INTRODUCTION

In recent years there has been increasing interest in the possibility of forecasting the profitability of oil-exploration ventures. Several authors have attempted to establish procedures for dealing with this intricate problem. The factor of risk is dominant in the economic evaluation of exploration ventures, and it must be emphasized in every calculation that is made.

Oil exploration involves financial investments for land acquisition, geological and geophysical surveys, and drilling that can be estimated on the basis of average costs for a program to evaluate the prospects of a given area. The return on these investments is entirely uncertain because an oil-exploration venture is not necessarily successful. Even if successful, parameters, such as size, porosity and permeability of the reservoir rock, gravity and composition of the oil, and other physical properties which combine to determine the optimum rate of production and the market value of the product are not predictable in advance.

To compare investments and income in a preliminary economic evaluation, it is necessary to express numerically the probability of discovering a certain hypothetical deposit. The risk factor therefore is involved in every decision in oil exploration. Here, risk-evaluation procedures developed in recent years are summarized and examined critically to formulate a new approach which, though modest in aim, has the merit of being applicable to practical cases. It is primarily adaptable to ventures outside the United States and southern Canada.

PROBABILITY OF SUCCESS IN DISCOVERING OIL

GRAYSON'S METHOD

Grayson (1960) described procedures for preliminary economic evaluation of investments and income. These are aimed at statistical expression of the ideas, facts, and objectives involved in a decision, under conditions of uncertainty, to drill an exploration prospect. The process is divided into three stages: (1) construction of a "payoff table" showing all combinations among possible decisions and possible results; (2) assignment of a numerical expression to the probability of success; and (3) entering the probability of each event in the payoff table, to determine a weighted average of the consequences, known as the "expected monetary value" of the act.

Possible decisions may be listed as:

1. Not to drill.
2. Drill, retaining 100 per cent interest.
3. Drill in a joint venture with 50 per cent interest held by partner.
4. Farm out, retaining a 1/8 override.
5. Farm out, coming back in for 50 per cent interest after payout.

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Possible results or "events" are the following:

1. Dry hole.
2. Successful well, finding 50,000 bbl.
3. Successful well, finding 100,000 bbl.
4. Successful well, finding 500,000 bbl.
5. Successful well, finding 1,000,000 bbl.

The two lists of possible acts and events are combined in a table. Grayson then embarks on an analysis of the probability factor. His exposition may be summarized under three headings.

1. Income from eventual sales:
It may be assumed (1) that the operator considers there is a 50 per cent probability that the crude oil price will rise 4 per cent per year; (2) that there is a 25 per cent probability that the price will remain unchanged during the next 5 years; and (3) that there is a 25 per cent probability that the price will fall 3 per cent per year. From this an "expected price" can be computed for each year. Given an initial price of $3.00/bbl, Table I shows how this is computed for the first 2 years; by the same procedure this may be extended into future years.

2. Discounting income:
If the future dollar were on hand today it could be invested and would earn profits during the next 10 years. By the tenth year that dollar, together with compound interest, would amount to a larger sum, say $1.63, if invested at 5 per cent. Therefore, a calculation of income should reduce the future dollars to equivalent present values. However, the discounted rate of return varies, depending on what the individual or company would do with the dollar if he had it today. If the investor believes he could obtain a higher rate than the cost of capital (say 5 per cent), he should discount the future dollar at a higher rate. This evaluation is obviously subjective.

3. Probability of success:
Exploratory success ratios covering large areas, such as an entire country, are misleading as a basis for assigning probabilities. Wells drilled to different objectives, located on the basis of a wide variety of geological and geophysical data, have discovered varying amounts of reserves. Most geologists regard local past experience (state, county, basin, or field) as pertinent to a decision on a particular venture. Even on this local basis there are still problems in making the statistics selective to the venture under consideration. The geologist concerned with the evaluation will incorporate all information: past local success ratios, direct evidence for the particular prospect, experience, and judgment when the probability assignment is made. Such probability assignments are known in decision theory as personal probabilities. Personal probabilities may vary widely, as does judgment. The operator must ensure that the evaluation is objective and not biased, either in favor of a high success ratio or the opposite. To obtain the expected monetary value of the various acts which the operator may select, the probability of each occurring is entered in the payoff table and multiplied by potential gain or loss if this event occurs.

In choosing the most profitable solution, additional factors must be taken into account. A 10 per cent probability of finding oil does not mean that one well out of 10 will be successful, but rather that after drilling a large number of wells (better, an infinite number of wells) an average of one successful wildcat out of 10 will be realized. The operator also must evaluate the availability of finance, and how much he is prepared to risk. Within the limits of these restrictive factors, the solution which the table shows to be the most profitable will be selected.

To the above brief summary of Grayson's philosophy the writer makes a basic objection. In determining the probability of certain events, and particularly the probability of success, Grayson recognizes the intrinsic difficulty of calculating "objective" figures and returns to subjective probability figures. This does not improve the understanding of the problem, and leaves the operator with the necessity of making a decision on the basis of subjective ideas. The only advantage attained is the possibility that these ideas may be expressed in tables and handled as numerical values. To the contrary, however, is the risk involved in the utilization of arbitrary data codified in the apparent objectivity of mathematics.

ALLAIS' METHOD

Allais (1957) approaches the problem of the probability of discovery according to a completely

Table I. INCOME PROBABILITY

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different philosophy. Though this study relates to exploration for mineral deposits, it is believed that it is analogous to hydrocarbon exploration. The study is long and complicated, and it attempts to solve an actual case. Two aspects of probability are considered. One is the probability of finding a deposit in an area of a particular size and the other the law of distribution of deposits with respect to value. The first, based on theoretical assumptions and supported by examples, is related to Poisson's law, and the second, after examination of statistical data, is based on log-normal law. Parameters are calculated from statistical data from France, North Africa, the United States, the western United States, and the entire world. In order to give a lower figure for ^sgr (standard d viation), world-wide data are refined by examining three deposits for each country, one producing 75 per cent of the entire output of the country, one producing 15 per cent, and one 5 per cent. Deposits with a yield of less than $500,000 per year are excluded. Finally, statistical data are examined to determine a value of density of mineralization (annual income per sq km output). The figure p, representing the probability of finding a deposit, is determined on the basis of the law of compound probability, using criteria peculiar to exploration for metallic mineral deposits, which cannot be extended to oil exploration.

Arbitrarily estimated numerical values are introduced into the calculation (based on the author's worldwide studies). These are averaged to determine an estimate for p from the number of deposits of a stated value in an area of certain size. The variability of p, calculated according to Poisson's law with a 95 per cent confidence interval, is applied to three cases: unfavorable, median, and favorable. The median of the complete log-normal distribution of the value of deposits is then calculated and a 95 per cent confidence interval is calculated around the lowest and the highest value. The lower extreme of the interval around the lowest value and the upper extreme of the interval around the highest value give the unfavorable and favorable hypotheses between which lies the median hypot esis. The same figures are recalculated by a log-normal law truncated between the limits of greatest practical interest, the lower limit being the minimum value for profitable exploitation of the area being studied and the upper limit the maximum value which may be reasonably expected.

The total value of discoverable deposits is then calculated for each case according to the law of distribution of probability. The value of deposits outside the limits of the truncated law also is calculated, to determine whether these may be ignored. Finally, expected income is evaluated both as a mean value and within the 95 per cent confidence interval. It will be found that in a specific percentage of cases exploration is not profitable, but that, on the the average, it promises a specific profit. If the known figure p is equated with e^-ux (where x = exploration expenditure), u is obtained and therefore the law of variation of p in relation to investment in exploration.

Allais attempts to establish a probability pattern by means of analytical examination of statistical experience. Consider whether a similar pattern may be extended to oil exploration, and whether data obtained from intensively studied areas may be extended to others which have not been explored. Such an extrapolation is not justifiable, because it is equivalent to application of parameters inferred from a sampling of universe U₁ to the study of universe U₂, which is not analogous. Actually, in oil exploration the probability pattern may be identified with two stratified stages. The first involves a series of second-degree systems (the different types of sedimentary basins) from which a certain number of first-degree systems (the basins) is extracted. The second s age requires that a certain number of "balls" (exploratory wells) be removed from each "urn" (type of trap). Thus the parameters for such a pattern are either calculated on the basis of worldwide data (difficult, and in practice impossible except by use of such arbitrary and rough approximations as to deprive the parameters of any meaning), or any attempt to treat the problem as one area on the basis of analogy with other areas must be given up. Worldwide data may be useful only to forecast average results over a large number of areas randomly spread to include the various types of basins. This is not of practical interest.

PROPOSED APPROACH TO PROBLEM

Exploration areas may be divided into three categories: (1) extensively exploited basins, (2) known producing basins, and (3) basins unexplored or explored without success to date.

1. Extensively exploited basins:
In extensively exploited basins a great amount of known data

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may be utilized for establishing statistically meaningful experience. First among the basic criteria is that, in such basins, the probability of discovery is a decreasing function of time. Therefore, instead of considering exploration as a whole, it is better to construct diagrams where frequency of discoveries is plotted versus time. Second, when historical experience may be considered reliable (the most favorable case being when the company concerned is the only holder of concessions in the area), the annual increase of reserves may be compared with the exploration effort incurred. Use of cumulative data facilitates extrapolation; it is probable that exponential functions with a variable base will be involved. The variation in the discovery cost per barrel can also be studied. Where possible, the index of correlation between expenditure and discoveries should be examined. From the procedures outlined, some data may be forthcoming having a probability significance to support an economic evaluation involving risk. Unfortunately, actual cases comparable with the hypothetical one outlined are rare. When such cases occur, an experienced operator may arrive at a relatively safe judgment, which hardly is modified by introducing mathematical-statistical procedures.

2. Known producing basins:
The second category of areas--sedimentary basins about which a great deal of data exist in certain parts, though extensive areas remain unexplored--may be analyzed fairly reliably if the parts of the basin which have been explored represent a good sampling of different structural subdivisions. If this is the case, f (frequency of discoveries) is approximately equal to p, and a confidence interval may be calculated. An approximate Bernouillian situation exists, except when dealing with particular objectives, for which a suitable probability pattern can be selected, according to geological considerations, keeping in mind the conceptual significance of each pattern. The figure p may be modified to relate to the discovery of a pool of a particular size. Naturally, the choice of data from his orical experience must be made carefully, and all data must be avoided where orderly exploration is inhibited by political, legal, or economic factors, or accelerated by the granting of large new concession areas.

This presentation of the problem could be disputed because in oil exploration the location of exploratory wells is not randomly distributed among the various possible traps or structural subdivisions of the basin. Oil companies, after completing a certain number of wells, may utilize the data obtained to concentrate their efforts along a specific trend. In that case they would operate in such a manner that probability of discovery follows the law:

[EQUATION]

in which r is the exploration effort in area A. The difficulty is still only apparent, because this behavior of probability is valid only for a company which has been operating for some time in the area. For a company engaged for the first time in exploration in a particular area the problem of preliminary evaluation is different. The procedure adopted by it would be comparable with that followed by other companies experienced in the area and on whose results the new venture is based. Thus, the respective results should be similar. For the comparison to be valid there should be similarity in the exploration effort and the technical resources of the new company and those on which the data are based.

Examine some concrete cases in which the determination of the figure p may be attempted. In Venezuela, during the years 1956-1958, 313 exploratory wells were drilled in the Eastern Venezuela basin (states of Anzoategui, Guarico, and Monagas), 120 of which were successful. In the interval ±1.28^sgr one has:(FOOTNOTE 3) 35%^lE p^lE 42%. In 1959, 61 exploratory wells were drilled, 25 of which were successful, giving a figure f=40.98%. In the Maracaibo basin (lake and land) during the 3 years 1956-1958, 143 exploratory wells were drilled, of which 67 were successful. In the interval ±1.28^sgr, one has: 42%^lEp^lE 52%. In 1959, 32 wildcats were drilled, 15 successful, giving a figure of f=46.87%. This is a noteworthy result because it deviates from the mean value by only 0.02 per cent, and falls into the interval ±0.005^sgr.

The case of Venezuela is a clear example of the care which must be taken when choosing reference

FOOTNOTE 3. A narrow interval of variability corresponding to a value 0.8 of the Sheppard integral function ^THgr (^lgr) has been chosen in order to have a figure for p which is not too vague. However, any criteria may be chosen, as the variability interval is a matter of opinion. Moreover, some error is introduced by the difference between p and (1 - p), but this is negligible due to the rough approximations of concern here.

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years. In fact, if the years 1956-1959 were compared with the year 1960, the results obtained would be: Eastern Venezuela basin: 36%^lEp^lE 42%, f (1960)=30.76%; and Maracaibo basin: 42%^lEp^lE 52%, f (1960)=32.35%. These large discrepancies have been explained by the exploring companies as due to thorough exploration of existing concessions and the failure of the government to grant new ones.

The same methods may also be used for development wells. Consider the case of development wells drilled in the fields of the Rio Magdalena basin of Colombia from 1949 to 1956.

Table

In attempting to forecast 1956 results with data from 1949-1955 within the interval ±1.28^sgr, one has: 92%^lEp^lE95%, and furthermore, f (1956)=91.04%. The actual figure is slightly lower than the forecast, but the forecast is still of interest. The discrepancy is due to the fact that in 1956 the value of p was reduced by the development of the Yaraquiri field (2 producing wells out of 3; f=66.67%), and the Sogamoso field (one unsuccessful well; f=0%). These new fields with low development success distort the 1956 figures. If these wells are excluded and a new calculation made the result is: 92%^lEp^lE 95%, with f (1956)=93.65%.

"Students t index" is another easy and well-known calculation which could be of some practical interest in similar situations. Suppose that an oil company has decided to start exploration in a particular basin, in which 138 exploratory wells, 51 successful, already had been drilled. Suppose also that the new oil company has drilled 11 exploratory wells, none successful. It is desirable to know the probability that such a result might be accidental. Therefore

[EQUATION]

from which ^sgr = 0.467

then: [EQUATION]

From tables of the function t the probability of an accidental result is: no success out of 11 wells is 1.2 per cent. If the exploration effort of the new oil company had been equal to that of the other oil companies, it would be in order to reexamine its geological concepts. If, on the contrary, the number of successful holes had been 2, 6, or 8, the respective probabilities that such results were accidental would be 24.4 per cent, 25.1 per cent, and 8.8 per cent. Of course such tests, and other more refined ones, have a limited significance and may be used only where exploration has reached an advanced stage.

Notice, incidentally, that the figure of the probability of success, if determined, could also serve another purpose. It is proved (De Guenin, 1959) that the optimum distribution of the exploration effort is obtained when the effort in every part of the area is proportional to the probability of success. If such a figure could be calculated for each kind of accumulation in a particular basin and having a sufficiently wide area to deal with, the optimum distribution of effort could be achieved.

In dealing with the second category of exploration areas, unfortunately, very particular cases were represented, so the problem stated at the beginning of this study still is not solved. Moreover, the procedures described are useful only for average forecasts concerning large areas, and their concrete interest is therefore basically limited.

3. Basins unexplored or explored without success to date:
Finally, the third category includes areas which are unexplored, explored without success to date, or for various reasons do not provide accurate data for statistical analysis. This is the general situation when starting an exploration venture, and it is here that the problem appears insoluble unless figures from areas considered geologically comparable are used. This is not advisable when numerical data are introduced. It results in vague and qualitative judgments (it is either very or slightly probable that a pool of a particular size could be discovered) which are unacceptable to this study because they cannot be introduced into the kind of economic evaluation which is desired here.

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ATTEMPT TO OVERCOME PROBLEM BY INTRODUCTION OF DEDUCTIVE REASONING TO PROBABILITY OF SUCCESS

The attempt to determine directly the probability of success clearly has shown the impracticality of the method which requires analysis of rarely existing and, if available, rarely understood data. As shown in the examples, the procedure described is suitable only where extensive exploration has provided a sound statistical base. In order to evaluate risk in a more practical way, it is necessary to start from data which may be reasonably, even if roughly, adopted for making an estimate.

Analyze again the nature of expenditures and income, and assume that an oil exploration venture is to be undertaken. A series of exploration expenditures must be anticipated which can be forecast with a fair degree of approximation. Assuming that an oil field is discovered, the development cost as well as income will be proportionate, on the average, to the size and production rate from the field. Even a preliminary knowledge based on the size of other fields in the same sedimentary basin would permit a generalized estimate from which hypothetical expenses and income could be calculated to determine the return on investment. This would still not solve the problem of evaluating risk. However, this reasoning may be reversed. If a 15 per cent profit is satisfactory, this figure should be introduced into the calculation. Assuming an oil field of a particular size, one may multiply exploration expenditure by a coefficient K, satisfying the general equation:

[EQUATION]

where R represents income, l production and transportation costs, c is the cost of development and producing facilities of the hypothetical oil field, and e is the exploration expenditure. This means that, if the venture is successful as postulated, the company realizes K times exploration expenditure, still obtaining a profit not lower than 15 per cent. Stated otherwise, a forecast of a probable 15 per cent profit can be made if probability of success is p = 1/K (p must be less than, or equal to, 1, but if p > 1 the venture, even if success is assured, cannot give a 15 per cent profit).

By repeating calculations according to different hypotheses a curve may be constructed in which the probability of success is plotted against field size, for a specified profit. Geologists will evaluate whether the following suppositions and figures are in the range of probability:

p = 0.1 for an oil field with 600 × 10⁶ bbl recoverable reserve;

p = 0.3 for an oil field with 300 × 10⁶ bbl recoverable reserve;

p = 0.8 for an oil field with 150 × 10⁶ bbl recoverable reserve.(FOOTNOTE 4)

At this point it is tempting to establish a discounted cash flow, and to base calculations on it.(FOOTNOTE 5) However, such a solution must be avoided as unrealistic. In fact, to be able to establish a cash flow an annual income must be assumed. This can only be hypothetical, as the rate of future production of wells cannot be assumed, including the number of years of steady production and the eventual rate of decline to depletion. The assumptions that could be made would be infinite and would involve varying times of productive life resulting in noncomparable discounted cash flows.

To present an example, from Figure 1, assume that 9 wells could produce 27,173 bbl a day (annual average) with a steady rate of production for 5 years and 8 months. After this period, production decreases according to an exponential law, falling by 1.5 per cent per month. If lifting costs are assumed to equal $1,296,000 per year, exploitation is profitable for 17 years. Table II is derived.

Apart from the arbitrary nature of the assumed conditions as to rate of production of the field, which imply knowledge of data on porosity, pressure, etc., an additional difficulty must be noted.

In order to obtain a second point in the required curve a different forecast of recoverable reserves of the field may be chosen. If it is assumed that reserves are half those of the example given above, three possibilities emerge: (1) the

FOOTNOTE 4. In the original text of the paper volumes of oil were expressed in cu m. for convenience of the American reader these have been converted to bbl, rounded off.

FOOTNOTE 5. It must be noted that a particular type of discounted cash flow would be obtained. It does not take into account profitability of an additional investment, but an investment for which the profitability is unknown. Current average rate of interest is introduced and present values of R, l, c, and e are evaluated. No profitability is calculated.

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Fig. 1. Typical exploration program for concession of assumed size of 52,000 sq km.

Table II. DISCOUNTED CASH FLOW FOR OPERATIONS DIAGRAMMED IN FIGURE 1

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same number of wells required to deplete reservoir, costs remaining the same; (2) half the number of wells required, production per well unchanged; or (3) any one of an infinite number of combinations of the two factors, production per well and number of wells.

Selection of the proper figure for possibility 3, which is what is required, still is not possible because the timing of exploration and initiation of production (Fig. 1) are undeterminable. If the initial production is 1,500 bbl a day per well, the limit of profitable production is at the end of the 20th year (13th year after production commences). There is then a point which cannot be plotted on the same curve as the one representing the foregoing table, in which different parameters were assumed. It must be borne in mind that a curve may be drawn only if two factors are variable; in this case, besides p, only one other factor can be variable, unless complicated nomographs are created in which many points must be determined.

The problem, simple at the beginning, now becomes complicated and its solution lengthy. Consider whether it is useful to calculate with apparent precision a forecast of discounted cash flow which can be based only on rough hypotheses, which are illustrated by the large number of contingencies in exploration. The discounted cash-flow method cannot be utilized for this purpose, because it requires hypotheses which are far too uncertain and would be devoid of meaning. Neither the number of wells nor the optimum well spacing can be forecast, as these are different for different reservoirs (Uren, 1956). However, it is possible to consider a field in terms of average annual production for a specified initial period, independent of variations in the production of individual wells. Also, in a particular stratigraphic and tectonic situation a fairly good estimate can be made of the expenditure required for the discovery of a field of a particular size. Therefore an attempt may be made to identify the limits within which some hypotheses may reasonably be formulated, and as a consequence what methods may be used to evaluate profit.

Estimates of the production from a hypothetical field are on the basis of average expectancy. For example, the program shown in Figure 1 may result in different amounts of recoverable reserves, which may be extracted in various numbers of years; i.e., with different rates of average annual production. With variation in reserves, development costs would be correspondingly variable. In this way a series of curves containing an "interval of forecast" of profitability can be obtained.

The calculations, being referred to average annual figures, are made for convenience by the method of Arps (1958).(FOOTNOTE 6) Though this method gives very approximate results, as is to be expected with this type of hypothetical calculation, it gives an adequate understanding of the problem without concealing serious logical and numerical errors in an apparently precise solution.

To consider an example, start with the program diagrammed in Figure 1, and assume the following possibilities: recoverable reserves 100-800 million bbl and exploitation period 10-20 years. A range of the number of development wells (say, 5-20) also could be assumed, but for the sake of simplicity is omitted, though it could

FOOTNOTE 6. For dealing with data of this kind, the formula devised by Arps is commonly used. Its expression is

[EQUATION]

where AARR is Average Annual Rate of Return, i is discount rate or cost of money, expressed in percentage units, E is undiscounted income, P is expenditure discounted for the first year, and D is the factor to be used to discount income for the first year.

This formula is not rigid, because by appropriately varying parameters it is possible to evaluate AARR for any other year but the first one. In practice this requirement is unnecessary and it is more convenient and easier to use the parameters shown above. The AARR thus obtained is a figure which, if the value i is deducted, gives an average rate of return on investment calculated from the first year onward. For example, AARR = 15 when i = 8 means a 7 per cent average rate of return on investment, provided that money is borrowed at 8 per cent. Coefficient D is calculated as the arithmetic mean of discount factors of the years in which income is earned (for further explanation see Appendix I). As a result, iD/(1 - D) is the sinking fund factor of that year which has as a discount facto the arithmetic mean of discount factors applicable to the series of earnings. Therefore, when undiscounted income is multiplied by iD/(1 - D) and is divided by the discounted value of investments, a profit is obtained which includes the cost of money. By deducting this cost, the average annual net return is obtained. This shows the limitations of the Arps formula, by implying identity between the discount rate and interest rate on the amount invested. On the other hand, the formula has the effect of distributing expenditure and income in equal annual shares over the period in which they occur (or, otherwise expressed, concentrates expenditure and income in a "mean year") and is suitable for calculation of average figures, which is what is required in this situation.

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be introduced without later complicating the concept. To show graphically a range of forecasts within maximum and minimum values it only is necessary to modify this range.

To cover different lengths of the exploitation period, three curves can be plotted, for 20, 15, and 10 years, respectively. To obtain the three curves, three points for each curve must be determined. Rates of 10×10⁶, 20×10⁶, and 40×10⁶ bbl average annual production may be chosen. These points cover the range of variability of recoverable reserves from 100 million bbl producible at a rate of 10 million bbl per year for 10 years, to 800 million bbl producible at a rate of 40 million bbl for 20 years. The same assumptions as to cash flow shown in Table II may be maintained, except that an average value must be assumed, independent of producing rate, for lifting and transportation costs. On an assumed basis of $0.16 per bbl, AARR figures fol ow:

Table

The results are plotted in Figure 2.

The determination of p is the next step. The values of AARR are obtained from the recognized formula:

[EQUATION]

where E is net income after royalty, lifting, and transportation costs, R is exploration costs, and C is cost of development and producing facilities. Both R and C are discounted figures. Suppose AARR = 15, and introduce a factor K, which must be multiplied by R; then:

[EQUATION]

By substituting numbers for the symbols, the unknown factor K is determined, which is the inverse of p, or the value of the probability of success satisfying the minimum acceptable return of 15 per cent. Notice that in case 3a, p would be greater than 1, the AARR being less than 15 per cent. The other figures for p are:

Case 1a   0.6902
Case 1b   0.2677
Case 1c   0.1203
Case 2a   0.8290
Case 2b   0.3076
Case 2c   0.1362
Case 3b   0.4036
Case 3c   0.1726

Another point for each of the three curves corresponding to p = 1 may be obtained if, in the above equation, K = 1 and E is unknown. The corresponding recoverable reserves are:

Case 1   About 160 million bbl
Case 2   About 133.3 million bbl
Case 3   About 110 million bbl

The results are plotted in Figure 3, and from a study of them values of p finally can be obtained. From the series of three curves it can be said that, under the assumed conditions, obtaining a 15 per cent rate of return involves a probability of 0.21^lEp^lE0.33 of finding a field with 334 million bbl of recoverable reserves, and a probability of 0.095^lEp^lE0.145 of finding a field with 666 million bbl of recoverable reserves, and so on. If, for example, the most realistic result of the exploration program expected is the finding of a 315-440-million-bbl field, which may be exploited in 15-20 years, the value of p should be between 0.18 and 0.33.

Geologists then must estimate whether these values are attainable. It is noted that the value of p thus obtained does not represent the probability of success of an exploratory well, but the probability of success of a program of geological and geophysical exploration (Fig. 1), including five exploratory wells. To find the probability of success of a single exploratory well p/5 may be calculated, although this gives a figure of doubtful value, because 1/5 of the geological and geophysical expenditure cannot be attributed to each exploratory well. Moreover, when the number of wells that must be drilled to discover a field increases, geological and geophysical expenditure does not necessarily increase proportionately. However, the p/5 figure may be taken as a rough average, in which the probability of success of the first well is slightly lower than p/5 and that of the fifth well is slightly higher than p/5.

With the above system an attempt has been made to give a rational solution to the problem of

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success probability. The writer believes it is worth while to determine whether the risk of exploration in a certain area is sufficiently low to justify the expectation of a minimum acceptable profit (15 per cent in the cases studied). There are two reasons for using Arps' formula in the study, one conceptual, as already explained, and the other a practical need for brevity.

Fig. 2. Average annual rate of return from fields with various volumes of recoverable reserves produced at different depletion rates.

Fig. 3. Probability of success in finding fields with various volumes of recoverable reserves, producible at different depletion rates.

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FORMULAS FOR RAPID USE IN ECONOMIC EVALUATION OF EXPLORATION AREAS

So far two requisites have been indicated for calculations if they are to be practical and meaningful: (1) they must involve the assumption of average data, and (2) they must be capable of rapid application. The first requisite has been obtained by introducing Arps' formula; the second is only partially achieved by using this formula, the calculation introduced in the example being based on an extremely simplified structure of income. In reality, when an oil concession is acquired, and especially when commercial production is found, it is necessary to envisage a complex series of regulations concerning rentals, royalties, and taxes, the calculation of which alone will determine the forecast of the rate of return on investment. To give weight to these factors necessitates some formulas adaptable to the various financial situations which may occur.

Such formulas provide a new procedure for making economic forecasts which, in many cases, may be more useful than the one already discussed. Once the economic phenomena are represented by symbols, the production that must be obtained to reach a minimum acceptable profit may be shown as an unknown. The usefulness of such an "inverse" calculation is evident, because it establishes a quick comparison between two or more areas and allows selection of the most profitable. At the same time determination can be made of the maximum bonus, whether in cash or other considerations, which can be offered the government or other owner of a property in order to obtain concession rights in a competitive situation.

Beginning with formulas for the calculation of payout time,(FOOTNOTE 7) a list of the factors to be considered, and their symbols, is made. All factors are known when making the forecast, with the exception of daily production P_p. The values of x and y are established at levels which the operator considers reasonable.

Symbols of factors are:

x Payout time (years)
y Annual average rate of return (AARR)
V_l Selling price ($/bbl)
s_e Exploitation expenses ($/bbl)
(A - B) Difference between posted price and realized price ($/bbl)
n Percentage of oil belonging to the government and/or partners in cases where profit split is applicable (FOOTNOTE 8)
r Rentals ($/year)
d Forecast development period (years) after discovery
d^prime Period after discovery before completion of development drilling and construction of pipelines and producing facilities, permitting initiation of production (years)
R Period of exploration before discovery (years)
S Exploration expense ($)
S^prime Expense for facilities outlined in d^prime ($)
C Cash bonus ($)
S_p (p+a) Development expenditure, function of a (dry hole expenses) and p (expense for producing wells); factor a is expressed as a percentage of p; S_p (p+a) is therefore a function of (p+a) ($)
Q An f(S) referred to the percentage of exploration which may be legally amortized or depreciated annually in calculation of the profit split (Q is the sum of the annual deductions) ($)
l Period during which Q can be amortized (years)
Q^prime An f[S_p(a)] referred to the percentage of dry hole expense which may legally be written off in the year incurred in calculation of profit split (Q^prime is the total of the annual amounts). If it must be depreciated instead of expensed, the amortization period l (years) is accounted for
Q^Prime An f[S_p(p)] expressing as a total of the annual amounts the percentage of expenditure for productive wells legally deductible in calculation of profit split
l^prime Period (years) during which Q^Prime can be amortized. If Q^Prime can be expensed, l^prime does not exist
K An f(R) to be used to discount S (see tables)
K^prime An f(R) used to discount S_p(p+a) (see tables)
K^Prime An f(R,d^prime) used to discount S^prime (see tables)
P_p The undetermined daily production rate which the field must provide to obtain the required income; symbol P_p is used to signify that production is given by p wells with an average output P

To proceed with the expression of the formulas, one determines the minimum required output to obtain payout within a reasonable period of time. Two cases may be considered, one in which (1) the government requires a specific percentage of crude oil as royalty; and the other in which (2) the government requires a profit split. The government may also require a change from the

FOOTNOTE 7. Payout time is the time necessary to recover all expenses, but does not discount the capital invested.

FOOTNOTE 8. In the present analysis, examples are given of a 50-50 profit split. In recent years the profit split in contracts with governments has tended toward a lower share for the company, and the matter is further complicated in OPEC-type arrangements under which the government requires royalty as well as a percentage of profits in the form of a tax from which royalty may or may not be excluded.

End_Page 2238------------------------------

first to the second case after production exceeds a particular limit. Then correct figures can be obtained only by means of the laborious cash-flow method applying the technique of trial and error. The possibility of this may be disregarded, as the calculation according to the profit split for other than short periods gives only a slightly lower figure. This is especially true if the forecast is required to determine the production rate required to obtain a given AARR (see later). In the calculation of payout time it is easy to evaluate which of the two above cases is the more onerous. Both calculations can be made rapidly and may be repeated, selecting the more conservative result.

In each of cases 1 and 2, two possibilities may be distinguished. In 1a and 2a production begins only after completion of all the wells necessary for exploitation and may be considered practically steady (in a forecast it is not important to consider the production from each well). In 1b and 2b production starts while development drilling is under way, and increases until the program is completed.(FOOTNOTE 9)

Following is an examination of each of the four cases.

1a. Royalty as percentage of crude, steady production:
This is the simplest case, in which a steady profit must equalize previous expenditure. The algebraic expression of this phenomenon is:

[EQUATION]

1b. Royalty as percentage of crude, production increasing as development drilling proceeds:
This calculation is more complicated because an increasing profit must equal expenditure, which may be divided into two parts, one being related to expenditure already made (S, S^prime, C), and the other increases during a certain period d. It is necessary to introduce a coefficient for increasing production, and also a coefficient for increment of S_p(p + a). The first coefficient must be an f(d) which reaches 1 when d^ne0 and (x - R) ^rarr ^infinity. It equals 1 when d = 0 and takes into account the constant growth of production.(FOOTNOTE 10) The second coefficient is equally an f(d) and must equal 1 when (x - R) ^gE d.

The resulting expression is:

[EQUATION]

As the calculation is referred to real time, and thus only to positive time, factor (x - R) - d must in every case be ^gE0. Therefore, as (x - R) < d cannot be negative, this factor must equal zero. It should be noted that the coefficient of S_p(p + a) is

[EQUATION]

because this factor has been transferred from the second to the first stage of the equation.

The equation expressed above concerns the case in which there is a minimum delay in the construction required to start production. If there is a delay in the construction of pipelines and producing facilities, or the related development program, thus delaying the initiation of production, a corrective factor must be introduced in the above formula.

The steady growth of production during d years may be considered as fractions of P_{p max}. The average annual figure of P_p will be:

Table

From the first (R + 1)^th year to the x^th year the sum of the first n odd numbers equals n². The cumulative production which gives the payout in the x^th year, taking into account the lost d^prime years, is the same which, without this delay, would give

FOOTNOTE 9. Obviously an increasing output per well is not what is referred to but to increasing production due to completion of new producing wells. Reference is made therefore to an average annual production which is not related to the decline curve of individual wells.

FOOTNOTE 10. Coefficient

[EQUATION]

may be simplified and reduced to

[EQUATION]

The first expression is chosen because it is more understandable, as will be seen when dealing with the correcting factor d^prime.

End_Page 2239------------------------------

the payout in the year x^prime. Therefore, if production is expressed as fractions of P_{p max} the following is obtained:

[EQUATION]

from which:

[EQUATION]

which provides the figure desired.

Next, the production should be determined which is necessary to reach, for example, payout in the twelfth year when d^prime = 2 and d = 5, i.e., the production necessary to reach payout in the year (12 - <fr>4</>10</fr>) = 11.6. The production figure thus determined is that necessary to reach payout at the end of the twelfth year, taking account of the production lost in the d^prime years.

It must be noted that the above exposition is valid when d^prime < d. If d^prime ^gE d, it is sufficient to insert R^prime = R + d^prime, and to use the formula concerning case 1a. This is an example of how formulas must be used, by taking account of the concrete meaning of each coefficient. Formulas may thus be modified to solve various kinds of problems.

2a. Fifty-fifty profit split is required, steady rate of production:
Here the payout equation becomes more complicated. It is necessary to introduce various factors (Q, Q^prime, Q^Prime, etc.) related to the proportion of expenditure which can be amortized and the amortization periods (l, l^prime, etc.). In case 2a it is assumed for example that exploration expenses may be written off in the period l, and cost of development wells (both dry and productive) in the period l^prime. Naturally, it is possible to simulate any other real case without changing the structure of the equation. It is sufficient to introduce all the necessary Q^prime and l^prime factors, and to cancel those that are unnecessary.

[EQUATION]

As with the factor [(x - R) - d], factors [(x - R) - l] and [(x - R) - l^prime] also must ^gE0.

2b. Fifty-fifty profit split required, production steadily increasing with continuous development drilling:
In this case other coefficients must be considered. In the example it is assumed that Q, f(S) can be amortized during the period l; that Q^prime, f(S_pa) can be deducted annually during the development period; and that Q^Prime, f(S_pp) can be amortized during period l^prime. Rentals are expensed. Any other fiscal conditions can be simulated with due modifications. The coefficient introduced for Q^Prime (share of cost of producing wells which can be legally amortized) is an f(d, l^prime) which must equal 1 when (x - R) ^gE (l^prime + d), the latter expression representing the final year of amortization.(FOOTNOTE 11)

To obtain realistic results, the factors may require these modifications:

a. (x - R) ^gE l^prime; unless this is the case, it is necessary to put x - R - l^prime = 0.

b. (x - R) ^gE l^prime + d; unless this is the case, it is necessary to put x - R - (l^prime + d) = 0.

c. (x - R) ^gE d; unless this is the case it is necessary to put x - R - d = 0.

This leads to the following equation:

[EQUATION]

FOOTNOTE 11. The coefficient introduced for Q^Prime is calculated not only to equal 1 when (x - R) ^gE (l^prime + d) but also to be equal to

[EQUATION]

The amortization rate is one d^th of the development expenses, when amortization is completed in l^prime years. The annual rate must be added during (x - R) years, and it is known that

[EQUATION]

but (x - R) < (l^prime + d), similar coefficients modify the figure

[EQUATION]

These coefficients are of course functions of d, l^prime, and (d + l^prime).

End_Page 2240------------------------------

One particular case is the calculation of payout time if the oil company decides to offer a share of participation to a government or another company. The value of S_p(p + a) must be modified to take account of the share of expenses to be paid by the second and/or third parties. From this figure a sum

[EQUATION]

must be deducted as the share of expenses eventually paid by the government and/or other company for exploration, pipelines, and other equipment during d^Prime years starting from year R. The participating factor ½ or its equivalent 100 - n/100 must be modified according to the percentage division of interest. Variations in Q, Q^prime, and Q^Prime will occur according to the particular agreement. Another variation is given by the introduction of factor d^prime, as already explained.

Beside calculation of payout time, calculation of the production necessary to obtain a specified AARR constitutes the other important element in the forecast (see Appendix I).

In cases 1a and 1b, if z is represented by the figure of iD/1 - D, for an example of 20 years of production the AARR obtained is:

[EQUATION]

where K = f(R), K^prime = f(R,d), and K^Prime = f(R,d^prime) are coefficients for discounting expenses which in a forecast may be reasonably distributed according to a constant annual rate. Coefficients K, K^prime, and K^Prime may be obtained from Table III. In cases 2a and 2b for an assumed 20-year exploitation period the AARR is:

[EQUATION]

It should be noted that the Arps formula must be applied to a steady rate of production. In the formulas presented, an error is introduced if production increases in the initial d years, but it is so slight that it may be ignored.

These equations are adaptable as may be required to various actual cases. In the case described of government participation it is sufficient to modify income and expenditure as already indicated and to introduce other factors K^tprime, K^qprime, etc., according to the procedures for reimbursement of expenditures already made. If factor d^prime is introduced, the fraction d^prime²/2d may be considered to represent lost production, as divisible by 20 years, and it follows that the coefficient

[EQUATION]

is transformed into

[EQUATION]

This correction may be overlooked if d^prime is small. If d^prime is large the coefficient cannot be overlooked, but if d^prime ^gE d it is more convenient to eliminate the coefficient and use a new figure R^prime = R + d^prime. If d^prime is small or if d^prime > d, coefficient

[EQUATION]

These two transformations differ in that the first one involves a different figure of iD/(1 - D) and extends the end of the 20-year period assumed for exploitation, and the second maintains the date of the end of exploitation but reduces the producing period to 20 - d^prime years. It should be noted that

[EQUATION]

is retained for greater clarity, though it may be simplified to

Table III. VALUES OF i - D/i - D AT 8 PER CENT SEMIANNUAL RATE OF INTEREST

End_Page 2241------------------------------

[EQUATION]

The numbers 20, 10, and 400 which are introduced into the two formulas are naturally selected for 20 years of exploitation at steady production. If a 15-year period of exploitation is assumed, the numbers should be 15, 7.5, 225, and so on.

The equations presented permit a quick calculation and they can be easily and quickly adapted to any type of agreement and to any area. In Appendix II some examples are presented to demonstrate the application of the formulas.

CONCLUSIONS

An examination has been made of procedures for effecting economic evaluations under conditions of uncertainty, as in the case of oil exploration. These procedures presuppose a subjective estimate or an objective understanding of the probability of success. Subjective estimates are shown to be of little value, and objective appraisal of risk is shown to have intrinsic difficulties. In most actual cases such difficulties cannot be overcome. Therefore, a different procedure has been suggested, i.e., to determine by analysis the value that the probability of success should have in order to be consistent with the economic goals of an oil operator. Using this procedure, geologists will be asked for their judgment, based on available technical and statistical data, of the probability of disc very within a particular field or area.

Formulas adaptable to a variety of circumstances have been presented to answer the question, "Once a minimum acceptable return is fixed as the goal of an oil project, what is the output required to obtain that return?" Thus stated, a concrete evaluation of the problem may be made. Highly speculative approaches have been avoided in favor of more modest objectives. Two procedures are suggested, and these may be combined to provide a complete picture of a particular situation. The calculations adopted are quickly made, so that use of the formulas is practical. Some suggestions also have been offered concerning possible studies based on historical data to determine the probability of success, but it is concluded that in practice such methods rarely can be relied on to solve the problem of risk.

APPENDIX I. COMMENTS ON ARPS' FORMULA

Using Arps' formula

[EQUATION]

and supposing the following situation: a period of x years exists in which expenses are incurred without any income, and starting from year x + 1 income begins and continues until year y; the symbols used are:

AARR = Annual average rate of return
i = Discount rate or cost of money
E = Undiscounted income
P = Expenses, including interest paid, until year x
D = Coefficient to be used to calculate discounted income at the beginning of year x + 1

Start by examining the case AARR = 0. This occurs when (A) E/P = 1, and when (B) D = 0. Case A represents the possibility of undiscounted income equaling expenses plus interest until year x. As income must be discounted to the same date, this means that, if AARR = 0, interest is lost at the rate i from year x + 1 until year y on the capital P, resulting in loss of interest on the investment from year 1 until year y. Case B represents a limit, which may be connected to case A because either (1) D^rarr0 because (y - x)^rarr ^infinity and i ^ne ^infinity, a transfer to the infinite of a finite income, with loss of interest, or (2) D^rarr0 because i^rarr ^infinity, interest being unrecovered.

Suppose that at the beginning of the year x + 1 discounted income equals investment plus interest until year x. Then:

[EQUATION]

If ED = E^prime = discounted income, then:

[EQUATION]

By deducting i the profit is found to be nil, as it must obviously be.

Table IV. COEFFICIENTS K, K^prime, AND K^Prime AT 8 PER CENT SEMIANNUAL RATE OF INTEREST

End_Page 2242------------------------------

Finally, suppose that discounted income at the beginning of year x + 1 exceeds investment plus interest until year x. Place E^prime/P = K > 1. Then:

[EQUATION]

Put

[EQUATION]

Then:

[EQUATION]

It is known that 0 ^lE D ^lE 1. Case D = 0 has already been discussed: case D = 1 may happen when i = 0 or i ^ne 0 and (y - x) = infinitesimal. If D = 1 the equation loses its meaning because income and investment, whether or not interest paid is included, may be directly compared. On the contrary, it is interesting to examine the meaning of the coefficient i/(1 - D) when 0 < D < 1. First examine what is D. If (y - x) = r, D is represented by:

[EQUATION]

Note in addition that if K^prime is constant while i increases, the ratio of discounted income to investment excluding interest must grow. By introducing the expression for D in i/(1 - D) the following expression is obtained:

[EQUATION]

This expression, by development and addition of binomials (1 + i)^r may be transformed as follows:

[EQUATION]

This increases as i increases, according to a law of compound interest related to constant shares during r year. This formula represents the capital recovery factor related to that year whose discounting factor equals the arithmetic mean of discounting factors of the r years. From this standpoint it is interesting to observe that if i^rarr0 the equation tends to

[EQUATION]

Thus, in the general case AARR is a number, and if the value of i is deducted from it the annual average rate of return on the investment is obtained. By following the inverse procedure and coming back to the original formula

[EQUATION]

it may be noted that, if

[EQUATION]

is constant, AARR must decrease when i increases. After some transformations which are not given here, in order to shorten the exposition, then:

[EQUATION]

This shows that iD/(1 - D) diminishes when i increases, and represents the sinking-fund factor related to that year whose discounting factor equals the arithmetic mean of discounting factors of the r years. In this case also it must be noted that if i^rarr0 the equation tends to

[EQUATION]

Practically, coefficients D and P are calculated in a different way. In fact, it is more convenient to discount all values at the year 1 and therefore, with P = discounted expenses at year 1, the formula is.

[EQUATION]

If, for example, i = 8 per cent, then i = 8. In this case, if AARR = 15, the annual average rate of return on the investment is 7 per cent (15 - 8 = 7), considering that money is borrowed at 8 per cent. In order to simplify calculations it is useful also to place E equal to income less exploitation and other expenditure during the period (y - x).

APPENDIX II. EXAMPLES OF APPLICATION OF FORMULAS

EXAMPLE No. 1

An oil company is examining concessions in three different countries and wishes to select only one for an exploration venture. It is necessary to determine which of the three areas is the most promising on the basis of a 20-year exploitation period. Operating conditions based on technical forecasts and careful examination of oil laws are as follows:

Lease A: royalty 12.5 per cent of crude produced.--

R = 5 years
S = $30,000,000
S^prime = $15,000,000
C = $8,700,000
S_p(p + a) = $28,000,000
d = 3 years
d^prime = 2 years
r = $640,000 after discovery
V_l = $1.85/bbl
s_e = $0.30/bbl
K = 0.832
K^prime = 0.659
K^Prime = 0.631
n = 12.5 per cent

Lease B: 50-50 profit split.--

R = 6 years
S = $20,000,000
S^prime = $60,000,000
C = $440,000
S_p(p + a) = $55,000,000
a = 10% of p

End_Page 2243------------------------------

d = 5 years
d^prime = 2 years
V_l = $1.80/bbl
r = $1,100,000 before discovery if annual exploration expenses do not exceed this amount; $1,700,000 after discovery, to be expensed
s_e = $0.30/bbl
Q = 100% of S + S^prime = $80,000,000
Q^prime = 100% of S_p(a) = $5,000,000
Q^Prime = 100% of S_p(p) = $50,000,000
l = 10 years
l^prime = 10 years
K = 0.800
K^prime = 0.523
K^Prime = 0.584

Lease C: royalty 15 per cent of crude produced.--

R = 5 years
S = $20,000,000 (including r)
S_p(p + a) = $77,000,000 for an offshore oil field
$38,500,000 for an oil field on land
d = 5 years
r = $1,000,000 (to be reduced to $250,000 if exploration continues after discovery with annual expenses not less than $750,000)
V_l = $1.20/bbl
s_e = $0.30/bbl
K = 0.832
K^prime = 0.565
n = 15%

In all cases the oil company assumes as reasonable a 12-year payout time and a 15 per cent AARR. Therefore:

Case A: Payout at end of 12th year:

[EQUATION]

From this, the production necessary is estimated to be about 37,800 bbl a day, or about 13,800,000 bbl a year.

Case A: AARR = 15 per cent. By transforming the known equation:

[EQUATION]

From this, the production necessary is estimated the same as above.

Case B: Payout at end of 12th year:

[EQUATION]

From this, the production necessary is estimated to be about 125,000 bbl a day, or about 46,625,000 bbl a year.

Case B: AARR = 15 per cent. By transforming the known equation:

[EQUATION]

From this, the production necessary is estimated to be about 85,000 bbl a day, or about 31,000,000 bbl a year.

Case C: Payout at end of 12th year:
Two possibilities are forecast, an offshore and an onshore oil field. These give, respectively:

[EQUATION]

From this, the production necessary is estimated to be about 32 million bbl a year for an offshore field and 20 million bbl a year for an onshore field.

Case C: AARR = 15 per cent. The same two cases are analyzed as follows:

[EQUATION]

From this, the production necessary is estimated to be about 22 million bbl a year for an offshore field and 14.5 million bbl a year for an onshore field.

If annual rate of return is considered more important than payout time, geologists will advise whether it is more or less probable to discover (a) about 12,500,000 bbl a year in lease A, (b) about 31,000,000 bbl a year in lease B, taking into consideration that the structure of the oil law delays payout in this case, or (c) about 12,500,000 bbl a year onshore or 22,000,000 bbl a year offshore in lease C.

EXAMPLE No. 2

Suppose that an oil company decides to choose lease A. In order to overcome competition the company studies the various advantages that may be offered to the government. The following possibilities are considered:

1. The company conforms to the minimum conditions of the law and offers no advantages to the government.
2. It offers a 50 per cent participation to the government

End_Page 2244------------------------------

in case of success, with the company bearing all exploration expenses, but without payment of a cash bonus.
3. It offers the same conditions as in case 2, and also a cash bonus.
4. As in case 3, but a 30 per cent participation is offered the government.
5. As in case 3, but a 20 per cent participation is offered.
6. A 50 per cent participation is offered, but with reimbursement of exploration expenses and a cash bonus including 8 per cent interest.
7. As in case 6, but 8 per cent interest is not required.
8. As in case 7, but reimbursement may be made in 5 years; interest payable from date of discovery.
9. As in case 8, but without interest.
10. A cash bonus of $12,000,000 is offered.
11. A cash bonus of $15,000,000 is offered.
12. A cash bonus of $10,000,000 is offered, plus $10,000,000 to be paid in addition when production of 12,500,000 bbl a year is reached.

From known equations we obtain, in bbl per year:

Table

To interpret these figures correctly, it must be remembered that in the case of government participation the profit decreases by 50 per cent in cases 2, 3, 6, 7, 8, and 9, by 30 per cent in case 4, and by 20 per cent in case 5, with the amount risked in exploration remaining unchanged, while investment for development and starting of production differs from case to case.

EXAMPLE No. 3

Suppose that an oil company, having evaluated concessions A, B, and C, decides to commit itself in all three areas, in order to have a greater probability of discovering at least one oil field. In this case, if only one discovery is made, the eventual profits must also cover expenditures made in the other two areas. Of course, payout time will be considerably delayed, but in this case a sure profit, though over a rather long term, is equally probable.

The production required is calculated to obtain a 10 per cent and a 15 per cent AARR in the following cases: (1) discovery in A; failure in B and C; (2) discovery in B; failure in A and C; and (3) discovery in C; failure in A and B.

It is probably unnecessary to make all these calculations. Geologists may estimate from previous calculations that concession C could hardly provide large oil fields even in the case of discovery. Thus, it could never match exploration expenditures of A and B. Consequently, it seems reasonable to consider only the first two cases. Therefore:

Case 1.--Discovery in A; failure in B and C.

AARR 15 per cent, 20,800,000 bbl a yr.
AARR 10 per cent, 15,120,000 bbl a yr.

Case 2.--Discovery in B; failure in A and C.

AARR 15 per cent, 53,000,000 bbl a yr.
AARR 10 per cent, 36,550,000 bbl a yr.

The consequences are very important, because if an oil field of the necessary size is discovered in the area of concession A, as in example No. 1, then it can also meet exploration costs in B and C. Similar considerations are valid as to concession B. Therefore, it is more profitable to take all three concessions rather than only one, in order to have a greater probability of success. In the future, consideration may be given to substituting concession C with a more promising one.

Other examples could be given, but it is sufficient to point out various alternatives which may be considered and graphically presented. For example, the terms offered may be plotted graphically for an area by competing oil companies, and after establishing an interval of profitability calculation may be made of the best terms to offer to overcome competition. Naturally, actual cases must be worked out as circumstances dictate.