Oil and Gas Exploration in the United States, 1998 

Oil 

Gas 

Both 

Dry 

Each dot represents 

1 square mile 

Probabilistic Method for Estimating 

Future Growth of Oil and Gas Reserves 

Chapter C of 

Geologic, Engineering, and Assessment Studies of Reserve Growth 

U.S. Department of the Interior 

U.S. Geological Survey 

U.S. Geological Survey Bulletin 2172–C 
Cover. This map represents historical oil and gas exploration and production data for the conterminous 

United States. It was derived from data used in U.S. Geological Survey Geologic Investigations Series I2582.* The map was compiled using Petroleum Information Corporation’s (currently IHS Corporation) 

database of more than 2.2 million wells drilled in the U.S. as of June 1993. The area of the U.S. was subdivided into 1 mi2 grid cells for which oil and gas well completion data were available. Each colored symbol 

represents a 1 mi2 cell (to scale) for which exploration has occurred. Each cell is identified by color as follows: red, a gas-producing cell; green, an oil-producing cell; yellow, an oil- and gas-producing cell; gray, a 

cell that has been explored through drilling, but no production has been reported. Mast and others (1998) 

gives details on map construction. 

*Mast, R.F., Root, D.H., Williams, L.P., Beeman, W.R., and Barnett, D.L., 1998, Areas of historical oil and gas exploration and production in the conterminous United States: U.S. Geological Survey Geologic Investigations Series I-2582, 

one sheet. 
Probabilistic Method for Estimating 

Future Growth of Oil and Gas Reserves 

By Robert A. Crovelli and James W. Schmoker 

Chapter C of


Geologic, Engineering, and Assessment Studies of Reserve Growth


Edited by T.S. Dyman, J.W. Schmoker, and Mahendra Verma


U.S. Geological Survey Bulletin 2172–C 

U.S. Department of the Interior 

U.S. Geological Survey 
U.S. Department of the Interior 

Gale A. Norton, Secretary 

U.S. Geological Survey 

Charles G. Groat, Director 

Version 1.0, 2001 

This publication is only available online at: 

http://geology.cr.usgs.gov/pub/bulletins/b2172-c/ 

Manuscript approved for publication July 25, 2001


Published in the Central Region, Denver, Colorado


Graphics by the authors and Gayle M. Dumonceaux


Photocomposition by Gayle M. Dumonceaux


Edited by L.M. Carter


Any use of trade, product, or firm names in this publication


is for descriptive purposes only and


does not imply endorsement by the U.S. Government

Contents 

Introduction .................................................................................................................................. 1


Acknowledgments........................................................................................................................ 1


Deterministic Method .................................................................................................................. 1


Input Data............................................................................................................................ 1


Calculations ........................................................................................................................ 2


Probabilistic Method ................................................................................................................... 2


Spreadsheet System .................................................................................................................... 6


Summary ........................................................................................................................................ 11


References Cited ......................................................................................................................... 12


Figures 

1.	 Spreadsheet illustrating program for estimating future oil and gas


reserve growth ........................................................................................................................ 3


2.	 Diagram showing left-triangular probability distribution of growth 


variable (dimensionless) ........................................................................................................ 4


3.	 Graph of probabilistic estimates of future reserve growth of Lower 48


United States gas fields ......................................................................................................... 12


III

Probabilistic Method for Estimating 

Future Growth of Oil and Gas Reserves 

By Robert A. Crovelli and James W. Schmoker 

Introduction 

In the United States, the estimated size (cumulative production plus remaining reserves) of oil and gas fields typically 

increases through time as fields are discovered, developed, and 

produced. (See, for example, Arrington, 1960; Attanasi and 

Root, 1994.) This phenomenon is usually referred to as reserve 

growth or field growth; the term “reserve growth” is used here. 

Reserve-growth patterns of individual fields are highly 

variable. Indeed, the sizes of some fields are observed to 

decrease through time. However, for United States fields as a 

whole, collective reserve growth is strongly positive and is a 

major component of remaining United States oil and natural-gas 

resources (Gautier and others, 1995; U.S. Geological Survey 

National Oil and Gas Resource Assessment Team, 1995; 

Schmoker and Attanasi, 1997). 

International oil and gas fields also show clear evidence of 

reserve growth (for example, Root and Attanasi, 1993; Oil & 

Gas Journal, 1996, p. 37; U.S. Geological Survey World Energy 

Assessment Team, 2000), even though criteria for estimating 

and reporting field sizes can be quite different from those of the 

United States. Worldwide, reserve growth has the potential to 

become important in the future, especially for gas, as demand 

increases and opportunities for new large-field discoveries 

decrease. Projections of the future reserve growth of known 

fields have thus become important, and in our view, necessary 

components of petroleum resource assessments. 

Many algorithms for estimating the future growth of known 

fields utilize the age of fields (years since discovery) as a predictive variable, on the assumption that age is a surrogate for the 

degree of the development activity that generates reserve 

growth. Two additional assumptions implicit in most reservegrowth models are as follows: (1) reserve growth of a field is 

proportional to its field size, and (2) patterns of past reserve 

growth provide some basis for forecasting future reserve growth. 

Schmoker and Crovelli (1998) presented an algorithm (a 

deterministic method) for estimating future reserve growth of oil 

and gas fields that incorporates the fundamental reserve-growth 

assumptions listed in the preceding paragraph, but which is programmed for a personal computer in the form of formulas for a 

Microsoft® Excel spreadsheet. Major advantages of this 

spreadsheet program include its simplicity and ease of use. 

However, like all other published reserve-growth models of 

which we are aware, this program (Schmoker and Crovelli, 

1998) generates single-value (point) estimates of future reserve 

growth, in contrast to estimates in the form of probability 

distributions. 

The purpose of this report is to explain an analytic probabilistic method and spreadsheet software system called probabilistic reserve growth spreadsheet (PREGS). The probabilistic 

method herein is a probabilistic extension of the deterministic 

method of Schmoker and Crovelli (1998). The PREGS method 

is based upon mathematical equations derived from probability 

theory. The PREGS spreadsheet can be used to calculate probabilistic estimates of reserve growth of oil and gas reserves. 

Acknowledgments 

The authors wish to acknowledge the helpful reviews of 

T.S. Dyman and Mahendra Verma of the U.S. Geological 

Survey. 

Deterministic Method 

The spreadsheet that implements the deterministic method 

(Schmoker and Crovelli, 1998) for estimating future growth of 

oil and gas reserves is shown in figure 1. The data used to illustrate the spreadsheet system are based on successive annual estimates made between 1977 and 1991 of the sizes of Lower 48 

United States gas fields. These data were compiled by the U.S. 

Department of Energy and summarized in table 2 of Attanasi 

and Root (1994). 

Input Data 

Column A of figure 1A and 1B identifies the age classes to 

which the input data of columns B and C apply, and is not used 

in the spreadsheet calculations. Age (column A) is defined as 

years since discovery. 

In the deterministic method, a growth function is composed of a set of growth factors. A growth factor is a dimensionless parameter, that is, a multiplicative constant representing 

proportional growth. The growth function of column B is composed of seventeen 10-year growth factors (rows 2 through 18). 

Each 10-year growth factor applies to fields of a particular age 

or range of ages, as indicated by column A. For example, the 

10-year growth factor of 1.529 in cell B7 applies to fields that 

are 5 years old. The size of fields 5 years of age would be multiplied by 1.529 to forecast the size of these same fields 10 years 

later, when they have become 15 years old. 

1

-------------------

whereP 

Rows 2 through 18 of column C (fig. 1A and 1B) list the 

volumes of petroleum that will be “grown” by the 10-year 

growth factors of column B. In this illustration, the values in 

column C are the totals of the estimated sizes of Lower 48 

United States gas fields as of 1977 for each of the 17 field-age 

classes of column A. For example, the size as estimated in 1977 

of Lower 48 gas fields 5 years of age (those discovered in 1972) 

is 5,400 billion cubic feet of gas (bcfg) (cell C7). 

Calculations 

The formulas shown in columns D through L (fig. 1A) are 

used to calculate future growth of the petroleum volumes tabulated in column C. The headings of columns D through L (row 

1) indicate that the spreadsheet program calculates reserve 

growth from 10 to 90 years beyond the date associated with the 

field-size estimates of column C, at 10-year increments. 

The spreadsheet algorithm can be explained by considering 

how reserve growth is calculated for the fields of a single age 

class. Consider the fields that were 5 years old (row 7) in 1977, 

when the field sizes of column C were estimated. To calculate 

the first growth increment of 10 years, the petroleum volume in 

cell C7 is multiplied by the 10-year growth factor that applies to 

5-year-old fields (cell B7). This multiplication is done in cell 

D7. In the example used here, the total estimated size of 5-yearold fields of 5,400 bcfg grows to 8,257 bcfg (fig. 1B, cell D7), 

after the first 10-year growth period. 

After 10 years of growth, fields that were 5 years old have 

become 15 years old. Therefore, to calculate the second growth 

increment of 10 years, the petroleum volume in cell D7 is multiplied by the 10-year growth factor that applies to 15-year-old 

fields, which is in cell B10. This multiplication is done in cell 

E7. The total estimated size of 5-year-old fields of 5,400 bcfg 

grows to 10,106 bcfg (fig. 1B, cell E7), after two 10-year growth 

periods. 

Fields that were 5 years old have become 25 years old after 

two 10-year growth periods. The 10-year growth factor that 

applies to 25-year-old fields is in cell B11 and the multiplication 

process continues. The estimated size of 5-year-old fields grows 

from 5,400 bcfg to 20,129 bcfg (fig. 1B, cell L7) after nine 10-

year growth periods. Reserve-growth calculations for the other 

age classes proceed similarly. 

Rows 2 through 18 of each column (except columns A and 

B) are summed in row 20 (fig. 1). Thus, cell C20 contains the 

total initial volume of all the fields that are to be grown 

(463,656 bcfg in this example); cell D20 contains the total volume of these same fields increased by 10 years of reserve 

growth (557,163 bcfg); cell E20 contains the total volume of 

these same fields increased by 20 years of reserve growth 

(629,585 bcfg), and so on. The total volume of petroleum 

attributable to reserve growth following each 10-year growth 

increment is calculated in row 22 by subtracting cell C20 (the 

total initial volume) from cell D20 (the total volume after 10 

years of reserve growth), from cell E20 (the total volume after 

20 years of reserve growth), and so on. In this example, total 

reserve growth in the 90 years between 1977 and 2067 is 

projected to be 480,816 bcfg (fig. 1B, cell L22). 

2 Probabilistic Method, Growth of Oil and Gas Reserves 

Probabilistic Method 

In the probabilistic method, a growth function is composed 

of a set of growth variables. A growth variable is a dimensionless random variable having a probability distribution. In contrast, a growth factor (previous section) is a multiplicative 

constant. It is important to the understanding of this paper to 

keep these two definitions in mind. Each 10-year growth variable, like the corresponding 10-year growth factor, applies to 

fields of a particular age or range of ages. The growth variable 

in the probabilistic method is related to the growth factor in the 

deterministic method as follows: the mean value of a growth 

variable is set equal to its corresponding growth factor. 

The mean-based left-triangular probability distribution (fig. 

2) is used here as a probability model for the random variable X: 

growth variable (dimensionless). The right-skewed shape of the 

left-triangular probability distribution is appropriate as a probability model for a growth variable, whereby high probability of 

low growth values tapers off to low probability of high growth 

values. The left-triangular probability distribution is described 

in Law and Kelton (1991, p. 516). The defining parameters of 

the mean-based left-triangular probability distribution are the 

minimum (a) and mean (μ). From the defining parameters, the 

standard characterizing parameters of the left-triangular probability distribution are obtained: minimum (a) and maximum (b). 

Namely, given the formula for the mean in terms of a and b 

μ X = a + (b – a) ⁄ 3 

We solve for the maximum, b, and get 

b = 3μ X – 2a 

The probability density function for the left-triangular distribution is given by 

– fx 2(bx -) () = a x

(b a) 2 ≤≤ b – 

The graph of the probability density function for the lefttriangular distribution of the growth variable is displayed in 

figure 2. 

The standard deviation is equal to 

σ – X = (b a) ⁄ (3 2) 

The fractiles can be computed from 

F 100 p = b – p ( b – a) 0 ≤≤ p 1

where ( X ≥ F100 p) = p 
1 age, yr grwth funct ESF known ESF + 10yr ESF + 20yr ESF + 30yr ESF + 40yr ESF + 50yr ESF + 60yr ES

2 0 4.460 2024 =C2*B2 =D2*B10 =E2*B11 =F2*B12 =G2*B13 =H2*B14 3 1 2.484 2794 =C3*B3 =D3*B10 =E3*B11 =F3*B12 =G3*B13 =H3*B14 4 2 2.077 3890 =C4*B4 =D4*B10 =E4*B11 =F4*B12 =G4*B13 =H4*B14 5 3 1.809 7009 =C5*B5 =D5*B10 =E5*B11 =F5*B12 =G5*B13 =H5*B14 6 4 1.696 5603 =C6*B6 =D6*B10 =E6*B11 =F6*B12 =G6*B13 =H6*B14 7 5 1.529 5400 =C7*B7 =D7*B10 =E7*B11 =F7*B12 =G7*B13 =H7*B14 8 6 1.449 5334 =C8*B8 =D8*B10 =E8*B11 =F8*B12 =G8*B13 =H8*B14 9 7-9 1.304 11402 =C9*B9 =D9*B10 =E9*B11 =F9*B12 =G9*B13 =H9*B14 

10 10-19 1.224 77058 =C10*B10 =D10*B11 =E10*B12 =F10*B13 =G10*B14 =H10*B15 11 20-29 1.167 105915 =C11*B11 =D11*B12 =E11*B13 =F11*B14 =G11*B15 =H11*B16 12 30-39 1.130 81263 =C12*B12 =D12*B13 =E12*B14 =F12*B15 =G12*B16 =H12*B17 13 40-49 1.100 47880 =C13*B13 =D13*B14 =E13*B15 =F13*B16 =G13*B17 =H13*B18 14 50-59 1.090 93233 =C14*B14 =D14*B15 =E14*B16 =F14*B17 =G14*B18 =H14*B18 15 60-69 1.080 14748 =C15*B15 =D15*B16 =E15*B17 =F15*B18 =G15*B18 =H15*B18 16 70-79 1.080 102 =C16*B16 =D16*B17 =E16*B18 =F16*B18 =G16*B18 =H16*B18 17 80-89 1.080 0 =C17*B17 =D17*B18 =E17*B18 =F17*B18 =G17*B18 =H17*B18 18 90+ 1.000 0 =C18*B18 = D18*B18 =E18*B18 =F18*B18 =G18*B18 =H18*B18 19 20 sum of columns =SUM(C2:C18) =SUM(D2:D18) =SUM(E2:E18) =SUM(F2:F18) =SUM(G2:G18) =SUM(H2:H18) =SUM(I2:I18) =SUM21 22 reserve growth =C20-C20 =D20-C20 =E20-C20 =F20-C20 =G20-C20 =H20-C20 =I20-C20 =23 A 24 25 26 age, yr grwth funct ESF known ESF + 10yr ESF + 20yr ESF + 30yr ESF + 40yr ESF + 50yr ESF + 60yr ES27 0 4.460 2024 9027 11049 12894 14571 16028 17470 28 1 2.484 2794 6940 8495 9914 11202 12323 13432 29 2 2.077 3890 8080 9889 11541 13041 14345 15636 30 3 1.809 7009 12679 15519 18111 20466 22512 24538 31 4 1.696 5603 9503 11631 13574 15338 16872 18391 32 5 1.529 5400 8257 10106 11794 13327 14660 15979 33 6 1.449 5334 7729 9460 11040 12475 13723 14958 34 7-9 1.304 11402 14868 18199 21238 23999 26399 28775 35 10-19 1.224 77058 94319 110070 124379 136817 149131 161061 36 20-29 1.167 105915 123603 139671 153638 167466 180863 195332 37 30-39 1.130 81263 91827 101010 110101 118909 128422 138695 38 40-49 1.100 47880 52668 57408 62001 66961 72318 72318 39 50-59 1.090 93233 101624 109754 118534 128017 128017 128017 40 60-69 1.080 14748 15928 17202 18578 18578 18578 18578 41 70-79 1.080 102 110 119 119 119 119 119 42 80-89 1.080 0 0 0 0 0 0 0 43 90+ 1.000 0 0 0 0 0 0 0 44 45 sum of columns 463655 557162 629584 697456 761286 814308 863299 46 47 reserve growth 0 93507 165929 233801 297631 350653 399644 

Geologic, Engineering, and Assessment Studies of Reserve Growth B Figure 1. Spreadsheet program for estimating future oil and gas reserve growth. 

A, Calculation formulas and example of input data (in this ca

gas fields). 

B, Results of calculations. See text for detailed explanation; grwth funct, growth function (dimensionless); ESF, estimated size of 

maining reserves); units for ESF and reserve growth in this example are billion cubic feet of gas. (Modified from Schmoker and Crovelli, 1998.) 

3

PROBABILITY DENSITY


p 

a μ F100p b 

GROWTH VARIABLE 

Figure 2. Left-triangular probability distribution of growth variable (dimensionless) where a: minimum, b: maximum, μ: mean, and 

fractile F 100p for 0≤ p ≤1. 

For example, the median (where p = 0.50) can be shown to be 

F 50 = a + 0.293(b a – ) 

Suppose we denote the parameter c: initial petroleum volume, that is, the volume of petroleum that will be “grown” (for 

example, column C of fig. 1). Then the random variable cX represents the grown petroleum volume (total estimated size) after 

the first growth period. It can be proved that the random variable 

cX is also distributed as a left-triangular probability distribution 

with characterizing parameters: minimum (ca) and maximum 

(cb). The mean, standard deviation, and fractiles of cX are the 

following: 

μcX = cμ X 

σcX = cσ X 

F 100 p = c[b – p (b – a)] 0 ≤≤ p 1

In general, the calculated future growth is based upon a 

series of multiplications by the appropriate growth variables, as 

in the case of the growth factors in the previous section and 

shown in figure 1A. Let us denote the random variable Yi : grown 

petroleum volume (total estimated size) after the ith growth 

period. The growth process can be described as a stochastic process consisting of the set of random variables {Yi

} where i = 0, 1, 

2, 3, ..., n. Note that Y0 represents the initial petroleum volume 

c. Also, Xi represents the growth variable that is applied in the ith 

growth period, where i = 1, 2, 3, ..., n. Therefore, we have the 

following sequence of grown petroleum volumes after i growth 

periods for i = 0, 1, 2, 3, ..., n: 

4 Probabilistic Method, Growth of Oil and Gas Reserves 

Y 0 = c 

Y 1 = Y 0 X 1 = cX 1 

Y 2 = Y 1 X 2 = cX 1 X 2 

Y 3 = Y 2 X 3 = cX 1 X 2 X 3 

... 

Yi = Yi – 1 Xi = cX 1 X 2 X 3 ... Xi 

... n 

Yn = Yn – 1 Xn = cX 1 X 2 X 3 ... Xn = c ∏ Xi 

i = 1 

This stochastic process can be viewed as an example of the 

law of proportionate effect. The law of proportionate effect as 

stated by Aitchison and Brown (1957) is: 

A variate subject to a process of change is said to obey the 

law of proportionate effect if the change in the variate at any 

step of the process is a random proportion of the previous value 

of the variate. 

The importance of the law is embodied in the following 

theorem: 

A variate subject to the law of proportionate effect tends, 

for large n, to be distributed as a two-parameter lognormal distribution, that is, asymptotically lognormally distributed in a twoparameter form. 

The lognormal probability distribution is a good 

approximate distribution for a product of independent random 
variables—the approximation becoming better with more vari- Notice that the total initial petroleum volume is 

ables (multiplicative central limit theorem for independent ran­ m m dom variables). Hence, the fractiles of Yi where i = 2, 3, ..., n S0 = ∑ Y 0 j = ∑ cj can be approximated by using the lognormal distribution. As 

derived in Crovelli (1992), the characterizing parameters of the j = 1 j = 1 

lognormal distribution, namely mu (μ) and sigma (σ), can be For example in the deterministic case, S0 corresponds to cell 

calculated from the mean μY and standard deviation σY of a log- C20 of figure 1. 

normal random variable Y as follows: Finally, the reserve growth after the ith growth period (Ri

) 

= (total grown petroleum volume after the ith growth period) – 2 

Y μ

2 

Y μ + 2 Y σ

(total initial petroleum volume). That is, 

μ = ln m m 

reserve growth(i) = Ri = Si – S0 = ∑ Yij – ∑ cj 

ln (σ

2 

Y / 2 Y μ + 1) 

j = l j = l 

σ = 

for i = 0, 1, 2, 3, ...., n 

Knowing the lognormal characterizing parameters, the Also, we have total reserve growth after the nth growth period = 

lognormal fractiles can be calculated from the formula reserve growth(n). 

F 100 p = eμ + zpσ 0 ≤≤ 1 A simplifying assumption for purposes of mathematical +	 p tractability will be made concerning the nature of the Yij series 

for j = 1, 2, 3, ..., m. The assumption of perfect positive correlawhere Z is a standard normal random variable and P(Z > zp) = p. tion is made for these random variables. Under the assumption 

of perfect positive correlation, the standard deviations and fractiles are additive. (The means are always additive.) That is, the 

The two fractiles of most interest in this report are mean, standard deviation, and fractiles of Si are 

m 

F 95 = e μ – 1.645σ and F5 = e μ + 1.645σ 

μi = ∑ μij 

j = 1 

At the core of the above stochastic process is the product of 

independent random variables in the form of YX, where Y repre- m 

sents a petroleum volume and X represents a growth variable. σi = ∑ σij 

The mean, standard deviation, minimum, and maximum of YX j = l 

are as follows: m 

μYX = μYμ X (F 100 p)i = ∑ (F 100 p)ij 0 ≤≤ p 1

j = 1 

σYX = σ

2 

Y σ

2 

X + σ

2 

Y 

2 

X μ + σ

2 

X 

2 

Y μ Because of the simplifying assumption, we have 

P F95)i ≤ (F5) {( i

} ≥ 0.90

Min(YX) = Min(Y)Min(X) 

Max(YX) = Max(Y)Max(X)	 The probability of Si being within the range from (F95)i to (F5)i 

is at least 0.90. 

Suppose there are m initial petroleum volumes, that is, we Also under the assumption of perfect positive correlation, 

are now interested in m volumes of petroleum that will be the mean, standard deviation, and fractiles of Ri

 are 

“grown.” For example in the deterministic case, figure 1 (col­ m m umn C) lists m = 17 initial petroleum volumes, two of which are 

zero. Let us denote the random variable Yij : jth grown petroleum μi = ∑ μij – ∑ cj


volume (total estimated size) after the ith growth period, where j j = 1 j = 1


= 1, 2, 3, ..., m. Note that Y0j represents the jth initial petroleum m


volume cj. Then the total grown petroleum volume after the ith σi = ∑ σij


growth period (Si) would be the sum j = l


m m m 

S p i =	 ∑ Yij for i = 0, 1, 2, 3, ...., n (F 100 p)i = ∑ (F100 p)ij – ∑ cj 0 ≤≤ 1 

j = 1 j = 1 j = 1 

≤ Si } 

Geologic, Engineering, and Assessment Studies of Reserve Growth 5 
Because of the simplifying assumption, we have 

P F95)i }≤ Ri ≤( (F5) {( i

} ≥ 0.90

The probability of Ri being within the range from (F95)i to (F5)i 

is at least 0.90. 

Spreadsheet System 

The analytic probabilistic method described in the previous 

section is incorporated into a spreadsheet software system called 

probabilistic reserve growth spreadsheet (PREGS). PREGS consists of a series of 10 panels in the spreadsheet. A panel is a set 

of approximately 11 columns of related calculations. 

Note that column A of Panel 1 in the probabilistic spreadsheet is the same as column A in the deterministic spreadsheet 

(fig. 1). Panel 1 includes the input of the defining parameters of 

the mean-based left-triangular probability distribution for each 

growth variable X in the growth function: mean (column B) and 

minimum (column D). Note that column B in the probabilistic 

spreadsheet is the same as column B in the deterministic spreadsheet. From the defining parameters, the maximum (column J) 

is computed, and then the other descriptive parameters: standard 

deviation (column C) and fractiles F95, F75, F50, F25, and F5 

(columns E through I). For example, the 10-year growth variable with parameters in cell B9 (mean of 1.529) through cell J9 

applies to fields that are 5 years old. The known estimated size 

of fields to be grown is also entered (column K). 

Note that column K in the probabilistic spreadsheet is the 

same as column C in the deterministic spreadsheet. For example, the size as estimated in 1977 of Lower 48 gas fields 5 years 

of age (those discovered in 1972) is 5,400 billion cubic feet of 

gas (bcfg) (cell K9). The total initial petroleum volume S0 is 

463,656 bcfg (cell K22). 

Panel 2 comprises the computed descriptive parameters of 

the left-triangular probability distribution for the grown petroleum volumes after the first growth period Y1j

, and the corresponding estimates of reserve growth R1 (row 24): mean, 

standard deviation, and fractiles. Note that column L in the 

probabilistic spreadsheet is the same as column D in the deterministic spreadsheet (fig. 1 (values, 1B)). For example, the total 

estimated initial size of 5-year-old fields of 5,400 bcfg grows to 

8,257 bcfg (mean estimate in cell L9, rounded) with a range 

from 5,617 bcfg (F95 estimate in cell O9) to 12,054 bcfg (F5 

estimate in cell S9), after the first 10-year growth period. The 

reserve growth is projected to be 93,507 bcfg (mean estimate in 

cell L24) with a range from 7,103 bcfg (F95 estimate in cell 

O24) to 217,794 bcfg (F5 estimate in cell S24), after the first 10-

year growth period. 

Panel 3 includes the computed descriptive parameters of 

the lognormal probability distribution for the grown petroleum 

volumes after the second growth period Y2j, and the corresponding estimates of reserve growth R2: mean, standard deviation, 

and fractiles. The mean (column U) and standard deviation 

(column V) are used to compute the characterizing parameters 

of the lognormal distribution: mu (column W) and sigma (column X), which, in turn, are used to compute the fractiles. Note 

that column U in the probabilistic spreadsheet is the same as 

A B C D E F G H I J K 

1 PREGS: Probabilistic Reserve Growth Spreadsheet R.A. Crovelli (Panel 1) 

2 Growth Function ESF,known 

3 Age,yr mean,E(X) SD(X) F100 F95 F75 F50 F25 F5 F0 (bcfg) 

40 4.46 2.446589463 1 1.26282747 2.390656309 4.040231611 6.19 9.058961439 11.38 2024 

51 2.484 1.049346463 1 1.112727158 1.596454902 2.30396061 3.226 4.456502536 5.452 2794 

62 2.077 0.761554003 1 1.081810747 1.43287192 1.94633799 2.6155 3.508526436 4.231 3890 

73 1.809 0.572049386 1 1.061453013 1.325156345 1.710851842 2.2135 2.884306302 3.427 7009 

84 1.696 0.49214632 1 1.052869341 1.279738957 1.611561041 2.044 2.621109006 3.088 5603 

95 1.529 0.374059487 1 1.040183737 1.212617684 1.464821538 1.7935 2.232136012 2.587 5400 

10 6 1.449 0.317490945 1 1.034106802 1.180463781 1.394527166 1.6735 2.045801643 2.347 5334 

11 7–9 1.304 0.214960461 1 1.023092356 1.122184832 1.267118616 1.456 1.7080706 1.912 11402 

12 10–19 1.224 0.158391919 1 1.01701542 1.090030929 1.196824243 1.336 1.521736232 1.672 77058 

13 20–29 1.167 0.118086832 1 1.012685603 1.067121273 1.146739503 1.2505 1.388972994 1.501 105916 

14 30–39 1.13 0.091923882 1 1.009875021 1.052250093 1.114228355 1.195 1.302793349 1.39 81263 

15 40–49 1.1 0.070710678 1 1.00759617 1.040192379 1.087867966 1.15 1.232917961 1.3 47880 

16 50–59 1.09 0.06363961 1 1.006836553 1.036173141 1.079081169 1.135 1.209626165 1.27 93233 

17 60–69 1.08 0.056568542 1 1.006076936 1.032153903 1.070294373 1.12 1.186334369 1.24 14748 

18 70–79 1.08 0.056568542 1 1.006076936 1.032153903 1.070294373 1.12 1.186334369 1.24 102 

19 80–89 1.08 0.056568542 1 1.006076936 1.032153903 1.070294373 1.12 1.186334369 1.24 0 

20 90+ 1 0 1 1 1 1 1 1 1 0 

21 

22 Sum of columns 463656 

23 

24 Reserve growth 0 

Panel 1 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: example of input data (in this case the 

same as that used in figure 1 and representing Lower 48 United States gas fields). ESF, estimated size of fields (cumulative production plus 

remaining reserves); units for ESF and reserve growth in this example are billion cubic feet of gas. In all panels, SD, standard deviation. 

6 Probabilistic Method, Growth of Oil and Gas Reserves 
L M N 0 P Q R S T 

1 (Panel 2) 

2 Estimated Size of Fields, +10yr 

3 mean,E(Y) SD(Y) F100 F95 F75 F50 F25 F5 F0 

4 9027.04 4951.897073 2024 2555.962799 4838.688369 8177.428781 12528.56 18335.33795 23033.12 

5 6940.296 2931.874018 2794 3108.959679 4460.494997 6437.265945 9013.444 12451.46809 15232.888 

6 8079.53 2962.445073 3890 4208.243807 5573.87177 7571.254781 10174.295 13648.16784 16458.59 

7 12679.281 4009.494146 7009 7439.724165 9288.020822 11991.36056 15514.4215 20216.10287 24019.843 

8 9502.688 2757.495829 5603 5899.226917 7170.377376 9029.576512 11452.532 14686.07376 17302.064 

9 8256.6 2019.921231 5400 5616.992182 6548.135495 7910.036307 9684.9 12053.53446 13969.8 

10 7728.966 1693.496699 5334 5515.925681 6296.593808 7438.407902 8926.449 10912.30597 12518.898 

11 14868.208 2450.979182 11402 11665.29904 12795.15145 14447.68645 16601.312 19475.42099 21800.624 

12 94318.992 12205.36449 77058 78369.17424 83995.6033 92224.88252 102949.488 117261.9506 128840.976 

13 123603.972 12507.28495 105916 107259.6084 113025.2167 121458.0612 132447.958 147114.4637 158979.916 

14 91827.19 7470.010387 81263 82065.4738 85508.99927 90545.53884 97109.285 105868.8959 112955.57 

15 52668 3385.627268 47880 48243.7046 49804.4111 52087.1182 55062 59032.11196 62244 

16 101623.97 5933.311788 93233 93870.39232 96605.53045 100605.9746 105819.455 112777.0762 118405.91 

17 15927.84 834.2728647 14748 14837.62265 15222.20576 15784.70141 16517.76 17496.05927 18287.52 

18 110.16 5.769991334 102 102.6198474 105.2796981 109.170026 114.24 121.0061056 126.48 

19 0 0 0 0 0 0 0 0 0 

20 0 0 0 0 0 0 0 0 0 

21 

22 557162.733 66119.24499 463656 470758.9301 501238.5804 545818.464 603916.0995 681449.9756 744176.199 

23 

24 93506.733 66119.24499 0 7102.930078 37582.58039 82162.46403 140260.0995 217793.9756 280520.199 

Panel 2 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after one growth 

period (in this application, 10 years). 

U V W X Y Z AA AB AC AD AE 

1 (Panel 3) 

2 Estimated Size of Fields, +20yr 

3 mean,E(Y) SD(Y) Mu Sigma F100 F95 F75 F50 F25 F5 F0 

4 11049.09696 6276.682824 9.17026423 0.528847333 2024 4025.121464 6726.573315 9607.162853 13721.33682 22930.38332 38511.37664 

5 8494.922304 3781.829456 8.956817213 0.42522153 2794 3855.776902 5826.768876 7760.617721 10336.2925 15619.98761 25469.38874 

6 9889.34472 3873.758204 9.127840092 0.37781761 3890 4945.978612 7138.022265 9208.111804 11878.5456 17143.08323 27518.76248 

7 15519.43994 5340.534566 9.593891156 0.334537145 7009 8464.003376 11712.51399 14674.86076 18386.44876 25443.22453 40161.1775 

8 11631.29011 3721.295844 9.312727251 0.312175971 5603 6628.924004 8976.0989 11078.11978 13672.39144 18513.52311 28929.05101 

9 10106.0784 2815.195707 9.183524988 0.2733765 5400 6209.406797 8097.164581 9735.409565 11705.10967 15263.64796 23357.5056 

10 9460.254384 2422.250171 9.123104777 0.25199117 5334 6054.63457 7733.081692 9164.611568 10861.14288 13872.03542 20931.59746 

11 18198.68659 3833.633531 9.787395261 0.208371994 11402 12640.02104 15474.55804 17807.86088 20492.98651 25088.55865 36450.64333 

12 110070.2637 18138.64562 11.59547717 0.163688898 77058 82967.88928 97260.75593 108605.4831 121273.4863 142165.2527 193390.305 

13 139672.4884 18170.54505 11.83866419 0.129548493 105916 111921.9369 126924.6017 138505.3436 151142.7254 171402.7719 220982.0832 

14 101009.909 10486.15497 11.51761415 0.10353505 81263 84736.2686 93697.94376 100469.9691 107731.443 119125.0789 146842.241 

15 57408.12 4989.92533 10.95417767 0.086756696 47880 49585.95504 53944.10302 57192.47848 60636.46276 65965.84842 79049.88 

16 109753.8876 8615.253872 11.60292439 0.078375602 93233 96181.90495 103787.3351 109417.3102 115352.685 124474.0139 146823.3284 

17 17202.0672 1275.100856 9.750045114 0.074023344 14748 15188.23984 16320.11018 17155.00272 18032.60609 19376.44661 22676.5248 

18 118.9728 8.818842373 4.776155168 0.074023344 102 105.0447833 112.8730159 118.647293 124.7169665 134.0112255 156.8352 

19 0 0 0 0 

20 0 0 0 0 

21 

22 629584.822 93749.62485 463656 493511.1061 563732.5044 620500.9894 685348.3796 796517.8676 1051250.7 

23 

24 165928.822 93749.62485 0 29855.10615 100076.5044 156844.9894 221692.3796 332861.8676 587594.7004 

Panel 3 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after two growth periods (in this 

application, 20 years). 

Geologic, Engineering, and Assessment Studies of Reserve Growth 7 
AF AG AH AI AJ AK AL AM AN AO AP 

1 (Panel 4) 

2 Estimated Size of Fields, +30yr 

3 mean,E(Y) SD(Y) Mu Sigma F100 F95 F75 F50 F25 F5 F0 

4 12894.29615 7477.014425 9.319607072 0.538392537 2024 4600.642965 7759.944813 11154.59803 16034.2709 27045.14527 57805.57634 

5 9913.574329 4547.942337 9.106160055 0.437035893 2794 4390.665155 6711.634977 9010.62806 12097.11469 18491.82645 38229.55249 

6 11540.86529 4691.430613 9.277182934 0.391066707 3890 5618.826997 8214.071293 10691.27144 13915.54577 20342.90877 41305.66248 

7 18111.18641 6526.802271 9.743233998 0.349431143 7009 9589.459134 13463.22815 17038.55503 21563.35422 30274.11177 60281.92742 

8 13573.71556 4575.927169 9.462070093 0.328086665 5603 7497.819537 10310.72279 12862.48344 16045.76939 22065.54577 43422.50556 

9 11793.79349 3511.142452 9.33286783 0.291413337 5400 6998.792341 9287.783755 11303.50157 13756.68848 18255.88495 35059.61591 

10 11040.11687 3052.93383 9.272447619 0.271452708 5334 6808.369309 8861.652107 10640.76458 12777.06115 16630.39499 31418.32778 

11 21237.86725 4983.832371 9.936738103 0.231529502 11402 14127.40875 17688.80476 20676.19057 24168.10307 30260.67017 54712.41564 

12 124379.3979 22918.75353 11.71439691 0.182728883 77058 90563.62294 108146.0216 122320.131 138351.9637 165212.1897 268812.5239 

13 153639.7372 22331.52479 11.93191251 0.144590903 105916 119857.6985 137924.0208 152042.0654 167605.2475 192868.6262 287276.7082 

14 110100.8008 13130.51068 11.60209034 0.118838183 81263 89913.33388 100910.9177 109326.0892 118443.0194 132930.1592 186489.6461 

15 62000.7696 6298.293463 11.02976885 0.101323502 47880 52213.25609 57611.46145 61683.3211 66042.97142 72870.99842 98021.8512 

16 118534.1986 11196.32452 11.67851556 0.094246818 93233 101060.9973 110745.8491 118008.9281 125748.3438 137799.0271 182060.9272 

17 18578.23258 1687.764019 9.825636291 0.090659711 14748 15938.63104 17405.32801 18502.04029 19667.85658 21477.72254 28118.89075 

18 118.9728 8.818842373 4.776155168 0.074023344 102 105.0447833 112.8730159 118.647293 124.7169665 134.0112255 156.8352 

19 0 0 0 0 

20 0 0 0 0 

21 

22 697457.5249 116939.0153 463656 529284.5687 615154.3143 685379.2151 766342.027 906659.2225 1413172.966 

23 

24 233801.5249 116939.0153 0 65628.56869 151498.3143 221723.2151 302686.027 443003.2225 949516.9662 

Panel 4 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after three growth periods (in this 

application, 30 years). 

AQ AR AS AT AU AV AW AX AY AZ BA 

1 (Panel 5) 

2 Estimated Size of Fields, +40yr 

3 mean,E(Y) SD(Y) Mu Sigma F100 F95 F75 F50 F25 F5 F0 

4 14570.55465 8559.402447 9.43852681 0.544483529 2024 5129.95109 8704.060422 12563.19529 18133.3617 30767.13074 80349.75111 

5 11202.33899 5236.062753 9.225079793 0.444517898 2794 4884.625271 7521.153003 10148.48582 13693.61379 21084.88549 53139.07796 

6 13041.17778 5423.597225 9.396102672 0.399410764 3890 6242.100139 9199.457716 12041.36004 15761.18463 23228.4565 57414.87085 

7 20465.64065 7584.625377 9.862153736 0.358744914 7009 10636.19784 15068.46855 19190.17554 24439.30091 34623.5415 83791.87912 

8 15338.29858 5335.818677 9.580989831 0.337989126 5603 8308.195523 11535.50743 14486.75164 18193.04216 25260.11484 60357.28273 

9 13326.98665 4125.686775 9.451787568 0.302518632 5400 7739.902884 10382.63534 12730.90229 15610.28274 20940.29804 48732.86611 

10 12475.33206 3606.924431 9.391367357 0.283341423 5334 7519.618745 9901.041557 11984.47519 14506.31682 19100.38933 43671.47562 

11 23998.79 5978.095229 10.05565784 0.245360347 11402 15553.48598 19737.68374 23287.16994 27474.97076 34866.28556 76050.25773 

12 136817.3377 26749.82856 11.80764523 0.193684181 77058 97639.24139 117842.2441 134275.0024 152999.2611 184657.0705 349456.2811 

13 167467.3135 26270.18389 12.0163887 0.155915158 105916 128015.7095 148940.4564 165444.1079 183776.4801 213815.5773 364841.4194 

14 118908.8649 15506.1954 11.67768152 0.129854694 81263 95231.92828 108029.4823 117910.5443 128695.3909 145989.866 231247.1611 

15 66960.83117 7661.421776 11.10536003 0.11404464 47880 55146.94318 61604.73053 66526.79169 71842.11301 80254.92905 121547.0955 

16 128016.9345 13841.22414 11.75410674 0.107806263 93233 106592.1807 118355.1724 127275.1732 136867.4422 151971.4636 225755.5497 

17 18578.23258 1687.764019 9.825636291 0.090659711 14748 15938.63104 17405.32801 18502.04029 19667.85658 21477.72254 28118.89075 

18 118.9728 8.818842373 4.776155168 0.074023344 102 105.0447833 112.8730159 118.647293 124.7169665 134.0112255 156.8352 

19 0 0 0 0 

20 0 0 0 0 

21 

22 761287.6065 137575.6495 463656 564683.7564 664340.2946 746484.8228 841785.3344 1008171.742 1824630.694 

23 

24 297631.6065 137575.6495 0 101027.7564 200684.2946 282828.8228 378129.3344 544515.7423 1360974.694 

Panel 5 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after four growth periods (in this 

application, 40 years). 

8 Probabilistic Method, Growth of Oil and Gas Reserves 
BB BC BD BE BF BG BH BI BJ BK BL 

1 (Panel 6) 

2 Estimated Size of Fields, +50yr 

3 mean,E(Y) SD(Y) Mu Sigma F100 F95 F75 F50 F25 F5 F0 

4 16027.61012 9490.864037 9.531775131 0.548257267 2024 5596.473391 9530.473999 13791.05029 19956.30732 33984.44963 104454.6764 

5 12322.57289 5825.661428 9.318328114 0.449132362 2794 5321.472761 8230.589361 11140.3409 15078.77382 23321.96385 69080.80135 

6 14345.29555 6048.973542 9.489350994 0.404540079 3890 6794.593921 10063.70816 13218.21384 17361.51071 25714.73425 74639.33211 

7 22512.20471 8484.630864 9.955402058 0.36444702 7009 11566.7133 16477.7253 21065.71376 26931.16241 38365.63461 108929.4429 

8 16872.12844 5980.68011 9.674238152 0.344035414 5603 9029.93001 12611.42317 15902.60405 20052.67862 28006.06598 78464.46755 

9 14659.68531 4644.234261 9.54503589 0.309259179 5400 8402.66789 11345.71255 13975.14802 17213.97058 23243.18475 63352.72594 

10 13722.86526 4072.493429 9.484615678 0.290527243 5334 8157.54352 10816.2003 13155.76939 16001.39269 21216.46889 56772.9183 

11 26398.669 6804.478012 10.14890616 0.253624559 11402 16843.05632 21546.38237 25563.12501 30328.68112 38797.78987 98865.33505 

12 149130.8981 30477.18466 11.89212143 0.202278445 77058 104754.3521 127489.0495 146110.9328 167452.8499 203794.9188 443809.477 

13 180864.6986 29948.49432 12.09197988 0.164466605 105916 136139.0831 159712.3056 178435.0373 199352.5948 233871.5805 452403.3601 

14 128421.5741 18068.39487 11.75327269 0.140007033 81263 101008.6186 115717.5924 127169.0641 139753.7793 160104.8613 286746.4798 

15 72317.69766 9110.45933 11.18095121 0.125482698 47880 58368.53075 65931.78109 71750.57915 78082.91423 88200.70579 150718.3984 

16 128016.9345 13841.22414 11.75410674 0.107806263 93233 106592.1807 118355.1724 127275.1732 136867.4422 151971.4636 225755.5497 

17 18578.23258 1687.764019 9.825636291 0.090659711 14748 15938.63104 17405.32801 18502.04029 19667.85658 21477.72254 28118.89075 

18 118.9728 8.818842373 4.776155168 0.074023344 102 105.0447833 112.8730159 118.647293 124.7169665 134.0112255 156.8352 

19 0 0 0 0 

20 0 0 0 0 

21 

22 814310.0396 154494.3559 463656 594618.8923 705346.3173 797173.4394 904226.6312 1092205.556 2242268.691 

23 

24 350654.0396 154494.3559 0 130962.8923 241690.3173 333517.4394 440570.6312 628549.5555 1778612.691 

Panel 6 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after five growth periods (in this 

application, 50 years). 

BM BN BO BP BQ BR BS BT BU BV BW 

1 (Panel 7) 

2 Estimated Size of Fields, +60yr 

3 mean,E(Y) SD(Y) Mu Sigma F100 F95 F75 F50 F25 F5 F0 

4 17470.09503 10412.73648 9.616251324 0.551352009 2024 6058.861975 10348.94716 15006.68914 21760.73717 37168.81487 132657.4391 

5 13431.60445 6408.943508 9.402804307 0.452904941 2794 5754.720069 8933.346723 12122.32784 16449.69537 25535.70469 87732.61772 

6 15636.37215 6667.406119 9.573827186 0.408724458 3890 7342.799501 10919.95252 14383.35891 18945.23014 28174.67828 94791.95177 

7 24538.30314 9374.12274 10.03987825 0.36908622 7009 12490.59723 17874.20941 22922.59193 29396.837 42067.26156 138340.3924 

8 18390.62 6617.731243 9.758714344 0.348946089 5603 9746.836281 13677.736 17304.36991 21892.60108 30721.88855 99649.87379 

9 15979.05699 5155.943086 9.629512082 0.31471296 5400 9061.67503 12300.50439 15207.01452 18800.3096 25519.9276 80457.96195 

10 14957.92314 4531.526609 9.569091871 0.29632598 5334 8792.33532 11723.70525 14315.41018 17480.05126 23307.91095 72101.60624 

11 28774.5492 7617.089101 10.23338236 0.260246852 11402 18129.14754 23341.21537 27816.4362 33149.69297 42680.11616 125558.9755 

12 161061.37 34022.95275 11.9677126 0.208940893 77058 111748.3596 136883.6318 157583.7911 181414.3218 222219.3803 550323.7515 

13 195333.8745 33966.26511 12.16757105 0.172594878 105916 144878.7523 171312.0638 192446.0348 216187.2052 255630.8341 560980.1665 

14 138695.3 20847.31159 11.82886387 0.149471392 81263 107257.0353 124010.3307 137154.5774 151692.0244 175385.9601 355565.6349 

15 72317.69766 9110.45933 11.18095121 0.125482698 47880 58368.53075 65931.78109 71750.57915 78082.91423 88200.70579 150718.3984 

16 128016.9345 13841.22414 11.75410674 0.107806263 93233 106592.1807 118355.1724 127275.1732 136867.4422 151971.4636 225755.5497 

17 18578.23258 1687.764019 9.825636291 0.090659711 14748 15938.63104 17405.32801 18502.04029 19667.85658 21477.72254 28118.89075 

18 118.9728 8.818842373 4.776155168 0.074023344 102 105.0447833 112.8730159 118.647293 124.7169665 134.0112255 156.8352 

19 0 0 0 0 

20 0 0 0 0 

21 

22 863300.9061 170270.2947 463656 622265.5075 743130.7976 843909.0419 961911.636 1170196.38 2702910.046 

23 

24 399644.9061 170270.2947 0 158609.5075 279474.7976 380253.0419 498255.636 706540.3803 2239254.046 

Panel 7 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after six growth periods (in this 

application, 60 years). 

Geologic, Engineering, and Assessment Studies of Reserve Growth 9 
BX BY BZ CA CB CC CD CE CF CG CH 

1 (Panel 8) 

2 Estimated Size of Fields, +70yr 

3 mean,E(Y) SD(Y) Mu Sigma F100 F95 F75 F50 F25 F5 F0 

4 18867.70263 11304.4517 9.691842501 0.55383099 2024 6508.019873 11142.92909 16185.03778 23508.67047 40251.1752 164495.2245 

5 14506.13281 6972.668595 9.478395484 0.455919525 2794 6175.887434 9615.250631 13074.19193 17777.43516 27677.72186 108788.446 

6 16887.28193 7264.71899 9.649418364 0.412062387 3890 7876.002344 11750.93763 15512.76269 20478.85997 30554.30862 117542.0202 

7 26501.36739 10232.51915 10.11546943 0.372779245 7009 13389.78751 19229.79574 24722.5096 31784.13798 45646.91414 171542.0866 

8 19861.8696 7232.162824 9.834305522 0.35284997 5603 10444.88143 14712.9692 18663.13603 23673.85139 33347.68794 123565.8435 

9 17257.38155 5648.841455 9.705103259 0.319036008 5400 9703.957061 13227.76078 16401.09303 20335.70586 27720.22287 99767.87281 

10 16154.55699 4973.267586 9.644683048 0.3009133 5334 9411.434549 12605.23502 15439.47852 18910.99187 25328.49756 89405.99174 

11 31076.51314 8397.009302 10.30897353 0.265458381 11402 19385.76856 25085.73367 30000.62616 35878.46311 46427.74762 155693.1296 

12 173946.2796 37906.3868 12.04330378 0.215397365 77058 119249.7265 146990.8996 169957.5162 196512.5555 242227.4512 682401.4518 

13 210960.5845 38359.78275 12.24316223 0.1803572 105916 154272.321 183799.6306 207557.1976 234385.6195 279246.4002 695615.4065 

14 138695.3 20847.31159 11.82886387 0.149471392 81263 107257.0353 124010.3307 137154.5774 151692.0244 175385.9601 355565.6349 

15 72317.69766 9110.45933 11.18095121 0.125482698 47880 58368.53075 65931.78109 71750.57915 78082.91423 88200.70579 150718.3984 

16 128016.9345 13841.22414 11.75410674 0.107806263 93233 106592.1807 118355.1724 127275.1732 136867.4422 151971.4636 225755.5497 

17 18578.23258 1687.764019 9.825636291 0.090659711 14748 15938.63104 17405.32801 18502.04029 19667.85658 21477.72254 28118.89075 

18 118.9728 8.818842373 4.776155168 0.074023344 102 105.0447833 112.8730159 118.647293 124.7169665 134.0112255 156.8352 

19 0 0 0 0 

20 0 0 0 0 

21 

22 903746.8076 183787.3871 463656 644679.2089 773976.6272 882314.5669 1009681.245 1235597.99 3169132.782 

23 

24 440090.8076 183787.3871 0 181023.2089 310320.6272 418658.5669 546025.2452 771941.9904 2705476.782 

Panel 8 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after seven growth periods (in this 

application, 70 years). 

CI CJ CK CL CM CN CO CP CQ CR CS 

1 (Panel 9) 

2 Estimated Size of Fields, +80yr 

3 mean,E(Y) SD(Y) Mu Sigma F100 F95 F75 F50 F25 F5 F0 

4 20377.11884 12272.04496 9.767433678 0.556298925 2024 6990.601948 11997.91547 17455.9122 25396.81759 43588.35947 203974.0783 

5 15666.62343 7585.321851 9.553986662 0.458914307 2794 6628.094435 10349.34404 14100.79788 19212.08726 29998.44113 134897.673 

6 18238.26448 7914.515312 9.725009541 0.415373493 3890 8448.296864 12645.38755 16730.84902 22136.23804 33133.4603 145752.105 

7 28621.47678 11167.35217 10.1910606 0.376436041 7009 14354.5654 20688.69537 26663.75962 34364.47123 49528.22028 212712.1874 

8 21450.81917 7901.730713 9.909896699 0.356711128 5603 11193.70553 15827.0123 20128.59458 25599.29265 36195.37056 153221.6459 

9 18637.97207 6186.619496 9.780694437 0.323301256 5400 10392.75149 14225.4713 17688.93243 21995.63896 30107.36194 123712.1623 

10 17446.92155 5455.572984 9.720274226 0.30543173 5334 10075.26843 13553.67815 16651.81046 20458.12129 27521.13191 110863.4298 

11 33562.63419 9249.790287 10.38456471 0.270569547 11402 20732.91485 26962.46076 32356.32213 38829.22968 50496.11158 193059.4808 

12 187861.9819 42159.39748 12.11889496 0.221665858 77058 127293.9963 157864.4716 183302.8455 212840.3739 263955.3643 846177.8002 

13 210960.5845 38359.78275 12.24316223 0.1803572 105916 154272.321 183799.6306 207557.1976 234385.6195 279246.4002 695615.4065 

14 138695.3 20847.31159 11.82886387 0.149471392 81263 107257.0353 124010.3307 137154.5774 151692.0244 175385.9601 355565.6349 

15 72317.69766 9110.45933 11.18095121 0.125482698 47880 58368.53075 65931.78109 71750.57915 78082.91423 88200.70579 150718.3984 

16 128016.9345 13841.22414 11.75410674 0.107806263 93233 106592.1807 118355.1724 127275.1732 136867.4422 151971.4636 225755.5497 

17 18578.23258 1687.764019 9.825636291 0.090659711 14748 15938.63104 17405.32801 18502.04029 19667.85658 21477.72254 28118.89075 

18 118.9728 8.818842373 4.776155168 0.074023344 102 105.0447833 112.8730159 118.647293 124.7169665 134.0112255 156.8352 

19 0 0 0 0 

20 0 0 0 0 

21 

22 930551.5345 193747.7059 463656 658643.9388 793729.5524 907438.0388 1041652.844 1280940.085 3580301.278 

23 

24 466895.5345 193747.7059 0 194987.9388 330073.5524 443782.0388 577996.8445 817284.0848 3116645.278 

Panel 9 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after eight growth periods (in this 

application, 80 years). 

10 Probabilistic Method, Growth of Oil and Gas Reserves 
CT CU CV CW CX CY CZ DA DB DC DD 

1 (Panel 10) 

2 Estimated Size of Fields, +90yr 

3 mean,E(Y) SD(Y) Mu Sigma F100 F95 F75 F50 F25 F5 F0 

4 22007.28835 13321.94045 9.843024855 0.558755959 2024 7509.10305 12918.59906 18826.57765 27436.41354 47201.37994 252927.8571 

5 16919.95331 8251.110163 9.629577839 0.461889672 2794 7113.63987 11139.62889 15208.01452 20762.24512 32512.70937 167273.1145 

6 19697.32564 8621.34853 9.800600718 0.418658412 3890 9062.566437 13608.16077 18044.58138 23927.32733 35928.77574 180732.6103 

7 30911.19492 12185.31631 10.26665178 0.380057655 7009 15389.74935 22258.80462 28757.43962 37153.40277 53736.43941 263763.1124 

8 23166.8847 8631.289763 9.985487876 0.360530937 5603 11997.03104 17025.88351 21709.12322 27680.56241 39283.5552 189994.841 

9 20129.00984 6773.268947 9.856285614 0.327510962 5400 11131.45409 15299.00734 19077.895 23790.17602 32697.08294 153403.0812 

10 18842.67527 5982.072594 9.795865403 0.309884284 5334 10787.09462 14574.13113 17959.33659 22130.84045 29900.33759 137470.6529 

11 36247.64493 10182.0432 10.46015589 0.275585934 11402 22177.13373 28981.44183 34896.99101 42019.99296 54912.41549 239393.7561 

12 187861.9819 42159.39748 12.11889496 0.221665858 77058 127293.9963 157864.4716 183302.8455 212840.3739 263955.3643 846177.8002 

13 210960.5845 38359.78275 12.24316223 0.1803572 105916 154272.321 183799.6306 207557.1976 234385.6195 279246.4002 695615.4065 

14 138695.3 20847.31159 11.82886387 0.149471392 81263 107257.0353 124010.3307 137154.5774 151692.0244 175385.9601 355565.6349 

15 72317.69766 9110.45933 11.18095121 0.125482698 47880 58368.53075 65931.78109 71750.57915 78082.91423 88200.70579 150718.3984 

16 128016.9345 13841.22414 11.75410674 0.107806263 93233 106592.1807 118355.1724 127275.1732 136867.4422 151971.4636 225755.5497 

17 18578.23258 1687.764019 9.825636291 0.090659711 14748 15938.63104 17405.32801 18502.04029 19667.85658 21477.72254 28118.89075 

18 118.9728 8.818842373 4.776155168 0.074023344 102 105.0447833 112.8730159 118.647293 124.7169665 134.0112255 156.8352 

19 0 0 0 0 

20 0 0 0 0 

21 

22 944471.6809 199963.1481 463656 664995.5121 803285.2446 920141.0195 1058561.908 1306544.323 3887067.541 

23 

24 480815.6809 199963.1481 0 201339.5121 339629.2446 456485.0195 594905.9084 842888.3233 3423411.541 

Panel 10 of PREGS. Probabilistic spreadsheet system for estimating future oil and gas reserve growth: after nine growth periods (in this 

application, 90 years). 

column E in the deterministic spreadsheet (fig. 1 (values, 1B)). 

For example, the total estimated initial size of 5-year-old fields 

of 5,400 bcfg grows to 10,106 bcfg (mean estimate in cell U9) 

with a range from 6,209 bcfg (F95 estimate in cell Z9) to 15,264 

bcfg (F5 estimate in cell AD9), after the second 10-year growth 

period. The reserve growth is projected to be 165,929 bcfg 

(mean estimate in cell U24) with a range from 29,855 bcfg (F95 

estimate in cell Z24) to 332,862 bcfg (F5 estimate in cell AD24), 

after the second 10-year growth period. 

Panels 4 through 10 are computationally similar in their 

composition to Panel 3. They compute the respective 

probabilistic estimates of reserve growth after the ith growth 

period for i = 3, 4, ..., 9. 

In the example used to illustrate PREGS, the system is 

applied to data based on successive annual estimates made 

between 1977 and 1991 of the sizes of Lower 48 United States 

gas fields. The results of this application (Panels 1–10) are sum	

marized in the graph displayed in figure 3. The probabilistic 

estimates of future reserve growth of Lower 48 United States gas 

fields are in the form of a mean estimate with a range from a low 

F95 estimate to a high F5 estimate for nine growth periods (10-

year increments). 

Summary 

The objective of this report is the development and descrip	

tion of a probabilistic method and spreadsheet system, called the 

PREGS system, for estimating future growth of oil and gas 

reserves. The primary advantages of the PREGS system are 

several-fold. The probabilistic method utilizes the same data as 

required by its deterministic predecessor in Schmoker and Crov	

elli (1998); most importantly, no additional data are needed. All 

the advantages of the deterministic system carry over to the 

probabilistic system. The mean estimates computed by the 

probabilistic system equal the point estimates computed by the 

deterministic system. Additionally, the probabilistic system 

also computes a range of low and high estimates. Many opera	

tional benefits accrue from incorporating an analytic probabilis	

tic method into a computer software spreadsheet. 

Geologic, Engineering, and Assessment Studies of Reserve Growth 11 
900 

800 

700 

600 

500 

400 

300 

200 

100 

0 

0 10 20 30 40 50 60 70 80 

F 5 

MEAN 

F 95 

YEARS IN FUTURE 

Figure 3. Graph of probabilistic estimates of future reserve growth of Lower 48 United States gas fields in the form of a mean estimate together 

with a range from a low F95 estimate to a high F5 estimate for nine growth periods (10-year increments). Units for reserve growth are trillion cubic 

feet of gas. This example is derived from the results shown in Panels 1–10. 

References Cited Law, A.M., and Kelton, W.D., 1991, Simulation modeling and analysis: 

New York, McGraw-Hill, Inc., 2nd ed., 759 p. 

Oil & Gas Journal, 1996, Oil and gas reserves, oil output rise in 1996: Oil 

& Gas Journal, v. 94, no. 53 (December 30), p. 37–38. 

Aitchison, J., and Brown, J.A.C., 1957, The lognormal distribution: Root, D.H., and Attanasi, E.D., 1993, A primer in field-growth estimation, 

Cambridge, Cambridge University Press, 176 p. in Howell, D.G., ed., The future of energy gases: U.S. Geological 

Arrington, J.R., 1960, Predicting the size of crude reserves is key to eval- Survey Professional Paper 1570, p. 547–554. 

uating exploration programs: Oil & Gas Journal, v. 58, no. 9 (February Schmoker, J.W., and Attanasi, E.D., 1997, Reserve growth important to 

29), p. 130–134. U.S. gas supply: Oil & Gas Journal, v. 95, no. 4 (January 27), p. 95–96. 

Attanasi, E.D., and Root, D.H., 1994, The enigma of oil and gas field growth: Schmoker, J.W., and Crovelli, R.A., 1998, A simplified spreadsheet 

American Association of Petroleum Geologists Bulletin, v. 78, no. 3, program for estimating future growth of oil and gas reserves: Nonp. 321–332. renewable Resources, v. 7, no. 2, p. 149–155. 

Crovelli, R.A., 1992, Probabilistic methodology for estimation of U.S. Geological Survey National Oil and Gas Resource Assessment 

undiscovered petroleum resources in play analysis of the United Team, 1995, 1995 National Assessment of United States oil and gas 

States: Nonrenewable Resources, v. 1, p. 153–162. resources: U.S. Geological Survey Circular 1118, 20 p. 

Gautier, D.L., Dolton, G.L., Takahashi, K.I., and Varnes, K.L., eds., 1995, U.S. Geological Survey World Energy Assessment Team, 2000, U.S. 

1995 National Assessment of United States oil and gas resources— Geological Survey World Petroleum Assessment 2000—Description 

Results, methodology, and supporting data: U.S. Geological Survey and results: U.S. Geological Survey Digital Data Series DDS-60, 

Digital Data Series DDS-30. version 1.1, multi-disc set. 

RESERVE GROWTH, IN TRILLION CUBIC FEET 

12 Probabilistic Method, Growth of Oil and Gas Reserves 

90