Petroleum-Resource Appraisal and

n

• An Application 

At «•-

to th€ JDenver basin

• Mathematieal Foundations

• An Application to Supply Modeling JL A: JL'. JL, f tj

GEOLOGIGAL PjROFE^SiONAI, PAFER U$8-A,
Petroleum-Resource Appraisal and 

Discovery Rate Forecasting 

in Partially Explored Regions

An Application to the Denver basin

By L. J. DREW, J. H. SCHUENEMEYER, and D. H. ROOT

Mathematical Foundations

By D. H. ROOT and J. H. SCHUENEMEYER

An Application to Supply Modeling

By E. D. ATTANASI, L. J. DREW, and]. H. SCHUENEMEYER

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1138-A, B, C

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON, B.C. : 1980
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

H. William Menard, Director

Library of Congress Cataloging in Publication Data

Petroleum resource appraisal and discovery rate forecasting in partially explored regions

Geological Survey Professional Paper 1138

CONTENTS: A. Drew, L. J., Schuenemeyer, J. H., and Root, D. H., An application to the Denver basin 

B. Root, D. H., and, Schuenemeyer, J. H., Mathematical foundations

C. Attanasi, B. E. D., Drew, L. J., and Schuenemeyer, J. H., An application to supply modeling 

Supt. of Docs. No.: I 19.16:1138

1. Prospecting-Denver Basin, Colo. and Wyo.- Mathematical models. 2. Petroleum-Denver Basin, Colo. and 

Wyo.-Mathematical models. I. Drew, Lawrence J. II. Series: United States. Geological Survey. 

Professional Paper 1138

TN271.P4P47 622M8282'09788 80-607018

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

Washington, D.C. 20402
Petroleum-Resource Appraisal and

Discovery Rate Forecasting

in Partially Explored Regions—

An Application to the Denver basin

By L. J. DREW, J. H. SCHUENEMEYER, and D. H. ROOT

GEOLOGICAL SURVEY PROFESSIONAL PAPER 113 8-A

A model of the discovery process that can be used to predict 

the size distribution of future petroleum discoveries 

in partially explored basins

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON, D.C. : 1980

CONTENTS

	Page

Abstract ________________________________________________ Al

Introduction______________________________________________ 1

Area of influence of a drill hole____________________________________ 1

Estimation of effective basin area________________________________ 2

Concept of discovery efficiency __ _____ ___ _ ____ __ ___ ____ _ ___ _ __ __ 5

Estimation of discovery efficiencies in the Denver basin ___ _ _____ ___ _____ _ 5

Predicting future discoveries within a size class __________________________ 7

Predicting future discoveries across size classes _______________________ 8

Discussion _________________________________________________ 10

References cited _________________________________________________ 11

ILLUSTRATIONS

Page

FIGURES 1-11. Graphs showing:

1. Average area exhausted per well (including both development and exploratory) for different target sizes and

number of wells _ _ _ _ _ _____ ___ _ _ _ _ ___ __ ____ __ __ __ ___ __ _ __ _ __—__ — — A2

2. Total area exhausted versus target size for selected numbers of wells ______ ———— —— ——— — __ — — 3

3. Area exhausted and effective basin size versus cumulative wells for 4.3-mi2 targets _______ ——— — _____ 4

4. Cumulative fraction of targets discovered versus fraction of basin exhausted for two different hypothesized

efficiencies __________________________________________ ——— ____ ———— -__ 5

5. Possible anomalous discovery rate that initially appears to have a discovery efficiency less than 1__ —— _— 6

6. Computed discovery efficiencies for classes 9 through 15 for years 1963-69 ____—__ — _ —— ——— -- — 6

7. Cumulative discoveries versus area exhausted for class size 9 (256,000 to 512,000 bbl) ____———__ ——— _ 6

8. Cumulative discoveries versus area exhausted for class size 10 (512,000 to 1,024,000 bbl)__ ————— ———— 7

9. Cumulative discoveries versus area exhausted for class size 12 (2,048,000 to 4,096,000 bbl) ________________ 8

10. Cumulative discoveries versus area exhausted for class size 13 (4,096,000 to 8,192,000 bbl) ——______ — __ 9

11. Seventeen successive predictions of the total volume of petroleum to have been discovered by the end of 1969 in

classes 9 through 15 ____________________________________________ — _———____ —— ___ 9

12. Histogram comparing actual versus predicted petroleum discovered within size classes 9 through 15 for discoveries

made between January 1, 1955, and December 31, 1969, based on data to December 31, 1954_________ —— _____ 10

13. Graph showing 22 successive predictions of the total volume of petroleum to have been discovered by the end of 1974 in

classes 9 through 15 _____________________________________________ ————— ______—— 10

14. Histogram comparing actual versus predicted petroleum discovered within size classes 9 through 15 for discoveries

made between January 1, 1956, and December 31, 1974, based on data to December 31, 1955__ —— _ —— _ —— __ 10

TABLES

Page

TABLE 1. Conversion factors used in computation for development well conversions _ _ ____ _ _ _ _ _ _ _ _ _ _—— — — — - A3

2. Expected number of discoveries per 1,000 square miles searched for efficiencies of 1 and 2.5 ______———__ — _— 5

3. Sample calculations for future discoveries in size class 12 _____________________________________—___ 8

PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE

FORECASTING IN PARTIALLY EXPLORED REGIONS 

AN APPLICATION TO THE DENVER BASIN

By L. J. DREW, J. H. SCHUENEMEYER, and D. H. ROOT

ABSTRACT

A model of the discovery process can be used to predict the size 

distribution of future petroleum discoveries in partially explored 

basins. The parameters of the model are estimated directly from the 

historical drilling record, rather than being determined by assumptions or analogies. The model is based on the concept of the area of 

influence of a drill hole, which states that the area of a basin 

exhausted by a drill hole varies with the size and shape of targets in 

the basin and with the density of previously drilled wells. It also uses 

the concept of discovery efficiency, which measures the rate of discovery within several classes of deposit size. The model was tested 

using 25 years of historical exploration data (1949-74) from the 

Denver basin. Prom the trend in the discovery rate (the number of 

discoveries per unit area exhausted), the discovery efficiencies in 

each class of deposit size were estimated. Using pre-1956 discovery 

and drilling data, the model accurately predicted the size distribution of discoveries for the 1956-74 period.

INTRODUCTION

Before the cost of petroleum obtained from future 

discoveries can be determined, a model of the discovery 

process must be used to predict not only the total volume of petroleum to be discovered with any given level 

of exploratory drilling but also the size distribution of 

these discoveries. This detailed information is required 

because, as the discovery process proceeds, the incremental size distribution of discoveries progressively 

contains a higher percentage of smaller deposits. Cost 

rises not only because progressively smaller volumes of 

petroleum are found per unit of exploratory effort but 

also because smaller deposits usually have higher per 

unit development costs.

The purpose of this investigation was to develop a 

model of the discovery process that will predict the incremental size distribution of future discoveries in any 

partially explored region as a function of exploratory 

drilling effort. In the construction of this model, a 

structure was specified whose parameters can be estimated directly from historical drilling and discovery 

data rather than by subjective judgment or the use of 

an analogy. The number of discoveries predicted in 

each size class is determined by the value of two parameters. The first parameter, which is commonly estimated for all size classes, is the effective basin area, 

which is that region within which drillers are willing 

to site wells. The second parameter, which is uniquely 

estimated for each size class, is the discovery efficiency 

parameter. This parameter measures the effectiveness 

of exploratory drilling relative to the outcome of a random search process. The higher the value of the discovery efficiency parameter in any given size class, the 

earlier most of the deposits in that class will be discovered.

The model of the discovery process described in this 

report is related to the model proposed by Arps and 

Roberts (1958) but differs in several significant aspects. First, the discovery efficiencies are estimated 

within each deposit size class rather than assumed to 

be a single efficiency for all size classes as determined 

by subjective judgment. Second, the effective basin 

area is estimated from discovery time series data, in 

contrast to the approach taken by Arps and Roberts in 

which the boundaries of the region to be explored were 

selected by expert judgment.

In the model used here, it was possible to estimate 

these parameters directly from discovery time series 

data by using the concept of the area of influence of a 

drill hole (Singer and Drew, 1976). Using this approach, it was possible to predict accurately the 

number of deposits to be discovered within a group of 

size classes in the Denver basin from only a small 

amount of historical data. The Denver basin was chosen to test this model because it contains only one 

major producing interval (the D and J members of the 

Cretaceous Dakota Sandstone), and so a reliable data 

base could be constructed within a reasonable period of 

time (Schuenemeyer and Drew, 1977b).

AREA OF INFLUENCE OF A DRILL HOLE

The area exhausted by a drill hole varies with the 

size of the targets that can occur in the basin. A basin 

may be fully explored with respect to large targets but

Al
A2 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

only 10 percent explored with respect to small targets. 

The shape of the targets also has an effect; for example, 

a pattern of exploratory and development holes may 

have fully searched a basin for circular targets having 

areas of 2 mi2 , although irregularly shaped targets also 

having an area of 2 mi2 are still undiscovered.

The amount of area exhausted by any given hole can 

range from a maximum of the area of the target being 

considered when the areas of influence of any previously drilled holes do not overlap to a minimum of 

zero when all points within the area of influence of the 

hole have been exhausted by previously drilled holes. 

Singer and Drew (1976) and Root and Schuenemeyer 

(1980) discuss this concept in detail, and Singer (1976) 

and Schuenemeyer and Drew (1977a) describe the 

computations. The degree to which the Cretaceous 

Dakota Sandstone interval in the Denver basin was 

physically exhausted by the end of each year within 

the period 1949-74 was calculated for targets up to 4.3 

mi2 in area (fig. 1). The levels of exhaustion shown are 

for elliptical targets with an axial ratio of 0.38, the 

mean axial ratio of deposits discovered during 1949- 

74. For example, the 1,096 wells drilled through the 

end of 1952 exhausted on the average 0.545 mi2 per 

well with respect to targets 1.09 mi2 in area. By the end 

of 1954, this average declined 12.4 percent to 0.478 mi2 

per well. This decline in the effective exhaustiveness of 

wells is a consequence of the progressive crowding of 

wells. For larger targets this crowding is even more 

severe. For example, for targets 4.3 mi2 in area, the 

corresponding area exhausted per well declines by 26 

percent from 1.71 to 1.26 mi2 per well during 1952-54.

In 1969 the Union Pacific Railroad released for exploration approximately 4,400 mi2 of virtually unexplored land in the basin (Oil and Gas Journal, 1969). 

The release and exploration of this large block of land 

during subsequent years are reflected in figure 1 by a 

slight increase in the level of the 1974 curve over the 

1968 curve. This increase is a result of the wider spacing of holes, which normally occurs during the early 

phase of exploration of any region.

ESTIMATION OF EFFECTIVE BASIN AREA

The effective basin area is defined as that part of a 

total basin within which drillers are willing to site 

wells. Given this definition, there is no reason to assume that the area searched will have any particular 

shape. One basin may be more or less evenly explored 

from rim to rim. In others, many wells may be clustered in small parts of the total basin area. In such 

basins, the effective basin area may not even be a con01 2345 

TARGET SIZE, IN SQUARE MILES

FIGURE 1.—Average area exhausted per well (including both development and exploratory) for different target sizes and number of 

wells.

nected region. 1 Using the concept of area of influence of 

a drill hole to measure the crowding of exploratory and 

development drilling, a method has been developed to 

estimate the effective basin area.

As a simplified example of this method, suppose that 

an irregular area is marked off on the floor, and then 

disks are placed at random inside the area so that they 

have as much chance of landing one place as another 

and they may overlap. After several disks have been 

placed, the boundaries of the area are erased, but the 

disks are left on the floor. The problem then is to calculate the area, B , inside the boundaries that have now 

been erased. Suppose the disks each have area "a." In 

the current analysis, the centers of the disks are the 

locations of the wells. The disk area is taken to be the

'A region is connected if and only if any two points in the region can be joined by a curve 

line that does not go outside of the region (region is in one piece).
AN APPLICATION TO THE DENVER BASIN A3

area of the largest target under consideration, and B is 

the effective basin area. The average target area corresponding to class 15, the largest deposit size under 

consideration, is 4.3 mi2 . Let the function A(n) measure the area covered by the first n disks. The value of 

this function will then be A(n)=na if none of the disks 

overlaps and A(n) < na when they do. Assume, furthermore, that the disks are small compared to B , the 

effective basin area. It can then be shown that even 

though A (n ) is a random variable, it grows in a regular 

fashion as n increases (Root and Schuenemeyer, 1980). 

On the average A(n} grows according to a negative 

exponential:

A(n) = B(l-e~na/B ). (1)

If an area A (n ) has already been exhausted, then the 

additional area that will be exhausted by m additional 

wells sited at random within B is given by

A(n+m)-A(n)=(B-A(n))(l-e-ma/B ). (2)

A proof of these identities is provided by Root and 

Schuenemeyer (1980).

In general, any arbitrary shape having the same 

area can be used without changing the rate of growth 

of A(n). In the following calculations, the targets are 

assumed to be elliptical. Given the area exhausted by 

the exploratory and development wells drilled through 

time (fig. 2), an estimate of the effective basin area for 

the largest class can be obtained in the case for which 

all wells are sited at random by solving equation 1 for 

B. In practice, however, development wells are not 

drilled at random but instead in a closely spaced regular pattern. In the Denver basin, most development 

wells have been drilled one-quarter mile apart. As a 

consequence, a development well cannot have the same 

average capacity to exhaust a region as an exploratory 

well. We therefore treat a development well as a fraction of an exploratory well. At any time, then, the 

equivalent number of exploratory wells that have been 

drilled is specified by

n = W+Dd(a) (3)

where

n = equivalent number of exploratory wells, 

W = number of true exploratory wells, 

D = number of development wells, and 

d(a) = conversion factor for development wells. 

a = target size

The conversion factor, d(a), is a function of the target 

size, the target shape, and the spacing of development 

wells. The value of this factor for very small targets is

0 .5 1 2 3 4 5 6 

TARGET SIZE, IN SQUARE MILES

FIGURE 2.—Total area exhausted versus target size for selected 

numbers of wells.

equal to one, and, as the target size increases, d(a) 

decreases toward zero. If the spacing between holes is 

one-quarter mile and the axial ratio is 0.38, then 

targets with an area not greater than 0.0187 mi2 will 

have d(a)-l. This factor is chosen so that two targetlike ellipses having the same orientation and centers 

one-quarter mile apart will have an area of overlap, 

averaged over all orientations of (l-d(a))a. Conversion factors for various sizes of ellipses with axial ratio 

0.38 are presented in table 1.

TABLE 1.—Conversion factors used in computation for development 

well conversions

Class

9

10 _

11 _ __ __ _

12

13 __________

14 ___ ___ _

15 __

Size (103 barrels)

__________ 256- 512

_ _ __ 512- 1,024

_ __ _ 1,024- 2,048

_ _ _ 2,048- 4,096

_ ___ 4,096- 8,192

_ _ __ 8,192-16,384

... _ ____ 16,384-32,768

Conversion 

factor d (a)

0.54

.45

.37

.31

.25

.23 -

.16

Average 

producing area

(mi2 )

0.339

.513

.753

1.091

1.670

2.070

4.300
A4 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

Given the number and location of exploratory and 

development holes drilled, we can now solve for the 

effective basin area using equation 4,

A(n) = (4)

For estimating the effective basin area, h is computed 

using d(a=4.3)=0.16, the conversion factor for the 

largest target size.

If an area A (h ) has already been exhausted within 

an effective basin area B, then the additional area to be 

exhausted with respect to a target of area a by an additional number of equivalent exploratory wells m is

-A(h)=(B-A(h))(l-e-"la'B ). (5)

The estimate ra is obtained in a manner similar to that 

of h, except that the future ratio of the number of development holes to exploration holes must be assumed. 

We assumed that the future ratio would be the same as 

the past ratio.

With the results of the calculations from equation 4,

both the actual area exhausted and the estimated effective basin area can be graphed as a function of 

cumulative wells drilled (fig. 3). The estimated effective basin area increased rapidly through the drilling 

of the first 9,000 wells (1949-1957). This rapid rate of 

growth implies that the drillers rapidly expanded their 

views of where deposits could be discovered in the 

basin. During the following 10 years, when an additional 8,300 wells were drilled, the effective area of 

the basin grew very little, which implies that exploration during this period was being confined, for the most 

part, to the region established by 1957. From 1969 

through 1973, the effective basin area again began to 

grow more rapidly. This growth was caused by the release in 1969 of approximately 4,400 mi2 in the Denver 

basin that had been held by the Union Pacific Railroad. 

The crowding that resulted in the change in slope in 

1957 recurred in 1973—implying that the area to be 

searched was beginning to be reestablished. It is 

reasonable to assume that, had such a large region not 

been withheld from exploration, the effective basin 

area curves would have shown only one change in 

slope, from growth to stabilization.

WELLS, IN THOUSANDS

FIGURE 3.—Area exhausted and effective basin size versus cumulative wells for 4.3-mi2 targets.
AN APPLICATION TO THE DENVER BASIN A5

CONCEPT OF DISCOVERY EFFICIENCY

Historical exploration records show that as exploration proceeds, the rate of discovery declines; that is, 

exploratory drilling begins in the more productive 

parts of a basin and later moves to the less productive 

parts of that basin. The concept of discovery efficiency 

is used to measure this phenomenon. A discovery efficiency of 1 is used as a reference point that defines the 

outcomes of a random search process. In such a search, 

the number of deposits discovered in each size class per 

unit area exhausted remains constant throughout the 

total exploration history of one region. For example, if 

the ultimate effective basin area were 10,000 mi2 and 

100 deposits existed in a certain size class at the start 

of the search process, we would expect 10 deposits to be 

discovered with the exhaustion of each 1,000 mi2 of the 

region.

If the exploration process were better than random, 

we would expect a larger number of deposits to be 

found during the earlier stages of exploration. For 

example, if the discovery efficiency were 2.5 in the preceding example, the exhaustion of the first 1,000 mi2 

would produce 23.2 discoveries versus 10 discoveries 

for the random search. The number of discoveries expected to be made in this example for each 1,000 mi2 

exhausted is shown in table 2. These values were computed from the following discovery process model:

(6)

where

f = the fraction of the deposits that have been 

found,

A — area that has been exhausted,

B = effective basin area, and

c = the discovery efficiency.

A graph of the cumulative number of discoveries as a 

function of area exhausted for two different efficiencies 

is schematically diagrammed in figure 4.

TABLE 2.—Expected number of discoveries per 1,000 square miles 

searched for efficiencies of 1 and 2.5

Area searched 

(mi

Expected number of discoveries 

per 1,000 mi2 searched

1,000 __________________

2,000 __ ___ __ _ ____.

3,000 _ _ ______________

4,000 __ _ — _ _ _____

5,000 ____________ _ __.

6,000 __________________

7,000 __________________

8,000 ——— _ _ ________

9,000 _________________

10,000 ______ ___ _____

Efficiency = 1

— — __ — - - — 10

_____________ _ 10

— - __ ___________10

— - _ ___________10

__________ _ __ _ _10

_ _ ____ __ _______10

————— _ ————— 10

— __ ____ ___ ___10

10

— ___ — _ -—10

Efficiency = 2.5

23.2

19.6

16.2

13.1

10.2

7.6

5.2

3.1

1.5

.3

1.0 r

0.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

FRACTION OF BASIN EXHAUSTED

0.9

FIGURE 4.—Theoretical graph of cumulative fraction of targets discovered versus fraction of basin exhausted for two different 

efficiencies.

While it is theoretically possible for a discovery efficiency to be less than 1, this value is of no relevance 

because it implies a steadily increasing discovery rate 

through the end of the exploration process; in the long 

run, drilling would terminate when the discovery rate 

reaches its peak! In the short run, however, a sudden 

geologic insight or chance fluctuation could cause the 

discovery rate to increase for a period of time (fig. 5). If 

the method for computing efficiencies described below 

results in an efficiency of less than 1, the efficiency is 

set equal to 1.

ESTIMATION OF DISCOVERY EFFICIENCIES IN 

THE DENVER BASIN

From the trend in the discovery rate (the number of 

discoveries per unit area exhausted), the efficiencies in 

each class of deposit size were estimated. In the estimation of the efficiencies it is necessary to choose a basin 

size. The assumption that the entire basin is available 

for drilling ignores two effects; namely, that exploration tends to follow past discoveries, and drillers tend 

to explore the shallower parts of a formation before the 

deeper. These effects combine to produce a greater 

crowding of wells than would occur with random drilling. This greater crowding results in an underestimation of the area that will ultimately be explored.

We have found a way to adjust the model that compensates for the fact that the area being considered for 

exploration grows through time. For estimating the
A6 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

AREA EXHAUSTED

FIGURE 5.—A possible anomalous discovery rate that initially appears to have a discovery efficiency less than 1.

efficiency of exploration for fields of each size class, the 

basin size used was the basin size estimated for that 

size class. For estimating future discoveries, the basin 

size used was that estimated for the largest size class. 

The theoretical description of the search of an expanding area is still under investigation.

Efficiencies were estimated annually2 for the years 

1952-69. Seven efficiency estimates (1963-69) for 

seven deposit size classes are given in figure 6. No 

trend with time was found within each series of estimates. No estimate was made for class 16 (32.7-65.5 

million barrels) because only one deposit was discovered in this, the largest class, and an estimate would be 

statistically meaningless. Efficiency increases as deposit size increases (fig. 6). Efficiencies less than 1 occurred in classes 9, 10, and 11, but for the purpose of 

estimating future discoveries, it was assumed that in 

the long run no efficiency can be less than 1. If the 

estimated efficiency was less than 1, then for purposes 

of prediction, it was set equal to 1.

The relation between the discovery efficiency and the 

shape of the cumulative discovery curve can be seen by 

comparing the efficiencies estimated in figure 6 with 

the discovery curves shown in figures 7-10. For example, the discovery efficiency within class 12 was estimated to be approximately 2.6, and the discovery curve 

has a markedly decreasing slope (fig. 9). An efficiency 

of this magnitude implies that when half the effective

'The details of the calculation procedure are presented in Root and Schuenemeyer (1980).

10 11 12 13 

DEPOSIT SIZE CLASS

14 15

FIGURE 6.—Computed discovery efficiencies for classes 9 through 

15 for years 1963-69.

110

100

90

80

CO

UJ

oc ui

g 7°

O

W 60

O

u. 50

O

£ 40

CO

5 30

D

Z 20

10

I I I I I I

0 640 1280 1920 2560 3200 3840 4480 

AREA EXHAUSTED, IN SQUARE MILES

FIGURE 7.—Cumulative discoveries versus area exhausted for class 

size 9 (256,000 to 512,000 bbl).

basin area has been exhausted, 84 percent of the deposits containing between 2.048 and 4.096 million 

barrels of recoverable petroleum will have been found. 

The efficiency of discovery for the deposits in class 9 

(0.256 to 0.512 million barrels) was approximately 1.4 

(fig. 7). This value means that deposits of this size were 

found at nearly a uniform rate through 1969, as the 

nearly straight cumulative discovery curve for this 

class of deposits shows (fig. 7). The discovery efficiency 

for the deposits in class 10 (0.512 to 1.024 million 

barrels) was estimated to be less than 1.0 (fig. 8) be-
AN APPLICATION TO THE DENVER BASIN A7

100

90 h

80

70

QL

§J 60

O

u 

w

O 50

u. 

O

(C

iu 40

00

5

30

20

10

600 1200 1800 2400 3000

AREA, IN SQUARE MILES

3600 4200 4800 5400

FIGURE 8.—Cumulative discoveries versus area exhausted for class size 10 (512,000 to 1,024,000 bbl).

cause the discovery rate increased rapidly during the 

exhaustion segment from 1,500 to 3,000 mi2 . As we 

have previously discussed, this phenomenon can be 

only temporary, and, as shown in figure 8, the discovery rate declined during the second half of the exhaustion sequence. No obvious geologic cause could be isolated to explain this temporary increase in the rate of 

discoveries.

PREDICTING FUTURE DISCOVERIES WITHIN A 

SIZE CLASS

The model of the discovery process shown in equation 6 was used to predict the ultimate number of deposits to be discovered and the number of deposits to be 

discovered for a given area exhausted. Each of these 

predictions is made within a deposit size class. The 

predictions made for some later segment of exploration 

are designed to answer the question: How many deposits may one expect to find by drilling a certain number 

of additional wells, given the previous exploration history?

In order to validate the model given in equation 6, 

resource appraisal estimates have been made within 

each deposit size class from the discovery time series 

for various years beginning in 1952. The last year of 

historical data used in the prediction is called the prediction year. The number of discoveries estimated from 

the end of the prediction year until 1969 and until 1974 

is compared with the actual number.

Before a forecast can be made, the following estimates are calculated: (1) the ultimate number of deposits within each size class; (2) the area that will be physically exhausted by the number of wells actually 

drilled between the time the forecast is to be made and 

the end of 1969 and 1974; and (3) the discovery efficiency.

The number of deposits estimated to have existed 

originally in each size class is obtained from equation 

6. Given the fraction of the effective basin area still 

remaining to be explored and the corresponding discovery efficiency for each class, the fraction of deposits 

remaining to be discovered is (1 —A/5 Y. The number of 

deposits then estimated to have occurred originally in 

each size class is obtained by dividing the number already discovered in a class by 1 — (1— A/BY. From this 

calculation we then can construct a resource appraisal 

for the basin for any segment of the exploration data 

time series.

The estimate of the basin area for class 15 is obtained from equation 4. Once the basin area has been 

determined, we then assume that the ratio of exploratory to development wells remains constant. Then 

we can calculate from equation 5 the additional area to
A8 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

44

40

36

32

to

UJ

E 28

UJ

>

O

0 to24 •**

IL

O 20

tc

UJ

o

I 16

z

12

1000 2000 3000 4000 5000 6000 7000

AREA EXHAUSTED, IN SQUARE MILES 

FIGURE 9.—Cumulative discoveries versus area exhausted for class size 12 (2,048,000 to 4,096,000 bbl).

8000 9000

be exhausted for a given number of additional wells. 

The basin size estimated for deposits of size class 15 is 

used in this calculation. Using these estimates, the 

number of discoveries expected within each size class 

from the end of the prediction year through 1969 and 

through 1974 are computed.

Sample calculations for future discoveries through 

1969 in deposit size class 12 are presented in table 3 for 

the prediction year 1957. In the computation of adjusted future wells, the number 2 appears as a divisor 

because the ratio of exploratory to development drilling has been found through time to be approximately 

equal to 1.

PREDICTION OF FUTURE DISCOVERIES ACROSS 

SIZE CLASSES

An estimate of the total volume of petroleum to be 

discovered was obtained by summing the future oil discovered over size classes 9 through 15 (deposits between 0.256 and 32.768 million barrels). This estimate 

was then compared with the volume actually discovered to test the validity of the prediction. In addition to 

estimating the total volume of petroleum to be discovered, it is necessary for purposes of economic analysis

TABLE 3.—Sample calculations for future discoveries in size class 12 

Basic Data

Description Symbol Value

Area exhausted to December 31, 1957 __ A (1957) 4,165 mi2 

Target area____________________________ a 1.09mi2

Estimated effective basin size __________S 10,624mi2

Correction factor for development wells __ d 0.31 

Discovery efficiency ____________________c 2.577

Number of discoveries

to December 31, 1957 __________ N 34 

Average volume of recoverable oil ____ 2.863xl06bbls 

Future wells from January 1, 1958,

to December 31, 1969 __________ 9,765

Calculations

„ 9765 + 9765(0.31) _ 00 _ Adjusted future wells = m = ————-——————- b,dyb

A(1969) = CB-A(1957))(l-e-";'</B ) + A(1957) = 7,273 mi2

(1957) Unexplored fraction of basin (1957) = 1 - A —— = 0.608

D

(1969} Unexplored fraction of basin (1969) = 1-A v " = 0.315

D

Undiscovered fraction of targets (1957) = (0.608)c = 0.277 

Undiscovered fraction of targets (1969) = (0.315)c = 0.051 

Fraction of targets discovered (1958-69) = 0.226

N Estimate of ultimate number of targets = = 47.0

J.—U.^ / /

Estimated number of discoveries (1958-69) = (47.0)x(0.226) = 10.6 

Future oil discovered (1958-69) = 10.6x2.863x!06 = 30.3 xlO6 bbls
AN APPLICATION TO THE DENVER BASIN A9

30

25

20

15

10

1000 2000 3000 4000 5000 6000 7000 8000

AREA EXHAUSTED, IN SQUARE IvHLES

9000 10,000

FIGURE 10.—Cumulative discoveries versus area exhausted for class size 13 (4,096,000 to 8,192,000 bbl).

to predict accurately the number of deposits to be discovered in each size class.

The prediction of the total oil to be discovered is the 

sum of seven quantities, each subject to independent 

random fluctuations that on the average tend to cancel 

each other; thus the total is a more stable quantity. On 

the average, estimated efficiencies tend to increase 

with deposit size class (fig. 6). Deviations from this 

trend are attributed to random fluctuations.

The prediction of the total amount of oil to be discovered by drilling 18,762 wells changes as the initial 

drilling and discovery sequence on which it is based 

changes (fig. 11). Thus, using only 5 years of data 

(1949-54), or 19 percent of the wells drilled through 

1969, the basin size and efficiencies are sufficiently 

well determined to permit accurate estimation of the 

oil to be discovered by the next 15,195 wells. In order to 

use this estimate for economic calculations, it is necessary to be able to predict the size distribution of deposits. Predicted and actual future discoveries agree 

closely across size classes using 1954 as the prediction 

year (fig. 12). This agreement is particularly good in 

the larger, more important size classes.

The calculations made through 1969 were repeated 

for the 22,577 wells drilled through 1974 (figs. 13 and 

14). During the 5-year period beginning in 1969, the

land released by the Union Pacific Railroad was explored. The late release of this much land might have 

caused the model slightly to underestimate the amount 

of oil to have been discovered through the end of 1974.

(/) DC QOC

Actual total discoveries

through 1969^

/-Predicted totel 

/ discoveries

Emulative discoveries

J_

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 

CUMULATIVE WELLS, IN THOUSANDS

20.0

FIGURE 11.—Seventeen successive predictions of the total volume of 

petroleum to have been discovered by the end of 1969 in classes 9 

through 15.
A10 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

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DEPOSIT CLASS SIZE

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FIGURE 12.—Histogram comparing actual versus predicted petroleum discovered within size classes 9 through 15 for discoveries 

made between January 1, 1955, and December 31, 1969, based on 

data to December 31, 1954.

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CUMULATIVE WELLS, IN THOUSANDS

FIGURE 13.—Twenty-two successive predictions of the total volume 

of petroleum to have been discovered by the end of 1974 in classes 9 

through 15.

DISCUSSION

In assessing the petroleum potential of an area as 

described above, no use was made of the geologic and 

geophysical knowledge acquired during the exploration of the area. Several reasons exist for not making 

direct use of such information. Geologic and geophysical data are difficult to assemble because they are scattered among many exploration companies and they are 

often confidential. Even if the data were gathered, a 

resource estimate for a partially explored area based 

upon them would have to be taken on faith because the 

data set would be too large and complex to communicate. Moreover, the reasoning by which one proceeds 

from geologic and geophysical knowledge of a basin to 

an estimate of the number and sizes of undiscovered

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125

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11 12 13 14 15

DEPOSIT SIZE CLASSES

FIGURE 14.—Histogram comparing actual versus predicted petroleum discovered within size classes 9 through 15 for discoveries 

made between January 1, 1956, and December 31, 1974, based on 

data to December 31, 1955.

fields and the number of wells required to find them is 

largely subjective and intuitive and hence difficult to 

expose.

Although direct use of geologic and geophysical information is not feasible, this information is contained 

implicitly in the record of drilling and discovery insofar 

as it affected the siting of wells. For example, in the 

Denver basin, the effective basin size is determined by 

geologists and is revealed through the siting of wells. 

The effective basin area was estimated here to be about 

10,000 mi2, or about one-fifth of the whole Denver 

basin. This area was deduced without directly examining the reports on prospects in the other fourth-fifths of 

the basin. The examination of those reports was left to 

those who were siting the exploratory holes.

From the discovery rate, one can estimate how well 

exploration prospects are evaluated. The decline in the 

discovery rate (quantity of oil found per unit area 

searched) gives a quantitative measure of the superiority of actual exploration over random drilling. This 

superiority, measured by the parameter "efficiency of 

exploration," shows to what extent the better prospects 

are drilled early. The efficiency of exploration in the 

Denver basin was found to be about two to four times 

better than that of random drilling in the search for 

targets in larger classes (2xl06bbl to 32xl06bbl) and 

to be about the same as that of random drilling in the 

search for small targets. Thus, both the area to be explored and the efficiency with which it is being explored can be deduced from the drilling and discovery
AN APPLICATION TO THE DENVER BASIN All

record. When the area to be searched and the efficiency 

of search are known, one can extrapolate the .past discovery record to obtain a reasonably accurate forecast 

of the number and sizes of oil fields that will be discovered by a given amount of future exploratory drilling. 

For the Denver basin, the discovery record of the first 

3,638 exploratory wells was extrapolated to provide 

forecasts of discoveries to be made by the next 7,929 

exploratory wells. The future discoveries were divided 

into six different size classes. In most of these size 

classes, close agreement was observed between the 

predicted and actual discoveries. Estimates of future 

oil discoveries are difficult to make in the larger size 

classes because of the paucity of data. However, at the 

time in the discovery sequence when sufficient data are 

available to estimate future oil discoveries, most of the 

large deposits have been discovered, and the interest is 

in predicting the number of moderate size deposits remaining to be discovered.

REFERENCES CITED

Arps, J. J., and Roberts, T. G., 1958, Economics of drilling for Cretaceous oil and east flank of Denver-Julesburg basin: American 

Association of Petroleum Geologists Bulletin, v. 42, no. 11, p. 

2549-2566.

Oil and Gas Journal, 1969, Union Pacific, Pan Am enter exploration 

pact in Rockies: Oil and Gas Journal v. 67, no. 35, p. 86.

Root, D. H., and Schuenemeyer, J., 1980, Petroleum-resource appraisal and discovery rate forecasting in partially explored 

regions—Mathematical foundations: U.S. Geological Survey 

Professional Paper 1138-B, p. B1-B9.

Schuenemeyer, J. H., and Drew, L. J., 1977a, An exploratory drilling 

exhaustion plot program: Computers and Geosciences, v. 3, no. 4, 

p. 617-631.

———— 1977b, Data reduction and analysis for a petroleum exploration model (abs.): American Association of Petroleum Geologists 

Conference, Exploration Data Synthesis, Tucson, March 1977, p. 

1.

Singer, D. A., 1976, Resin—A Fortran IV program for determining 

the area of influence of samples or drill holes in resource target 

search: Computers and Geosciences, v. 2, no. 2, p. 249-260.

Singer, D. A., and Drew, L. J., 1976, The area of influence of an 

exploratory hole: Economic Geology, v. 71, no. 3, p. 642-647.

Petroleum-Resource Appraisal and 

Discovery Rate Forecasting

in Partially Explored Regions— 

Mathematical Foundations

By D. H. ROOT and J. H. SCHUENEMEYER

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1 138-B

The basic mathematical properties of a model 

of the discovery process used to predict 

the size distribution of future petroleum 

deposits in a partially explored region

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON, D.C. : 1980

CONTENTS

Page

Abstract _________________________________________________ Bl

Introduction._____________________________________________ 1

Estimation of basin size ________________________________________ 2

Practical difficulties and modifications ______________________________ 4

Efficiency of exploration __________________________________________ 5

Results of simulation __________________________________________ 7

References cited _________________________________________ 9

ILLUSTRATION

Page

FIGURE 1. Estimates of effective basin size from simulated data____________________________________ B8

TABLE

Page

TABLE 1. Estimated median efficiencies _____________________________________________________ B7

III

PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE 

FORECASTING IN PARTIALLY EXPLORED REGIONSMATHEMATICAL FOUNDATIONS

By D. H. ROOT and J. H. SCHUENEMEYER

ABSTRACT

A stochastic model of the discovery process has been developed to 

predict, using past drilling and discovery data, the distribution of 

future petroleum deposits in partially explored basins, and the basic 

mathematical properties of the model have been established. The 

model has two exogenous parameters, the efficiency of exploration 

and the effective basin size. The first parameter is the ratio of the 

probability that an actual exploratory well will make a discovery to 

the probability that a randomly sited well will make a discovery. The 

second parameter, the effective basin size, is the area of that part of 

the basin in which drillers are willing to site wells. Methods for 

estimating these parameters from locations of past wells and from 

the sizes and locations of past discoveries were derived, and the properties of estimators of the parameters were studied by simulation.

INTRODUCTION

This paper presents the mathematical derivations 

for the estimators of the parameters used in a model of 

the discovery process proposed by Drew, Schuenemeyer, and Root (1980). The model was designed to 

summarize the past exploration and discovery experience in a region in order to predict the quantity of 

recoverable oil that future exploration could be expected to find. The model was developed because a 

significant amount of the petroleum remaining to be 

discovered in the United States and elsewhere is in 

regions where there has already been some exploratory 

drilling.

During exploration, a large amount of data is generated, including seismic surveys, magnetic surveys, 

geologic maps, well logs, discovery sizes and locations, 

and drilling dates and locations of discovery wells and 

dry holes. Because it is not feasible to use all the available data, a subset must be selected. The data incorporated in the model were the dates and locations of past 

exploratory wells together with the locations of discovery wells and the sizes of any deposits that were found. 

The geologic and geophysical data are thus implicitly 

included in that they guide the drilling and discovery 

process.

If exploration were completely random, then the 

probability of a given oil field being found by a single 

exploration well is the ratio of the area of the field to 

the area being searched. A model developed by Arps 

and Roberts (1958) assumes that the probability of an 

exploratory well discovering a given oil field is the 

product of a constant, the efficiency of exploration, and 

the ratio of the area of the field to the search area. In 

that model the measure of exploration was the number 

of exploratory wells. The model proposed by Drew, 

Schuenemeyer, and Root (1980) is similar to the model 

of Arps and Roberts but uses a different measure of 

past exploratory effort. Rather than the number of exploratory wells, it uses the cumulative area exhausted 

by these wells (Singer and Drew, 1976). The area 

exhausted by a well is related to the area and shape of 

the deposits under consideration. For example, for circular deposits having a radius of 1 mi, a dry exploratory well tells the explorer that the center of such a 

target cannot lie within 1 mi of the well and 3.14 mi2 

have been exhausted so far as such deposits are concerned. A pattern of dry wells will exhaust, with respect to such targets, an area equal to the area covered 

by all circles of radius 1 mi centered at each well. Because these circles may overlap, the exhausted area 

may be less than the number of wells times the area of 

a circle. If the deposits are elliptical having a major 

axis of, say, 4 mi and a minor axis of 2 mi, then a dry 

well exhausts all of the area within I mi of the well, 

none of the area more than 2 mi away, and partially 

exhausts the area between 1 and 2 mi from the well. A 

deposit centered within 1 mi from the well would have 

been hit regardless of its orientation, assuming certainty of recognition. If the center of the deposit is between 1 and 2 mi from the exploratory well, the deposit 

would have been hit for some orientations but not for 

others. The degree to which a point is exhausted is the 

probability that a randomly oriented deposit centered 

at the given point was hit. This concept of area exhausted can be used as a measure of the extent of exploration for targets of any given size and shape.

Bl
B2 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

When an exploratory well is successful, then any 

point within 2 mi of the discovered deposit would have 

been partially exhausted for elliptical targets 4 mi by 2 

mi. The degree of exhaustion at a point is the probability that a randomly oriented 4-mi by 2-mi ellipse 

centered at the point would have intersected the discovered deposit. This example illustrates that the area 

exhausted is a function of the size and shape of the 

deposits being considered and that more area is 

exhausted for larger deposits than for smaller ones. In 

the analysis of the discovery data, deposits are divided 

into size classes. All deposits within a given size class 

have approximately the same area, and each of these 

classes is analyzed independently. The following discussion focuses on elliptical deposits of the same size 

and shape.

If exploratory wells are randomly sited, then the petroleum richness of the area exhausted remains approximately equal to the richness of the unexplored 

area so that the fraction of all the deposits that are still 

undiscovered remains approximately equal to the fraction of the search area that has not been exhausted. If 

the exploration is better than random, then the petroleum richness of the area exhausted is greater than 

that of the area remaining to be searched.

In order to predict future discoveries, it is necessary 

to know the ultimate area of exhaustion that may be 

expected. This search area, defined as that part of the 

basin where operators are actually willing to site wells, 

is called the effective basin, and it may be only a small 

fraction of the entire basin. Because the size of the 

effective basin is an unknown quantity that typically 

grows through time, it was estimated from growth of 

the physical exhaustion time series.

ESTIMATION OF BASIN SIZE

We assume that within a large geologic basin there 

is a smaller effective basin to which the search for oil is 

restricted. Depth is not considered; the targets and the 

basin are assumed to be two-dimensional. The oil fields 

are assumed to be nonoverlapping ellipses all of the 

same size and shape inside the effective basin. The 

orientation of the targets, the angle between the major 

axis of the ellipse and the east-west line measured 

counter clockwise from the east-west line, may be 

anywhere between 0° and 180°. The search consists of 

the searcher selecting a point (drilling a well) in the 

effective basin. If the point is in a target, then the 

searcher is told the location of the boundaries of the 

target. This information approximates what would 

be learned in practice by subsequent development 

drilling.

There is an (x, y) coordinate system on the basin 

having x increase to the east and y increase to the 

north. The position of a target is completely specified 

by three coordinates (x,y, 0), where x and y specify the 

location of the center and 8 gives the angle between the 

major axis of the ellipse and thejc-axis of the coordinate 

system measured counterclockwise from thex-axis. We 

define a search solid, S, to be all points (x,y, 6), where 

(x, y) is a point in the geologic basin and 8 is between 0° 

and 180°. Then any target in the geologic basin corresponds to a unique point in S and vice versa. We can 

visualize S as a cylinder 180 units high and having a 

base of the same size and shape as the geologic basin. 

Thus the 0-axis is vertical, and the x-y plane is horizontal.

When a search point (x',y') in the effective basin is 

selected and is not in a target, then every point (x,y, 8) 

in S that corresponds to an ellipse in the geologic basin 

that contains the search point, (x', y'), is eliminated 

from further consideration. The set of points in S condemned by an unsuccessful search point at (x',y') can 

be visualized as the volume swept out by a horizontal 

target-size ellipse centered at (x',y', 0°) with major axis 

lying along the jc-axis as it rises to (x',y', 180) while 

rotating 1°, counterclockwise from above, for each degree it rises. Each point in this volume corresponds to 

an ellipse in the geologic basin that would contain (x', 

y'). When a search point is successful, then the points 

in S that are eliminated are all those points corresponding to ellipses that would intersect the target 

which was found. The introduction of the space S has 

changed the problem from search points looking for 

area targets to search volumes in S looking for point 

targets inS. The cumulative volume inS that has been 

eliminated by successful and unsuccessful search 

points in the effective basin, EB, is measured in acredegrees. This unit, the acre-degree, is the unit in which 

we will measure the extent of exploration.

Inside the search volume, S, we define the effective 

search volume, ES, as all points (x, y, 8) in S where (x, 

y) is a point in the effective basin. The size of the effective basin is not known to the data analyst. We will 

give a method for estimating the effective basin size 

from well locations under simplifying assumptions and 

then describe how the method is modified to fit more 

realistic assumptions. At first, it is assumed that wells 

are sited randomly in EB according to a uniform probability distribution. From this assumption we derive 

an estimator of effective basin size which is then 

modified to account for the fact that development wells 

are always close to existing wells.

Let EB be the effective basin, let ft be the Lebesgue 

measurable sets on EB, and assume EB is endowed 

with the uniform probability measure. Let fl =
MATHEMATICAL FOUNDATIONS B3

EB xEBx. .., a countable product of copies of the effective basin. Let f3m be the smallest <r field containing all 

sets of the form AjX. . .xAm xEB. . . where each A, is a 

Lebesgue measurable set in EB. Let /30 be the trivial <r 

field on n. A function defined on O can be fim measurable only if it is dependent on at most the first m coordinates of a point of O. LetdjJ = 1, 2,... be a sequence of 

positive numbers. Let D} (P) denote a disk of area d3 

centered at the point P. If (Pl5 P2v . .) is a point in O, 

define a(j,n; Px , P2j . . .) to be the total area covered by 

DjCPj),. . ., Dj(Pn ). Because a(j,ri) is a continuous function of PI, ., Pn it is measurable with respect to fin .

Using the notation just defined we now establish the 

basic theorem for the estimation of effective basin size. 

Theorem: If d}-^0 as 7—»°° and n and j are made to 

approach infinity in such a way that ndj-^A where A is

an arbitrarily selected positive number and if EB is a 

bounded set whose boundary has zero area, then

EB(l-e-A/EB > 1 in probability. )

The symbol EB is used for both the effective basin and 

its area.

Proof: The proof proceeds by first showing that 

E(a(j,n))^>EB (l-e~A/EB ) and then showing that 

a2(a(j,rc)}-»0. Let a sequence 13,7'= 1,2. . . . of positive 

numbers be defined by the equation 7rrf=dj. Define 

EB(rj+) to be all points (x,y) for which there is a point 

(x',y') in EB such that (x'-x)2 + (y'-y)2 ^. Define 

EB(r} —) to be all points (x,y) inEB for which there is no 

point (*',/) outside ofEB such that (x'-x)2 + (y'-y)2 ^

Thus EB(rj+) is EB swelled by 15 and EB(r} -) is EB 

shrunk by r5 . Define dEBfa) by dEB(rf) = EB(r.)+) - 

EB(rj — ). Thus dEB(r}) is all points within 73 of the boundary ofEB. Let fj be a function defined over EB(rj +} by

l

forP inEB (^~

area forP in

Then/j(P) is the probability that a randomly located 

disk Dj will cover the point P. Let S(j ,n\P ±,P 2, . . . ) be 

the set in EB(rj+) actually covered by the first n disks 

Dj(Pi), . . . , Djifn). We now establish an inequality by 

successive conditioning. From the definition of conditional expectation we have, letting S = S(j,n-l\

E(a(j,n)

since fj

(1)

S

we have

S

Substituting this expression for the integral yields 

E(a(j,n) \ Pn-i)^ dj + (l-dj/EB)a(j,n-l). (2)

Next we condition both sides with respect to /3n _2 ; recall

that

E(E(a(j,n)

(I - dj/EB)E(a(j,n - 1)|A,_2). (3) 

Applying our inequality (1) to E(a(j,n-l)\(3n -2) gives

E(a(j,n-l)\pn-2 ) ^ dj + (l-d} /EB)a(j,n-2). (4) 

Combining (3) and (4) gives

=dj+(l-dj/EB)dj+(l-dj/EB)2a(j,n-2).

Continuing this process through n—2 more conditionings gives

£(a(7») = E(a(j,n)\ 00) & 4 S (l-dj/EB) k (5)

&=o

and

K=0 }

(l-dj/EB) k =EB-EB(l-dj/EB) n+l. (6)

Because we assume that nd} —»A we have that the 

R.H.S. of (6) approaches EB-EB e~A/EB . Thus we may 

conclude

!TMLE(a(jsi» (7)

when ndj — »A as n,j -^><x>

Referring back to equation 1, we now seek a lower

bound for / £(P)dP.

o

Recall that j is d} IEB on EB(r5 -) and it takes smaller 

values on dEB(r-} so we assume that as much of 

S(j,n - 1) is in d EB (73) as possible to find a lower bound. 

Thus

S

fi(P}dP 

SftEBfy-) SftdEBdj) Sr\EB(r } -)

The area of SO>-l)nEB(7--) is at least a(j,n-l) 

-dEB(r j), where dEB(r j) is used to denote the area of 

dEB(ij). Thus/ j(P)dP ^ (a(j>-l) -

o

which yields the inequality,
B4

E(a(j,n) !#,_,) «s

dj + a(j,n-I) - (a(j,n-l) - dEB(rj))dj/EB

=dj + a(j,n-I)(I-dj/EB) + (dEB^dJEB. 

As before we condition both sides on (Bn -2 and obtain

PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

n

2

k =

|j3w _2 ) ^ dj + (l-dj/EB)E(a(j,n-I)\pn -2 ) 

(dEB(rj))dj/EB ^ d} + (1-dj/EBKd, + (l-dj/EB) x

dj + (l-dj/EB)dj + (dEB(rj))dj/EB

+ (l-dj/EBfa(j,n-2)

+ (l-dj/EB) (dEB(rj))dj/EB.

We repeat this process for j3w _3, ;3W _4 , . . . ,j30 to obtain 

E(a(j,n))

n-l

j 2 (l-dj/EB) k

n-l

+ (dEB(rj))dj/EBx 2 (l-dj/EBy. (8)

Since the boundary of ES has 0 area and since EB is 

a bounded set it follows that dEB(r]) -»0 as /j -»0. Hence 

as TIJ' -»e» the second term in (8) goes to 0. The limit of 

the first term was calculated before (see equation (6)) to 

be EB(l-e-A/EB ). 

Hence we have shown that

lim (9)

when ndj —> A.

Combining (9) and (7) we have that 

E(a(j,n)) ^EB(l-e-A/EB ) when nj ^oo and nd } ^A. 

To complete the proof of the theorem, it is sufficient 

to show that the variance of a(j,n), cr2 (a(j») ->0 as nj 

^•oo and nd) -> A. This will be done by establishing

(10)

cr2 (aO») =

We decompose the function a(j,n) — E(a(j,n}) into a 

sum of orthogonal functions

By definition of variance

Because the n summands are orthogonal, we have that

We will have completed the proof of (10), and hence 

of the theorem, when we have shown that

Let (Plf . . . , Pk -lt Pk , Pk+l, . . . , Pn , • • • ) = P and 

(P,, . . . , P*-!, P^, Pfc+1 , . . . , PBf . . . ) = P' be two points

in ft which agree in each of the first n coordinates except for the kth . Then aO»(P) - a(j,«)(P') ^ rf, because they differ only in the position of a single disk of 

area dj. Hence

E(a(j,n)\(Bk )(P1 ,...,Pk )\ 

Hence it follows that

This last inequality completes the proof of (10). The

theorem then follows from Chebyshev's inequality.

End of proof.

Corollary: If, in the above theorem, the sets Dj are replaced by ellipses of area dj all having the same orientation, then the conclusions of the theorem still hold if

the major axis of Dj — »0 asj — »°°.

Proof: The only modification required in the theorem is

to let /j be equal to one-half of the major axis of Dj.

We can derive an estimator for EB from the above 

theorem and corollary. When n points in EB have been 

selected at random, disks of area d are centered at each 

point and the total area covered, a(n), is measured. The 

estimate, EB, forEB is then chosen to satisfy the relation

a(n) =EB (l-e-nd/£B ). (11)

PRACTICAL DIFFICULTIES AND 

MODIFICATIONS

There is the practical problem of what size disk to 

use. If d is too small, then a(n) = nd from which it 

follows that K& = 0° . On the other hand, if the disks 

are too large, then because EB>a(n)> d, we will get 

unreasonably large estimates for the effective basin 

size. The problem of disk size is further complicated by
MATHEMATICAL FOUNDATIONS B5

the effect of target size on the effective basin size. For 

example, a very large area may be sparsely drilled for 

very large targets, whereas the search for small targets 

might be restricted to a smaller area. The assumption 

of random location of wells is not applicable to development wells, which are always sited near existing 

wells, so the disks around development wells cover 

very little additional area compared to what the average randomly located disk would cover.

In applying the estimator, EB, denned by equation 

(11) in the analysis of the discovery of targets of a particular size class, d was taken to be the area of the 

targets in question, a(n) was taken to be the acredegrees in the search solid which had been exhausted 

divided by 180, and n in the exponent was taken to be 

the number of wildcat wells plus a fraction, a, of the 

number of development wells. The fraction, a, is chosen 

to satisfy the condition that ifD1 andD2 are target-size 

and target-shape ellipses having the same but randomly selected orientation and centers 0.25 mi apart 

(40-acre spacing), then (1 + a)d is the expected total 

area covered by D^ and D2 . Thus a is between 0 and 1 

and decreases as target size increases.

EFFICIENCY OF EXPLORATION*

We assume that there are N target ellipses all of the 

same shape and area inEB and that they correspond to 

N points inES, the effective search solid. Let T{ be the 

fraction o{ES that had been exhausted at the time of 

the ith discovery. Thus Q<Tl<T2< . . . <TN<1. Because 

the search process is random, the T{ are random. We 

wish to find the joint probability distribution of these AT 

random variables as a function of the efficiency of exploration. We say that exploration is c times as efficient as random if an infinitesimal amount of search is 

c times as likely to make a discovery as if it were 

purely random. Specifically, if "a" of ES has been 

searched and N-k + l targets remain to be found, then 

as the fraction of ES searched increases from "a" to 

"a+Aa," the probability of discovering exactly "j" 

targets in the case of a purely random search is

Aa 1— a—Aa\ N-k+iFrom this expression we now find the density of Tk in 

the case of purely random search. Let/j^a \Tk ^ = ak - v } 

be the conditional density of Tfe , given that the k — 1 

discovery occurred atafe _!. Clearly/£(a |Tfc _! = afc _!)=0 

for a<afe _! or a>l. 

Suppose afe_!< a < 1; then

*A part of the results in this and the next section appeared in Schueneme>er and Root 

(1977) and are reproduced here with permission from American Statistical Association.

fk (a Tfc _! = afc _i)

= lim 1 P[a<rfc < a+Aa | Tk ^ = afc _J 

Aa—»0 Aa

= lim 1 P[Tk < a+Aa | Tk > a and 

Aa—»0 Aa

= lim 1

Aa-*0 Aa 

/1-a-Aa

V !-« t

]

V PfT "> n T — n 1 XN x 1 x I/. --"^ Cc- x fr _ i — CZlr _ 1 1 |_ n. ft. i ft. 1J

.7 = 1 V J A 1-a /

)

W-A- + l-j / \ A'-fc + l 

x 1 1-a l

X Vl-Oft-J

N-k + l

= N-k + l fl-a

1-a 

We isolate the factor

lim 1_ P[Tk < a+Aa | Tk>a and Tfc _ t = a^ 

Aa—»o Aa

and call it the instantaneous rate of discovery for the 

& th discovery, given that the next discovery is the & th 

discovery. In purely random exploration, this instantaneous rate is

N-k + l 

1-a

We say that exploration is c times as efficient as 

random exploration if the instantaneous discovery rate 

for the &th discovery is

c (N-k + l\ i ————— i • V I-" /

The instantaneous discovery rate for the kth discovery 

determines the conditional density of Tk given Tk -i. 

For if^fc (a JTfc-i) is the conditional density for Tk given 

TVi and Afc (a|Tfc _i) is the instantaneous discovery 

rate, then hk and gk must satisfy the relation

hk (a Tk -J =

whenever a > Tfc _i. Thus hk andgfe each determine the 

other. Using the above notation we now establish the
B6 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

main theorem from which the estimator of the efficiency is derived.

Theorem: If the efficiency is c > 0, then 7\,. . . ., TN 

have the same joint density as the order statistics of 

N independent random variables Xi, . . , XN taking 

values on the interval [0, l] and having the density

Moreover if Tk < a < Tk+l for some fixed "a," then 

k

has the same density as the sum of k independent random variables Zlt . . ,Zk each having the density

ce 0 <z < - In(l-a). 

otherwise

Proof: Let Xlf . . , XN be independent random variables 

with common density

{ C \^ -L J(f) \J *^~*. jC ^-* J_ 

0 otherwise

and distribution function F(x) = l-(l-^)c . 

Let yi? Y2 ,. . , YN be their order statistics, that is, Yl is 

the minimum of Xlt X2 , . . , XN and Y2 is the next larger 

and so forth, YN being the maximum of Xlt . . , XN .

In order to show that Ylt . . , YN and 7\,. . , TN have 

the same joint density, it is sufficient to show that

and

= c(N-k + l)

the instantaneous discovery rate for the & th discovery.

lim J_ 

Aa— »o Aa

Yk >a and Yk ^

_ lim 1. P[Yk<a+i\a and Yk >a and Yk -i<a] 

~ Aa P[Yk >a and 5V,<a]

lim J_ 

Aa—»o Aa

N-k + l

N-k + l\ x 

7

lim 1 

Aa—»o Aa

(N-k + l) f(a) (N-k + l) cd-af 1 c(Nl-F(a) (l-a)r 1-a

Hence Ylt .., YN and Tlt . . , TN have the same joint 

density. The last part of the theorem follows from the 

fact that Z, has the density of -ln(l-Xt ) given that 

Xt < a and -Sl/iCl-Tj) has the same probability

all Ti<a 

distribution as -Sl^(l-F,-) which has the same

all Y( <a 

probability distribution as — 2 l/i(l— X,-).

allX( <a 

The joint density of Tl9 . . . , TN is given by

N 

f(ai,..aN ) =N\ CN 77 (l-a,)'" 1

The conditional density of 7\, . . , Tfc , given that there 

are exactly £ discoveries in the exploration of "a" of the 

search solid, is given by

1 aA 

a a

1, . . ,Q,\-)ClClk +i . . . Udff

.< a and7\.+1 >a]

Two important features of this joint density are that 

it is independent of AT and that a sufficient statistic for 

"c" is given by

77 (1-a,.).

Hence - ^ In (1-a,) is also a sufficient statistic

for c given that there are k discoveries. From the previous theorem we can calculate that
MATHEMATICAL FOUNDATIONS B7

E _ 1 - 1

l-(l-a)c

1 + (l-ctf Ire(l-a)

We can thus obtain a point estimate, c, for c based on 

di, . . ,ak by choosing c so that

~ T *

Such a c will exist only if

and this inequality need not always hold.

In the practical problem of oil exploration (see Drew 

and others, 1980) it is reasonable to assume that c^l. 

Therefore, in the analysis of oil exploration data we 

take c to be the solution of (12) if (12) has a solution and 

if that solution is greater than or equal to 1; otherwise 

c is taken to be 1.

The unconditional joint density of the areas 

exhausted by the time that "a" of the effective search 

solid has been exhausted is

klck

J 

77

h(a v . . ,ak ) =

and its integral over all («!,.., ak ) is the probability of 

exactly k discoveries. If one attempts to find maximum 

likelihood estimates simultaneously for "N" and "c," 

the maximum can occur at Af = + °° and c = 0. To avoid 

this, it is necessary to include in the model the condition that the point targets in the effective search solid 

cannot be too close together because they must correspond to nonoverlapping ellipses in the effective basin. 

Because this condition is difficult to work with, it is 

decided to estimate the efficiency first by the method 

described above and then to choose ft to maximize JV

k

(l-d-ary ((l-a^f ~fe , which meanstf is the smallest 

integer such that

RESULTS OF SIMULATION

A simulation study was conducted to investigate the 

behavior of the estimated effective basin size. The well 

locations were chosen randomly within the basin, and 

the basin size was estimated using equation (11). Circular disks were assumed; the ratio of the area of a 

single target to the basin size was 1 to 2,500. The results for three replications are presented in figure 1; 

basin size estimates are plotted for every tenth well. 

The estimated basin size is infinite until disks overlap.

In order to study the distribution of c as estimated by 

equation (12), a simulation experiment was conducted 

for various values of c and N. Estimates for c were 

calculated at 0.1, 0.2,. . . , 0.9 fraction of the area 

searched, given that at least one target had been discovered. The median efficiencies from these simulations are presented in table 1. Each simulation was 

replicated 100 times; however, the median estimates 

corresponding to c=2 or c=4 sometimes are based on 

fewer replications when the area searched is less than 

0.4 since at least one discovery is required for an estimate. The estimated efficiency was set to zero when 

discoveries were present, but no solution to (12)

TABLE 1.—Estimated median efficiencies

N> k -I.

0.1 _

.2 ___

.3 _

.4 _

.5 _

.6 _

7

.8 _

.9 __

0.1 _

.2 _ _

.3 _

.4 _

.5 _

.6 ___

7

.8 ___

.9 _

0.1 _

.2

.3 _

.4

.5 _

.6 _

7

.8

.9 ._

Efficiency = 1.0

Fraction of Medians 

area searched N = 50

.__ — ___ _ ________ 0.33

.88

1.17

. __ ____________________ .83

_ ___ _____________ _ ___ .97

— _ _____ _ ___ - __ .99

_________________________ .94

_________________________ .92

_ _ _ _______ 1.01

Efficiency = 2.0

Fraction of Medians 

area searched N = 20

_ ______ _ ___ _____ _ 1.27

_______ ___ — ___ ____ 1.33

— _ _ _ _ ________ ___ 2.70

_ __ _ _ _ _ 2.08

_ _ _ _ 1.95

_ _ _ _ 2.02

________ ____ ___ __ ___ 2.17

— — _____ — _ ___ - 2.16

____ _ _ ____ ___ ____ 2.05

Efficiency = 4.0

Fraction of Medians 

area searched N = 3

— _ _____ . _______ 5.30

— _ . ___________ 3.31

4 4S

3.80

. ___ _____ ____ __ 4.02

4.25

_ _____ _ __ — _ _ 4.19

4.31

4.33

Medians

N = 70

0.00

.45

1.12

.94

.96

1.13

1.06

1.03

1.01

Medians

N = 30

2.30

1.83

2.36

2.17

2.11

1.91

2.01

1.98

1 97

Medians

N = 5

6.18

3.64

4.21

4.39

3.84

4.21

4.41

4.35

4.37

Medians

N = 90

0.82

1.07

.92

1.08

1.04

1.04

1.00

1.10

1.03

Medians

N = 40

0.00

1.95

1.81

2.16

2.20

2.02

1.98

2.02

2.03

Medians

N = 10

3.22

4.73

4.29

4.47

4.32

4.53

4.52

4.55

4.54
SIZE BASIN ESTIMATED 

*d 

e?

sr

1 8 -

TO (D 

s. & 

Si 

ri co 

m

C/5

ivsivHddv QNV AHaAOOsia 3LLVH 
MATHEMATICAL FOUNDATIONS B9

existed. The fractions of the area searched were generated from 1 — (1 — Uj)l/c where a, is uniform (0,1), 

i = l,2, . . . ,N. Smaller values ofN were associated with 

larger c's because in petroleum exploration the larger, 

less numerous deposits tend to be found with higher 

efficiency.

REFERENCES CITED

Arps, J. J., and Roberts, T. G., 1958, Economics of drilling for Cretaceous oil and east flank of Denver—Julesburg basin: American Association of Petroleum Geologists Bulletin, v. 42, no. 11, p. 

2549-2566.

Drew, L. J., Schuenemeyer, J. H., and Root, D. H., 1980, Petroleumresource appraisal and discovery rate forecasting in partially 

explored regions—An application to the Denver basin: U.S. 

Geological Survey Professional Paper 1138-A, p. A1-A11.

Schuenemeyer, J. H., and Root, D. H., 1977, Computational aspects 

of a probabilistic oil discovery model: Proceedings of the Statistical Computing Section, American Statistical Association, p. 

347-351.

Singer, D. A., and Drew, L. J., 1976, The area of influence of an 

exploratory hole: Economic Geology, v. 71, no. 3, p. 642-647.

Petroleum-Resource Appraisal and

Discovery Rate Forecasting

in Partially Explored Regions —

An Application to Supply Modeling

By E. D. ATTANASI, L. J. DREW, and J. H. SCHUENEMEYER

GEOLOGICAL SURVEY PROFESSIONAL PAPER 113 8-C

Investigation of field exploration behavior 

with distributed lag models and a discovery 

process model, and application of models to 

predict forthcoming reserves

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1980

CONTENTS

Abstract _________________________________________________ Cl

Introduction__________________________________—_________—— 1

Previous studies________________________________————_—————_ 2

Operator behavior at the field level ____________________——————————— 3

Firm behavior________________________________________ 4

Formation of expectations ___ _____ _______ _____________ _______ 4

Distributed lag models ____________________________—________ 5

Operator drilling behavior _____________________________________ 7

Empirical results ________________________________________ 8

Application of the discovery process model predictions _______———___————— 11

Supply of reserves ___________________________________________ 15

Conclusions ___ ___ _ _______ _____ ___________________________ 18

References cited __________________________________________ 19

ILLUSTRATIONS

Page

FIGURE 1. Shapes of normalized time profiles for weights associated with values of the lag variable _______———_____ C7

2. Temporal effects of a one-time unit change in the value of total discoveries on wildcat drilling ———————_____ 11

3. Temporal effects of a one-time unit change in the value of total discoveries on drilling expenditures __________ 11

4. Residuals for well-drilling models ____________________ _____ _______________________ 14

5. Residuals for well-drilling expenditures models _____________________________———————_____ 15

TABLES

Page

TABLE 1. Estimated coefficients of infinite lag models ——_______________—————_____—————————————___ C9

2. Coefficients of polynomial distributed lag models semiannual observations _____________———————_____ 10

3. Coefficients of polynomial distributed lag models quarterly data ________———_____————————_—_—— 10

4. Forecast performance of integrated system — ————————____— —— ——————————————————————————— 16

in

PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE

FORECASTING IN PARTIALLY EXPLORED REGIONS—

AN APPLICATION TO SUPPLY MODELING

By E. D. ATTANASI, L. J. DREW, and J. H. SCHUENEMEYER

ABSTRACT

This study examines the temporal properties and determinants of 

petroleum exploration for firms operating in the Denver basin. Expectations associated with the favorability of a specific area are 

modeled by using distributed lag proxy variables (of previous discoveries) and predictions from a discovery process model. In the second 

part of the study, a discovery process model is linked with a behavioral well-drilling model in order to predict the supply of new reserves.

Results of the study indicate that the positive effects of new discoveries on drilling increase for several periods and then diminish to 

zero within 2% years after the deposit discovery date. Tests of alternative specifications of the argument of the distributed lag function 

using alternative minimum size classes of deposits produced little 

change in the model's explanatory power. This result suggests that, 

once an exploration play is underway, favorable operator expectations are sustained by the quantity of oil found per time period rather 

than by the discovery of specific size deposits. When predictions of 

the value of undiscovered deposits (generated from a discovery process model) were substituted for the expectations variable in models 

used to explain exploration effort, operator behavior was found to be 

consistent with these predictions. This result suggests that 

operators, on the average, were efficiently using information contained in the discovery history of the basin in carrying out their 

exploration plans. Comparison of the two approaches to modeling 

unobservable operator expectations indicates that the two models 

produced very similar results. The integration of the behavioral 

well-drilling model and discovery process model to predict the additions to reserves per unit time was successful only when the quarterly predictions were aggregated to annual values. The accuracy of 

the aggregated predictions was also found to be reasonably robust to 

errors in predictions from the behavioral well-drilling equation.

INTRODUCTION

Interest in the state of the U.S. domestic petroleum 

industry has resulted in a critical examination of the 

usefulness of current economic models. There is some 

question as to whether these models can provide useful 

information aboout petroleum supply availability, 

supply price sensitivity, and the effects of alternative 

policy options on future supply possibilities. The performance of current models in terms of predicting future oil and gas discoveries and supply has thus far 

been disappointing (MacAvoy and Pindyck, 1975). The 

poor performance of these models may be caused in

part by their almost universal reliance on aggregated 

data, which obscures the effects of physical exhaustion 

on the supply response.

Although a major reason for constructing empirical 

economic models is to predict price responsiveness of 

future supply, the historical data upon which current 

models are based generally do not contain sufficient 

price variation to accurately predict future pricesupply responses within a region. The price variations 

present in available data relate to the quality of crude 

oil rather than to the incremental costs of exploration 

and development of individual deposits. Moreover, the 

effect of price changes on expected supplies is 

moderated by the level of resource depletion for a particular basin. Because of the potential interaction between price and resource depletion, the use of highly 

aggregated data will likely lead to spurious correlation 

between changes in supply and incremental price 

changes. Very few attempts have been made to model 

economic behavior at the level of the exploration play. 

In this report, an exploration play is defined as the 

increase in wildcat drilling attributed to and following 

the discovery of a significant (large) deposit in a formation that was not known to yield significant amounts of 

oil. Although the major obstacle to modeling has been 

the lack of basin-specific exploration data, many workers believe that field behavior is too erratic or unsystematic to model successfully.

This study has two parts. The first discusses the 

temporal properties of operators' exploration behavior 

for the Denver basin. In particular, the field behavior 

of operators in terms of responses (wildcat drilling and 

drilling expenditures) to new discoveries is examined 

and then compared with results from another basin. 

Properties of operators' behavior include the duration 

of the response and the nature of new deposits that 

must be discovered in order to sustain favorable 

operator expectations associated with a given area. 

The stochastic process characterizing the temporal distribution of previous discoveries is used as a basis for 

modeling operator expectations associated with the

Cl
C2 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

value of deposits remaining to be found. The stochastic 

nature of this scheme is compared with the process 

characterizing the predictions of the value of remaining deposits calculated with discovery process models 

similar to the ones proposed by Arps and Roberts 

(1958) and Drew, Schuenemeyer, and Root (1980).

In the second part of the study, the discovery process 

model is linked with the drilling model in a recursive 

system of equations in order to predict the supply of 

new reserves. Whereas the purpose of the behavioral 

drilling model is to explain operator action within a 

given time period, the discovery process model predicts 

the number and size distribution of new discoveries as 

a function of the number of wells drilled. By linking 

the two models, we can predict the quantity of new 

reserves discovered per unit time. These predictions 

are compared to the actual sequence of discoveries.

PREVIOUS STUDIES

Industry-level petroleum supply models have historically treated exploration as a process using wildcat 

wells that generates proved reserves. As such, little 

effort was devoted to formulating testable behavioral 

hypotheses or extending the theory of the firm to include the exploration function. In early models, no consideration was given to integrating the role of the initial resource base into the analysis. Because of geologic 

differences from area to area and very limited historical data series, the results of those industry-level 

studies have been somewhat disappointing. Data have 

been aggregated to the extent that significant behavioral and physical relations are concealed by the model 

specifications. Moreover, although almost all of these 

models have attempted to determine the price sensitivity of exploration and petroleum supply, price has 

varied little within specific geologic regions during the 

time periods chosen for analysis. Alternatively, 

engineering-type process models (National Petroleum 

Council, 1972; Federal Energy Administration, 1974; 

1976) designed to forecast oil and gas supply have 

frequently reduced exploration behavior to mechanical 

rules, assuming, for example, that firms' exploration 

expenditures are a certain percent of the previous 

period's net profits. Rather than investigating each of 

the models in detail, the following discussion focuses 

on motivation and differences in approach.

Perhaps the first widely publicized econometric 

study of oil and gas exploration was by Fisher (1964). 

Fisher, who noted that the supply of exploratory drilling differs greatly from the supply of new discoveries, 

used a three-equation model to attempt to explain (1) 

the annual number of wildcat wells drilled, (2) success 

ratio (proportion of wildcat wells drilled which resulted

in a discovery), and (3) the average size of the discovery. Fisher distinguished between exploration at the 

intensive and extensive margins. Exploration at the 

extensive margin yields discoveries that are characterized by a relatively low frequency and a large size, 

whereas exploration at the intensive margin yields 

relatively small discoveries that occur with greater 

frequency. Consequently, Fisher specified both the success ratio and the discovery size as a function of economic variables and also asserted that short-term reaction to increases in price results in a shift of exploration to the intensive margin.

In the three-equation model, the number of exploratory wells drilled is specified as a function of the 

price of oil, geophysical crew time, and lagged values of 

the average size of oil and gas discoveries, depth, the 

success ratio, and regional dummy variables. The 

equation for the success ratio includes the price of oil, 

geophysical crew time, and lagged values of the success 

ratio, depth, and average size of oil and gas discoveries. 

For the final equation, the average-size oil discovery is 

specified as a function of the price of oil and the previous period's values of the average size of oil and gas 

discoveries and the success ratio. In order to increase 

the number of degrees of freedom, short-time series 

data for large, geologically heterogeneous areas were 

pooled. Intercept dummy variables were included to 

account for differences in individual areas. Although 

the model specification was not derived from a firm's 

decision process, the explanatory power of the estimated equations was adequate. For predictive purposes, however, the price responsiveness of the new 

discoveries is unlikely to be the same for all the areas 

from which the data were taken. Furthermore, variations in the historical price data used by Fisher were 

the result of differences in the quality of the oil rather 

than incremental production costs. Erickson and 

Spann (1971), working to model the supply of natural 

gas, elaborated on Fisher's original formulation by including an equation for the average annual size of gas 

discoveries. The functional form of the gas supply 

model is similar to the functional forms used by Fisher 

in modeling oil supply.

MacAvoy and Pindyck (1973, 1975) modeled and integrated the components of natural gas demand and 

supply. Like Fisher, they modeled the supply of new 

discoveries by predicting the number of wells drilled, 

success ratios for oil and gas exploration, and the expected size of oil and gas discoveries for individual Petroleum Administration Districts. The supply of reserves may be calculated from the success ratio, wells 

drilled, and expected size. Both the success ratio and 

the expected discovery size represent (decline) extrapolations from a calculated reference size and success
AN APPLICATION TO SUPPLY MODELING C3

ratio. These variables are also sensitive to field prices 

and signify whether firms are operating at the intensive or extensive margins. The well-drilling equation is 

based upon the expected returns and the variance of 

returns which are, in turn, calculated from the expected size of discovery and the success ratio. In particular, the annual number of wells drilled is a function 

of the petroleum district dummy variables, expected 

returns, the variance of returns, and a drilling cost 

index. Although recent revisions (Pindyck, 1978) of the 

model are logically similar to previous versions, the 

success ratio and expected size of oil discoveries were 

made independent of price and dependent upon regional physical characteristics. These revisions led to ( 

reductions in previous estimates of the price responsiveness of new oil discoveries.

Two other approaches to modeling new discoveries 

were made by Khazzoom (1971) and Epple (1975). 

Khazzoom (1971) estimated the volume of gas discovered from the two-period ceiling averages for the price 

of natural gas, price of crude oil, price of natural gas 

liquids, and previous period's volume of gas discovered. 

The mgdel does not explicitly include a variable associated with the exhaustion of undiscovered deposits. 

Using a unique approach, Epple (1975) considered exploration as a production process that used wells and 

oil-bearing land as inputs. Exhaustion of the oilbearing resource is explicitly considered to be represented by the productivity and input cost of oil-bearing 

land. The model attempts to predict the value of new 

discoveries of crude oil and natural gas by using a joint 

production function. Specifically, Epple assumes a form 

for the production function, 1 then derives the firm's optimizing conditions under the assumption that it 

maximizes the net present value of exploration effort. 

Parameters of the oil-bearing land supply function are 

estimated from aggregated U.S. oil and gas exploration 

statistics. Although the analytical approach that Epple 

takes is novel and some aspects of resource exhaustion 

are considered, the use of aggregated data and the absence of detail relating to the spatial distribution of 

deposits restrict the applicabilty of the model.

Along similar lines, Uhler (1976) developed a 

stochastic production function for the discovery of new 

petroleum reserves. The marginal exploration cost 

function, derived from the production function, includes the following variables: an index of field knowledge, an index of physical exhaustion, and wells 

drilled. One distinguishing feature of the analysis is 

that it considers a disaggregated area. Because the 

study concentrates on the marginal cost of new discov-

'The supply of oil-bearing land is specified asL = R eTM where R is unit rent, w is past 

exploratory effort; with r > O and B > O and defined as estimated parameters and e is the 

Naperian constant.

cries, it does not attempt to explain the behavioral determinants of wildcat drilling.

Cox and Wright (1976) abstracted data from the exploration process by considering all drilling expenditures as investment in reserves, which are 

treated as inputs in the production of crude oil. The 

objective of their study was to link investment in reserves to government policies such as import quotas, 

prorationing, and special income tax provisions. Although the empirical results are impressive, the authors do not consider exploration investment apart 

from reservoir development investment. Furthermore, 

they do not consider the influence of uncertainty or 

resource exhaustion in their investment decision 

models.

In summary, economists appear to have taken two 

approaches to modeling firm exploration behavior. For 

the first approach, the number of prospects, labor, and 

capital services serve as inputs to the petroleum reserve production process called exploration. For the 

second, reserves are considered an input to the crude 

oil production process. Exploration is then regarded as 

similar to firm investment in physical capital equipment. To some extent, both approaches are correct, 

depending upon the nature of the firm under consideration. Firms engaged primarily in exploration regard 

discovered reserves as the end product, whereas vertically integrated firms might view reserves as inputs. 

Exploration can be regarded as an investment or input 

in the production of an inventory of reserves that results in crude oil production. As an intermediate product, reserves will undergo additional modification in 

the production process in order to generate final products.

OPERATOR BEHAVIOR AT THE FIELD LEVEL

With a few exceptions, the studies just reviewed 

have been concerned with modeling exploration behavior when data are highly aggregated. Frequently, data 

limitations necessitate such spatial and temporal 

aggregation. However, the testable behavioral hypotheses generated from these models are, in general, quite 

limited. In order to examine operator behavior, in more 

detail, data specific to the Denver basin are used in this 

study. In this area deposits were usually found in a 

single formation; the only exploration play that occurred during te period examined was stratigraphic in 

nature. Because the initial discoveries were made earlier than the beginning of the historical time series of 

data used in this study, the behavior examined relates 

to exploration at the intensive margin. With the exception of the Union Pacific Railroad acreage (Drew and 

others, 1980), mineral rights were regularly bought 

and sold. The following discussion concerns the explo-
C4 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

ration decisionmaking of firms, specifically, how they 

formulate and respond to expectations of undiscovered 

deposits remaining in the basin.

FIRM BEHAVIOR

Investment in production of petroleum reserves may 

take several forms. Additional reserves can be developed by extension drilling (leading to revised estimates 

of reserves), investment in improved recovery methods 

(enhanced recovery) from existing reservoirs, and 

wildcat drilling for the discovery of new deposits. 

Within the individual firm there are tradeoffs in costs 

that must be considered when using any one of these 

three sources of additional petroleum (White and 

others, 1975). For example, development drilling can 

be done with borrowed funds, whereas only equity capital can be used to pay for exploratory drilling.

The theory of firm behavior under production uncertainty has only recently been treated extensively in the 

economic literature (see Fama, 1972; Leland, 1974). If 

it is assumed that firm managers attempt to maximize 

the net present value of the firm (perhaps in terms of 

outstanding capital stock), a capital market valuation 

model might be posited. For purposes of explaining 

overall firm exploration decisions, such a model may be 

particularly applicable because generally only equity 

capital is used for exploration expenditures. Among 

the alternatives available to the firm for obtaining additional reserves—that is, wildcat drilling, extension 

drilling, and enhanced recovery—wildcat drilling 

clearly is the most uncertain. Firms commonly restrict 

their exploration activities to a small set of areas 

where they have had previous experience or where the 

firm plans a prolonged exploration program. For a 

given area, early exploration investment can be viewed 

as investment in the current set of prospects that provides additional information about other prospects in 

the area. The decision regarding when and in what 

areas to initiate an exploration program and at what 

times to allocate expenditures between exploration, extensions, and enhanced recovery are made at the firm 

level. However, the decision on when and where to drill 

strategic prospects in an area is frequently the responsibility of the field managers (Kaufman, 1903). This 

description of the optimizing process of a firm is nontechnical. For a technical discussion of the optimal exploration, production, and capital investment policy 

when the firm operates in geologically diverse regions, 

see Attanasi (1978).

At the field level, the operator is faced with the problem of allocating a fixed amount of funds to a number of 

prospects over a given planning period. For larger 

firms operating in many areas, a marginal amount of 

exploration funds might be reallocated within the typical planning period. The optimal search policy for a 

given set of prospects may be expressed as a function of 

the expected profit, marginal opportunity costs, and 

expectations associated with the size (and spatial) distribution of remaining deposits (Attanasi, 1978). Field 

expectations are formed when operators make exploration decisions based upon unknown or uncertain parameters. For the shortrun model considered here, unobservable operator field expectations appear to be the 

most difficult components of the determinants of exploration to measure.

FORMATION OF EXPECTATIONS

Expectations are formed when economic agents (firm 

managers or consumers) are required to make decisions based upon the unknown value of a particular 

variable. The expectations or predictions that are 

formed are then used in the decision process. In tests of 

the theory of the formation of expectations in economic 

models, the modeler may attempt to gather data directly from economic decisionmakers or construct a 

prediction scheme based upon observable characteristics of the specific decision problem such as market 

prices or output. Traditionally, these prediction 

schemes, although typically rather simple, rely almost 

entirely upon past values of the variables to be 

forecasted. A serious limitation of nearly all the existing expectations models is that their estimates are 

formed without consideration of the decisionmaker's 

criterion function. Embodied in the decisionmaker's 

criterion function are attitudes toward risk and consequences of under- and over-estimating the value of the 

uncertain variable. Originally, expectations models 

were developed in connection with agricultural price 

supply responses (Ezekiel, 1938; Nerlove, 1958). Farmers must commit land and labor to the production of 

specific crops before the market price of the crop is 

known. The farmer may base his predictions of the 

current period's price on the previous period's realized 

price. In particular, suppose pf is the expectations variable and pt is the observed value at time t. The static 

expectations formulation is defined as

pf = Pt-i- (1)

If the current period's expectation is a weighted combination of the realization and expectation of the previous period, that is,

pf = pf_! + a (p,-i - pf_j) 0 < a < 1, (2)

then expectations are said to be formed adaptively. In 

the equation above, a is the coefficient of adaptation 

and controls the degree to which expectations will con-
AN APPLICATION TO SUPPLY MODELING C5

form to the previous period's realization. Expectations 

are said to be regressive if

pf= (3)

with a,> a2 > a3 the weights are positive and approach 

zeros with the passage of time. Alternatively, expectations are extrapolative, if

and

pf =

p =

•>t-\ - Pr-a) + a-z (Pt-2 ~ P^-a) + • •

-i + (a2 -ai)pV2 + (as-«i)A-3 + • • • -,(4b)

where the weights a, are also positive and decline. In 

terms of the final coefficients of the lagged terms of pt, 

the coefficients (equation 4b) are negative and approach zero. Turnovsky (1969) has shown how a 

Bayesian updating scheme for uncertain values of the 

distribution parameters of the stochastic decision variables results in a general expectations formation 

scheme yielding as special cases static, extrapolative, 

and adaptive expectations.

As suggested by these formulations and others frequently found in the literature, the estimated form of 

the unobservable expectations variable is based on the 

autoregressive process2 of the historical realizations of 

the decision variable. In fact, a relatively unrestrictive 

definition of rational expectations is that the generated 

predictions and realizations follow the same autoregressive scheme. A reason for this definition is that 

autoregressive forecasting schemes provide optimal 

predictors for a wide range of stochastic processes. 

However, these predictors are still constructed without 

recourse to the decisionmaker's criterion function. 

Moreover, the criterion functions for the prediction and 

decision problems will, in general, differ because the 

consequences of errors in the estimates differ. Distributed lag models are frequently applied when the unobservable expectations variable is approximated by the 

autoregressive formulation of the historical realizations of the variable of interest. If it is assumed that 

the decisionmakers have knowledge of the underlying 

stochastic structure that generates the time series of 

the variable to be forecasted, then the distributed lag 

proxies are equivalent to calculating the conditional 

expected value of the uncertain variable (Pesando, 

1976).

DISTRIBUTED LAG MODELS

Unobservable expectations variables are frequently 

modeled with a distributed lag function of the histori2 The term autoregressive process used here refers to a model that can be expressed as a 

finite linear aggregate of previous values of the process. The nature of the stochastic process 

might correspond to an autoregressive model, moving average model, or combination of the 

two models. For more detailed definitions, see Box and Jenkins (1970, p. 7-11).

cal realizations of the variable of interest. Simply, distributed lag models provide the basis for analyzing the 

temporal response of the dependent variable (yt ) to a 

change in a specific independent variable (xt). In particular for any given T, it is assumed that

yt = c + W(Xt +WiXt-i + , . . . , + wTxt-T + et, (5) 

where dyt = w. j = o, 1, . . . , T

and et is a stochastic error term.

Two fundamental problems arise if one were simply 

to apply ordinary least squares to equation (5) to estimate all H/S. First, if T is large, there may be too many 

parameters, leaving few degrees of freedom to make 

statistical inferences about parameter values with any 

degree of confidence. Second, there is likely to be a high 

degree of collinearity among the lagged values of the 

independent variables. Consequently, in order to conserve available degrees of freedom and improve efficiency of the statistical estimates, a lag generating 

function is chosen to correspond to a given time profile 

that characterizes the response of the dependent variable to changes in a particular independent variable. 

By assuming a specific analytical form for the generating function, the number of parameters required to be 

estimated can be made quite small. For example, suppose the decisonmaker's expectation xf is adjusted 

adaptively as an observation of a realization xt-i is 

made, that is,

•V'l* __ «i

•A'f J

(6)

where 0 < a < 1. If X is defined by X = 1— a, then the 

expectations variable may be calculated by the following equations:

x* = (1-X) Xt-i

xf = (1- (7)

Hence, given the derived structure, a single parameter 

X can be used to generate values of the unobservable 

expectations variable from the historical data associated with xt .,. Suppose it is assumed that

= * (8)

Then recursively substituting (7) into (8) and the lagged form of (8) into the resulting equation, the estimation form of equation (8) is

yt =(l-k)xt-l (9)
C6 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

or using the lag operator L the equation is

y,=

(1-XL)

Consequently, by deriving the specific form of the lag 

generating function, the estimation of the distributed 

lag structure of the model can be reduced to a small 

number of parameters.

Jorgenson (1966) has shown that any arbitrary lag 

function can be approximated by the rational form3

=

B(L)

(10)

m

where A(L) =2

i = 0

B(L) =

A rational lag function means that the function can be 

expressed as the ratio of two polynomial (lag) functions. That is, any finite or infinite response function 

can be approximated by the rational form (10). For 

example, a form of the geometric function results when 

A(L) = a I andB(L) = 1 - \L, that is,

= («/)

(1-XL)

(11)

Alternatively, by specifying the denominator in (10) to 

be equal to a constant (unity), a finite distributed lag 

function results. In general B(L) of higher degree than 

2 or 3 might be difficult to identify uniquely. The 

statistical methods used to estimate such models are 

complex because the estimation routines are highly 

nonlinear. In addition, the distribution of the descriptive statistics are known only for large samples 

(Dhrymes, 1971).

In terms of temporal behavior, the implications of 

particular distributed lag schemes are not always obvious from the estimated structural equation form. For 

model comparison and selection, certain standard 

properties of the implied function, which are more intuitively interpretable, are frequently considered. Two 

such properties are the normalized time profile of the 

distributed lag response function and the average lag 

length. The time profile represents the temporal distribution of the effects on the dependent variable of a unit 

change in the independent variable. The general shape 

of the time profile indicates how rapidly and in what 

direction a change in the independent variable affects 

the dependent variable. Changes in the units of measurement of either the dependent or independent vari3The general form of the rational distributed lag function may be derived from an extension of the adaptive expectations hypothesis (see Dhrymes, 1971).

ables affect the shape of the time profile. Consequently, 

when the time profiles for alternative distributed lag 

models are compared, they must be taken to use the 

same units of measurement. The time profile may exhibit a variety of forms (fig. 1). Some common ones 

include an exponentially decaying function, an inverted "V" shape or a function with weights oscillating 

between positive and negative values. In some cases, 

the general shape of the time profile can be inferred 

from the functional form of the distributed lag model. 

However, the time profile weights can always be calculated using the estimated function and tracing the effects on the dependent variable of a unit change in the 

specific independent variable.

Another property of distributed lag functions that 

may serve as a basis for comparison is the average lag 

for an n period lag function, which is calculated as

n

e = t=Q twt ; 

n wt

(12)

where t is the period subscript and w is the associated 

weight. The average lag 0 turns out to be a simple 

weighted average of the time periods where the 

weighting is proportional to the time profile weights. 

The average lag reflects how the various values of the 

time profile weights are temporally distributed. Values 

of the average lag must be interpreted in relative 

terms. That is, larger values of 0 indicate that much of 

the distributed lag effect is felt at larger values of t, 

and smaller values of 0 indicate that the weights associated with earlier lag periods are relatively larger 

than for the later ones.

The time profile and average lag represent properties of the distributed lag model that are easily interpreted. Frequently, an individual researcher may 

have subjective prior knowledge of the form for the 

time profile and attempt to use this prior knowledge to 

evaluate the empirical model. For example, suppose a 

distributed lag model is estimated representing the effects of increases in advertising on product sales. One 

might reasonably expect the time profile to indicate the 

effect of a one-time increase in advertising to be exponentially decaying, but it would be unreasonable to 

expect the function to indicate explosive or 

monotonically increasing effects. The time profile provides a means whereby the estimated distributed lag 

model can be evaluated for consistency with expected 

economic behavior.

OPERATOR DRILLING BEHAVIOR

Assuming the type of decentralized decisionmaking 

that was previously described, the field manager's re-
AN APPLICATION TO SUPPLY MODELING C7

-0.1 -

-0.2 -

-0.3

45678

LAG PERIOD

10 11 12

FIGURE 1.—Shapes of normalized time profiles for weights associated 

with values of the lag variable. Analytical specifications of the 

distributed lag functions are adjacent to each of the time profiles 

with L defined as the lag operator.

source allocation problem might then be specified as a 

sequential decision or adaptive control problem. Without going into an involved mathematical derivation, 

the optimal shortrun search strategy might be 

specified as a function of profit expectations, perceived 

field risks, and expected opportunity costs of foregone 

alternatives. Field managers are assumed to use available information to predict the spatial and size distribution of undiscovered deposits. The optimal search effort might be specified as a function of expected profit ir 

and expectations associated with deposits left to be 

found x* • Depending on the nature of the area, a variable k might also be included to index the current state 

of field knowledge relating to the original distribution 

of deposits and extent of physical exhaustion. The optimal search effort may be specified by the following 

function:

yt=f(i*,Xi,kt\ (13)

To estimate a specific functional form of equation 

(13), the variables Jrt and kt must be defined. It seems 

reasonable to assume that there is a lag between the

firm's exploration allocation decisions across regions 

and the reporting of profits from which irt is assumed to 

be estimated. In the empirical models to be presented, 

the expected profits variable was computed as the lagged value of returns per wildcat well. Returns per exploratory well were calculated by multiplying the 

quantity of the dollar value of oil found per successful 

wildcat less the cost of the well by the success ratio. 

This variable was assumed to reflect the general level 

of a basin's exploration profitability. The variable kt is 

assumed to reflect the relative degree of knowledge 

operators possess about a particular set of targets and 

the extent of physical exhaustion. This relative index 

of experience and exhaustion k was calculated on the 

basis of the weighted average of the cumulative 

number of wells drilled in a specific target area over 

the previous three periods. 4 The index as defined would 

have greater significance and intuitive appeal if the 

area under consideration exhibited several plays or 

target formations. Because a large block of acreage in 

the Denver basin was restricted from exploration, the 

index was calculated on the basis of areal extent withheld rather than on the basis of target formation (see 

footnote 4). The index specified in this fashion captures 

the effects of sampling deposits without replacement 

and also will increase as economically exploitable deposits are exhausted. The wildcat drilling data and reserves data were obtained from the Well History Control File of Petroleum Information, Inc. Drilling costs 

were calculated from various annual issues of the Joint 

Association Survey of the U.S. Oil and Gas Producing 

Industry (JAS) (American Petroleum Institute, 1953, 

1955-56, 1959-73). The study area was partitioned 

into 88 smaller units (of 625 mi2 or 1,619 km2) for purposes of assigning target depths for wildcat wells 

drilled within each cell. Well costs were calculated 

from the inferred target depth, and costs per foot were 

taken from the JAS annual summaries for Colorado. 

Although some natural gas deposits were found, 

operators were searching predominantly for new oil 

deposits. The oil found was of a relatively uniform 

quality. During the period from 1949 to 1972, prices 

were relatively stable and ranged from $2.63 per barrel 

to $3.46 per barrel. However, during 1973 prices for 

new oil increased to over $8.90 per barrel. Initially, the 

general estimated form of equation (13) was

+ ykt +€t. (14)

4If P, is the proportion of wells drilled in the i"1 formation or sub basin at time t and C, is 

the cumulative number of wells drilled in the i th formation or subarea, then the experiencem 3 

exhaustion index is defined by k, = 2 O, • U/3K 2 p'-t). The Denver ba- i=l j=l

sin had two subareas and a single producing formation. The areas were partitioned into 

acreage excluded from exploration and held by the Union Pacific Railroad; other acreage 

corresponded to the rest of the basin which was open for exploration.
C8 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

where f(Dt) represents the general lag structure and x*< 

= f(Dt). The variable yt represents search effort expended as measured by either wildcat wells drilled or 

estimated drilling expenditures, Trt-z is the expected 

value of returns per wildcat well lagged two periods, 

Dt is the value of new discoveries equal to or greater 

than a given minimum size class of deposit for a 

specific time period, and kt is the index of field experience described earlier. The lag function of the dollar 

value of the oil found in certain size classes of deposits 

per time period is assumed to reflect changes in the 

operator's expectations regarding the distribution of 

deposits remaining to be found in the basin. The variable Dt was defined with respect to a minimum size 

class of deposits for two reasons: Larger deposits generally have lower production costs; moreover, the occurrence of larger deposits is frequently taken to be 

indicative of the presence of deposits of a similar size. 

Estimated empirical models are presented in the following two sections. In the first section, alternative 

forms of the lag function were tested; the final model 

took the form of a finite lag function with lag weights 

constrained to lie in a second-order polynomial. Inferences about operator behavior are made from the estimated models. On the basis of these results, inferences 

about operator behavior observed in the Denver basin 

are compared with inferences about behavior observed 

in the Powder River basin (Attanasi and Drew, 1977). 

In the next section, the estimated quarterly model is 

compared to the model estimated from semiannual observations for consistency in interpretation. The discovery process model similar to the model presented in 

Arps and Roberts (1958) is then used to calculate the 

value of deposits remaining to be found. The performance of this type of model and its apparent efficiency 

in the use of information contained in the drilling data 

suggest that the model predictions are equivalent to 

the expected value of the uncertain variable. A more 

restrictive definition of a rational expectation is that it 

efficiently uses all available information about the uncertain variable. It would therefore be useful to compare the statistical properties of the ad hoc distributed 

lag model with the performance of an empirical drilling model that uses the (rational) predictions of the 

value of deposits remaining to be found calculated with 

the discovery process model.

EMPIRICAL RESULTS

Several forms of the distributed lag function f(Dt) 

were fit to the historical drilling data in order to arrive 

at a reasonable description of the discovery process. In 

evaluating alternative empirical models, several 

criteria are frequently applied. The model structure 

and estimated coefficients should first be consistent

with economic theory. Other criteria relate to the standard statistics that describe the quality of the fit of the 

model, that is, coefficient "t" statistics, coefficient of 

determination and the standard error of the regression. 

Finally, the predictive performance of the model can be 

examined by using part of the historical sample to estimate the model and the remainder of the sample to 

compare model forecasts with actual sample values.

The specific lag forms that were tested included the 

geometric, second-order rational, and finite polynomial 

distributed lag functions. The geometric lag structure 

is derivable if it is assumed that expectations are 

formed adaptively.5 If, in addition to an adaptive expectations hypothesis, it is assumed that reactions to 

changes in expectations exhibit some inertia—that is, 

only a partial immediate adjustment (see Johnston, 

1972, p. 302- 303)6—a second-order rational lag model 

results. Motivation for the finite polynomial lag function was not based on any specific set of assumptions 

that explain the process by which expectations are 

formed. However, the empirical results for the finite 

lag distribution will provide a useful comparison to the 

results obtained for alternative infinite lag forms, that 

is, the geometric and second-order rational functions.' 

An infinite lag distribution means that the effect on 

the dependent variable of a change in the lag variable 

continues over an infinite number of time periods.

The parameters for the geometric and second-order 

lag functions were obtained using a nonlinear iterative 

regression technique that yields maximum likelihood 

estimates (Pierce, 1971). For the initial test runs, Dt 

was defined as the aggregate value of discoveries per 

time period of deposits of at least 500,000 barrels of 

reserves. Prior to carrying out the estimation procedures, the data were tested for seasonal variations, 

which were removed from the quarterly data using the 

Census X-11Q procedures (Shiskin and others, 1967). 

Table 1 presents the parameters estimates for models 

based on semiannual and quarterly observations. Similar results were obtained when Tft-\ was used in place of 

77f_ 2 and when alternative assumptions were imposed 

on the error term. Comparisons of the coefficients of the 

lag terms associated with the models estimated from

5For this derivation see the preceding discussion and Attanasi and Drew (1977, p. 57), The 

estimated form of the geometric lag function was

-+ ftir,_.2 + yk,+ft ,

(1-XL)

where y,, Dt, J7,_2, *, were defined earlier and ft, ft, X, ft, and y are estimated coefficients. L 

is the lag operator. The stochastic component of the model is embodied in «, and was 

permitted to have autoregressive and moving average components.

"The second-order rational lag function that was estimated took the following form:

y, = ft +

(1-X,L+XZL2)

+«„

where the variables y,,D,, 7r,_2, k, were denned earlier and the estimated coefficients are ft, 

ft. ft, *», *«. and y. L is the lag operator. The error term e, was assumed to have autoregressive and moving average components.
AN APPLICATION TO SUPPLY MODELING C9

TABLE 1.—Estimated coefficients of infinite lag models

[Numbers in parentheses are 'V" statistics, and A2 is the coefficient of determination]

Geometric: 

Drilling:

Semiannual _.-488.0 0.00102 0.852

(4.1) (8.2) (52.5) 

Quarterly____--17.5 .00039 .899

(1.4) (11.7) (100.2) 

Drilling expenditures:

Semiannual__-2768.0 .00450 .881

(4.6) (7.9) (68.8)

Quarterly_____-40.4 .00161 .923 

(.7) (12.2) (130.6)

-0.0173 0.066 0.735

(.8) (5.8)

-.0052 .0098 .806

(1.0) (6.7)

-.0690 .320

(.7) (5.7)

-.0241 .0258

(1.0) (3.6)

.766

.830

Second order: 

Rational Lag 

Drilling: 

Semiannual

Quarterly

-—-394.4 0.00064 

(3.5X3.9) 

-IB 3 00019

(1.3)(4.6)" 

Drilling expenditures: 

Semiannual .. -2370.0 .0028 

(3.9X3.1) 

Quarterly . 28.4 .00096

(.5X3.9)

1.284 

(7.9) 

1.491 

(10.2)

1.270 

(5.8) 

1.363 

(7.0)

-0.389 

(2.6) 

-.543 

(3.8)

-.354 

(1.8) 

-.411

(2.2)

-0.0266 

(1.3) 

-.00324

(.7)

-.100 

(-1) 

-.0192

(.8)

0.0589 

(5.4) 

.0101 

(7.6)

.290 

(5.0) 

.0262

(3.6)

0.766 

.840

.766 

.843

the semiannual and quarterly data show surprising 

similarity. Parameter coefficient estimates of the expected profits and the field exhaustion variables which 

are negative and positive, respectively, are inconsistent with prior theoretical expectations. The negative 

expected profit coefficient is, in part, due to multicollinearity between 7rt andDt . Because the empirical estimates were inconsistent with the theoretical restrictions, both infinite lag forms were rejected.

There seems to be no a priori reason for restricting 

the distributed lag function to an infinite lag distribution. A finite lag model was estimated in which the lag 

weights or coefficients were constrained to lie within a 

polynomial function (Almon, 1965). As discussed earlier, behavioral interpretations associated with the 

derivation of such lag models are limited. The form of 

the model used to study wildcat drilling expenditures

is

m

yt = fa + fa 7T,_2 + .2o o>t-iDt-i + ykt + et. (15)

Coefficients c^., are the weights associated with (t-i)th 

period and reflect the temporal effects of a change in 

DM on operators' exploration effort. Assuming that a 

finite lag model is appropriate, it is important to determine the appropriate lag length for Dt.t and the degree of the polynomial containing the lag weights.

Owing to the presence of serial correlation in the 

data, the application of ordinary least-squares regression procedures in the linear equations would have resulted in inefficient parameter estimates. The 

Cochrane-Orcutt procedure, which is described in detail in Kmenta (1971, p. 287-289), was used to generate efficient estimates. This procedure amounts to 

the iterative application of ordinary least squares on 

data that have been transformed using an estimate of 

the serial correlation coefficient p. Ordinary least

squares is initially applied to the raw data to obtain an 

estimate of p from the residuals. The data are 

transformed, and least squares is applied to the 

transformed data. A new estimate of p is obtained from 

the residuals of the second regression. If xt is the tth 

observation of the variable, then the transformed variable xt is defined by xt =xt-pxt-i- This iterative procedure continues until there is no significant change in p.

The consequence of understating or overstating the 

true lag structure is the introduction of a specification 

error (Schmidt and Waud, 1973). If the finite lag model 

weights are interpreted as representing the temporal 

response of operators' search effort to new discoveries, 

then certain restrictions should be placed on the lag 

weights to be consistent with economic theory. For 

example, the weights would be difficult to interpret in 

this fashion if they oscillated between positive and 

negative signs, or if they monotonically increased as 

the lag period became more distant. Given that the 

pattern of the coefficients conforms to certain prior restrictions, Schmidt and Waud (1973) suggest that appropriate lag length be determined after estimating 

the model for a number of different lengths and that 

the final choice be made on the basis of maximizing the 

adjusted coefficient of determination or minimizing the 

standard error of the regression. In this study, the lag 

structure was determined by searching over various 

combinations of lag lengths, and the choice of the final 

form was based on minimizing the standard error of 

the regression equation. In this search, Dt.j represented 

the value of new discoveries found in deposits of at 

least 500,000 barrels. Estimates of coefficients for the 

finite distributed lag models are presented in tables 2 

and 3. For the model based on semianual data, the 

appropriate choice of lag length was five periods7, with 

the coefficient for the fifth period restricted to zero. The 

appropriate lag length for the model based on quarterly observations was found to be 11 periods with the 

coefficient for the final period restricted to 0. When the 

initial period is also taken into consideration, the empirically determined lag lengths for the two models 

(semiannual and quarterly) are consistent. The time 

profile indicated by the lag weights have the shape of 

an inverted "V".

Using the semiannual data, the finite lag models 

were reestimated with Dt redefined to represent the 

total value of new discoveries of deposits greater than 

2.5 million barrels and 5 million barrels. A comparison 

of the estimated models and associated statistics across

7 One problem that arises in determining the appropriate lag length and that has not been 

addressed in the literature is whether the models of varying lag length should be estimated 

using the maximum sample size or using identical sample sizes. For the model specifications 

that were tested, the five-period model estimated from the maximum sample size had the 

minimum standard error of the regression, and the seven-period model had the minimum 

error when alternative models were estimated using identical samples.
CIO PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING 

TABLE 2.—Coefficients of polynomial distributed lag models, semiannual observations

[Specifications A and B are based on 46 observations from exploration activity in the Denver basin. Specification C is based on 29 observations from exploration activity in the Powder River 

basin. R*, adjusted coefficient of determination; D.W., Durbin-Watson statistics; S.E., standard error of regression. Numbers in parentheses are absolute values of "t" statistics]

A .

R

C __

A „

B „

c

__ _213.9

(2.2) 

.. .184.6

(1.5) 

——47.5

(5.4)

2098.0

(3.2) 

__ 1577.0

(2.1) 

. __ 932.4

(7.8)

0.0340

(.6) 

.0330

(.6) 

.0054

(2.4)

0.5180

(1.8) 

.5090

(1.7) 

.0410

(1.6)

-0.271

(1.1) 

-.227

(.8) 

-.061

(1.5)

-1.420

(1.0) 

-.976

(.7) 

-1.740

(3.8)

0.279

(1.6) 

.309

(1.6) 

.192

(6.8)

-0.072

(.1) 

.272

(.3) 

.253

(.7)

0.579

(3.5) 

.600

(3.5) 

.302

(11.3)

0.744

(.7) 

.994

(1.0) 

1.271

(3.8)

Wells

(4.1) (4.2) (.4)

(4.1) (4.2) (.2)

(11.2) (2.7) (5.0)

Expenditures

(1.1) (1.2) (2.1) 

1.189 .858 .092

(1.3) (1.4) (.6) 1.313 .381 -.545

(4.3) (.9) (4.2)

0.005

(.2) 

-.051

(4.9)

0.080

(.5) 

-.571

(4.4)

-0.001

(.2) 

-.006

(1.4)

0.036

(1.5) 

.077

(1.5)

-0.012

(.2) 

.117

(5.1)

-0.257

(.9) 

.936

(3.4)

0.756

.749

.940

0.756

.756

.764

2.3

2.3

2.1

2.6

2.4

2.2

60.5

61.4

13.6

325.8

325.9

160.1

TABLE 3.—Coefficients of polynomial distributed lag models, quarterly data

[R2 , coefficient of determination; D.W., Durbin-Watson statistic; S.E., standard error of

regression]

Drilling

ft.

ft -— Y

We

CUg

fl2 ....

D.W. —

S.E. —

Coefficients

.... . -112.7

.0096

___ .017

... _ .110

___ .184

.240

- _ . .276

_ - .293

___ .292

- __ .271

___ .232

___ .174

.-— .096

— — .912

— „. 17.45

"t" statistics

2.2 

1.7 

.7 

.3 

2.7 

3.2 

4.0 

4.3 

4.4 

4.4 

4.4 

4.4 

4.4 

4.4

Drilling expenditures

Coefficients

801.0 

.0275 

-.0546 

.126 

.401 

.620 

.781 

.884 

.930 

.919 

.850 

.724 

.540 

.299 

.912 

1.98 

94.89

"t" statistics

2.9 

.9 

1.7 

.4 

1.3 

2.1 

2.4 

2.5 

2.6 

2.6 

2.6 

2.5 

2.5 

2.5

these deposit size categories indicated little difference 

in terms of the overall model form or the explanatory 

power of the model specifications. 8 It might reasonably 

be concluded that operator expectations regarding the 

value of remaining deposits were sustained by the 

quantity of oil found per time period rather than by the 

discoveries of specific size deposits. This finding is consistent with results of a similar study carried out for 

the Powder River basin (Attanasi and Drew, 1977). 

Both the profit and field experience exhaustion variables are statistically more significant in the expenditures equation than in the drilling equation.

In order to determine if the experience-exhaustion 

index exhibited systematic intertemporal effects, a lag 

structure was imposed on the variable kt , and the equation was reestimated. Results of these specifications 

are presented in table 2. None of the lag coefficients 

was significant in both the drilling and drilling expenditures equations, nor was there any improvement 

in the explanatory power of the equations.

8 The adjusted coefficients of determination (R2) for a five-period lag when the minimum 

size classes of deposits are defined as 2.5 million and 5 million barrels are 0.7434 and 0.7455 

for well drilling and 0.7131 and 0.7550 for well drilling expenditures, respectively.

Neither drilling or drilling expenditures are perfect 

measures of exploration effort. Drilling activity, although easily interpreted in terms of sampling effort, 

cannot differentiate between the wells drilled under 

varying depths or qualitative degrees of difficulty. Although drilling expenditures may more completely reflect the economic dimension of search effort, they cannot fully represent the temporal behavior because the 

costs of land acquisition and preliminary geophysical 

work are not included. The negative coefficients associated with the first periods of the distributed lag 

models based on semiannual observations appear to be 

the result of two factors. First, drilling and drilling 

expenditures do not reflect the proper magnitude of 

search effort in the early stages of exploring a prospect. 

Second, adjustments in onshore exploration activity 

would be more readily captured if observations were 

associated with shorter time periods, as shown by the 

model based on quarterly data.

The estimated responses can also be compared to results obtained for exploration activity in the Powder 

River area (specifications C, table 2). In the study of 

that area, the geometric, second-order rational lag and 

finite lag models were also estimated. The model that 

most appropriately described the data was again found 

to be a finite lag distribution. The empirically determined lag length for the Denver basin was five 

semiannual periods with the fifth-period coefficient 

constrained to zero. The appropriate lag structure for 

the Powder River basin was determined to be four 

periods with the coefficient for the fourth period unconstrained. Major differences between the estimated response functions are related to the differences in the 

magnitudes of the lag function weights and the 

significance of the field experience-exhaustion variable. The lag coefficient on the expenditures equations 

are comparable. However, the values of the lag coefficients for the drilling equation for the Powder River 

basin area are about half of those for the Denver basin.
AN APPLICATION TO SUPPLY MODELING Cll

This result, in part, reflects the fact that wells were two 

or three times as expensive to drill in the Powder River 

basin. Because there were three exploration plays and 

four different target formations in the Powder River 

basin, the field experience-exhaustion index varied 

widely. The estimated coefficients are consistent with 

this observation and indicate that exploration effort in 

the Powder River basin was significantly influenced by 

the current state of experience or exhaustion. Differences in the estimated relations appear to be reasonably consistent with differences in economic and 

geologic factors that are reflected in the specific basin's 

exploration history.

In order to compare the implied temporal properties 

of the alternative lag schemes, the time profiles of the 

geometric, second-order rational lag and polynomial 

(finite) distributed lag schemes are presented in figures 

2 and 3. Infinite lag functions consistently overesti40

(/) 35

I

< 30 

C/)

O

f 25

O 

LU 

CC.

Z 20

15

S2

LU 10

0.00039

Finite lag model

56789

LAG PERIOD

10 11 12 13 14

FIGURE 2.—Temporal effects of a one-time unit change in the value 

of total discoveries (in thousands of dollars) of at least 500,000 

barrels on wildcat drilling. Specifications of the lag function are 

adjacent to each of the time profiles with L defined as the lag 

operator.

17

16

15

14

C/) _ X 13

O 12 

LU

g 11

Z 

D 10

X 

-^ 9

C/)

X 7

S2

LU 6

456789

LAG PERIOD

10 11 12 13

FIGURE 3.—Temporal effects of a one-time unit change in the value 

of total discoveries (in thousands of dollars) of at least 500,000 

barrels on drilling expenditures. Specifications of the lag function 

are adjacent to each of the time profiles with L defined as the lag 

operator.

mate operators' responses9 because, in the infinite lag 

models, the negative coefficients on the profit variables 

(table 1) inflate the role of the lag function in inducing 

exploration effort.

APPLICATION OF THE DISCOVERY PROCESS MODEL 

PREDICTIONS

A fundamental premise of the approach to field exploration modeling taken here is that operators are 

induced to explore as long as there are high expectations associated with the value of remaining deposits. 

In order to test this premise, a discovery process model 

was used to provide predictions of the undiscovered recoverable resources as a function of the cumulative 

number of wells drilled. Two discovery process models 

that are very similar in their initial assumptions and 

that have been estimated for the Denver basin are the 

models described in Arps and Roberts (1958) and Drew, 

Schuenemeyer, and Root (1980).

• These comments are relevant if the distributed lag model is interpreted to represent 

operators' temporal response functions to new discoveries.
C12 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

Both models proceed from the assumption that the 

probability of finding a deposit in a particular size class 

is proportional to the number of undiscovered deposits 

of that size and proportional to the ratio of the area of 

the typical deposit of the deposit class to the relevant 

area of the basin under exploration. The model of Arps 

and Roberts (1958) specifies that the rate of discovery 

(within a size class of deposits) declines exponentially 

as a function of the number of exploratory holes drilled. 

Alternatively, Drew, Schuenemeyer, and Root (1980) 

use a somewhat different functional form to express 

the decline rate. They are able to estimate objectively 

the discovery efficiencies and relevant basin size, 

whereas Arps and Roberts used prior subjective information to make these estimates. The discovery process 

model described below is similar to the Arps-Roberts 

model in functional form, although the data used to 

estimate the model are more recent and accurate. Because of its computational simplicity, the model of Arps 

and Roberts was used instead of the model by Drew, 

Schuenemeyer, and Root (1980). Both models appear to 

predict the historical discovery sequence quite well 

(Drew, Schuenemeyer, and Root, 1980). 10

One function of the discovery process model is to provide a framework for predicting future recoverable resources as a function of the cumulative number of wells 

drilled. An important component of the model is the 

estimate of the ultimate or total number of deposits 

within the particular size class of deposits, given as 

Fj(w) where i denotes the deposit size class and u is an 

arbitrary large number of wells (perhaps infinite) that 

exhausts or finds all the deposits in the ith size class. 

This estimate positions the extrapolation of the discovery decline curve. Given the parameters of the discovery process model, historical data may be used to obtain an estimate ofFt (u). During the discovery history 

of the basin, estimates of F,(w) may be made and 

operator exploration behavior tested in order to determine (1) if exploration is induced by the expectations 

associated with the remaining deposits and (2) if exploration is sustained by expectations attached to specific 

size classes of deposits or the cumulative value of remaining deposits.

The Arps-Roberts discovery process model has the 

following form:

Fi(w)=Fi(u)(l-e~0tW) (16)

where FI (w) is the predicted number of deposits within 

size class i, Ft (u) is the ultimate number of deposits in 

the class, w is the cumulative number of wildcat wells,

10 Root and Schuenemeyer (1980) show the Arps and Roberts (1958) model and the limit 

discovery process model of Drew, Schuenemeyer, and Root (1980) to converge in the limit.

and 0, is a constant that includes the basin size, the 

area of the typical deposit from a given size class of 

deposits, and the exploration efficiency (Arps and 

Roberts, 1958, p. 2563). The basin size B was taken to 

be 5,700,000 acres (approximately 23,068 km2) (Arps 

and Roberts, 1958). The exploration efficiency was assumed to be 2 for all size classes of deposits, and the 

area At associated with a deposit size class i was taken 

from Drew, Schuenemeyer, and Root (1980). n More 

specifically for the ith deposit size class,

(17)

B

Although exploration efficiency does vary across size 

classes of deposits, tests of the model using the constant 2 for all deposit size classes did not produce any 

appreciable loss in predictive accuracy. Arps and 

Roberts (1958) indicate that predictions of the model 

are equivalent to finding the (mathematical) expected 

value of the number of discoveries to be made with a 

given search effort. Consequently, when this discovery 

process model is used to predict the value of undiscovered deposits, the specific forecasts will be referred to 

as "rational" predictions or "rational" expectations.

The "rational" predictions of the number of remaining deposits for each size class of deposits was generated in the following manner. An estimate of Ft(u) 

was made by calculating

(18)

where Ft(w) is the number of deposits already found 

with w wells within the ith size class. The number of 

deposits remaining to be found is the difference between Fi(u) and FI(W). By assuming a current price of 

oil and an average size to deposits within each size 

class, the value of oil remaining to be found in the 

basin is predicted after summing the values across size 

classes. The estimation value of remaining deposits is

Deposit size

in 

millions of barrels

Area in acres

per square kilometer

(values in parentheses

are in square kilometers)

0.0 - 0.002 _________-_______________ 2.500 (.010)

.002- .004.._______________________ 3.900 (.016)

.004- .008_________________________ 6.100 (.025)

.008- .016________________-____-__ 11.700 (.047)

.016- .032_________________________ 19.400 (.079)

.032- .064_________________________ 34.000 (.138)

.064- .128—————————— ——— 54.700 (.221)

.128- .256_______________________ 97.300 (.394)

.256- .500 _________________________ 216.960 (.878)

.500- 1.00 -_____-__________-______ 328.320 (1.329)

1.00 - 2.00 _______-_______——_____- 481.920 (1.950)

2.00 - 4.00 _______________________ 697.600 (2.823)

4.00 - 8.00 _______-_______________-1068.800 (4.325)

8.00 -16.00 __________-_____________1324.800 (5.361)

16.00 -32.00 _________________________2752.000 (11.137)

32.00 -64.00 ______________-_______6400.000 (25.900)
AN APPLICATION TO SUPPLY MODELING C13

updated each period using both the most recent discovery data and the data from previous periods. This 

scheme for generating "rational" expectations can naturally be interpreted to suggest that each period 

operators using available information sequentially 

update their estimates of the value of undiscovered deposits remaining in the basin. Empirical tests that are 

carried out are related to how closely levels of exploration effort correspond to the predicted value of remaining deposits and whether exploration is correlated 

more with predictions for specific size classes of deposits or with the cumulative value of deposits remaining. 

The models explaining exploration behavior were 

specified to include the lagged profit term and the predicted value of undiscovered deposits. Because the discovery process model implicitly takes into consideration the effects of physical exhaustion, the variable kt 

was omitted from the model specifications. Estimates 

of Fi(u) can be quite erratic when there are relatively 

few wildcat wells drilled in the basin. Consequently, 

the data set that was used included the period from the 

third quarter in 1951 through 1973. Again, the 

Cochrane-Orcutt procedure was applied to handle 

problems of autocorrelation (Kmenta, 1971). The estimated equations and the statistics describing the quality of the estimates are presented in the following 

equations: 

for drilling,

yw = 14.160+ 0.00934^_2 + 0.10172 x? (19) 

(0.4) (1.4) (3.3)

#2=0.885 S.E.=20.72 Durbin-Watson statistic = 

1.45;

for drilling expenditures,

ye = -49.010+ 0.02559^^ t + 0.58802 x? (20) 

(0.4) (1.7) (3.6) (3.6)

#2=0.898 S.E. = 104.36 Durbin-Watson statistic = 

1.72.

In these equations R2 is the adjusted coefficient of determination, S. E. is the standard error of the regression, and the numbers in parentheses below the coefficients are the associated "t" statistics. The variable xf 

represents the total value (across all classes) of deposits remaining to be found in the basin. Regressions 

using particular size classes of deposits for the variable 

associated with operators' expectations were also carried out. Although they produced slight differences in 

explanatory power, the improvements were not 

significant. From these results it can be concluded that, 

with reference to exploration, operators appear to react

to the total perceived economic value of undiscovered 

deposits rather than to the presence of a specific size of 

deposits.

Predictions generated by the discovery process model 

could also be used as arguments in a distributed lag 

model. This use might be rationalized by assuming 

that the current estimates of the value of undiscovered 

deposits to which operators react are weighted averages of previous estimates. Several model 

specifications of this type have been estimated. However, the results indicated no substantive improvement 

in the statistics describing the quality of the models. In 

some cases, the signs of the lag coefficients were difficult to rationalize. Consequently, the model estimates 

are not presented or discussed in the subsequent 

analysis.

The qualitative statistics that describe the fits of .the 

ad hoc distributed lag model and the model basedf on 

the predictions of the discovery process model are comparable. It would be of particular interest to determine 

if the predictive performances of the models are similar. For a situation of equal reliability, the use of the 

simpler model is to be preferred since it has a somewhat less ad hoc foundation and appears to use available observations for parameterization efficiently. The 

motivation for comparing the performance of the 

models beyond their respective qualitative statistics is 

to identify systematic relative weaknesses. Procedures 

for the comparative evaluation of econometric models 

have only recently been discussed in detail in the literature (Dhrymes and others, 1972). During the process 

of constructing the present models, several evaluation 

criteria have already been applied. These criteria concern how well the model structure conforms to accepted 

economic theory and if the estimates are theoretically 

consistent. Other techniques compare the pattern of 

the model predictions with the historical data. One 

method is to examine whether the model predicts the 

turning points that actually occur in the dependent 

variable. In this regard, the performance of both 

models was relatively poor. Generally, the predicted 

turning points were lagged one period from the historical turning points. For the well-drilling equations, 

both models predicted only 2 of the 27 turning points 

correctly; for the expenditures equations, the "rational" model predicted 5 of 26, and the distributed lag 

model predicted 3 of 26 correctly. One explanation for 

the relatively large number of turning points in the 

data is that the data appear to contain a residual seasonal variation that was not removed.

Another technique frequently used in model evaluation is to examine the pattern in the residuals. Figures 

4 and 5 plot the residuals of the two models over a 

common sample period. The plotted residuals do not
C14 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

_ 

LLJ

60

50

40

30

20

10

0

-10

-wo -20 x-

-30

-40

-50

-60

o o 

x x

EXPLANATION 

x Residual of "rational" model

o Residual of distributed lag model

54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

YEAR

FIGURE 4.—Residuals for well-drilling models.

indicate systematic variation in either figure or for 

either model specification. The presence of a systematic 

component in the residuals would indicate that the 

model was misspecified. That is, if a significant variable was omitted from the specification, then a pattern 

in the residuals would correspond to systematic variations of the omitted variable. The most extreme residuals are accounted for by the "rational" model. Although the residuals vary widely for drilling from the 

third quarter of 1968 through 1970, the residuals for 

the drilling expenditure models indicate less erratic 

behavior.

Economic theory rarely specifies the functional 

forms that are used in applied econometric modeling. 

Consequently, in the process of model selection, several 

alternative functional forms are generally examined, 

and the final form is chosen on the basis of goodness of 

fit. This procedure is not entirely satisfactory, because 

specification errors resulting from incorrect or omitted 

variables are still possible. Specification errors may 

lead to biased estimates and predictions. A technique 

used to detect specification errors is to test for stability 

or structural changes in the model parameters during 

the original sample period (Jorgenson and others, 

1970; Dhrymes and others, 1972). In particular, the 

full data sample is split into two subsamples or periods,

and the models are reestimated. The resulting parameters are tested to determine if statistically significant 

changes have taken place in their values.

The procedure used in detecting structural changes 

is the following: Suppose Q2 is the sum of squared residuals associated with each subsample. Further, letQx 

be the sum of squared residuals of the regression based 

on the pooled data. Under the hypothesis that the sets 

of regressors are equal for the two subperiods and the 

pooled sample, the test statistic is distributed according to theF distribution and is calculated (Chow, 1960)

as

Q2/(m+n-2kY

(21)

with degrees of freedom (k,m+n—2k). The variables m 

and n are the number of observations in each subsample, and k is the number of parameters that must be 

estimated in each model.

For the distributed lag models, the two subperiods 

were from the fourth quarter of 1952 to the first quarter of 1962 and from the second quarter of 1962 to the 

fourth quarter of 1973. The sample periods used for 

tests of the "rational" model were from the fourth quarter of 1953 to the third quarter of 1962 and from the
AN APPLICATION TO SUPPLY MODELING C15

O

I

O

LJJ

400

350

300

250

200

150

100

50

£ 0

QC

3

o 

o

o o x

x o o ^

-50

-100o

-150

-200

-250

o o 

o

°0

0 0

-300

x o

I

EXPLANATION 

x Residual of "rational" model 

o Residual of distributed lag model

54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73

YEAR

FIGURE 5.—Residuals for well-drilling expenditures models.

fourth quarter of 1962 to the fourth quarter of 1973. 

The level of type I error was set at 0.05; that is, one 

could expect to reject the null hypothesis about 1 time 

in 20 even if it were true. Rejection of the null 

hypothesis would indicate temporal parameter 

changes that might be the result of model specification 

error. For the distributed lag model the "F" statistics 

were 1.34 and 0.58 for the drilling and drilling expenditures equations, respectively. The model based on 

the predictions of the discovery process model had F 

statistics of 1.57 and 0.74 for the drilling and drilling 

expenditures equations, respectively. Thus there were 

no instances when the test statistic exceeded the corresponding critical level and the null hypothesis was 

not rejected, and the results of the tests for structural 

change provided no evidence that specification errors 

might be present in either set of models.

In conclusion, the test results comparing the distributed lag model and the model based upon predictions 

from the discovery process models indicated little difference in the potential predictive performance of the 

models. This finding is significant for three reasons.

First, it appears that the predictions of the discovery 

process model contain most, if not all, of the information included in the distributed lag functions of Dt and 

the experience-exhaustion variable kt . Because of its 

relative simplicity, it may be more appropriate to use 

this model instead of the distributed lag model in 

applied situations. Second, if the discovery process 

model efficiently uses available information to generate predictions, then the specific distributed lag 

proxy used here does seem to capture operator expectations. Third, the reasonably high explanatory power of 

the "rational" model appears to indicate that the 

operators are using information contained in the 

discovery history of the basin efficiently and operating 

in a fashion consistent with these predictions.

SUPPLY OF RESERVES

The behavioral drilling models presented in the preceding sections appear adequate to explain the historical pattern of operator behavior in the basin. As such, 

they served as a basis for making inferences concerning the determinants of firm behavior. The estimated
C16 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

behavioral models may also be applied in conjunction 

with the discovery process models to forecast the temporal pattern of new reserves, given certain assumptions about crude oil prices and drilling costs. The 

usefulness of forecasts of future reserves from undiscovered deposits is obvious. Furthermore, the generation of the forecasts by the analytical model allows the 

predictions to be conditioned on prices, well costs, and 

the current state of physical exhaustion. However, the 

accuracy of the forecasts is generally constrained by 

the nature of the models and the relevant behavioral 

and technical relations.

The estimated discovery process model mechanically 

provides predictions of the number of deposits within 

each size class as a function of the cumulative number 

of wildcat wells. To obtain a prediction of the reserves 

found for a given increment in wells, the field size (in 

barrels of oil) is multiplied by the incremental value in 

the predicted number of deposits. By linking the discovery process model to the behavioral drilling model, 

the predicted number of wildcat wells (per unit time) 

can be used to generate a forecast of the amount and 

value of reserves forthcoming. Because of the nonlinear nature of the discovery process model, the linking of it and the behavioral drilling model must be 

relatively simple. Consequently, the integrated model 

is recursive in nature.

Although the discovery process model is independent 

of time, the behavioral drilling model is not. The drilling model was estimated with quarterly data, and consequently forecasts of the number of wildcat wells are 

also on a quarterly basis. Moreover, the model is 

specified so as to describe shortrun operator decisions. 

In general, the ability of econometric models to provide 

accurate forecasts deteriorates rapidly as the number 

of periods that the forecasts are made into the future 

increases. The extent of the deterioration can, to some 

degree, be determined by carrying out simulation experiments with the integrated equation system. In particular, these tests attempt to determine how rapidly 

the accuracy of the forecasts deteriorate, the robustness of the predictions of one endogenous variable 

when errors are in the other endogenous variable, and 

whether forecast errors are compensating or noncompensating as the results from individual time periods 

were aggregated.

A measure used to determine the quality of the integrated model's forecasting performance was the rootmean-square (RMS) prediction error (Thiel, 1966). It is 

defined as

/Tn

RMS= A / n 2 (Pi-Atf, 

V i=l

(22)

where Ai is the actual value of the predicted variable 

and Pi is the prediction. Also used as measures of performance were the mean error (over several forecast 

periods) and the ratio of the RMS prediction error to 

the actual mean value of the variable predicted.

The integrated model is specified as a recursive equation system. Because of its simplicity, the "rational" 

well drilling specification was used rather than the distributed lag model. Variables that are endogenous or 

determined within the system are

yt = number of wells drilled in period t and

Rt = reserves discovered in period t. 

Variables that are taken as given or determined 

exogenously are

cwt = cost per well in period t,

pt = price of crude oil, and

U0 — the initial value of undiscovered deposits. 

Among the predetermined variables that are taken as 

given are the initial values of the lagged endogenous 

variable yt-i and Tft.2. The discovery success ratio is not 

predicted, and consequently the profit variable was redefined to have the following form:

--CWt-2, (23)

where Rt ,pt, cwt , and yt were defined as above. In the 

original estimation of the "rational" model, Ft (u) was 

reestimated each period as new information became 

available or as discoveries were made. Reestimation of 

Fi(u) each period for the integrated model would result 

only in the initialized value of the variable. Therefore, 

the final empirical estimates of the ultimate numbers 

of deposits for each size class were used. Because of the 

redefinition of the profit variable and the use of the 

final empirical estimates forFj(u), the "rational" well 

drilling equation was reestimated. The new equation is 

of the following form:

= 8.386+0.06339^-2+ 0.10358 

(0.2) (1.7) (3.0)

(24)

#2 =0.881 S.E.=21.1 Durbin-Watson statistic = 1.53,

where the numbers in parentheses below the model 

coefficients are the "£" statistics, R2 is the adjusted coefficient of determination, and S. E. is the standard 

error of the regression equation. Using the CochraneOrcutt procedure to handle serial correlation, the 

first-order correlation coefficient was estimated to be 

0.814. The predictive form of equation (24) (Kmenta, 

1971) is

y,=0.81ty«_1 +(l-0.814)8.386+0.06339(ir,_2 -0.814w<: _3) 

+0.10358(xf-0.814xf-i). (25)
AN APPLICATION TO SUPPLY MODELING C17

The predicted reserves found per time period, Rt , is 

calculated by taking the cumulative number of wells to 

period t-1, calculating the predicted (total) reserves, 

adding the number of wells drilled in period t to the 

cumulative well count, recalculating the predicted (total) reserves, and taking the difference between two 

total reserve predictions. The value of undiscovered 

deposits at the beginning of time t+l was forecast by 

taking the difference between the predicted amount of 

discovered reserves at t and the sum of Ft(u) across 

classes and multiplying the result by the crude oil price 

at the time. Predictions of yt , Rt-i, and x*t+i are employed in (23) and (25) to generate the predicted 

number of wells for period yt+i, which in turn is used to 

predict Rt+i.

The historical period from the first quarter of 1960 to 

the fourth quarter of 1968 served as the base period for 

examining the integrated model's forecasting ability. 

This period was chosen because the estimated ultimate 

number of deposits in each size class had stabilized 

(Drew and others, 1980). The test period was cut off at 

the end of 1968 in order not to include the perod when 

the Union Pacific Railroad acreage began to be drilled. 

We believed that the initiation of drilling on the new 

acreage represented an entirely exogenous influence, 

and the model could not be expected to reproduce it.

In order to determine how the accuracy of the forecasts was affected by the length of the forecast period, 

that is, how far ahead forecasts should be made, three 

sets of simulations were generated for the 9-year 

period. These include forecasts of 4, 8, and 12 periods 

(quarters) ahead. For example, during the 36 periods 

there were nine sets of 4-period forecasts. Using 

8-period forecasts, four sets were generated, and for the 

12-period forecasts, three sets were generated. Quarterly observations, particularly for new discoveries, 

have substantial stochastic components. Therefore, it 

is also of interest to determine if prediction errors during the forecast period offset each other when results 

for the time periods are aggregated.

Results of the experiments are presented in table 4. 

The table indicates the RMS prediction error, the mean 

(actual-predicted) prediction errors over the forecast 

time period, and the mean value of the actual variable 

that is being forecasted. The pattern of the prediction 

errors for wells indicates the degree by which 

lengthening the forecast period increases the RMS 

prediction errors and mean prediction error. The overall mean values of the RMS to the actual variable 

value are 0.141, 0.240, and 0.257, and the ratios of the 

value of mean period prediction error to the mean actual values are 0.033, 0.162, and 0.189 for the 4-, 8-, 

and 12-period forecasts, respectively. The obvious effect of increasing the forecast period is the accumulaTABLE 4.—Forecast performance of integrated system

Dependent 

Variable

Forecast 

length' Period

A 1 __ .

2 __ .

3 ... ...

4 . .

5 .

6

7 ......

8 - .....

9 _ ...

Mean Values

B 1 ..

2 ... ...

3 ......

4 _ ...

Mean Values

C 1 .

2 ......

3 ... ...

Mean Values

RMS 

errors2

7.551 

12.795 

13.183 

8.509 

5.227 

20.195 

12.703 

9.783 

19.146

12.121

22.953 

19.434 

13.424 

28.126

20.984

20.438 

15.643 

29.878

21.986

Wells

Mean 

errors3

3.589 

-9.213 

.868 

5.915 

3.679 

14.769 

9.698 

6.882 

-10.847

2.816

14.986 

12.399 

5.803 

23.671

14.215

13.099 

11.746 

23.761

16.202

Reserves

Actual RMS Mean Actual 

mean errors errors mean

109.250 

92.250 

110.000 

90.500 

86.250 

89.250 

68.750 

54.250 

70.750

85.694

100.750 

100.250 

87.750 

61.500

87.563

103.833 

88.667 

64.583

85.694

2.103 

3.183 

3.036 

2.969 

1.928 

1.927 

3.381 

1.298 

.585

2.268

2.729 

2.987 

1.915 

2.549

2.545

2.860 

2.345 

2.102

2.436

0.786 

-.025 

-.573 

.277 

.330 

1.346 

-.519 

-.526 

-.458

.071

0.776 

.059 

.805 

-.275

.341

0.341 

.753 

-.141

.318

3.763 

3.161 

4.111 

2.555 

2.096 

1.142 

2.374 

2.069 

1.837

2.567

3.462 

3.333 

1.619 

2.221

2.659

3.678 

1.931 

2.093

2.567

'Forecast lengths A, B, and C are 4, 8, and 12 quarters, respectively. 

'Root-mean-square value is given by ( N 2 (P, - A( )2)8-5, where Pt is the predicted value and At is the actual value.

'Mean error is based on actual minus predicted value.

tion of prediction errors. It does not appear that the 

percentage errors increase linearly with the forecast 

period. To some degree, it appears that errors will be 

offsetting if the forecast period is taken to be sufficiently short, as evidenced by the relatively low mean 

error to actual value of 0.033 for the 4-period forecast 

simulation. As the forecast period is increased, the 

number of wells predicted appears to be systematically 

overestimated.

Results for the predicted amount of reserves found in 

new deposits are also presented in table 4. The discovery process model, used to predict the number of deposits and reserves forthcoming, produces relatively 

smooth estimates of forthcoming reserves. However, 

the actual number and sizes of new deposits, particularly for as short a period as a quarter, are highly erratic in nature. As a result, individual prediction errors 

as measured by the RMS prediction errors can be expected to be relatively large. However, if the model is 

operating reasonably well, in aggregating the predictions over time, errors will be compensating so that the 

mean prediction error should be relatively small. Results of the simulations appear to be consistent with 

this conjecture. The RMS prediction errors are very 

large, even for the set for which only 4-period forecasts 

were made. However, the mean prediction errors taken 

over the entire period are relatively small. The ratios 

of the mean prediction error values to the actual mean 

values are 0.028, 0.128, and 0.124 for the simulations 

based on the 4-, 8-, and 12-period forecasts, respectively. Individual errors in predicting forthcoming
CIS PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

reserves tend to offset each other as the errors are 

aggregated over time. The supply of reserves for increments in the drilling rate are too random to be predicted on a quarterly basis. Aggregation of quarterly 

forecasts to obtain annual values results in reasonably 

accurate predictions. The positive values of the mean 

forecast error for the 8- and 12-period simulation experiments does not necessarily mean that forecasts of 

the discovery process models are generally biased. The 

positive mean prediction error is probably the result of 

the overestimates for the number of predicted wells 

drilled as discussed earlier.

In summary, the prediction errors of the behavioral 

well-drilling model seem to be biased positively. However, the magnitude of the RMS and mean errors are 

quite comparable to the performance of well-drilling 

equations reported in other studies (MacAvoy and Pindyck, 1975) for similar forecast periods, that is, 4-8 

periods ahead. The relatively high RMS errors for the 

additions to reserves appear to be more the result of the 

erratic or stochastic nature of the historical arrival of 

discoveries than systematic bias in the discovery process model. The relatively small mean prediction 

errors, particularly for the 4-period forecasts, indicate 

that if the quarterly forecasts of additions to reserves 

were aggregated, perhaps on an annual basis, and 

compared to the historical realizations, the RMS prediction errors would also be much smaller. The relatively small mean prediction error associated with the 

additions to reserves seems to suggest that the forecasts are reasonably robust even though there were 

errors in predictions of the number of wells to be 

drilled.

Although the linking of the discovery process model 

with the behavioral well-drilling model produced an 

analytical means of translating the forecasts of reserves per unit exploration effort, that is, wells drilled 

to reserves per unit time, several limitations of the 

analysis should be kept in mind. First, the economic 

model was specified to describe operator field decisions 

that are short term in nature. That is, the nature of 

decisions that are modeled are marginal adjustments 

in the rate of exploration rather than decisions to enter 

or exit a geologic basin. Second, because of the shortrun nature of the models, it would be misleading to 

attempt to draw general conclusions about the effects 

of a general price change on drilling activity within the 

basin. That is, a general price change would induce 

some longrun adjustments to take place in the firm's 

internal allocation of resources across several regions. 

The type of price change that the behavioral welldrilling model might more appropriately capture corresponds to a change in the relative price of oil, for 

instance, the price of oil found in the Denver basin as

opposed to another basin. In order to predict the effects 

of a general change in the price of oil on drilling behavior for a particular basin, the behavioral well-drilling 

equation should be respecified to reflect the firm's longrun decisions and include a variable that would denote 

the firm's alternative exploration opportunities in 

other geologic basins or its alternative opportunities 

for obtaining additional reserves.

CONCLUSIONS

The purpose of this study was to examine operator 

exploration behavior at the field level. In the first part 

of the study an empirical model was specified and estimated. Distributed lag proxy variables were used to 

model operator expectations associated with the distribution of deposits remaining to be found in the basin. 

The estimated drilling models permitted inferences to 

be made about operator field behavior. First, if the 

form of the distributed lag function is interpreted to 

represent the operators' responses to new discoveries, 

then the nature of function is important. The appropriate form of the function was a finite polynomial lag 

function with an inverted "V" shape; that is, the effects 

of new discoveries increased for several periods and 

then diminished rapidly to 0 within 2Mi years after discovery of the deposit. The independent variable used in 

the distributed lag model was the total value of deposits found in a given period with a minimum size of 

500,000 barrels. Second, tests using alternative 

minimum size classes of deposits, that is, 2.5 million 

and 5 million barrels, in the distributed lag function 

produced no improvements in the fits. This result implies that, once an exploration play is underway, 

operator field expectations are sustained by the quantity of oil found per time period rather than by the 

discovery of specific size deposits. Comparison of these 

results and the regression coefficients showed them to 

be quite consistent with the results of a similar previous study based on the discovery history of the Powder 

River basin (Attanasi and Drew, 1977).

In the second part of the study, the discovery process 

model was applied to generate sequential predictions of 

the value of undiscovered deposits, which were used in 

the behavioral well-drilling model in place of the distributed lag operator-expectations proxy variable. The 

discovery process model efficiently uses information 

contained in the historical time series to estimate the 

future discoverie's. Consequently, it was assumed that 

such predictions would closely correspond to "rational" 

expectations where "rational" is used in the sense that 

all available information in the historical data is used 

efficiently in the estimation process. Using the predictions from the discovery process model as the unobservable expectations variable associated with the
AN APPLICATION TO SUPPLY MODELING C19

value of undiscovered deposits, the behavioral welldrilling equation was reestimated. The estimated "rational" model explained historical operator behavior 

well. This result seemed to indicate that operators, on 

the average, were efficiently using information contained in the discovery history of the basin and were 

behaving in a fashion consistent with these predictions. Comparison of the performance of the distributed 

lag model with that of the "rational" model indicated 

little differences in the predictive performance of the 

two models. The predictions of the discovery process 

model, used as "rational" expectations, appear to contain most if not all of the information included in the 

distributed lag functions and experience-exhaustion 

variable.

In the final section of the study, the behavioral 

well-drilling model and the discovery process model 

were integrated into a recursive equation system. The 

purpose of linking the models was to gain the ability to 

forecast additions to reserves in the time domain or on 

a unit time basis. The forecast performance of the integrated system was tested by examining the accuracy 

for various lengths of forecast periods. As might be 

expected, as the number of forecast periods increased, 

the accuracy of the forecasts deteriorated. However, 

the level of accuracy was still comparable to levels attained in other studies. Because of the stochastic nature of the discovery process, aggregation of quarterly 

predictions of additions to reserves to annual values 

tended to increase accuracy as measured by the mean 

error of prediction. The accuracy of the aggregated 

predictions appeared to be reasonably robust to the 

presence of prediction errors in the behavioral welldrilling equations. Finally, the integrated model provides for the first time a means of explicity incorporating physical and economic exhaustion in the generated 

profit predictions and predictions of the value of remaining deposits into a behavioral model that describes the exploration effort (wells and expenditures) 

of operators.

REFERENCES CITED

Almon, Shirley, 1965, The distributed lag between capital appropriations and expenditures: Econometrica, v. 33, no. 1, p. 176-196.

American Petroleum Institute, 1953, 1955-56, 1959-73, Joint Association Survey of the U.S. Oil and Gas Producing Industry: 

Washington, B.C., American Petroleum Institute.

Arps, J. J., and Roberts, T. G., 1958, Economics of drilling for 

Cretaceous oil on the east flank of the Denver-Julesburg basin: 

American Association of Petroleum Geologists Bulletin, v. 42, 

no. 11, p. 2549-2566.

Attanasi, E. D., 1978, Firm size and petroleum exploration, Council 

of Economics of American Institute of Mining, Metallurgical, 

and Petroleum Engineers, Annual Meetings, Denver, Proceedings: p. 57-62.

Attanasi, E. D., and Drew, L. J., 1977, Field expectations and the

determinants of wildcat drilling: Southern Economic Journal, v. 

44, no. 1, p. 53-67.

Box, G. E. P., and Jenkins, G. M., 1970, Time series analysis: San 

Francisco, Holden-Day, Inc., 553 p.

Chow, G. C., 1960, Tests of equality between sets of coefficients in 

two regressions: Econometrica, v. 28, no. 3, p. 591-605.

Cox, J. C., and Wright, A. W., 1976, The determinants of investment 

in petroleum reserves: American Economic Review, v. 66, no. 1, 

p. 153-167.

Dhrymes, P. J., 1971, Distributed lags: Problems of estimation and 

formulation: San Francisco, Holden-Day, Inc., 414 p.

Dhrymes, P. J., Howrey, E. P., Hymans, S. H., Kmenta, Jan, Learner, 

E. E., Quandt, R. E., Ramsey, J. B., Shapiro, H. T., and Zarnowitz, Victor, 1972, Criteria for evaluation of econometrica 

models: Annals of Economic and Social Measurement, v. 1, no. 3, 

p. 272-328.

Drew, L. J., Schuenemeyer, J. H., and Root, D. H., 1980, Petroleumresource appraisal and discovery rate forecasting in partially 

explored regions; Part A, An application to the Denver basin: 

U.S. Geological Survey Professional Paper 1138-A, 11 p.

Epple, D. N., 1975, Petroleum discoveries and government policy: An 

econometric study of supply: Cambridge, Mass., Ballinger, 139 p.

Erickson, E. W., and Spann, R. M., 1971, Supply response in a regulated industry: The case of natural gas: Bell Journal of Economics and Management Science, v. 2, no. 1, p. 94-121.

Ezekiel, M. J., 1938, The cobweb theorem: Quarterly Journal of Economics, v. 53, no. 1, p. 255-280.

Fama, E. F., 1972, Perfect competition and optimal production decisions under uncertainty: Bell Journal of Economics and Management Science, v. 3, no. 1, p. 151-174.

Federal Energy Administration, 1974, Project independence blueprint: Final task force report: Oil: Possible levels of future production: Washington, D.C., U.S. Government Printing Office, 

162 p.

———1976, National energy outlook: Washington, D.C., U.S. Government Printing Office, 190 p.

Fisher, F. M., 1964, Supply and costs in the U.S. petroleum industry: 

Baltimore, Johns Hopkins University Press, 177 p.

Johnston, J. J., 1972, Econometric methods [2d ed.]: New York, 

McGraw-Hill, 437 p.

Jorgenson, D. W., 1966, Rational distributed lag functions: 

Econometrica, v. 34, no. 1, p. 135-149.

Jorgenson, D. W., Hunter, J. S., and Nadiri, M. L, 1970, The predictive performance of econometric models of quarterly investment 

behavior: Econometrica, v. 38, no. 2, p. 213-224.

Kauftnan, G. M., 1963, Statistical decision and related techniques in 

oil and gas exploration: Englewood Cliffs, Prentice-Hall, 307 p.

Khazzoom, J. D., 1971, The FPC Staffs econometric model of natural 

gas supply in the United States: Bell Journal of Economics and 

Management Science, v. 2, no. 1, p. 51-93.

Kmenta, Jan, 1971, Elements of econometrics; New York, MacMillan, 655 p.

Leland, H. E., 1974, Production theory and the stock market: Bell 

Journal of Economics and Management Science, v. 5, no. 1, p. 

125-144.

MacAvoy, P. W., and Pindyck, R. S., 1973, Alternative regulatory 

policies for dealing with the natural gas shortage: Bell Journal 

of Economics and Management Science, v. 4, no. 1, p. 454-498.

———1975, The economics of the national gas shortage (1960-1980): 

Amsterdam, North-Holland, 259 p.

National Petroleum Council, 1972, U.S. energy outlook: Washington, D.C., U.S. Government Printing Office, 260 p.

Nerlove, M. L., 1958, Distributed lags and estimation of long-run 

supply and demand elasticities—theoretical considerations: 

Journal of Farm Economics, v. 40, no. 2, p. 301-311.
C20 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING

Pesando, J. E., 1976, Rational expectations and distributed lag expectations proxies: Journal of the American Statistical Association, v. 71, no. 1, p. 36-42.

Pierce, D. A., 1971, Fitting dynamic time series models—some considerations and examples: Annual Symposium on the Interface 

Between Computer Science and Statistics, 5th Stillwater, Okla., 

Proceedings, p. 1-2.

Pindyck, R. S., 1978, Higher energy prices and the supply of natural 

gas: Energy Systems and Policy, v. 2, no. 2, p. 177-209.

Root, D. H., and Schuenemeyer, J. H., 1980, Petroleum-resource appraisal and discovery rate forecasting in partially explored regions; Part B, Mathematical foundations: U^S. Geological Survey Professional Paper 1138-B, 9 p.

Schmidt, P. J., and Waud, R. N., 1973, The Almon lag technique and 

monetary versus fiscal policy debate: Journal of the American 

Statistical Association, v. 68, no. 1, p. 11-19.

Shiskin, Julius, Young, A. H., and Musgrove, J. C., 1967, The X-ll 

variant of the Census Method II seasonal adjustment program: 

Washington, D.C., U.S. Department of Commerce, 66 p.

Thiel, Henrie, 1966, Applied economic forecasting: Amsterdam, 

North-Holland, 474 p.

Turnovsky, S. J., 1969, A Bayesian approach to the theory of expectations: Journal of Economic Theory, v. 1, no. 3, p. 220-227.

Uhler, R. S., 1976, Costs and supply in petroleum exploration—the 

case of alberta: Canadian Journal of Economics, v. 9, no. 1, p. 

72-90.

White, D. A., Gerrett, R. W., Marsh, G. R., Baker, R. A., and 

Gehman, H. M., 1975, Assessing regional oil and gas potential, 

in Haun, J. D., ed., Methods of estimating the volume of undiscovered oil and gas resources: Tulsa, American Association of 

Petroleum Geologists, p. 143-159.

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