variate x, which could conceivably attain any 

value within a certain range LLxLW. By analysis of the outcomes of the same estimation 

process as applied to a suite of well-explored 

geologic domains { Ai}, one could construct a 

probability function f (x) so that 

p(x:::::...O)= Lx f(x)dx. 

0 

Alternatively, one could estimate f(x) by using 

a model that involves subjective probabilities. 

Another way of looking at the problem is to 

aim at the estimation of limits for x and for 

the range of x, within assigned probabilities. 

In practice, it would be more meaningful, as we 

have explained, to target on certain limits of 

x, rather than ori f(x). 

What use is to be made of a petroleumresource assessment is one of the crucial questions. Some of the possibilities are that: 

1. A bank is to decide on the financing of an 

exploration and development project. 

2. An oil company is to decide on beginning an 

exploration project. 

3. An oil company is to decide on continuing an 

exploration project. 

4. A government is to define a short-term national energy policy. 

5. A government is to define a mid-term national energy policy. 

6. A government is to define a long-term national energy policy. 

Each application involves: (1) a decisionmaker who will use the resource estimate and 

(2) a specific decision that must be made and 

that hinges on the resource estimate. Both are 

10 

important in defining the nature of the petroleum-resource assessment reqp.ired. 

For certain decision, one needs a floor, or 

lower bound, for the resource estimate that has 

a high probability of being either fulfilled or 

overpassed. A bank, for example, w~n require 

almost a 100 percent certainty. By using the 

language of mathematics, this can be explained 

as follows: 

For deciding on the financing of a E"pecific investment for resource exploration anti development, the bank needs to know the am"lunt xL of 

the resource so that the relationship 

LLxLxL 

will be fulfilled with a low probability q.. This 

probability q determines the marginal risk of 

losing the investment. ConV'OOtional financing, 

moreover, would require (1) that tl~~ bulk of 

the investment be recoverable on the lH'.sis of an 

amount of resources that .is almost certain to 

exist and (2) that only a marginal fraction of 

the investment be risked. Such an estimation 

problem could be described by a probability density distribution such that 

p(x:::::...L)=1, 

and 

tt 

p (LLxLu) = £ fL(x) dx. 

L 

A government, as another example, when 

analyzing the various outcomes of r.lternative 

energy postures will need an indication of the 

minimum amount that with reasonable certainty can be counted on to be available from 

the domestic sources. 

An economic forecaster may want to focus 

his discussion on a most likely value of the recoverable resources. Here, a word of caution is 

in order. The "most likely" value should be 

sought in most resource assessrr.ents, but 

seldom is it really available. What we often 

find is an intermediate value that has been 

termed "the most likely value." 

For other decisions, one needs a roof, or 

upper bound, for the recoverable resources 

characterized by a low probability of being surpassed. When appraising, for examph,- the possible impact of a new energy resource, one may 

want to know how large its contribution to the 

total energy supply could be. It may be enough 

to know that the upper bound is at the most a 
few percent of the total energy supply; however, if the calculated percentage turns out to 

be substantial, then a subsequent estimation 

aimed at the midpoint of the likely outcomes 

may be in order. 

An important bias arises out of competitive 

considerations. For instance, when appraising 

the petroleum prospects of specific undrilled potential areas throughout the world, an oil company may not want to encourage its competitors by aiming its appraisal at the upper range 

of the possible outcomes; rather, it may want 

to aim at the lower range while still staying 

within the range of what the company considers viable. 

Now that there is widespread recognition of 

the energy crisis, national economic planners 

have been asking how much oil is left in their 

country. They want to know this with precision, 

in order to m·ake firm policy decisions. We meet 

here several problems, one of which is to make 

the planners understand that counting barrels 

of undiscovered recoverable oil in the ground 

i:s not as simple as counting sheep gra.zing on 

the land. Let us examine what kinds of answers 

are both viable and useful to economic planners. 

One needs to distinguish the various types of 

countries as to their initial petroleum posture, 

namely: (a) no exploration and no development; (b) unsuccessful exploration; (c) incipient exploration and incipient development; (d) 

limited exploration and limited development; 

and (e) intensive exploration and intensive development. 

For countries groups (a) , (b) , and (c) , the 

first legitimate question for the planner is 

whether to act so as to permit exploration to 

proceed. In this decision problem, one has to 

compare the two alternatives : 

Go : that is, proceed with exploration, 

and 

No Go: that is, no exploration allowed. 

This is not an unlikely situation. In fact it is 

the type of decision confronting national planners in some developing nations having a petroleum potential. What they must weigh here is 

the (1) economic prize that might be obtained 

if exploration turns ()Ut to be successful against 

(2) the costs of exploring plus possible indirect 

economic losses incurred because of having proceeded with the exploration. For those coun11 

tries, estimates at the center of the range and 

at the upper bound would be most useful to 

them in making, their decision. 

For countries in group (d), howev~r, further 

limited exploration and developme:"t may be 

discouraged by an estimate that is aimed at the 

lower end of the range. 

A country in group (e) that has had intensive exploration and development may want, in 

the first place, an estimate of the w.inimum of 

recoverable resources that are left in order to 

proceed cautiously. For comprehenfive contingency planning beyond this minimum, they 

would need also the most likely outcome and 

the upper bound. 

The requirements on which resourceappraisal methods should be judged r:re several: 

(1) the consistency with observations; (2) the 

degrees of objectivity and, its courterpart, of 

subjective judgment; (8) the stabil~ty of forecast; (4) reproducibility; and (5) the adequacy, as to the true answer. 

Often, forecasting methods haVf~ been adv·anced because they satisfy ·some of these criteria, but they fail to demonstrate adequacy. 

Objectivity and reproducibility,. for instance, 

are not enough. The finl:ll criterion is adequacy, 

which is difficult to achieve and eve11 more difficult to demonstrate. 

PROBLEMS WITH HISTORICAL 

PROJECTION MODELS 

LINEAR. "DR.ll..LING INPUT-PETROLEUr·-r OUTPUT" 

MODEL 

A linear input-output model of pe+roleum exploration and development may be used to gage 

the economic impact of the finding rate. 

The input into the model is a schedule of exploratory drilling per year, and the main output is the .amount of oil or gas discovered per 

year. Such a model has been used in the United 

States, for example, by the N a tiona' Petroleum 

Council (1973) and the U.S. Federal Energy 

Administration (1974). 

One ·assumption in this model is that the 

amount discovered per year is simply proportional to the drilling per year. The amount of 

oil or gas discovered per foot drrled can be 

taken to be a parameter that is a function of 

the cumulative drilling and that var:~~ from re-
gion to region. Another basic assumption made 

i-s that the production per year is a constant 

fraction of the proven reserves. 

The calculation starts with the reserves at 

the beginning of the period of projection and 

updates them year by year as the result of the 

drilling discoveries and the yearly production 

withdrawals. A separate tally is kept of primary, secondary, and tertiary reserves. 

The ba·sic equations for this model would be 

as follows: 

aRn=p.lmD,_"A. +aR' n+ aR" ,+aR" I,+£, (2) 

P,+l = /R, (S) 

and 

where 

aR,. === (recoverable) proven reserve established during the n-th year in a 

region, 

Dn-"A. = explomtory drilling footage undertaken during then-,\ year, 

m == finding rate in barrels of oil or thousand cubic feet of gas per foot 

drilled, where m is a function of 

lDn, 

P.1 = primary recovery factor, 

A == lead time in years, 

aR' = revisions and extensions of earlier pri- , mary proven reserves on n-th year, 

aR" = actual increment to secondary proven 

" reserves on the n-th year, P.2 = secondary recovery factor, 

aR"' = actual increment to tertiary proven re-

, serves on the n-th year, p.a ==tertiary 

recovery factor, 

£ = revisions to prior computations of 

aR', ~R", and aR"', 

P, =production in n-th year, 

I = fraction of the current reserves to be 

produced each year, and 

R,. = recoverable proven reserve as of the 

end of the n-th year. 

The starting conditions are the set of the 

Ro values as of the beginning of the period of 

projection. The parameters P.1t p.2, p.a, m, /, and 

A are assumed to be known in the model and 

would be estimated by studying the historical 

data for the region. The ·schedule_ of primary, 

secondary, and tertiary recovery factors is also 

assumed for the model. 

Then an exploratory drilling schf\~ule is assumed for the region ; that is, the fo'ltage Dn to 

be drilled in each region in each ye.1.r. 

Equations (2), (3), and (4) expr~s the aproaches used by the National Petrol~um Council (1973) for oil and gas calculatio:'ls. For the 

oil model it is assumed that a cert:'.in proportion orf associated and dissolved gas will be 

found per barrel of oil reserve discovered, and 

this gas will be passed to the gas me iel. A constant fraction of the amount of all ~xploratory 

drilling is assumed to be allocated to gas exploration. 

Up to this point the big uncertainty is the 

role of the market price in t\le assumed drfiling schedule. Actual economic costs of various 

oils vary widely, and because some producers 

would lose money at the average price, they 

would not have undertaken the assumed scheduled exploration and development. ~,he market 

price required to yield the assum ~d drilling 

schedule should be higher than tl~ ·average 

cost, or, said another way, the actual drilling 

schedule and petroleum outcome would be 

sm·aller than indicated by the above model if 

the market price is the same as the average 

cost. 

What is needed, then, is a way to p,-ee8timate 

the supply curves for oil and for ge.s--that is, 

the amounts of oil and gas that wo~dd become 

available ·as a function of their market price. 

This preestimate thus requires consideration of 

the variety of the individual development projects ranked as to their p·rofitability. 

The exploratory effort Dn in a givE''l province 

on a given yearn can be viewed as the initial 

point of an investinent decision. Let us define 

this as ·an "n-project." The exploratory effort 

Dn would result in the discovery of a proven 

reserve AR, +"A where A is an exploration lead 

time, but only if the industry actuslly decides 

to make the necessary investments. An entrepreneur would decide whether to undertake the 

project, depending on his assessm~nt of the 

project's financial outcome. For this decision, 

the discounted cash-flow method, having an assumed discount rate, r, can be used. This is the 

procedure adopted, for example, in Project Independence (U .is: Federal Energ7" Admin., 

1974), which can be used to develo11 a mathematical formulation. 

12 
Let 11Pn(,.) be the annual production to be 

obtained from an n-project, ,. being the time 

counted from its initial year n. For a given discount rate r, there would be a minimum unit 

price Pn ( r) of the oil, or gas, that the entrepreneur would require for deciding to make the 

investment on then-project. This unit price is 

a function of the discount rate r and also of the 

specific n-project considered. As a basic characteristic of an n-proj ect, we can take the finding 

rate m. Hence, we can write 

Pn (r) =P (m,r). (5) 

That is, given the finding rate m and the discount rate r, the minimum required price can 

be calculated. 

Consider now the supply situation in a given 

yearn as being the result of all prior n-projects 

undertaken. For a unit market price U we can 

ask what would be the supply P ~ o.f oil, or gas, 

that would be available. The supply P,.,, would be 

the sum of the productions for the year n of all 

prior n-proj ects that would be undertaken. The 

supply for a given year would thus consist of 

the addition of the contributions drawn various 

years after discovery from a set of n-proj ects, 

that, is of several "vintages." The question, 

therefore, is which n-projects would have been 

undertaken. 

If the future-price expectation of all entrepreneurs prior to year n had been U, then all 

those n-projects for which 

U(m,r)LU (6) 

would have been carried out, and the supply 

would be 

T=L 

Sn= ~ 8m,p11Pn(n-T), (7) 

T=n 

where the operator, Sm. P' is equal to 1 if (6) is 

satisfied, and equal to 0 if it is not. 

The finding rate is defined as the ratio 

&Rn+'A. 

m 

Dn 

(8) 

For a given region, this finding rate may ~ary 

in some manner as a function of the cumulative 

drilling, that is m=/(SDn). 

Now I would like to discuss whether the value 

Sn above, as a function of U, would provide an 

estimate of the supply curve. The most critical 

input data to the calculation of this estimate of 

the supply curve appears to be the finding rate. 

We have 

&Rn+x=mDn. (9) 

First, the future-price expectation in the past 

may be different from the actual pric~ at a later 

time. If the future-price expectation in the past 

would have been higher than the actual current price, then the actual current supply would 

be higher than indicated by the eq'lation (7) 

because more entrepreneurs would have decided, some mistakenly, to carry out their nprojects. On the other hand, if the f'lture-price 

expectation in the past would have been lower 

than the actual current price at a later time, 

then the actual supply would be lower than indicated by (7) because less n-projects would 

have been carried out. If however, it is as-

·sumed that the future-price exJ)E'~tation of 

entrepreneurs is correct, then the above difficulty disappears. 

Second, ann-project would generally encompass several petroleum fields of various degrees 

of profitability, yet all of them are linked in 

this simple model as one project. This naturally 

tends to blur the significance of the supply 

curve as calculated above. 

Now we assume that the producthn rate obtained from tilln+>. decreases exponertially with 

time and that the decay constant is T 1, so that 

&Pn(T)=(&Rn+>./Tt)e-r!f'l_ (10) 

Hence 

&Pn(T)=m(D,JTt)e-r/Tt, (11) 

The development and exploitatio:'l of an nproject gives origin to a sequence of annual 

investments, expenditures, and i n c o m e s 

throughout the life of the project. In the discounted cash-flow method, the cas:'l proceeds 

and cash outlays are discounted to the initial 

time. If the discount rate is assumei, then the 

unit required price to make the prt~sent value 

of the proceeds equal to the present value of 

the outlays can be calculated. 

For the discounted cash flow of a given nproject, and taking the U.S. type of taxation 

as an example, we can use the expre~,~ion 

F=(1-a) (1-p) USAPre-rr 'T 

+r (Ea+Eb)-Ilo-lE,.e-r,. 'T 

-Sy{ (1-a) (1-p)paP.,. 'T 

13 
where 

F = present value of the discounted cash 

flows, 

U = unit price of oil (or gas), 

P.,. = production per year on .,. year of project, 

a = ad valorem tax, 

{3 = royalty rate, 

E a = expensed items : dry holes + 8 o/o of 

successful wells + lease rentals + 

overhead, 

Eb =tax credits: £% of successful wells + 

environment and safety + gas plant 

+ lease equipment} 

y = corporation tax rate, 

Eo= initial cash expenditures, 

E.,. = cash expenses on .,. year of project, 

., = effective depletion rate for .,. year of 

project, as a fraction of the net revenue, 

o =depreciation for.,. year of project, and 

r = annual discount rate. 

The first term in the summation of expression (12) corresponds to the present value of 

the net revenues ; the second term, to tax 

credits; the third, to the initial expenditure; 

the fourth, to the present value of the annual 

expenses; and the last, to the present value of 

the income taxes. 

u 

Equation (12) is to be solved for U, namely 

Eo+Y~ (E.,----q--8) e-r-r_y (Ea+Eb) ,. 

y(1-a) (1-,B)~~P.,e-r.,. 

(13) 

'I' 

Some of the quantities in the numerator of (13) 

correlate strongly with D, and others, with aR, 

but the predominant effect appears to be a 

linear dependence of Eo on D. Moreover 

~P.,=m(D/T ) e-r/Tl (14) 

because of equation (11), and thus ~P., is also 

proportional to D. Therefore, the factor D tends 

to cancel out in ( 13) , and the price U would 

turn out to be inversely proportional to the finding rate m, namely 

c 

U=-, (15) 

m 

where c is a constant which depends on the 

investment and opel"'ating expense coefficients, 

tax rates} and production decay constant T 1 • 

14 

DRILLING FINDING RATE 

The density of drilling, that is, t:'le number 

of exploratory wells drilled per squr.re mile of 

prospective area, is a useful indicat1r of what 

needs to be done in young petroleum provinees, 

as, for instance, in most developing countries. 

Because of this, it is necessary to review the 

relationship between wells drilled and petroleum found. Total footage is a mearure of the 

amount of drilling. 

A common assumption is that t1'~ average 

amount of proven reserve found each year is 

proportional to the amount of explorr'""tory drilling in that year. Moreover, it has be~n assumed 

that this finding rate, in barrels of oil or thousand cubic feet of gas per foot, decre<:~.ses as the 

cumulative drilling increases. Offhard, this appears to be a reasonable assumption. 

Why should the finding rate decrer;se steadily 

as the cumulative drilling increases? It cannot 

be simply because the oil resources in a region 

are being depleted as the exploratory drilling 

increases because the finding rate could remain 

constant or even increase during n10st of the 

exploratory phase of a region and then drop 

rapidly to zero as the limits of the resource 

base are finally approached. 

It has been claimed that the finding rate 

should decrease because the larger fi,~lds will be 

discovered first. Of course, this is what one 

would like to have happen, but thr. record of 

exploration in basin after basin reveals that 

this is not so. The discovery of giant fields typically occurs some 30 years after exphration begins in a region. There does not arpear to be 

a dominating "pickle-barrel effect-" to thus 

justify a decrease of the finding rate. 

Maximum depth of drilling has be~n increasing worldwide. For given basins and time lapses 

of 10-20 years, the average drilling depth may 

show a trend of increase that would introduce 

a decreasing trend of the finding rate with 

time. The year-to-year fluctuations of the finding rate, however, can be considerably larger 

than the effect per year of the basic trend due 

to drilling depth. 

As an example, let us examine how the finding rate behaves for the National Petroleum 

Council (U.S.) regions (table 2). For the 

period 1956-70, the finding rates for regions 

1, 2> 2A, 6, 6A, 7, 8, 9, 10, and 11 vary rather 
randomly from year to year and do not demonstrate any trend of decreasing values as drilling footage increases. On the contrary, for regions 1, 2A, the aggregate of 8, 9, and 10, and 

11, the basic trends indicate that the finding 

rate increases as cumulative drilling footage increases. Only for regions 3, 4, and 5 is a declining trend indicated. 

TABLE 2.-New oil in place (bbl/ ft) added per foot of exploratory drilling 

[From Natl. Petroleum Council, 1973, p. 212-222] 

U.S. region 

Year 1 2 2A 3 4 

1956 0 191 33 93 94 

1957 0 163 13 188 115 

1958 0 186 38 227 107 

1959 214 143 34 113 86 

1960 3,968 222 522 159 100 

1961 2,178 151 774 138 99 

1962 0 545 365 69 54 

1963 43 263 1,206 71 42 

1964 366 1,360 163 56 52 

1965 2,975 226 1,063 40 71 

1966 2,256 40 1,262 80 40 

1967 4,492 93 1,867 112 69 

1968 11,233 411 640 68 75 

1969 128 43 1,695 45 25 

1970 463 77 0 44 41 

ECONOMIC ROLE OF THE FINDING RATE 

To gage the quantitative importance of the 

finding-rate function on the supply curve, let us 

assume a linearly decreasing function, namely 

m=mo-hQ, (16) 

where mo is the finding rate at the initial point 

considered, Q is the cumulative exploratory 

drilling measured from the initial point, and h 

is a constant. 

The supply S is obtained by integrating the 

outcome of the drilli,ng effort, namely 

Ql Ql 

S=J mdQ=j (mo-hQ)dQ, (17) 

0 0 

or 

(18) 

The cutoff point of the drilling, namely the Ql 

value, would depend on the economics. The cumulative amount of drilling Q determines m by 

(16), and m determines the minimum required 

price from equation (15). Thus 

(19) 

and 

(20) 

15 

5 6 6A 7 8,9,10 11 

201 86 1,239 97 106 (] 

254 92 1,027 79 90 0 

274 80 4,037 61 90 (] 

329 103 17,451 65 132 (] 

314 87 75 112 (] 

273 146 76 68 (J 

249 135 751 92 75 22. 

213 81 460 72 84 (J 

149 102 323 76 82 138 

184 93 257 83 69 569 

134 62 487 55 81 (J 

136 71 282 46 106 (J 

95 41 391 41 144 16 

87 59 415 51 149 129 

121 77 1,423 78 175 72 

So that the supply S is 

mo2 c2 

8=- (21) 

2h 2hU12 

Equation (21) demonstrates that the supply 

functionS has an asymptote. No tnatter how 

large the price ul the quantity s V10Uld be no 

larger than m0

2 /2h. As U 1 increases. S increases 

at a IS'lower and slower P'ace to its ultimate 

asymptotic value. The asymptotic value of S 

varies with the square of the initial finding rate 

and also is inversely proportional to the downward slope of the finding-rate line. 

A quantitative assessment of the role of the 

finding-rate function, which I have made briefly 

here, reveals that the finding rate plays a major 

role in defining the supply curve for petroleum. 

A decreasing finding rate, even a mild linear 

decrease as cumulative drilling increases, imposes a definite roof (asymptotic v2lue) to the 

supply curve. No m~atter how high the price 

goes, the supply does not go above it. As the 

price increases, the supply increases at a more 

and more sluggish pace. 

NOTE ON THE LOGISTIC METHOD OF PROJECTION 

Application of the logistic function to the 

estimation of petroleum resourcE*' has been 

made by M. K. Hubbert in an extensive series 
of papers since 1956 or so (see, for example, 

Hubbert, 1969). A similar approach is the use 

of the Gompertz curve (Moore, 1965). These 

methods have the advantage of their simplicity 

and of being based on published statistical data. 

They are intended to be used in areas where 

exploratory drilling has reached a mature 

stage. 

Thes·e methods have been called "mathematical" because they use a formula to fit the cumulative-production and proven-reserve data. In 

fact they ·are empirical booause no theory justifies the use of either the logistic function or 

the Gompertz curve. 

When these methods are used, it is difficult 

to recognize when the final maximum has been 

reached. The hi·storical oil-production curves of 

several countries show well-defined maxima. 

Had the logistic, or Gompertz, projection 

method been used, there would have been a 

gross underestimation of the undiscovered oil 

resources. For example, the annual oil-production curve for Mexico in the period 1918-32 has 

a maximum in 1921; for Austria in the period 

1946-61, a ·single m·aximum in 1955; for France 

in the period 1946-73, a maximum in 1965; for 

Rumania in the period 1918-47, a maximum in 

1936; and for the USSR in the period 1918-45, 

a maximum in 1941. 

These methods provide a firm estimate of the 

minimum amount of petroleum that ought to be 

found if the industry continues doing what it 

has been doing in the past-what is already in 

the bag, so to speak. They cannot) however, 

predict new plays in a basin. 

The application of these method's is limited 

to mature petroleum regions and thus cannot 

be used in most of the partially explored and 

unexplored petroleum-p·rospective ·areas of the 

world. 

NOTE ON RANDOM SEARCH 

In petroleum resource appraisal, two basic 

questions are to be answered: (1) How much 

petroleum is there? and (2) where is it? If we 

assume the answer to the fi~st of these questions, we have a problem of search only. 

Some random-drilling models purport to 

·show that the outcome of past exploration is 

to be explained by random drilling. But even 

a ·superficial acquaintance with the history of 

16 

petroleum exploration reveals that such is not 

the case. The overwhelming majority of exploratory drilling decisions are made on the basis 

of all information ava.Hable to the decisionmaker, and the choices of· exploratory ·sites are 

those that currently appear to be mo.qt promising. The aggregate nature of the dabl used and 

the naivete of some decision models u~P.d might 

explain why this exercise seems to p:':"osper. 

A major logical difficulty in the mc<ielling of 

past exploration is the introduction of a Sunday 

morning quarterback or post-morteir point of 

view. When exploration begins in a new province, it does so with no assurance that commercial accumulations exist and with no ~ s,surances 

as to the ·size of such accumulations. Tf, in fact, 

there ·are, let us say, 100 fields in a province at 

the beginning of the exploration, a search 

strategy cannot alSSUtne this but must assume 

the variety of possible ultimate out.comes : 0 

field, 1 field, 2 fields ... n fields. On the other 

hand, if we assume that at the beginning of 

the exploration, we are provided with a statement s·aying that somewhere in the basin are 

100 basins and that we ought to find them, the 

situation would be different. Then we could engage in a systematic campaign of grid drilling, 

using closer and closer grids, until the: fields are 

found. Thus, a model of random or gr~d drilling 

loses meaning when examined against the exploration that has to unfold the plot. 

In order to explain the outcome o:t: exploratory efforts, one must take into consideration 

both the search str:ategy of predrilling surveys 

(geology and geophysics) and the e:r:plor·atory 

drilling st~ategy. By trying different composite 

strategies the petroleum industry ha~ come to 

settle on quite different strategies for the above 

two phases of exploration. 

Predrilling surveys, not being so 1nuch hindered by logistic limitations as drilling, can 

systematically co~er wide-ranging areas. The 

purpose of this predrilling phase i.s to find 

likely traps. A probability of trap recognition 

could be associ,ated with the outcom~ of each 

predrilling exploratory method. Whether the 

behavior of predrilHng surveys could be simulated by a random search is not clea.r. 

Explo~atory drilling, on the other 1' and, cannot roam freely over the prospect.ive area. 

Logistic considerations of the explo1·atory ac-
tivity itself constrain its freedom of movement. 

Moreover, logistic cons.ide~ations of the petroleum industry itself ·also modify and restrict 

the choice of the areas for more intensive exploratory drilling. As soon as a discovery is 

made, a rea.s,sessment has to be m:ade about the 

exploratory I development drilling strategy. Because of the above considerations, it would be 

uneconomic.al to assume a random exploratory 

drilling strategy. Explorntory drilling is focused on specific traps; as soon as a discovery 

is made, the drilling is concentrated on trap's 

that belong to the same geologic family, or what 

is called a play. While a systematic search is 

being made for a given type of trap, unforeseen 

conditions may be met which m~ay lead to a 

random disturbance, which in turn may trigger 

another .sequel of exploratory drilling. 

The result of this dual strategy can be gaged 

by a simple model. Let us assume that a number dn1 of exploratory holes ·are drilled after 

traps of type i; then the number dT of traps 

discovered is 

ASj dT=aidni+~(l-aj) (Nj-Tj)--dnj, (22) 

j s 

where 

ai=probability of correct recognition of 

trap type i, 

aj=probability of correct recognition of 

trap type j, 

ASj=average area of trap type j, 

S=total search area, 

dnj=number of exploTatory holes drilled 

after traps of type j, 

Nj=total number of traps type j, other than 

i, which exist in the regions, and 

Tj=Cumulative number of traps of type j 

discovered. 

If n1 and nj are independent variables, then 

the solution of equation (22) is 

ASj(l-aj) 

T=a1n1+l[Nj-k exp {- nj}]+c, 

j s 

(23) 

where c and k are constants of integration. 

The above formula reveals the two elements 

of the explomtory drilling outcome namely, a 

hit with constant probability a1 represented by 

the first term on the right side, plus a term 

with a negative exponential represented by the 

second term. 

If we were to focus only on the rate of discovery of traps type j, accidentall~r while running after type i, one gets 

dT kAS(l-a) AS(l-a)n 

=----exp { }. (24) 

dnj S S 

H. W. Menard and George Sharrr.an (written 

com.mun., 1975) have proposed a model of 

search ·which in effect takes into ~<·eount only 

the second term of 'equation (22). ~hat is, they 

assume a purely random search. 

SKETCH OF A PROBABILISTIC MODEL OF 

PETROLEUM-RESOURCE ASSESSMENT 

The petroleum-prospective regjons of the 

world consist of a ·set of sedimerta.ry basins 

{Bi}. Each basin may be segmented in geologic 

compartments {Vu}, characterized by a homogeneous probability space as to ,size and location 

of 'a given type of petroleum trnp TLj· The Vi,J 

may overlap in part with each other, and the 

union of the { V1,j} may be sm,aller than the 

space occupied by B1. 

The exploration strategy aims at. identifying 

these geologic compartments, wll ich provide 

lanes for further discoveries. The probability of 

discovery of a field is enhanoed whr"l the definition of a geologic compartment is established. 

Exploration then can move to a l'igher probability p·ath. 

Some of the ·statistical distribution functions 

of the T1,j may be correlated or fun~tionally dependent on those of other Ti,kJ brt many are 

totally independent. Therefore, tll e statistical 

properties of some Ti,j may be inferred statisr 

tically from thos.e of known Ti,j, but there are 

others for which the known Ti,j will provide no 

information. 

For .a given Bb the domains of some of the 

vi,j may have been defined in pa~ or in total 

by the exploration effort. That is, an element 

vu of Vi,j would be known. The ultimate recoveflable resources 1·i,j from vi.j 1nay be estimated from field-production data, success ratios, and even logistic or Gompertz type of 

projections. 

17 

For those already identified Vu having a 

known element vi.j, an estimate of the recoverable resources r 1,j in Vi,j may be made : 

1. By sealing 1·i.j in terms of the significant 

dimensional parameters of Vi.j and Vi.j, Or 
2. By an econometric analysis based on the dimensional parameters, which would allow 

expression of the results in terms of their 

statistical significance, or 

3. By formulating a field of probability occurrences over Vi,j based on an extension of 

what may be known in vu. 

For those Vi,j that are already identified but 

that have no known element, an estim,ate of the 

recoverable resources would have to be made 

by comparison with other Vi.j and by the introduction of a subjective judgment of the field of 

trap occurrences throughout Vi,j· 

For the remaining parts of Bi where no Vi,j 

have been identified a broader range of estimates would have to be made based on more 

subjective judgments as to the likelihood of 

18 

occurrence of fields and the statistical properties of their ,size and locational parameters. 

REFERENCES CITED 

Hubbert, M. K., 1969, Energy resources, Chap. 8 of 

Resources and man: San Francisco, W. H. Freeman, 

and Co., p. 157-242. 

McKelvey, V. E., 1972, Mineral resource ertimates and 

public policy: Am. Scientist, v. 60, no. 1, p. 32-40. 

National Petroleum Council, 1973, U.S. energy outlookOil and gas availability: [Washington, D.C.], 768 p. 

Moore, d. L., 1965, Analysis and projection of the historic pattern of supply of exhaustible natural resources: Operations Research Soc. America, 27th 

Natl. Mtg., Boston, Mass., 29 p. 

U.S. Federal Energy Administration, 197 4, Oil: Possible 

levels of future production-Project Independence 

Blueprint, Final, Task Force Report: Washington, 

D.C., U.S. Govt. Print. Off.~ variously paged. 

i:r U.S. GOVERNMENT PRINTING OFFICE: 1975 0-211-317/1