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Faced with diminishing supplies of domestic crude oil and increased demand for energy, the US has come to rely on imported crude and domestic supplies of natural gas. During the last decade US natural gas production has risen about 10%, but it still does not succeed to meet demand. This supply shortage has resulted in significant increases in natural gas prices. We believe that developing a reliable method to forecast US natural gas production rates and reserves will benefit gas producers, consumers and policy makers. This paper presents one methodology for developing forecasting models for predicting U.S. natural gas production, proved reserves, and annual depletion to year 2025 using a stochastic (time series) modeling approach. The methodology is not mechanistic. A mechanistic model would examine individual geologic settings, exploration success, the physics of gas production, and the rate of exploitation for provinces, basins, and reservoirs. However, to do so would result in an extraordinarily massive model that would be difficult, if not impossible, to develop and use. Instead we used a simpler approach which takes advantage of established trends in easily obtained published data. Various time series models were tested and validated using data that are not used in the mathematical development of the models. Having adequately validated these time series models using historical data we believe that they can be used to make at least short time forecasts. Comparison of results of this study with other published forecast is also presented. Our forecasts show that U.S. gas production rate will maintain a production plateau of 18.7 Tcf/yr from 2005 to 2008 after which gas production will increase gradually from 19.0 Tcf/yr in 2010 to reach 22.5 Tcf/yr in 2025. We predict that U.S. gas production will have an average annual rate of increase of 0.5% from 2005 to 2015 after which the average annual rate of change of production will increase to 1.2% for the period of 2015 to 2025. Our forecasts also show that U.S. gas depletion rate will increase from 10.6%/yr in 2005 to 13.4%/yr in 2025. While the reserves discovery rate increases, U.S. gas proved reserves are predicted to increase from 197 Tcf in 2005 to 215 Tcf in 2010. Afterwards, the gas proved reserves will increase at an annual rate of 1.3% and are expected to reach 263 Tcf in 2025.
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Petroleum Economics
Time Series Modeling for U.S. Natural Gas Forecasting
Time Series Modeling for U.S. Natural Gas ForecastingTime Series Modeling for U.S. Natural Gas Forecasting
Time Series Modeling for U.S. Natural Gas Forecasting
Saud M. Al-Fattah
Address: Petroleum Engineering Application Services Department, Saudi Aramco, P.O. Box 8349, Dhahran 31311, Saudi Arabia
E-mail: saud.fattah@aramco.com
Published: 28 April 2006 Received: 06 January 2006
E-Journal of Petroleum Management and Economics 2006, ISSN: 1718-7559. Accepted: 27 April 2006
This article is available from: http://www.petroleumjournals.com/
© 2006 Al-Fattah; licensee Petroleum Journals Online.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by-nc-nd/2.0/),
which permits unrestricted use for non-commercial purposes, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
AbstractAbstract
Abstract
Faced with diminishing supplies of domestic crude oil and increased demand for energy, the US has come to
rely on imported crude and domestic supplies of natural gas. During the last decade US natural gas production
has risen about 10%, but it still does not succeed to meet demand. This supply shortage has resulted in
significant increases in natural gas prices. We believe that developing a reliable method to forecast US natural
gas production rates and reserves will benefit gas producers, consumers and policy makers.
This paper presents one methodology for developing forecasting models for predicting U.S. natural gas
production, proved reserves, and annual depletion to year 2025 using a stochastic (time series) modeling
approach. The methodology is not mechanistic. A mechanistic model would examine individual geologic
settings, exploration success, the physics of gas production, and the rate of exploitation for provinces, basins,
and reservoirs. However, to do so would result in an extraordinarily massive model that would be difficult, if
not impossible, to develop and use. Instead we used a simpler approach which takes advantage of established
trends in easily obtained published data.
Various time series models were tested and validated using data that are not used in the mathematical
development of the models. Having adequately validated these time series models using historical data we
believe that they can be used to make at least short time forecasts. Comparison of results of this study with
other published forecast is also presented. Our forecasts show that U.S. gas production rate will maintain a
production plateau of 18.7 Tcf/yr from 2005 to 2008 after which gas production will increase gradually from
19.0 Tcf/yr in 2010 to reach 22.5 Tcf/yr in 2025. We predict that U.S. gas production will have an average
annual rate of increase of 0.5% from 2005 to 2015 after which the average annual rate of change of production
will increase to 1.2% for the period of 2015 to 2025. Our forecasts also show that U.S. gas depletion rate will
increase from 10.6%/yr in 2005 to 13.4%/yr in 2025. While the reserves discovery rate increases, U.S. gas
proved reserves are predicted to increase from 197 Tcf in 2005 to 215 Tcf in 2010. Afterwards, the gas
proved reserves will increase at an annual rate of 1.3% and are expected to reach 263 Tcf in 2025.
Introduction
IntroductionIntroduction
Introduction
The time-series analysis can be classified as descriptive,
inferential, predictive, and control. Stochastic or time
series models are the models for time series that are
needed to achieve optimal forecasting. They are used
to compute the probability of future values lying
between two specified limits.
Time series analysis
accounts for the fact that data points taken over time
may have an internal structure (such as
autocorrelation, trend or seasonal variation) that
should be accounted for. The usage of time series is
two-fold: obtaining an understanding of the underlying
forces and structure that produced the observed data,
and fitting the model for forecasting. There are many
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methods that are used to model and forecast time
series. The most common classes of time series
methods are the autoregressive integrated moving
average (ARIMA) models, developed by Box and
Jenkins [1,2], and often referred to as Box-Jenkins
models. The ARIMA approach has become popular in
many areas and research practice confirms its power
and flexibility. The basic assumption of ARIMA
methodology is that the past behavior of the actual
time series will continue into the future. The ARIMA
method is appropriate only for a stationary time series,
meaning that the mean, variance, and autocorrelation
of the series should be approximately constant
through time. It is also assumed that the values of the
estimated parameters are constant throughout the
series. The following section describes the Box-Jenkins
models.
Moving Average Model
Moving Average ModelMoving Average Model
Moving Average Model
The moving average (MA) model describes a time
series that is a linear function of the current and
previous random shocks (ε). The random shocks are
also called errors, residuals or a white noise process.
A time series,
x
t, is said to be a moving average
process of order
q
, MA(
q
), if
qtqtttt
x=
εθεθεθε
2211 , …….. (1)
where
x
t is the current value of time series data; εt, εt-1,
εt-2, εt-q the current and previous errors or random
shocks; and θ1, θ2, θq the moving average model
parameters.
Autoregressive Model
Autoregressive ModelAutoregressive Model
Autoregressive Model
A time series is said to follow an autoregressive (AR)
model of order
p
if the current value of the series can
be expressed as a linear function of the previous values
of the series plus a random shock term. The general
equation of an autoregressive model of order
p
, AR(
p
),
can be written as
tptpttt xxxx
εφφφ
++++= 2211 , … (2)
where φ1, φ2, φp are the autoregressive model
parameters.
Autoregressive Moving Average Model
Autoregressive Moving Average ModelAutoregressive Moving Average Model
Autoregressive Moving Average Model
While the AR and MA models can be used for many
data sets, they are not adequate for some data, and a
more general set of models is needed. The
autoregressive moving average (ARMA) models
contain both AR and MA processes. Alternatively, a
time series
x
t is said to follow an autoregressive
moving average model of orders
p
and
q
, ARMA(
p
,
q
), if
x
t satisfies the following difference equation
qtqttt
ptpttt xxxx
=
εθεθεθε
φ
φ
φ
2211
2211 ……… (3)
Note that ARMA(
p
,0) = AR(
p
), ARMA(0,
q
) = MA(
q
),
and ARMA(0,0) = WN (i.e. white noise process).
Autoregressive Integrated Moving Average Model
Autoregressive Integrated Moving Average ModelAutoregressive Integrated Moving Average Model
Autoregressive Integrated Moving Average Model
To model a given time series with the ARMA process,
the series must be stationary. This means that both the
expected values of the series and its autocovariance
function are independent of time. In addition, the
series must have stabilized variance and constant mean.
Most time series are nonstationary but some can be
transformed to a stationary series by differencing. This
process is often used to remove the trend, seasonality,
and periodic variations of the series, thus rendering the
nonstationary time series stationary. The differenced
time series,
w
t, can then be analyzed and modeled like
any other stationary time series. After modeling the
differenced time series the output series is
transformed back to the original raw data,
x
t, by
reversing the order of differencing. Inclusion
of differencing in the formulation of the ARMA model
results in the autoregressive integrated moving average
(ARIMA) model. An ARIMA model predicts a value in a
response time series as a linear combination of its own
past values, past errors, and current and past values of
other time series. The order of an ARIMA model is
usually denoted by the notation ARIMA(
p
,
d
,
q
), where
p
is the order of the autoregressive component,
d
the
order of the differencing, and
q
the order of the
moving-average process. Therefore, all models
discussed previously are subsets of ARIMA models.
Mathematically, the ARIMA model is written as
tt B
B
w
ε
φ
θ
µ
)(
)(
+= , ……………………………... (4)
where
w
t = is the response series or a difference of the
response series,
µ = a constant or intercept,
B
= the backshift operator (i.e.
Bx
t =
x
t-1),
φ(
B
) = the autoregressive operator, represented as a
polynomial in the backshift operator: φ(
B
) =1-φ1
B
-…-
φp
B
p,
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θ(
B
) = the moving-average operator, represented as a
polynomial in the backshift operator: θ(
B
) =1- θ1
B
-…-
θq
B
q, and
εt = the random error or shock.
Data Sources
Data SourcesData Sources
Data Sources
The data used to develop the time series model for
U.S. gas production were obtained from
Twentieth
Century Petroleum Statistics
[3-4] for the period
1918-97; the 1998-2004 production data were from
the
Energy Information Admin.
(EIA) [5]. Proved gas
reserves for the period 1949 to 1998 were from the
Oil and Gas Journal
(
OGJ
) database [6]; the 1999-2003
gas reserves data were from the
Energy Information
Admin.
Development of Models
Development of ModelsDevelopment of Models
Development of Models
The ARIMA modeling procedure involves three stages:
model identification, estimation of model parameters
and diagnostic checking, and forecasting [1,2,7]. This
paper discusses three stochastic models; namely,
annual gas depletion, gas proved reserves and
production. The next section will discuss in more
detail the development of the annual-depletion-rate
model to illustrate the modeling procedure. Similar
procedure and analysis on the other models were
applied and implemented. Depletion rate measures
how fast reserves are being depleted each year at that
year’s production rate. It is calculated as the annual
production during the year divided by the reserves at
the beginning of the year expressed in percentage.
Model Identification
Model IdentificationModel Identification
Model Identification
The first step in the development of the gas-depletion-
rate model was model identification. As mentioned
earlier, the Box-Jenkins approach requires that the
input series be stationary. This means that the series
should have a constant mean, variance, and
autocorrelation through time. We used three function
plots to check for stationarity: the autocorrelation
function (ACF), partial autocorrelation function
(PACF), and inverse autocorrelation function (IACF).
The autocorrelation function measures the degree of
correlation between neighboring time series
observations at a specified number of samples apart,
called the lag. If the series is nonstationary, its
autocorrelation function will usually decay slowly.
Therefore, the series first needs to be differenced until
it is stationary. This may also require log or Box-Cox
transformations [8] to stabilize the variance of the
series. A rapid decrease in the ACF plot indicates that
the series is stationary. The partial autocorrelation
function measures the degree of association between
time series observations when the effect of other time
lags on the response time series is held constant. The
PACF is an extension of ACF and it is used to examine
serial dependencies for individual lags. The inverse
autocorrelation function plays almost the same role as
the PACF but generally indicates subset and seasonal
autoregressive models better than the PACF. If the
series is nonstationary, the IACF has the
characteristics of a noninvertible moving average and
therefore decays slowly. The IACF may also be useful
for detecting over-differencing. If the data have been
over-differenced, the IACF looks like the ACF from a
nonstationary process.
0
2
4
6
8
10
12
14
1945 1955 1965 1975 1985 1995 2005
Time, year
Gas Depletion Rate, %/yr
Fig. 1
Fig. 1Fig. 1
Fig. 1-
--
-Annual gas depletion rate (1949-2003). Data from ref. 3, 4, 5, and 6.
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Fig. 2
Fig. 2Fig. 2
Fig. 2-
--
-Autocorrelation plots of raw data of gas depletion rate.
We first plotted the data of gas depletion rate to
check the existence of any trend, pattern, or periodic
variations. The data, as shown in Fig. 1
Fig. 1Fig. 1
Fig. 1, show a trend
of linear relationship of gas depletion rate with time.
Fig. 2
Fig. 2Fig. 2
Fig. 2 shows the ACF plot of the data. Each bar
represents the value of the correlation coefficient at a
given lag. The overlaid lines represent confidence limits
computed at plus and minus two standard errors. The
ACF shows high correlation coefficients through most
lags, with up to lag 4 the bars lie outside the
confidence limits, indicating that the data are
nonstationary. The slow decline of the ACF suggests
that first differencing may be warranted. Fig. 3
Fig. 3Fig. 3
Fig. 3 is a plot
of the gas depletion rate after the first difference
transformation was performed, showing the removal
of the trend and stabilization of the variance. To test
this differenced data for stationarity, the
autocorrelation plots were examined as shown in Fig.
Fig. Fig.
Fig.
4
44
4. The ACF plot of the differenced series shows a
rapid decrease and has small correlation coefficients
within the confidence limits for all lags throughout,
indicating that the series is stationary. In addition, the
PACF and IACF plots show that the correlation
coefficient of the first lag decreased substantially with
all values confined by the confidence limits of
stationarity series.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1945 1955 1965 1975 1985 1995 2005
Time, year
First Difference of Depletion Rate, %/y r
Fig. 3
Fig. 3Fig. 3
Fig. 3-
--
-First difference of annual gas depletion rate.
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Fig. 4
Fig. 4Fig. 4
Fig. 4-
--
-Autocorrelation plots of first difference of gas depletion rate.
In addition, we also used other methods to check for
stationarity using the stationarity tests: the white-noise
test, and the unit-root test. The white-noise test is an
approximate statistical test of the hypothesis that none
of the autocorrelations of the series up to a given lag is
significantly different from zero. In other words, it
tests the hypothesis that the residuals are
uncorrelated. The hypothesis is rejected if the p-value
for the test of the autocorrelations is very small at the
0.05 significance level, which means that the series is
nonstationary. Significance probability values greater
than 0.05 through all given lags indicate that the series
is a white noise. Fig. 5
Fig. 5Fig. 5
Fig. 5 shows a plot of the white-noise
test of the differenced gas-depletion rate. The plot
shows that the autocorrelations of Lags 1 through 24
are not significantly different from zero at the 0.05
level of significance, meaning that the differenced series
has a stable variance and constant mean.
The other stationarity test is the unit-root test. A time
series having a unit root is nonstationary. The unit-
root test tests the hypothesis that the time series
exhibits a lag
d
unit root against the alternative
stationary series. The hypothesis is rejected when the
test statistic gives a small probability value. The results
of the unit-root test for the differenced data of gas
depletion rate are shown in Fig. 5. The test statistics
shown by the plot indicate that the series is likely
stationary at the 0.05 significance level for the unit
root of all lags, except for Lags 3 and 4 at which the
series is stationary at the 0.1 significance level.
Fig. 5
Fig. 5Fig. 5
Fig. 5-
--
-White noise and unit root tests of first differenced gas depletion rate data.
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Table 1-Parameter Covariances Of The Gas Depletion Model, Arima(1,1,2).
Parameter Constant p(1) q(1) q(2)
Constant
p(1)
q(1)
q(2)
0.005230
0.054137
0.041358
-0.002451
0.054137
0.269466
0.266664
0.054137
0.041358
0.266664
0.285943
0.041358
-0.002451
0.054137
0.041358
0.037786
Table 2-Parameter Correlations Of The Gas Depletion Model, Arima(1,1,2).
Paramet. Constant p(1) q(1) q(2)
Constant
p(1)
q(1)
q(2)
1.000000
-0.174319
1.000000
0.960668
0.536509
0.960668
1.000000
0.397882
-0.174319
0.536509
0.397882
1.000000
Estimation of Model Parameters
Estimation of Model ParametersEstimation of Model Parameters
Estimation of Model Parameters
The next step of model development of time series
was estimating model parameters and diagnostic
checking. We specified different combinations of
orders of the ARIMA model in the time series
software to estimate the parameters of each specified
model. The parameters were estimated using function
minimization procedures, so that the sum of squared
residuals is minimized. Two criteria were used to judge
the goodness of fit of the ARIMA model: the Akaike
information criterion (AIC), and the root mean square
error (
R
MSE). The model that has the smallest AIC and
R
MSE is chosen. We also used the mean absolute-
percentage error (MAPE) and coefficient of
determination (
R
2) to complement the AIC and
R
MSE
criteria and to have a better judgment of the accuracy
of the model.
In this study, several selections of potential ARIMA
models have been fitted to the annual gas depletion
data. Analysis of statistics shows that, among all models
considered, the ARIMA(1,1,2) model has the smallest
AIC value (-103.14), the lowest
R
MSE (0.3709), lowest
MAPE (3.2911), and highest
R
2 (0.983). These results
indicate that the ARIMA(1,1,2) model gives better
results of accuracy and goodness of fit than other
models. Therefore, we chose the ARIMA(1,1,2) model
as the appropriate model for forecasting the gas-
depletion rate.
The ARIMA(1,1,2) model contains one autoregressive
parameter and two moving average parameters, which
were computed for the series after it was differenced
once. The model also includes a constant or intercept
since it has an autoregressive parameter. The estimates
of the model parameters, as determined by fitting the
time series data, are intercept = 0.13820, θ1 = 0.40338,
θ2 = 0.08182, and φ1 = 0.63371. Substituting these
parameters in Eq. 4, the mathematical form of the
ARIMA(1,1,2) model is
( )
(
)
( )
t
B
BB
t
xB
ε
63371.01
2
08182.040338.01
1382.01
+=
…….…(5)
Tables 1 and 2
Tables 1 and 2Tables 1 and 2
Tables 1 and 2 present the covariances and
correlations of the model parameters.
Diagnostic Checking
Diagnostic CheckingDiagnostic Checking
Diagnostic Checking
We checked significance tests for parameter estimates
of the ARIMA(1,1,2) model for unnecessary terms
included in the model. The tests indicate that the
model parameters are significant enough to be
included in the model. For example, the autoregressive
parameter of the model has a very small significance
probability of 0.0007, which means that it is unlikely to
reject the hypothesis that the autoregressive
parameter is significant. The tests for white noise
residuals, as described in the model identification stage,
were also examined; the tests indicated that the
residual series contains no additional information that
might be used by a more complex model. Figs. 6 to 8
Figs. 6 to 8Figs. 6 to 8
Figs. 6 to 8
show the scatter plot, normal distribution plot, and
normal probability plot, respectively, of residuals of the
ARIMA model developed for the annual-depletion rate.
These figures show that the residuals are fairly
scattered and they are not systematically distributed
across the series, indicating that the residuals are
uncorrelated and approximately normally distributed.
This serial independency also suggests that the ARIMA
model is adequate.
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-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1945 1955 1965 1975 1985 1995 2005
Time, year
Residual Error, %/yr
Fig. 6
Fig. 6Fig. 6
Fig. 6-
--
-Residual plot of gas depletion rate model.
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Residuals
0
2
4
6
8
10
12
14
16
18
No of observations
Expected Normal
Fig. 7
Fig. 7Fig. 7
Fig. 7-
--
-Normal distribution plot of residuals for the gas depletion rate model.
-1.0 -0.8 -0.6 - 0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Residuals
-3
-2
-1
0
1
2
3
4
Expected Normal Value
Fig. 8
Fig. 8Fig. 8
Fig. 8-
--
-Normal probability plot of residuals for the gas depletion rate model.
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0
2
4
6
8
10
12
14
16
18
20
1950 1960 1970 1980 1990 2000 2010 2020 2030
Time, year
Annual Depletion Rate, %/yr
Actual
Predicted
High
Low
Mostly
Fig. 9
Fig. 9Fig. 9
Fig. 9-
--
-Prediction model of gas depletion rate, 1949-2025. Historical data from ref. 3, 4, and 5.
Forecasting
ForecastingForecasting
Forecasting
Having estimated its parameters and tested for its
adequacy of fit, we used the ARIMA model in Eq. 5 to
produce forecasts of future values of the annual
depletion rate and to generate confidence intervals for
these forecasts. Fig. 9
Fig. 9Fig. 9
Fig. 9 shows the forecasting model of
the annual-depletion rate to the year 2025. The figure
shows the 95% upper and lower confidence limits. The
upper and lower 95% confidence can be considered as
high and low cases of forecasts, respectively, that the
annual-depletion data would follow these trends. The
most likely forecast is the one produced directly from
the specified ARIMA model, Eq. 5, and is represented
by the solid line for all forecasting models as shown in
Fig. 9. The forecast in Fig. 9 shows that the annual
depletion rate of reserves will increase at an average
annual rate of 1.1% through the forecast period. The
gas depletion is predicted to continue to increase from
annual rate of 10.7% in 2005, 11.3% in 2010, 12% in
2015, 12.7% in 2020, to a rate of 13.4% in 2025. The
high case of forecast shows an increase in the annual
gas depletion rate from 11.7% in 2005 to 17.4% in
2025. The low case, however, shows a steady decrease
of depletion rate from 9.7% in 2005 to 9.3% in 2025.
Analysis of Results
Analysis of ResultsAnalysis of Results
Analysis of Results
The same procedures that we used to develop the
annual-gas-depletion model were followed and applied
on the development of models for gas proved reserves
and gas production. Summary of results of statistical
analysis of the developed models of gas depletion,
proved reserves and gas production is given in Table 3
Table 3Table 3
Table 3.
Fig. 10
Fig. 10Fig. 10
Fig. 10 shows the forecasts of the prediction model for
the U.S. gas proved reserves from 1949 to 2025, and
Figs. 11 and 12
Figs. 11 and 12Figs. 11 and 12
Figs. 11 and 12 depict the residual error distribution of
the model. Figs. 11 and 12 show that the model
residuals are evenly scattered and they are not
systematically distributed indicating that the residuals
are uncorrelated and normally distributed. The U.S. gas
proved reserves forecasting model was developed with
the subset Log_ARIMA(2,2,1) model. This model
shows excellent agreement with actual data, Fig. 10.
The model has AIC value of 199.98, RMSE 6.355, R2
0.979, and MAPE 2.064, Table 3. It predicts that US gas
proved reserves will increase at an average annual rate
of 1.5% through the forecast period. The gas proved
reserves is predicted to increase at an average annual
rate of 1.8% from 197 Tcf in 2005, to 215 Tcf in 2010.
Thereafter, the gas proved reserves will increase at a
rate of 1.4% per year to reach 232 Tcf in 2015, 248 Tcf
in 2020, and 263 Tcf in 2025. The high forecast case
predicts 215 Tcf of proved reserves in 2005 and 278
Tcf in 2010. The low forecast case predicts proved
reserves of 179 Tcf in 2005 and 152 Tcf in 2010.
Table 4-Statistical Analysis of U.S. Natural Gas Time-Series Models.
Model
ModelModel
Model
Type
TypeType
Type
Fore
ForeFore
Forecast Period
cast Periodcast Period
cast Period
AIC
AICAIC
AIC
R
RR
RMSE
MSEMSE
MSE
MAPE
MAPEMAPE
MAPE
R
RR
R2
22
2
Gas production_1 ARIMA(2,1,1) 1949-2025 -47.10 0.6286
2.6960
0.9793
Gas production_2 ARIMA(1,1,1) 1970-2025 -24.33 0.6796
2.5697
0.7913
Gas depletion rate ARIMA(1,1,2) 1949-2025 -103.14
0.3709
3.2911
0.9829
Gas proved reserves
Log ARIMA(2,2,1)
1949-2025 199.98 6.3549
2.0638
0.9789
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0
50
100
150
200
250
300
350
400
1950 1960 1970 1980 1990 2000 201 0 2020 2030
Time, year
Proved Reserves, Tcf
Actual
Predicted
High
Low
Mostly
Fig. 10
Fig. 10Fig. 10
Fig. 10-
--
-Prediction model of gas proved reserves, 1949-2025. Historical data from ref. 6.
Expec ted Normal
-0.12
-0.11
-0.10
-0.09
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
Upper Boun daries (x<=boundary )
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
No of obs
Fig. 11
Fig. 11Fig. 11
Fig. 11-
--
-Normal distribution of residuals of proved reserves model.
-0.10 -0.08 - 0.06 -0.04 - 0.02 0.00 0.02 0.04 0.06 0.08
Residuals
-4
-3
-2
-1
0
1
2
3
Expected Normal Value
Fig. 12
Fig. 12Fig. 12
Fig. 12-
--
-Normal probability plot of prediction residuals of the gas proved reserves model.
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To develop a forecasting model for future U.S. gas
production using univariate time-series analysis, we
used gas production data from 1918 to 2004. Analysis
of the production data indicates that the time series is
nonstationary even after applying different
transformations. Consequently, we chose to split the
data into two data sets. Because of its exponential-like
trend, we chose the first data set from 1918 to 1970,
and the second data set from 1971 to 2004. The first
data set was then transformed using the first
difference, and stationarity tests showed that the series
is a white noise, indicating it is stationary. We
performed the difference of the first difference
transformation on the second data set, attempting to
render the series stationary. However, time-series
analysis of the second data set indicated that the series
is a random walk suggesting that the data series is
nonstationary and errors are correlated. Therefore,
using the ARIMA model to directly forecast gas
production using only time-dependent gas production
data could not achieve time-independent errors over
the entire production history from 1918 to 2004. The
probable reason is that the series of gas production
from 1974 to 2004 was influenced by economic factors
and political events. These factors and events were
discussed in Ref. 9.
However, we developed two ARIMA models of gas
production. The first model uses production data from
1949 to 2000, and the second model uses production
data from 1970 to 2000. We spared data from 2000 to
2004, which are not used in the model development,
for validation of both production models. The
prediction of gas production from the first model is
shown in Fig. 13
Fig. 13Fig. 13
Fig. 13, which was developed using
ARIMA(2,1,1) model. This model is in good agreement
with actual data as shown in Fig. 13. The model
statistics, Table 4, shows that it has AIC value of -47.1,
RMSE 0.6286, R2 0.979, and MAPE 2.696. Figs. 14 and 15
Figs. 14 and 15Figs. 14 and 15
Figs. 14 and 15
show the autocorrelation and partial autocorrelation
functions of the residuals of the first gas production
model. The ACF and PACF plots show a decrease and
small correlation coefficients within the confidence
limits for all lags throughout, indicating that the series
is stationary. The normal distribution and normal
probability plots of the residuals of the gas production
model are given in Figs. 16 and 17
Figs. 16 and 17Figs. 16 and 17
Figs. 16 and 17, indicating that the
model errors are uncorrelated and approximately
normally distributed. The first production model
predicts that the U.S. will produce at a production rate
of 18.7 Tcf/yr from 2005 to 2008 after which
production will increase from 19 Tcf/yr in 2010 to 22.5
Tcf/yr in 2025. The high case forecasts an increase in
gas production from 19.8 Tcf/yr in 2005, 23.4 Tcf/yr in
2010, to 34.1 Tcf/yr in 2025. On the other hand, the
low case predicts a decline in gas production from 17.6
Tcf/yr in 2005, 14.5 Tcf/yr in 2010, to 10.8 Tcf/yr in
2025.
0
5
10
15
20
25
30
35
40
1950 1960 1970 1980 1990 2000 2010 2020 2030
Time, year
Gas Production, Tcf/yr
Actual
Predicted
Mostly
High
Low
Fig. 13
Fig. 13Fig. 13
Fig. 13-
--
-Prediction model_1 of U.S. natural gas production, 1949-2025. Historical data from ref. 3, 4, and 5.
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Autocorr elation Function
Gas Produc tion Model(1949-2025): ARIMA (2,1,1) residuals;
(Standard errors are w hite-noise estimates)
-1.0 -0.5 0.0 0.5 1.0
0
15 -.150 .1130
14 -.049 .1144
13 -.072 .1157
12 -.112 .1171
11 +.041 .1185
10 +.110 .1198
9 -.064 .1211
8 +.170 .1224
7 +.053 .1237
6 -.045 .1250
5 -.113 .1263
4 -.069 .1275
3 +.169 .1288
2 -.070 .1300
1 +.002 .1312
Lag Corr. S.E.
0
9.84 .8294
8.08 .8852
7.89 .8505
7.51 .8224
6.59 .8315
6.47 .7745
5.63 .7761
5.35 .7195
3.42 .8435
3.24 .7783
3.11 .6829
2.31 .6793
2.01 .5700
.29 .8646
.00 .9874
Q p
Conf. Limit
Fig. 14
Fig. 14Fig. 14
Fig. 14-
--
-Autocorrelation function of residuals of U.S. natural gas production model_1, 1949-2025.
Partial Autoc or relation Function
Gas Produc tion Model(1949-2025): ARIMA (2,1,1) residuals;
(Standard errors ass ume AR order of k-1)
-1.0 -0.5 0.0 0.5 1.0
0
15 -.145 .1348
14 -.043 .1348
13 -.077 .1348
12 -.045 .1348
11 -.046 .1348
10 +.098 .1348
9 -.060 .1348
8 +.200 .1348
7 +.070 .1348
6 -.084 .1348
5 -.090 .1348
4 -.080 .1348
3 +.170 .1348
2 -.070 .1348
1 +.002 .1348
Lag Corr. S.E.
Conf. Limit
Fig. 15
Fig. 15Fig. 15
Fig. 15-
--
-Partial autocorrelation function of residuals of U.S. natural gas production model_1, 1949-2025.
-2.5 -2.0 -1.5 - 1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Residuals
0
2
4
6
8
10
12
14
16
18
20
22
24
26
No of observations
Expected Normal
Fig. 16
Fig. 16Fig. 16
Fig. 16-
--
-Normal distribution plot of residuals of the gas production model_1, 1949-2025.
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-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
Residuals
-3
-2
-1
0
1
2
3
4
Expected Normal Value
Fig. 17
Fig. 17Fig. 17
Fig. 17-
--
-Normal probability plot of residuals of the gas production model_1, 1949-2025.
0
5
10
15
20
25
30
35
1970 1980 1990 2000 2010 2020 2030
Time, year
Gas Production, Tcf/yr
Actual
Predicted
Mostly
High
Low
Fig. 18
Fig. 18Fig. 18
Fig. 18-
--
-Prediction model_2 of U.S. natural gas production, 1970-2025. Historical data from ref. 3, 4, and 5.
Autocorr elation Function
Gas Produc tion Model(1970-2025): ARIMA (1,1,1) residuals;
(Standard errors are w hite-noise estimates)
-1.0 -0.5 0.0 0.5 1.0
0
15 -.153 .1246
14 -.137 .1278
13 -.038 .1310
12 -.140 .1341
11 -.073 .1371
10 +.013 .1400
9 -.143 .1429
8 +.213 .1457
7 +.045 .1485
6 +.043 .1512
5 -.067 .1539
4 -.100 .1566
3 +.040 .1591
2 -.015 .1617
1 -.025 .1642
Lag Corr. S.E.
0
8.15 .9174
6.64 .9477
5.48 .9629
5.40 .9433
4.30 .9602
4.02 .9465
4.01 .9107
3.00 .9342
.87 .9967
.77 .9928
.69 .9835
.50 .9733
.09 .9924
.03 .9843
.02 .8797
Q p
Conf. Limit
Fig. 19
Fig. 19Fig. 19
Fig. 19-
--
-Autocorrelation function of residuals of U.S. natural gas production model_2, 1970-2025.
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Partial Autoc or relation Function
Gas Produc tion Model(1970-2025): ARIMA (1,1,1) residuals;
(Standard errors ass ume AR order of k-1)
-1.0 -0.5 0.0 0.5 1.0
0
15 -.194 .1715
14 -.195 .1715
13 -.048 .1715
12 -.088 .1715
11 -.089 .1715
10 +.011 .1715
9 -.156 .1715
8 +.216 .1715
7 +.054 .1715
6 +.036 .1715
5 -.071 .1715
4 -.098 .1715
3 +.039 .1715
2 -.016 .1715
1 -.025 .1715
Lag Corr. S.E.
Conf. Limit
Fig. 20
Fig. 20Fig. 20
Fig. 20-
--
-Partial autocorrelation function of residuals of U.S. natural gas production model_2, 1970-2025.
The gas production data from 1970 to 2000 was best
fitted with the ARIMA(1,1,1) model. Data from 2000
to 2004 was spared to validate the prediction model.
This second model of production gives good match
with actual production data as shown in Fig. 18
Fig. 18Fig. 18
Fig. 18. The
model statistics as presented in Table 4 shows that the
second model of gas production has AIC value of -
24.33, RMSE 0.6796, R2 0.791, and MAPE 2.570. The
ACF and PACF plots of the residuals of the second gas
production model
are illustrated in Figs. 19 and 20
Figs. 19 and 20Figs. 19 and 20
Figs. 19 and 20.
The ACF and PACF plots show a decrease and small
correlation coefficients within the confidence limits for
all lags throughout, indicating that the production data
series is stationary. Figs. 21 and 22
Figs. 21 and 22Figs. 21 and 22
Figs. 21 and 22 are
the normal
distribution and normal probability plots of the
residuals of the second gas production model, showing
that the model errors are uncorrelated and
approximately normally distributed. The forecast of
the second production model predicts that the U.S. gas
production will decline at an average annual rate of
0.5% from 2005 to 2025. U.S will produce at a
production rate of 18.8 Tcf/yr in 2005, 17.9 Tcf/yr in
2015, and 17.0 Tcf/yr in 2025. The high case forecasts
an increase in gas production from 19.9 Tcf/yr in 2005,
21.4 Tcf/yr in 2010, to 22.8 Tcf/yr in 2025. In contrast,
the low case predicts a decline in gas production from
17.6 Tcf/yr in 2005, 15.2 Tcf/yr in 2010, to 11.2 Tcf/yr
in 2025. All these forecasts inherit uncertainties which
are bounded with ±95% confidence limits.
-2.0 -1.6 -1.2 -0.8 - 0.4 0.0 0.4 0.8 1.2 1.6 2.0 2.4
Residuals
0
1
2
3
4
5
6
7
No of observations
Expected Normal
Fig. 21
Fig. 21Fig. 21
Fig. 21-
--
-Normal distribution plot of residuals of the gas production model_2, 1970-2025.
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-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
Residuals
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Expected Normal Value
Fig. 22
Fig. 22Fig. 22
Fig. 22-
--
-Normal probability plot of residuals of the gas production model_2, 1970-2025.
0
5
10
15
20
25
30
35
40
1950 1960 1970 1980 1990 2000 2010 2020 2030
Time, year
Gas Pro duction, Tcf/yr
Actual
Predicted
Fig. 23
Fig. 23Fig. 23
Fig. 23-
--
-Hybrid prediction model of U.S. natural gas production. Historical data from ref. 3, 4, and 5.
However, since we had earlier determined that both
depletion rate and reserves discovery rate were
stationary time series we used these two ARIMA
models to forecast gas production by multiplying
depletion rate and gas proved reserves. It is the
product of these two time series that determined the
stochastic gas production forecast in Fig. 23
Fig. 23Fig. 23
Fig. 23. The
forecast of U.S. gas supply by this new hybrid model
shows an increase in production with an average
growth rate of 2.6%/yr. It predicts gas production rate
of 21.1 Tcf/yr in 2005, 24.3 Tcf/yr in 2010, 31.4 Tcf/yr
in 2020, and 35.1 Tcf/yr in 2025. This model of gas
production is, however, more optimistic than the
other two production models.
The future of U.S. natural gas supplies will depend
mainly on domestic production of unconventional gas,
natural gas production from Alaska, and imports of
liquefied natural gas (LNG). Because of technology
advancement and increase in natural gas prices, natural
gas production from unconventional sources (i.e. tight
sands, shale, and coalbed methane) is expected to
increase more rapidly than conventional production.
The decline in U.S. conventional natural gas production
is caused by the increase in costs of the exploration
and development of resource depletion.
Validation of Produc
Validation of ProducValidation of Produc
Validation of Production Models
tion Modelstion Models
tion Models
We used production data from 2000 to 2004, which
are not used in the models development, to validate
the prediction models and examine the accuracy of the
forecasts. It appears that the production data (2000-
2004), which are spared for validation and not used in
the model development, matches very well the
forecasts of both the first and second production
models as shown in Figs. 24 and 25
Figs. 24 and 25Figs. 24 and 25
Figs. 24 and 25. Both models of gas
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production show excellent agreement with actual data
for 2000, 2001, and 2004, with a slight overestimation
of production data for years 2002 and 2003. After
successful validation of both production models and
because of limited data, we retained the validation data
(2000-2004) to the original models to produce the
forecast
Although the five year validation of the models with
historical data was very satisfactory it remains to be
seen if longer term forecasts have validity. It seems
reasonable that, when the modeling method used in
this paper ceases to provide acceptable accuracy, then
it’s time to update the models with newer data and
continue the forecasting process.
0
5
10
15
20
25
30
1990 1992 1994 1996 1998 2000 2002 2004
Time, year
Gas Production, Tcf/yr
Actual
Predicted
Validation
Fig. 24
Fig. 24Fig. 24
Fig. 24-
--
-
Validation of U.S. natural gas production model_1, 2000-2004.
0
5
10
15
20
25
30
1990 1992 1994 1996 1998 2000 2002 2004
Time, year
Gas Production, Tcf/yr
Actual
Predicted
Validation
Fig. 25
Fig. 25Fig. 25
Fig. 25-
--
-
Validation of U.S. natural gas production model_2, 2000-2004.
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0
5
10
15
20
25
30
35
1950 1960 1970 1980 1990 2000 2010 2020 2030
Time, year
Gas Production, Tcf/yr
Actual
Model_1
Model_2
EIA2005-Reference Case
EIA2005-Restricted Case
Fig. 26
Fig. 26Fig. 26
Fig. 26-
--
- Comparison of forecasts of U.S. natural gas production, 2004-2025.
Table 5-Comparison Of Forecasts Of Natural Gas Production.
Year
This Study
Model_1
This Study
Model_2
EIA Reference
Case
EIA Restricted
Case
2010 18.966 18.308 20.420 20.260
2015 19.912 17.866 20.770 20.440
2020 21.133 17.425 21.890 19.810
2025
22.462
16.983
21.830
19.070
Comparison of Production Forecasts
Comparison of Production ForecastsComparison of Production Forecasts
Comparison of Production Forecasts
We compared the results of production forecasts of
this study with two cases of the 2005 forecast of EIA
[10]. The EIA published two cases of gas production
forecast, namely, the reference case and the restricted
supply case. The reference case assumes that gas
production from Alaska pipeline will be on stream by
2016, new LNG expansion and regasification terminals
will be built, and technological advancement for oil and
gas exploration and development. Conversely, the
restricted natural gas supply case has the following
assumptions: (1) Alaska natural gas pipeline is not built
before 2025, (2) no new LNG regasification terminals
are built during the forecast but all proposed
expansions at existing LNG terminals are constructed,
and (3) the future rate of technological progress for oil
and gas exploration and development is 50% the
historic rate.
Fig
FigFig
Fig. 26
. 26. 26
. 26 shows the comparison of forecasts of gas
production between this study and EIA. The first and
second models of this study predict that U.S. will
produce at 19.0 Tcf/yr and 18.3 Tcf/yr, respectively, in
2010. This prediction is slightly lower than the EIA
forecast which predicts 20.4 Tcf/yr for the reference
case and 20.3 Tcf/yr for the restricted case. In 2025,
the first model of this study shows the highest forecast
of 22.5 Tcf/yr where the second model gives the
lowest forecast of 17.0 Tcf/yr. The EIA predicts gas
production of 21.8 Tcf/yr in 2025 for the reference
case, and 19.1 Tcf/yr for the restricted case. The
forecast of the first model looks in a closer agreement
with the reference case of EIA than other forecasts,
Table 5
Table 5Table 5
Table 5.
Conclusions
ConclusionsConclusions
Conclusions
This paper presented a methodology for developing
stochastic time series ARIMA models. We developed
ARIMA models for U.S. natural gas depletion, proved
reserves, and production. Having adequately tested
and validated these time series models using historical
data we believe that they can be used to make at least
short time forecasts. The following conclusions seem
warranted from our study.
Our forecasts show that U.S. gas production rate will
maintain a production plateau of 18.7 Tcf/yr from 2005
to 2008 after which gas production will increase
gradually from 19.0 Tcf/yr in 2010 to reach 22.5 Tcf/yr
in 2025. We predict that U.S. gas production will have
an average annual rate of increase of 0.5% from 2005
to 2015 after which the average annual rate of change
of production will increase to 1.2% for the period of
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2015 to 2025. Our forecasts also show that U.S. gas
depletion rate will increase from 10.6%/yr in 2005 to
13.4%/yr in 2025. While the reserves discovery rate
increases, U.S. gas proved reserves are predicted to
increase from 197 Tcf in 2005 to 215 Tcf in 2010.
Afterwards, the gas proved reserves will increase at an
annual rate of 1.3% and are expected to reach 263 Tcf
in 2025.
Acknowledgements
AcknowledgementsAcknowledgements
Acknowledgements
The author would like to thank Saudi Aramco
management for their permission to publish this paper.
Many thanks to Dr. R. A. Startzman, Texas A&M
University, for his valuable comments and suggestions.
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SI Metric Conversion Factors
SI Metric Conversion FactorsSI Metric Conversion Factors
SI Metric Conversion Factors
bbl x 1.589 873 E-01 = m3
ft3 x 2.831 685 E-02 = m3
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