Mathematical Geology, Vol. 21, No. 5, 1989

Petroleum Exploration Models--Estimation and Applications ~ Zbigniew ~Eucki2 and Zbigniew Szkutnik 3

Three statistical models of oil and gas exploration prevail in the literature: field size distribution, "exploration function," and "effectiveness of exploration" models. They may have many functional forms and may be used for prediction and planning results of future exploration activity. A review of American and Soviet literature of exploration models is presented with special emphasis on functions suitable for countries which, like Poland, are not rich in oil and gas. Two new procedures are proposed: a method for assessing field size distribution and a method for simultaneous interval estimation of parameters of the exploration function. The methods complement existing ones as they are suited to areas that are not rich in hydrocarbons. Examples of application to exploration of two Polish petroliferous basins are given.

KEY WORDS: Explorationmodels; resource evaluation; maximumlikelihood estimates.

INTRODUCTION Many deterministic and probabilistic mathematical tools exist for decision-making in petroleum exploration. Among them, three statistical models of exploration process prevail in the literature: field size distribution, "exploration function," and "effectiveness of exploration" models. Regarding field size distribution, various models are proposed: unimodal log-normal distribution and several amodal distributions including exponential, Pareto, and Zipf's law. Whereas economic truncation occurs in countries rich in oil and gas, the sampling distribution should be close to the parent distribution of oil and gas fields in countries like Poland. A numerical example supporting this hypothesis and a forecast of the number and size of future discoveries in two Polish basins are given. ~Manuscript received 22 March 1988; accepted 6 October 1988. 2Institute of Industrial Engineering, Academyof Mining and Metallurgy, ul. Gramatyki 10, 30067 Cracow, Poland. 3Institute of Mathematics, Academyof Mining and Metallurgy, AI. Mickiewicza30, 30-059 Cracow, Poland. 495 0882-8121/89/0700-0495506.00/1 © 1989 International Association for Mathematical Geology

496

-Eucki and Szkutnik

The so-called "exploration function" is the relationship between exploration result and effort. The function permits estimation of ultimate exploration effect and of decay rate of exploration efficiency. The known exploration functions and many of the estimation methods have disadvantages including the fact that only point estimates are obtained. So, a maximum likelihood method for estimating parameters and finding confidence sets is proposed here and applied to the Polish oil industry to yield a prediction of the results of future exploration activity and the decay rate of exploration effectiveness. R E V I E W OF E X P L O R A T I O N M O D E L S Field Size Distribution

Schuenemeyer and Drew (1983) have presented a r6sum6 of main achievements of geostatistics in field size distribution pattern. Beginning with the classic work of Krige (1951) who proposed a log-normal field size distribution, this work culminated in papers of Schuenemeyer and Drew (1983) and Attanasi and Drew (1985). In the first of these two papers it was stated that "the parent population of the oil and gas fields in any region is a monotone decreasing function of size." The second paper showed that log-normal distribution "should be regarded as the end result of an economic filtering (truncation) process." The new step is represented by the Lee and Wang (1983a, 1983b, 1985) feedback procedure. In the Soviet literature, works of Baymukhametov and Sattarov (1975) and Kontorovich and Detain (1977) are considered classic. The filtering process has been described by Shpilman (1982) who introduced a concept of an "exploration filter" having a geological character. A value of the so-called "filter parameter" decreases in time, and geologists always omit a part of deposits, looking for fields larger than the value of the "filter parameters." The filtering process terminates when an economic limit is approached. Soviet authors have come to similar conclusions as their U.S. colleagues: the distribution of discovered fields is log-normal and its asymmetry increases in time tending to an amodal distribution which describes the parent population. A consequence of that rule is a strict order of discovering individual field size classes and decreasing average size of discovered fields, which finally approaches the average for the parent population. The largest field in a region is discovered between the fifth and the twentieth discovery (Barouch and Kaufman, 1978; Eremenko et al., 1979). Kontorovich et al. (1985) reported the following empirical relationship that determines the moment when all deposits of ith size class are discovered: T~i = 0.857 - 0.092 ln0i

(1)

Petroleum Exploration Models

497

where Tdi is dimensionless time, expressed as a ratio of reserves discovered to total reserves in a given region; 0 = lO00Q/Qmaxis normalized field size; Q is field size; Qmaxis size of the largest field in a region; and ln0g is logarithmic mean of normalized field sizes in the ith class. A practical application of various field size distribution models, including log-normal, gamma, exponential, Pareto, and Zipf's law, should give (1) total number of fields in a region, (2) number of fields in each size classes, (3) number of remaining fields to be found in each class, (4) total reserve volume in a region, and (5) ultimate reserve volume. To estimate the parent distribution for a region, some data must be assumed or prepared, like total recoverable reserves, size of largest field, size of smallest commercial field, value of the parameter for the exploration filter, etc., to say nothing of type of distribution. The simplest procedure is that based on Zipf's law; to know the size of the largest field in a basin and a commercial field size limit (Ivanhoe, 1976) is enough. Another procedure consists in fitting the theoretical Zipf's field size sequence to the empirical sequence of discovered fields arranged from the greatest to the smallest one. Both approaches have, however, many disadvantages, as has been described (Schuenemeyer and Drew, 1983). Other distribution models are not commonly used for the purposes mentioned above. The problem is that these models must be based on sampling data that are not consistent with the parent population of fields. Consequently, the total number of fields and total reserves are estimated in the U.S. from "exploration functions." This approach requires, however, that numerous fields be discovered in each size class. Thus, this method is not suitable for regions that are poorly explored or not rich in hydrocarbons. More suitable for small basins or basins that are not rich is a model given by Kontorovich and Detain (1977) in which the parent population is approximated with the Pareto distribution. The model input data include: total reserves recoverable, size of the smallest commercial deposit, and sequence of all deposits discovered. This model is considered in a later paragraph. Exploration Function

The so-called "exploration function" is the relationship between exploration effect and effort. Exploratory effect may be represented by total hydrocarbon reserves discovered, total number of fields discovered, degree of area recognition (e.g., dimensionless time, Td), etc. Also several measures of ex-

498

~Lucki and Szkutnik

ploratory effort may be used: total expenditure for exploration, total number of wells drilled, total meters drilled, number and meters of exploratory wells, etc. In general, two types of exploration functions exist: (A) Functions that approximate the exploration process from the beginning. (B) Functions that describe the exploration process from the point of its maximum effectiveness. Type A function is not commonly used in the United States. An exploration function of this type has been described by Schoeffler (1971) who deduced the character of the relation "degree of area recognition vs. total expenditures" from a process of geological data acquisition and associated it with the cumulative distribution function of the normal law. Such exploration function has an inflection point (Fig. 1). A similar function has been presented by Krylov (1982)

R = Ru I 1

-

-

(D D2 - D1 + 1) e -(D- DI)/D21

(2)

where R is total exploratory effect, R, is ultimate exploratory effect, D is cumulative exploratory effort, D1 is total exploratory effort from the beginning of geological exploration to the first discovery, and D2 is total exploratory effort from the first discovery to the point of maximum effectiveness (inflection point in Fig. 1). Coordinates of the inflection point have been analyzed by numerous Soviet authors, e.g., Eremenko et al. (1979), Babalyan and Glazunov (1982), Krylov (1982), and Buyalov and Korchagin (1985). They stated that the inflection point

•"

"on of model(3)

o~ o

~

lnflection point o

Cumulative exploratory

effort

Fig. 1. Explorationfunctionwith an inflectionpoint•

Petroleum Exploration Models

499

occurs at Td between 0.10 and 0.30, rarely between 0.30 and 0.40, the actual value depending on geological conditions. Type B function, called also "discovery process model" for well-explored areas, is known as the model of Arps and Roberts (1958) or the model of Drew et al. (1979). Numerous applications may be found in the U.S. literature (Arps et al., 1971; Rapoport and Grender, t977; Attanasi et al., 1981; Jones et al., 1982a, b; Schuenemeyer and Drew, 1983; Attanasi and Drew, 1985; Collins, 1985; etc.) and also in Soviet papers (Baymukhametov and Sattarov, 1975; Balasanov and Kamenecki, 1977; Nesterov et al., 1977; Selicki, 1978; Krylov, 1982; Leybson, 1982; Shpilman, 1982; etc.). Exploration functions for well-explored areas may be divided into two groups: (B1) Models that use cumulative drilling (meterage or footage) and reserve volume as measures of exploratory effort and effect. (B2) Models that use number of wells drilled and number of fields discovered as respective measures. The general analytical form for all these models is (3)

R = R . ( t - e -At')

where R is total exploratory effect obtained from assumed zero point, R, is ultimate exploratory effect related to zero point (parameter to be estimated), A is parameter to be estimated, in many papers split into several coefficients, and D is cumulative exploratory effort from assumed zero point. The zero point (coordinate origin) may correspond to the point of maximum exploration effectiveness or may be shifted forward in time (Fig. 2). Bal-

o

•

Zero point may be shifted Forward in time

J

D

Inflection point

Cumulative

exploratory

effort

Fig. 2. Exploration function for well-explored area.

500

-Eucki and Szkutnik

asanov and Kamenecki (1977) stated that such parallel shifts of the coordinate system do not change the real (nonrelated) estimates of R, and A. Thus, the exploratory efforts and effects need not be totalled from a strictly determined time. This is useful because, in many countries, reliable data before 1945 are not available. A practical application of exploration or discovery function permits estimates of (1) Ultimate exploratory effect corresponding to infinite exploratory effort. (2) Partial exploratory effect due to determined exploratory effort done in a given exploration phase. (3) Effectiveness of exploration in a given exploration phase to be made. The full range of possible applications has been described widely by Arps et al. (1971). The input information is usually historical data relating to reserves

or fields discovered and wells drilled, both from the zero point to the moment at which the analysis is made. Applications of the exploration function encountered in the literature often have a disadvantage consisting in point estimation of parameters which, in turn, implies point estimates or predictions for oil and gas recovery, investment and production costs, revenues, etc. This gives no information about reliability of the results obtained. In order to remove this disadvantage, a new method of estimating parameters R, and A that gives confidence sets for predicted values (ultimate reserves, exploration efficiency, etc.) is proposed. For a discussion of alternative approaches, see Wiorkowski (1981). Effectiveness

of Exploration

The effectiveness of exploration, say E, is expressed as exploration effect per unit of exploration effort, e.g., tons per meter, number of fields per 1,000 wells, etc. The exploration effectiveness is related closely to the sequence of discoveries. The value of E increases at the beginning of the exploration, reaches a maximum when the largest fields are being discovered, and then declines exponentially. This fact has been confirmed in numerous papers with historical data (e.g., Hubbert, 1967; Arps et al., 1971; Haun, 1975; Eremenko et al., 1979; Attanasi et al., 1981; Babalyan and Glazunov, 1982; Leybson, 1982; Korchagin, 1984; Buyalov and Korchagin, 1985). An analytical expression of exploration effectiveness for a whole exploration process was given by Krylov (1982) (Fig. 3) E -

Ru D - Dt e - c o - D~)/D2

D2 D2 the symbols having the same meaning as in Eq. (2).

(4)

Petroleum Exploration Models

501

E c-

O e-

O Cumulative

exploratory

effort

Fig. 3. Function of exploration effectiveness.

In the Soviet literature, the change of E as a function of dimensionless time Td is widely discussed. For example, Eremenko et al. (1979) presented the relationship E

=

CTd e-(ATd+BT2)

(5)

that behaves similarly to relation (4). The quantity Td has the same meaning as in Eq. (1), and A, B, and C are parameters. For well-explored areas, a model of exploration effectiveness may be obtained by differentiating Eq. (3): E = A R , e-AD = A ( R ,

-R)

(6)

Thus E is proportional to the volume of resources to be discovered. This model covers only the declining part of the curve (Fig. 3). A practical method of efficiency prediction by straight-line approximation of segments on the declining part of the efficiency curve also has been proposed (Baturin et aI., 1982). From Eq. (6), an interpretation of parameter A may be deduced. It determines the decay of exploration effectiveness caused by a unit of reserves discovered and is measured in m-~ when meterage drilled is used as a measure of exploratory effort. Functions determining E may be used in practice to allocate exploration expenditures among various basins with the help of such mathematical tools as linear, nonlinear, or dynamic programming. Concluding the review of exploration models, the practical meaning of parameters R, and A of the most popular exploration function given by Eq. (3) must be emphasized. The parameters forecast the two most important quantities for decision-making in the oil industry: (1) amount of oil and gas in the area and (2) decrease in effectiveness per unit of reserves discovered.

502

~Lucki and Szkutnik

E S T I M A T I O N OF E X P L O R A T I O N M O D E L S Exploration Function

Denote by { Di } and { Ri } the time series of cumulative drilling and cumulative discoveries, respectively. Parameters R. and A of the model R

i

=

R,(1

-

e -ADi) + Ei,

i = 1..... n

(7)

are to be estimated, where { Ci } is a sequence of random variables with expectations zero. Arps et al. (1971) used, in a similar context, a least-squares method for fitting a curve of some more general form. A scanning procedure was applied for finding solutions of the respective minimization problem. Several other authors also use the least-squares techniques without giving any analysis of statistical properties of the solutions. If Ru obtained from the model is to be interpreted as ultimate discoverable resources at infinite drilling effort, clearly a statistical estimation problem is involved and not a curve-fitting problem. Hence a different method is needed. Our maximum likelihood approach leads to asymptotically normal estimates and also gives confidence sets for parameters. The maximum likelihood approach was used also by Balasanov and Kamenecki (1977). They gave, however, no confidence sets for parameters. Compare also Lee and Wang (1985), and Wiorkowski (1981). Assume that { e; } is a sequence of independent and identically distributed ff~ (0, cr)-random variables, where o is a parameter to be estimated. The loglikelihood function then becomes log L = - n log o - n log ~

[R i

- (202) -1 ~

--

Ru(1 - - z A ) ] 2

i=l

where Z/ = e -D',

i = 1. . . . . n

(8)

Putting 0 log L / O R u = 0 log L / O A = 0 log L / O a = O, the estimate .,t of A is obtained as a numerical solution (e.g., by a bisection method) of the equation

i~l

i=1

i=1

i=1

Estimates/~, of R, and ~ of o are then given by n

=

Z i=l

-

z

(1

-

i n

?~2 = n - 1 ~=, [Ri - /~u(1 - Z~)] z i=

(10)

Petroleum Exploration Models

503

Let } = (.0., A, 6 2) and ~" = (R., A, a z) be the vector of true parameters. Maximum likelihood estimators are known to be jointly asymptotically normal and efficient in the sense of attaining the Cramer-Rao lower bound for the covariance matrix

where I(~') is the information matrix (Theil, 1971). This gives the following asymptotic variances for .0u and A:

~r2

k (Di Z ~ )

Var.0. = _~.02 i=1 ~.2

Var,4 = -

2

n

Z (1 - Z j ) 2

A i=1

(11)

where i~l

- Z~d)2 ~ , (DiZdi)2 - .0~

i=

i=, DiZAi (1 - Z~)

Using the Bonferroni inequality (Seber, 1977, Chapter 5) the asymptotic simultaneous (1 - o0-confidence intervals for R~ and A are

e(g.l-~ where Ru = .0, - U~/4 (Var _0,) 1/2 ~" = /~ + U~/4 ( V a r / ~ ) l / 2

(12)

A_ = d - U~/4 (Var,q)'/2 = d + U~/4 (Vat A) 1/2 and U,~/4 is defined by O(U~/4) = 1 - c~/4, qS(.) being the standard normal distribution function. Hence, with asymptotic probability not smaller than 1 - c~, the curve R = R , (1 - e -A°) lies between two curves: R = R ~ ( 1 - e -d° ) and R = Ru(1 - e -~D) The procedure may be summarized as follows: (1) For a given sequence { D i }, compute sequence { Z i } according to Eq. (8). (2) Compute d from Eq. (9) and then .0~ and 62 by formulas (10). (3) Assume ¢ and find the confidence set according to Eqs. (11) and (12). Note that if estimation of Ru only is required, U~,/4 may be replaced by U,~/2 in the formulas for Ru, Ru, resulting in a (1 - c0-confidence interval [Ru, Ru] for R~ only, which is more accurate.

504

~Eucki and Szkutnik

Field Size Distribution Kontorovich and Detain (1977) proposed to approximate the field size distribution with the truncated Pareto distribution with the density

f(Q[h)

=IC(Q-X-R=X)

elsewheref°rl <_ Q _ < R .

where C is a normalizing factor, R. is ultimate recoverable reserves (input data), and X is a parameter of the distribution (X > 2). This leads to formulas for the number of fields and reserves in the size class Q1 < Q < Q2:

n(Q1, Q2) = 2(2 - X)R.[R.(Q2 x , - x _ QI- x) - (1 - X)(Q2 - Q,)] (1 - X)[ZR.(R.Xz - x _

R(QI, Q2)

:

1) - ( 2 -

X)(R 2 -

1)]

R"[2RX" (Q~-X _ Q2-×) _ (2 - X)(Q 2 - Q~)] 2R 2 (R.2 - x - 1) - (2 - X ) ( R .2 - 1)

Denote by 3, the sample median for the set of discovered field sizes. In estimating parameter X, the following "quantile law" (e.g., Shpilman, 1982) will be used: All deposits greater than (3-4)3, have been discovered already. Here, X may be estimated in the following manner: (1) For a sequence of deposits discovered, find 3,. (2) Find R v, the total reserves in discovered deposits greater than 47. (3) Find X as a numerical solution (e.g., by the bisection method) of the following equation implied by Eq. (13) and the "quantile law" X Rv[2R.(R .2 - h - 1) - (2 - X)(R 2 - 1)]

= R.{ZRX.[R 2-x - (43,) z - x ] - (2 - X)(R] - 163,2)} This value of X may be used in formulas (13) to obtain estimates for N(Q1, Q2) and R(Q1, Q2)The input value of R. may be obtained by a method described in the preceding paragraph. Note that reserves in the described model are measured in units such that the size of the smallest commercial deposit is equal to one.

E X A M P L E S FROM POLAND Two well-explored oil and gas basins from Poland were chosen to illustrate applications of the analytical approaches described in the previous section:

Petroleum Exploration Models

505

(1) The Carpathian Basin, comprised mainly of Tertiary formations, is one of the first basins in the world to be drilled as oil was discovered there in 1853. The first major discovery after the Second World War was made in 1954. (2) The Forecarpathian Basin is comprised mainly of Miocene formations. Exploration activity was started in 1947, and the first major field was discovered in 1958. Preliminary Study First, a study was made in order to check the conformity to general exploration rules. The total numbers of discovered fields are 71 and 58 in Carpathian and Forecarpathian Basins, respectively. The Forecarpathian field size distribution (Fig. 4) shows a good conformity of empirical data to a theoretical cumulative curve of log-normal distribution; but the left tail suggests an amodal distribution. Methods of graphic presentation and computation were taken from Megill (1971). Regarding the discovery sequences, the rule mentioned earlier has been confirmed: (1) In the Carpathian Basin, the largest field discovered in the 19th century was the 12th in succession.

0 "10

FOREC ~RPATHIAN BASIN

S

so

~o

. eo

!

j

/

90

Q

[

gS~ ¢

98

(202

o.os

0.2

,i

J

~

l

2o Q Field size ('~Oeron)

Fig. 4. Field size distribution for the Forecarpathian Basin.

506

-t~ucki and Szkutnik Q

FORECARPA THIAN BASIN

=o

.3

5O

u~

Discovered fields Fields to be discovered

-o Lt.

20

\

10

3 2

1.5 1 I

2

5

lo

20

so

loo

Field rank

Fig. 5. Oil and gas fields of the Forecarpathian Basin fitted to Zipf's law.

(2) The largest field discovered in that basin in the 20th century was the 8th in succession. (3) In the Forecarpathian Basin, the largest field was discovered somewhere between the 19th and 25th in succession. Prediction based on Zipf's law (Fig. 5) for fields greater than 0.1 million ton of oil equivalent gives the ultimate reserves and number of fields shown in Table 2.

Exploration

and Effectiveness

Functions

The B1 type function given by Eq. (3) has been used for both basins. Total oil and gas drilling meterage was used as a measure of exploratory effort because drilling activity classification corresponding to that used in the U.S. is not common in Poland. The aggregation of exploratory and development drilling will understate the exploration efficiency (when compared to areas where disaggregation is possible), but should not influence the estimates of undiscovered resources.

507

Petroleum Exploration Models

Both models, estimated with the method described in the previous section (Fig. 6 and 7), are shown below.

R, from exploration function, millions of tons of oil equivalent (MTOE) Total ultimate reserves including those discovered before zero point (MTOE) A, decay coefficient of exploration effectiveness, [m -~]

Carpathian Basin

Forecarpathian Basin

28.63 __+ 3.09

244.58 _+ 44.79

48.44 ___ 3.09

245.01 ± 44.79

1.318 +_ 0.293

0.291 _+ 0.077

The confidence interval for R, in the older basin is smaller than that in the younger basin. To explain the opposite behavior of the confidence interval for A, note that, for the older basin, data from the last part of the exploration curve which has small curvature is used and, hence, is less sensitive to changes of A.

I CARPATHIAN BASIN 1956-1985

R

60

U,]tima,. . . . . .

o

"o

~es,=~&44+``0` &096 ,

SO

40

- - - - 7 7

~-;Z-----~

I

IS

"~

r

r

30

20

10

0

Q$

I0

20

2.5 3.0 3.5 D Cumulati~ drilling activity (qO6rn)

Fig. 6. Exploration function for the Carpathian Basin. Historical data are shown as well as upper and lower limits of confidence for a significance level of 0.1. Year 1953 is assumed as zero point. Because it is an old basin, the zero point may be shifted backward in time also and no great change in the forecast will occur,

508

~'ucki and Szkutnik

'~~

: z .~ ~

~li

o

~

~~ \

\ '\.. .

~~

o

i~

/ UO,1gO,~

p4~JOAO,3S!pSO^JOSOJ $06 p~Jo )1,0

~

~

,..; ,~.

°°°

"~ "~

Petroleum

Exploration

Models

509

On the other hand, data concerning the younger basin come from the part of the exploration curve which is more sensitive to changes of A because of its greater curvature. Thus, A may be estimated more precisely. Note also that some data were discarded in both cases to shift the zero point beyond the inflection point (even if temporary). According to our experience, only the first inflection point must be discarded when several such points occur (cf. Fig. 7). The actual and estimated values of exploration effectiveness for successive millions of meters, including those drilled in the future, are given in Table 1. Number of Fields and Their Distribution Estimates of total ultimate reserves and the algorithm based on the model of Kontorovich-Demin were used to predict the number of fields and their distribution by size for both basins (Table 2). Zipf's law leads to an overestimation of larger size classes as it is too stiff to take the discovered sample of fields into account, whereas this is possible when the Kontorovich-Demin approach is used. CONCLUSIONS This report demonstrates methods for predicting results of future exploration activity which are useful for regions and countries not rich in hydrocarbons. T a b l e 1. E f f e c t i v e n e s s o f E x p l o r a t i o n "

Successive millions

Carpathian Basin

Forecarpathian Basin

of meters drilled f o r oil a n d g a s

Actual

Estimated

Actual

--

31.15 b

1

1 8 . 0 4 I'

2

22.39

20.79

67.30

Estimated -53.35

3

2.80

5.61

37.16

39.90

4

--

1.50

25.00

29.83

5

--

0.40

--

22.30

6

--

0.11

--

16.68

7

--

--

--

12.47

8

--

--

--

9.24

9

--

--

--

6.68

10

--

--

--

4.69

11

--

--

--

3.16

12

--

--

--

1.97 n

a M e t r i c t o n s o f oil e q u i v a l e n t p e r m e t e r o f d r i l l i n g . t~Values o c c u r r i n g b e f o r e t h e m a x i m u m e f f e c t i v e n e s s , n o t c o v e r e d by t h e models.

510

~Eucki and Szkutnik

Table 2. Estimates of Ultimate Number of Fields Carpathian Basin

Forecarpathian Basin

Reserves, million tons

Number of fields

Reserves, million tons

Size class, million metric tons of oil equivalent

KontomvichDemin °

Zipf

KontorovichDemin

Zipf

KontorovichDetain a

Zipf

KontorovichDemin

Zipf

0.1-2.1 2.1-4.1 4.1-6.1

121-158 1.4-1.3 0.4

119 3 2

32.5-39,8 4.2-4.1 2.1-2.0

36.2 7.0 8.5

353-773 7-8 2

499 12 4

104.9-200.8 18.7-22.9 10.5-11.4

160.3 35.2 20.2

6.1-8.1 8.1-10.1 10.1-12.1

0.2 0.1 0.06

---

1.4-1.2 1.0-0.9 0.8-0.6

-8.2 --

1 0.6 0.4

2 1 1

7.2-7.3 5.4-5.2 4.4-4.0

14.0 8.7 10.5

12.1-14.1 14.1-16.1 16.1-18.1

0.05 0.03 0.02

-

0.6-0.5 0.5-0.4 0.4-0.3

-

0.3 0.2 0.1

1 1

3.6-3.2 3.1-2.6 2.7-2.2

13.1 17.4

18.1-20.1 above 20.1

0.02 0.01

--

0.3-0.3 1.5-1.4

--

0.1 0.07

2

2.4-1.9 37.3-28.3

78.5

Total

123-160

125

45.3-51.5 b

59.9

365-786

523

1

Number of fields

-

200.2-289.8 b 357.9

aFractions should be treated as probabilities, bValues obtained from exploration functions.

N e w t e c h n i q u e s for e x p l o r a t i o n p l a n n e r s are p r o p o s e d f o r u s e in allocating exp l o r a t i o n e x p e n d i t u r e s . U s i n g interval e s t i m a t e s f o r p a r a m e t e r s m a k e s the analysis m o r e reliable. W h e n v a r i o u s e x p l o r a t i o n f u n c t i o n s ( d i s c o v e r y m o d e l s ) are a p p l i e d , a t t e n t i o n s h o u l d b e p a i d to t h e i r v a l i d r a n g e as m o s t o f t h e m do not reflect t h e p r e - i n f l e e t i o n b e h a v i o r o f a b a s i n . T h e e x a m p l e s s h o w n d e m o n s t r a t e that the m e t h o d s p r o p o s e d g i v e results c o i n c i d i n g w i t h e m p i r i c a l data a n d are u s e f u l in p r a c t i c e .

ACKNOWLEDGMENTS W e are g r a t e f u l to the e d i t o r a n d r e v i e w e r s f o r r e v i e w i n g this m a n u s c r i p t and providing many valuable c o m m e n t s and suggestions. T h i s r e s e a r c h w a s s u p p o r t e d b y the Institute o f M i n e r a l R e s o u r c e s E c o n omy, Polish A c a d e m y of Science (Grant CPBR-1.7).

Petroleum Exploration Models

511 REFERENCES

Arps, J. J. and Roberts, T. G., 1958, Economies of Drilling for Cretaceous Oil on East Flank of Denver-Julesburg Basin: Am. Assoc. Petr. Geol. Bull., v. 42, p. 2549-2566. Arps, J. J., Mortada, M., and Smith, A. E., 1971, Relationship between Proved Reserves and Exploratory Effort: J. Petr. Teehnol., v. 23, p. 671-675. Attanasi, E. D. and Drew, L. J., 1985, Lognormal Field Size Distributions as a Consequence of Economic Truncation: Math. Geol., v. 17, p. 335-351. Attanasi, E. D., Garland, T. M., Wood, J. H., Dietzman, W. D., and Hicks, J. N., 1981, Economics and Resource Appraisal--The Case of the Permian Basin: J. Petr. Technol., v. 33, p. 603-616. Babalyan, G. G. and Glazunov, V. I., 1982, Forecasting of Deep Oil Well Drilling Effectiveness: Neftegaz. Geol. Geofiz., No. 2, p. 5-8 (in Russian). Balasanov, V. A. and Kamenecki, S. G., 1977, Some Problems of Forecasting Methodology of Oil Field Discoverable Reserves: Ekon. Neff. Promst., no. 9, p. 26-28 (in Russian). Barouch, E. and Kaufman~ G. M., 1978, The Interface Between Geostatistical Modeling of Oil and Gas Discovery and Economics: Math. Geol., v. 10, p. 611-627. Baturin, J. N., Morozova, M. N., and Sokolova, T. V., 1982, Relationship between Exploratory Drilling Volume and Discovered Reserves: Neftegaz. Geol. Geofiz., no. 12, p. 1-2 (in Russian). Baymuldlametov, K. S. and Sattarov, M. M., 1975, About Some Probabilistic-statistical Model for Forecasting of Oil Reserves Increase: Geol. Neff. Gaz., no. 3, p. 27-32 (in Russian). Buyalov, N. L and Korchagin, V. I., 1985, Structure of Oil and Gas Drilling Activity as a Function of Reserves Recognition Degree: Neftegaz. Geol. Geofiz. Burene, no. 7, p. 1-4 (in Russian). Collins, D. R., 1985, Changes in Drilling Density and Discovery Rates Through Time: Math. Geol., v. 17, p. 375-393. Drew, L. J., Root, D. H., and Bawiec, W. J., 1979, Estimating Future Rates of Petroleum Discoveries in the Permian Basin: SPE Paper No. 7722. Eremenko, N. A., Krylov, N. A., Kuvykin, J. S., and Stasenko, V. V., 1979, Forecasting Method of Petroleum Exploration Effectiveness: Geol. Neft. Gaz., no. 1, p. 7-13 (in Russian). Haun, J. D. (Ed.), 1975, Methods of Estimating the Volume of Undiscovered Oil and Gas Resources: AAPG Stud. Geol., no. 1., 204 p. Hubbert, M. K., 1967, Degree of Advancement of Petroleum Exploration in USA: Bull. Am. Assoc. Petr. Geol., v. 51, p. 2207-2215. Ivanhoe, L. F., 1976, Oil/Gas Potential in Basins Estimated: Oil Gas J., no. 49, p. 154-155. Jones, D. A., Buck, N. A., and Kelsey, J. H., 1982a, Choosing an Optimum Exploration Strategy: World Oil, v. 195, p. 71-86. Jones, D. A., Buck, N. A., and Kelsey, J. H., 1982b, Model to Evaluate Exploration Strategies: Bull. Am. Assoc. Petr. Geol., v. 66, p. 317-331. Kontorovich, A. E. and Detain, V. I., 1977, Estimation Method of Number and Size Distribution of Oil and Gas Fields in Large Oil and Gas Basins: Geol. Nell Gaz., no. 12, p. 18-26 (in Russian). Kontorovich, A. E., Detain, V. I., and Strakhov, I. A., 1985, Rules Governing the Discoveries of Oil and Gas Fields of Various Size in Oil and Gas Basins: Geol. Geofiz., no. 11, p. 3-16 (in Russian). Korchagin, V. I., 1984, Forecasting the Number and Size of Oil and Gas Deposits: Geol. Neft. Gaz., no. 12, p. 30-33 (in Russian). Krige, D. G., 1951, A Statistical Approach to Some Basic Mine Valuation Problems on the Witwatersrand: J. Chem. Met. Min. Soc. S. Afr., v. 52, p. 119-139. Krylov, N. A., 1982, About the Notion "Prospecting Orientation" in Oil and Gas Fields Exploration: Geol. Neft. Gaz., no. 11, p. 17-22 (in Russian).

512

4~ucki and Szkutnik

Lee, P. J. and Wang, P. C. C., 1983a, Probabilistic Formulation of a Method for the Evaluation of Petroleum Resources: Math. Geol., v. 15, p. 163-181. Lee, P. J. and Wang, P. C. C., 1983b, Conditional Analysis for Petroleum Resource Evaluations: Math. Geol., v. 15, p. 349-361. Lee, P. J. and Wang, P. C, C., 1985, Prediction of Oil or Gas Pool Sizes when Discovery Record is Available: Math. Geol., v. 17, p. 95-113. Leybson, M. G., 1982, A Method of Long-Range Petroleum Exploration Planning: Geol. Neff. Gaz., no. 11, p. 12-17 (in Russian). Megil, R. E., 1971, An Introduction to Exploration Economics: Petroleum Publishing Co., Tulsa, 159 p. Nesterov, I. I., Shpilman, V. I., Myasnikova, G. P., Plavnik, G. I., Zavgorodnaya, L. D., Krasnova, G. N., and Sudat, L. G., 1977, New Characteristics of Potential Resources: Geol. Neff. Gaz., no. 12, p. 26-32 (in Russian). Rapoport, L. A. and Grender G. C., 1977, The Impact of Assumptions About Oil Exploration on Projections of Long-Range Energy Supplies in the United States: Math. Geol., v. 9, p. 429439. SchoetIler, J., 1971, La "bonn~ technique" et le hasard: Rev. AFTP, no. 207, p. 15-20. Schuenemeyer, J. H. and Drew, L. J., 1983, A Procedure to Estimate the Parent Population of the Size of Oil and Gas Fields as Revealed by a Study of Economic Truncation: Math. Geol., v. 15, p. 145-161. Seber, G. A. F., 1977, Linear Regression Analysis, Wiley, New York, 480 p. Selicki, A. G., 1978, Forecasting of optimum oil exploration volume: Geol. Neff. Gaz., no. 11, p. 10-14 (in Russian). Shpilman, V. I. 1982, Quantitative Forecast of Oil and Gas Occurrence: Nedra, Moscow, 214 p. (in Russian). Theil, H., 1971, Principles of Econometrics, Wiley, New York, 736 p. Wiorkowski, J. J., 1981, Estimating Volumes of Remaining Fossil Fuel Resources: A Critical Review: J. Am. Stat. Assoc., v. 76, p. 534-559.