Estimation of Natural Gas Production, Import and Consumption in Brazil Based on Three Mathematical Models

doi:10.4236/nr.2012.32007

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Natural Resources, 2012, 3, 42-47

http://dx.doi.org/10.4236/nr.2012.32007 Published Online June 2012 (http://www.SciRP.org/journal/nr)

Estimation of Natural Gas Production, Import and

Consumption in Brazil Based on Three Mathematical

Models

Antonio Carlos Gracias1,2*, Sérgio Ricardo Lourenço1, Marat Rafikov1

1Engineering, Modeling and Applied Social Sciences Center (CECS), Federal University of ABC, Santo André, Brazil; 2Department

of Mathematics, University Center of FEI, São Bernardo do Campo, Brazil.

Email: *[email protected]

Received January 6th, 2012; revised March 2nd, 2012; accepted March 10th, 2012

ABSTRACT

A mathematical model capable of providing a forecast of future consumption and import of natural gas is essential for

the planning of the Brazilian energy matrix. The aim of this study is to compare three mathematical models, logistic

model or model of Verhulst, exponential model or the model of Malthus and the model of von Bertalanffy to analyze

the possibilities of these models to describ e the evolution of production, import and consumption of natural gas in Bra-

zil, from data provided by the energy balance of the Ministry of Mines and Energy (MME) from 1970 to 2009. A pro-

jection of the production and the import of natural gas up to 2017 is made with the models studied in this article and

compared with the Brazilian Ten-Year Plan for Expansion of Energy (PDE). At the end of this paper a comparison with

the Hubbert model for Brazilian natural gas production is made. These data were adjusted to use the differential equa-

tions which describe th e models of population growth. All the computer work used in this article: graphics, resolution of

differential equations, calculations of linearization and the least squares fitting was prepared in the software MatLab.

The results obtained by means of graphs show that the population dynamics models (logistic, exponential and von Ber-

talanffy) can be applied in modeling the production, import and consumption of natural gas in Brazil.

Keywords: Natural Gas; Mathematical Modeling; Logistic Model; Exponential Mod e l; Model of Von Bertalanffy

1. Introduction

The federal government, through a policy of development,

aims to extend the participation of natural gas in the en-

ergy matrix from 2% to 12% in the next ten years [1]. A

mathematical model capable of providing a forecast of

future consumption and impo rt of natural gas is essential

for the planning of the Brazilian energy matrix.

The incr ease in supply of el ectric energy in Brazil will

only occur with the generation of heat by natural gas.

However, the Brazilian reserves of natural gas have two

main characteristics: 80% is associated gas and 55% is

located in deep water. In consequence, the supply of natu-

ral gas is influenced and very dependent on oi l product ion.

World consumption of natural gas will increase at a

rate of 2.3% per year un til 2025 [2]. Th is increase in g lo-

bal consumption will be felt also in Brazil, with the in-

creasing participation of natural gas in the energy ma-

trix, especially after the crisis in the electrical sector in

2001. The supply of natural gas has been dependent on

Bolivia, since the end of the construction of the pipeline

in 1999. This dependence has reached 50% in 2006 [3],

which led to supply problems due to the political crisis

faced by Bolivia in 2006. These problems caused a crisis

in supply of gas in Brazil, which led Petróleo Brasileiro

(PETROBRAS) to review the dependencies on Bolivian

gas aiming to reduce the dependence to the maximum of

22% until 2016 [3].

Around the world, the use of natural gas is growing

both in industry and in transport and the generation of

electricity with the use of thermoelectric power plants for

a number of reasons, including price, environmental

concerns, fuel diversification, issues security, deregula-

tion and economic growth worldwide marketing. Studies

show the relationship between natural gas consumption

and economic growth [4].

2. Description of the Models

The dynamics of populations [5] deals with changes in

time and space the densities and sizes of populations.

The study of population dynamics is not restricted only

to the understanding of the variation in the number of

*Corresponding a uthor.

Estimation of Natural Gas Production, Import and Consumption in Brazil Based on Three Mathematical Models 43

individuals of a given population, but also in the study of

biological control of pests [6], the growth strategies of

animals [7] strategies and growth of cities. The models

dealing with population growth are the models logistic,

exponential and von Bertalanffy. The application of these

models in the study of production and import of natural

gas is possible because the data presented in Tables 1

and 2 show characteristics of population growth.

2.1. Exponential Models

The exponential m odel i s the simplest m odel that describes

the population growth of some species. It is represented by

a differential equation of first order establishing the rate

of change of population in relation to time. The differen-

tial equation for this model is:

dN = rN

dt (1)

where dN/dt is the rate of the populational change and r

is the rate of population growth r = (1 +



) and



is the

rate population growth average



= (Nt/N0)1/2 – 1. Nt is

the population after a period of t years in relation to ini-

tial population N0.

The solution of the Equa tion (1) is:

N(t)N e (2)

where N0 is the initial population.

Table 1. Production of natural gas.

Year Production (106 m3) Year Production (106 m3)

1970 1264 1990 6279

1971 1178 1991 6597

1972 1241 1992 6976

1973 1180 1993 7355

1974 1488 1994 7756

1975 1625 1995 7955

1976 1642 1996 9156

1977 1808 1997 9825

1978 1933 1998 10788

1979 1899 1999 11898

1980 2205 2000 13283

1981 2475 2001 13998

1982 3030 2002 15568

1983 4013 2003 15792

1984 4902 2004 16971

1985 5467 2005 17699

1986 5686 2006 17706

1987 5781 2007 18152

1988 6076 2008 21593

1989 6105 2009 21142

Source: MME, 2010.

2.2. Logistic Model

The logistic model assumes that a population will grow

to a maximum limit, i.e., the p opulation tends to stabilize.

This stability of the population in the logistic model is

related to the ability to support the way that people live.

The differential equation for this model is:

dN N

rN 1

dt K













(3)

where r is the rate of population growth and K is the

level of the population saturation.

The solution of the Equation (3) is obtained by the

method of separab le variables [8]:



0rt

N(t)

KNe



 (4)

where N0 is the initial population.

The logistic model [9] was used to model annual and

seasonal natural gas consumption for residential and com-

mercial sectors in Iran. The logistics parameters were es-

timated using optimization techniques as NLP (nonlinear

programm i ng) and GA (genetic algorithm).

2.3. Von Bertalanffy Model

The von Bertalanffy model is the logistic model modified

to model fish weight growth [10]. The differential equa-

tion for this model is:

1/3

dN N

rN 1

dt K























 (5)

where r is the rate of population growth and K is the

level of the population saturation.

The solution of the Equation (5) is obtained solving

Bernoulli equation [8]:

1/3

03K

N(t)K11()e































(6)

where N0 is the initial population.

2.4. Hubbert Model

The Hubbert model was developed by M. K. Hubbert in

1956 to project the discoveries and production in US-48

in the USA [11]. The equation for th e Hubbert model [12]

for the annual production P is simple when related to the

annual peak production Pm occurring at year tm:



2Pm

P(t) 1cosh bttm



(7)

where b is 5/c, P is production in t, Pm is production at

peak, tm is the peak period and c is the duration of the

half life from a cut-off at 0.027 Pm.

Estimation of Natural Gas Production, Import and Consumption in Brazil Based on Three Mathematical Models

3. Methodology

The study was conducted from data on production, import

and consumption of natural gas from the energy balance of

the Ministry of Mines and Energy [13] described in Ta-

bles 1-3. Ta ble 1 presents data fro m the production, Table

2 presents data from the import and Table 3 presents data

from the consumption of natural gas in B razil.

To use the models applied to population dynamics in

the production, import and consumption of natural gas

the data were adjusted to use the differential equations

which describe the models. Some parameters of the dif-

ferential equations were linearized and others obtained by

the least squares fitting.

4. Results and Discussions

The results for production, import and consumption of

natural gas will be presented separately. In each set of

data, production, import and consumption of natural gas,

we applied the best model to describe the data. In the

final section a comparison with the model used in the Ten

Year Energy Plan (PDE) 2008-2019 [14] was ma de.

4.1. Production of Natural Gas

The results for the exponential model with the data of

Table 1 are presented in Figure 1. For the exponential

model to average growth rate of production



is given

by:



21142 15001





(8)

where: N0 = 1500 × 106 m3 of gas.

4.2. Import of Natural Gas

Brazil began to import natural gas in 1999, therefore,

there were few points to apply the von Bertalanffy model,

as Table 2. The same procedure done for the production

of natural gas was applied to the import of natural gas.

For the von Bertalanffy model shown in Figure 2, the

value of K used was 11,330 × 106 m3 of gas.

Table 2. Import of natural gas.

Year Import (106 m3)

1999 400

2000 2.211

2001 4608

2002 5269

2003 5055

2004 8086

2005 8998

2006 9789

2007 10334

2008 11314

2009 8543

Source: MME, 2010.

Table 3. Consumption of natural gas.

Year Energy

(106 m3)Industry

(106 m3)Home

(106 m3) Other

(106 m3)Total

(106 m3)

1970 74 3 0 3 80

1971 93 12 0 21 126

1972 100 22 0 51 173

1973 98 23 0 77 198

1974 137 163 0 81 381

1975 149 173 0 92 414

1976 146 183 0 148 477

1977 160 312 0 123 595

1978 156 294 0 333 783

1979 161 311 0 386 858

1980 188 363 0 452 1003

1981 197 381 0 322 900

1982 391 413 0 482 1286

1983 489 449 0 801 1739

1984 628 519 0 877 2024

1985 911 680 0 948 2539

1986 1050 871 0 1037 2958

1987 1062 1131 1 1108 3302

1988 935 1198 0 1191 3324

1989 894 1246 0 1268 3408

1990 859 1535 2 1018 3414

1991 768 1617 5 1068 3458

1992 840 1806 6 1043 3695

1993 974 1947 6 1089 4016

1994 1025 2025 20 1193 4263

1995 989 2353 30 1063 4435

1996 1199 2860 52 983 5094

1997 1226 3194 72 916 5408

1998 1471 3133 81 1054 5739

1999 1696 3517 87 1015 6315

2000 2278 4343 79 1265 7965

2001 2419 5141 114 1576 9250

2002 2722 6343 140 2067 11272

2003 2938 6658 154 2438 12188

2004 3168 7572 196 2729 13665

2005 3500 8209 206 3129 15044

2006 3712 8595 217 3556 16080

2007 4013 9149 236 3841 17239

2008 5227 9605 251 3469 18552

2009 5414 8137 260 3118 16929

Source: MME, 2010.

The Figure 2, shows that the import of gas tends to

stabilize according to what was described by Santana et

al. [3], especially with the reduction of dependence on

Bolivian gas.

4.3. Consumption of Natural Gas

The natural gas consumption in Brazil is primarily Indus-

trial as Figure 3. In 2007, industrial use accounted for

54% of the Brazilian natural gas demand [15]. The par-

ticipation of home consumption in Brazil’s energy matrix

Estimation of Natural Gas Production, Import and Consumption in Brazil Based on Three Mathematical Models

is very small when compared with countries such as Po-

land [16], Iran [9] and Turkey [17]. Aydinalp et al. [18]

used neural network (NN) to model residential energy

consumption in Canada.

then stay at around that level until 2013, when it will

begin a further period of increase until 2016. The impor-

tation of natural gas from Bolivia and Argentina is ex-

pected to stabilize at a level of 30 MMm3/day.

For the logistic model shown in Figure 4, the terms K

and r of Equation (4) were obtained by least squares fit-

ting of the Equation (3). The value of K used was 50,000

× 106 m3 of gas.

The Figure 5 shows the forecast of natural gas pro-

duction until 2017 from exponential model. Comparing

the results in Table 4 with the results of the exponential

model, we found out that between 2009 and 2017 the

points are near the projection.

4.4. Projection of Production and Import of

Natural Gas by 2017 with the Use of

Population Dynamics Models

The Figure 6 shows the forecast of natural gas import

until 2017 from von Bertalanffy model. Comparing the

results in Table 4 with the results of the von Bertalanffy

model, it’s possible to see that results are close to the

projection. The results from the logistic model showed

30.9 MMm3/day in 2015 and the value of Table 4 for the

same period was 30.1 MMm3/day.

Table 4 represents the forecasted daily production and

importation of natural gas in the PDE from 2008 to 2017.

Total production (TP) is expected to increase year-on-

year until 2010, reaching a level of 95 MMm3/day. It will

Figure 1. Production of natural gas as function of time with

the exponential model. Figure 2. Import of natural gas as function of time with the

von Bertalanffy model.

Figure 3. Consumption of natural gas in Brazil.

Estimation of Natural Gas Production, Import and Consumption in Brazil Based on Three Mathematical Models

The Figure 7 shows the Hubbert curve for Brazilian

gas production together with the exponential model. Bra-

zil’s peak production was forecasted by the Hubbert

curve to occur in 2022 at 32,380 MMm3.

consum

tion unit: 10

Figure 4. Total consumption of natural gas as function of

time with the logistic model.

Figure 5. Forecast of the natural gas production in the pe-

riod 2007 to 2017.

Figure 6. Forecast of the natural gas import for the period

from 2007 to 2017.

Table 4. Projection of production and import of natural gas.

Year Production (MM m3/day) Import (MM m3/day)

2007 49.766 29.1

2008 60.971 30.1

2009 77.475 30.1

2010 95.354 30.1

2011 95.015 30.1

2012 97.164 30.1

2013 101.509 30.1

2014 113.696 30.1

2015 129.240 30.1

2016 139.501 30.1

2017 140.144 30.1

Source: PD E, 2009.

Figure 7. Hubbert curve for Brazilian gas production.

5. Conclusion

This study examined the possibility of using biomathe-

matical models in the construction of developments in

production, import and consumption of natural gas in

Brazil. It was concluded that it isn’t appropriate to use

only one model to describe the three sets of data involved

in this article. In fact each set of data is better described

by a certain model. For instance the exponential model is

the best one to describe the data of natural gas production,

the von Bertalanffy model is the indicated one to describe

one data on import of natural gas and finally the logistic

model is the best one for describing the data on con-

sumption of natural gas. Therefore, the initial results

presented here show that the models used in the study of

population dynamics can be used to study the production,

consumption and import of natural gas in Brazil.

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