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Department of Economics Discussion Paper 2002-08
The Cost of Lifting Natural Gas in Alberta: A Well Level Study
Matthew Foss, Daniel Gordon and Alan MacFadyen May 2002
Department of Economics University of Calgary
Calgary, Alberta, Canada T2N 1N4
This paper can be downloaded without charge from http://www.econ.ucalgary.ca/research/research.htm
The Cost of Lifting Natural Gas in Alberta: A Well Level Study
by
Matthew Fossa) Daniel V. Gordonb) Alan MacFadyenb)
Abstract The economics literature on natural resource depletion and petroleum/gas extraction is extensive. Much less research has been devoted to empirical modeling and measurement of the cost characteristics of lifting natural gas at the well level. The contribution of this paper is to add to empirical understanding of the cost characteristics in lifting natural gas at the well level using data for wells in Alberta, Canada. We are particularly interested in measuring for common factors affecting production costs across reservoirs and also for individual well effects. Information on extraction cost at the well level can be used in calculating projected cost in gas production and combined with expected drilling and infrastructure cost allow a total cost forecast for the project. Keywords: Natural Gas, Cost Characteristics, Well Level JEL classification: Q4 a) CERI, Canadian Energy Research Institute, Calgary, Alberta Canada T2L 2A6
b) Department of Economics, University of Calgary, Calgary, Alberta Canada T2N 1N4
The Cost of Lifting Natural Gas in Alberta: A Well Level Study
I. Purpose
The economics literature on natural resource depletion and petroleum/gas
extraction is extensive. Sweeney (1993) describes the basic model for resource depletion
based on maximization of present value of profits subject to a time path of extraction.
This model makes clear that in resource depletion marginal cost has two components, the
marginal extraction cost and the marginal user cost. The marginal extraction cost is the
cost of lifting the next unit of gas from the well. The marginal user cost is the opportunity
cost of producing now rather than waiting to produce at some point in the future. In
extracting natural gas, production now reduces pressure in the well increasing the cost of
extraction in the future. Current extraction reduces future profits.i Kuller and Cummings
(1974) provide a detailed characterization of the many facets of marginal cost in a
petroleum production model. The basic Sweeney model has been generalized by
Krautkraemer (1998) to incorporate reserve additions and thus, explicitly account for
exploration in the stock formulation of the model.
Some research has been devoted to empirical modeling and measurement of the
cost of lifting natural gas at the well level. Chermak and Patrick (1995) using data for gas
wells in Wyoming and Texas found that operating costs are determined by the quantity of
gas lifted and by reserves remaining in the well. In this paper, our contribution is to add
to empirical understanding of the cost characteristics in lifting natural gas at the well
level using data for wells in Alberta, Canada. We are particularly interested in measuring
for common factors affecting production costs across reservoirs and also for individual
well effects. Using well-level data allows us to avoid the problems of aggregation
common in regional gas analysis.ii Information on extraction costs at the well level can be
used in calculating projected costs in gas production and combined with expected drilling
and infrastructure costs to allow a total cost forecast for the project. Knowledge of the
cost function is also important for measuring costs associated with different output levels
and for determining when production is no longer economically viable and the well
should be abandoned.iii
The paper is organized as follows. In Section II, a brief
survey of the characteristics and nature of natural gas is
presented. Section III describes the basic economic model of
gas extraction and the cost estimation model. In Section IV, the
data and empirical cost estimates are reported. Section V is a
conclusion.
II. Nature of Natural Gas
Conventional natural gas is a depletable resource found in underground
reservoirs. To accumulate in a reservoir an original source of carbon and hydrogen,
buried by porous rock, must have undergone sustained heat and pressure and eventually
natural gas formed that migrated through the porous rock until entrapped in reservoir
rock. The larger the pores in the rock the greater the porosity (i.e., the greater the amount
of fluids that the rock could contain) and permeability (i.e., the easier it is for the gas to
flow). Natural gas is primarily composed of methane with lesser concentrations of ethane,
propane and butane.iv
Hydrocarbons within a reservoir are driven by pressure. Pressure increases with
the depth of the reservoir due to gravity. Gas has a tendency to expand and being trapped
in the reservoir adds to the pressure. When water is present it moves into pores in the
rock vacated by gas further enhancing pressure. Pressure is the key to gas production.
The difference in pressure between the well bore and the reservoir acts to lift or push the
gas to the surface. The flow of petroleum into the well bore and up to the surface reduces
the pressure differential between the reservoir and the well bore and eventually the flow
decreases and production slows.v
Porosity and permeability differ across reservoirs and within the reservoir itself.
As well, the volume of the reservoir rock, the depth at which natural gas is found, the
sulphur content, the amount of associated water, the presence of other hydrocarbons and
the pressure of the gas are all variable factors that differ across reservoirs. These variable
factors affect the value of the natural gas and the cost of production.
Often it is necessary to assist the flow of gas by fracturing the reservoir rock
around the well bore (Berger and Anderson, 1992). Injecting acid or other fluids
containing a propping agent into the reservoir does this. The acid dissolves part of the
formation around the well bore, making existing pores larger. Injecting fluids causes the
formation to crack. The propping agent prevents the formation from collapsing back.vi In
some cases, if the reservoir size is large, compressed gas is added to the reservoir to
enhance pressure levels.
The fact that deposits of petroleum are underground implies that natural gas
stocks are not known with certainty. Exploration to locate deposits is a fundamental
activity in this industry. Once a reservoir of petroleum has been found estimates of the
size can be made and potential production determined. Investment in development of the
formation is based on expected profits. Importantly, development of the reservoir
improves estimates of stock size and production potential. After the well has been
developed, changing price expectations can impact production and may warrant well
abandonment. But with government regulation requiring special sealing of abandoned
wells (and thus costly for the firm to reopen the well) it is not uncommon to observe
producing wells with operating costs exceeding revenue (Brennan and Schwartz, 1985).
The nature and characteristics of the reservoir determines the rate of flow of gas
from a well. Lifting gas too rapidly can pull excess water into the reservoir as gas is
extracted. Excess water can impede the flow and reduce the amount of total gas
recoverable from a reservoir. The efficient rate of gas flow is set by engineering
standards and allows for the greatest rate of flow without damaging the reservoir. The
spacing of wells is also important for gas production and is set by regulation.vii The rate
of gas flow is subject to external factors such as government regulations and pipeline and
processing plant capacity. After lifting to the surface gas is processed to remove foreign
components (e.g., sulphur, water and other hydrocarbons) and shipped by pipeline to
markets.
III. Modelling Depleteable Resources
Economic models of depleteable natural resources start from the proposition that
there exists a fixed stock of the resource. Production overtime reduces the amount of
stock. Depleting the resource today generates an opportunity or user cost because there
would be less resource available in the future. A profit maximising firm will account for
user cost in decision making. The user cost of not extracting the resource is measured as
the increase in the present value of future profits. Sweeney (1993) represents the user cost
of production in a resource model to maximise the present value of profits by choosing a
time path of extraction. Let the cost equation at time t be a function of the extraction
rate q and the stock remaining at the end of the period , or . Also, let the
interest rate be defined as r and the price of the resource at time t as . The firm is
assumed to be a price taker and extraction takes place over a known time period with
terminal time denoted as T. The resource owners problem is to maximise discounted
profits subject to the resource constraint or
)( tC
1−tRt )R,q(C ttt 1−
tp
(1) ∑=
−−−=
T
t
rttttt e)]R,q(Cqp[
01Π
subject to , for all t. The solution to the problem is a straightforward
application of Lagrangian multipliers (
ttt qRR −= −1
)tλ with first order conditions
11
0
−
−− ∂
∂+=
>+∂∂
=
t
trtt
ttt
tt
RC
e
,qifqC
p
λλ
λ (2)
The terminal conditions 0=TT Rλ must also be satisfied. The first expression in
Equation (2) makes clear that price is equal to the sum of marginal extraction cost and the
marginal user cost. In natural gas extraction, marginal user cost is associated with the
decline in pressure as gas is lifted, which makes production in the future more costly.
Krautkraemer (1998) generalizes the model by incorporating reserve additions
(RA) into the stock equation and explicitly accounting for the cost of reserve additions
in the maximization problem. The profit-maximizing problem is now written as )C( RA
(3) ∑=
−− −−=
T
t
rtt
RAtttttt e)]RA(C)R,q(Cqp[
01Π
subject to , for all t. Nothing fundamental has changed with the
addition of reserve additions but solving the first order conditions provides a rule for
exploration or
tttt qRARR −+= −1
t
RAt
t
tt RA
CqC
p∂∂
=∂∂
− (4)
Equation (4) states that a resource firm will incur the cost of exploration until the stated
marginal conditions are satisfied. Importantly for this study, Equation (4) makes clear
that extraction costs are an important determining factor for the levels of both current
production and exploration and shows the importance of empirically measuring costs of
extraction.
A number of empirical studies have been formulated based on this fundamental
economic model of resource depletion (Livernois, 1987; Livernois and Uhler, 1987;
Griffin and Jones, 1989; Helliwell, et. al., 1989). However, Chermak and Patrick (1995)
appear to be the first to model and estimate a cost function for natural gas extraction at
the individual well level. Chermak and Patrick argued that cost (C) of lifting natural gas
is influenced by quantity (q) of gas extracted, the reserves (R) remaining, a time (t) trend
to capture macro (e.g., inflationary) effects and the number of months since
commencement of production (pm), which is a measure of the age of the well. Although a
variety of functional forms where tested they argued that the basic log-log model was
preferred on statistical grounds and written as
ttpmttRtqot pmlntlnRlnqlnCln εβββββ +++++= (5)
where ln is the logarithmic transform and tε is a random error term. The log-log model
has the advantage that the estimated beta coefficients can be interpreted as elasticities
measuring the partial influence of each independent variable on costs
Chermak and Patrick extended the model somewhat to include variables defining
ownership, geographic location, rock formation and specific well characteristics. They
argued that well ownership impacts costs through differing levels of efficiency; that
locationviii is important because cost factors can be specific to each region; and that
different geological formations may be more costly to produce from than others.
Chermak and Patrick also postulated that it is possible for individual wells to be quite
unique in their production costs as would be the case, for example, if reservoirs differ
significantly in characteristics. Two wells within the same reservoir could have quite
different costs if well spacing in the pool is not even, or if the pool has heterogeneous
reservoir characteristics.
The data they used in estimation represented 451 monthly time series
observations covering twenty-nine tight gas wells. They report results showing operating
cost inversely related to reserves, and directly related to quantity, with marginal cost of
production decreasing in quantity of gas extracted. The month of production variable
showed inconsistent results over the different models estimated. The ownership variable
proved statistically important and this was explained by different accounting and business
practises for the different firms.
The Chermak- Patrick approach to cost modelling is followed in this paper and
applied to a cross-section, time-series panel data set for natural gas wells in Alberta. The
panel nature of the data lends itself to econometric estimation to account for both
individual fixed well effects as well as common factors across wells impacting cost of
production
IV. An Empirical Cost Model for Alberta Gas Wells
The model developed in this paper is premised on the position that natural gas
firms attempt to minimize the cost of lifting natural gas. In the very short run or
immediate run period, the firm must operate within a fixed input setting i.e., the cost of
lifting gas depends on the economic and physical input factors in place during the period
of production. A longer run approach to cost modelling would allow for variable input
relationships. The data available for our analyses would not allow for this longer run
approach and, consequently, the focus of the study is on the immediate cost period.
The data used in measurement represents monthly observations on operating
costs, reserves, production month, depth of well, geological zone and reservoir pool for
twenty-two natural gas wells in Alberta over the period 1/96 to 12/98. Some wells report
no gas extraction in some months and these observations have been deleted from the data
set. The panel data set represents a total of 608 observations. Table 1 provides a
description of the variables showing mean and range of the data on an annual basis. There
is substantial variation in all variables defined as indicated by the range of standard
errors. Of particular note is the large variation in reserve size and depth of well. To
provide some information on the statistical relationship across variables Table 2 shows a
matrix of pair-wise correlations between variables. Note the strong positive correlation
between quantity of gas extracted and the remaining reserves reflecting the fact that
higher reserves support higher production levels. Operating cost is the sum of many
different types of expenditures related to gas operation and this aggregate sum is
described in an appendix.
The estimating equation is a regression of operating costs on variables defined in
Table 1, written in general form as,
),,.,( εdmrqfC = , (6)
where C is operating cost, q is gas extracted, r is reserves, t is a time variable, m is
monthly age of well, d is depth of well and ε is a random error term. We start the
empirical analysis by testing for functional form of the cost equation using a Box-Cox
transformation and then testing for fixed versus random-effects in the panel data set.
A non-linear Box-Cox transformation is applied to each variable used in the
regression equation, where the transformation takes the form λ
λ 1−ix (Greene, 2000). The
non-linear transformation requires a maximum likelihood estimation procedure where λ
and the vector of regression coefficients are measured based on maximising the log of the
likelihood function. The Box-Cox transformation allows testing the functional form of
the cost equation by measuring the value of the parameter, λ. If λ equals 1 the data
support a model linear in the variables and if λ equals 0 the data support a model where
the variables are linear in logarithmic form. A Wald test procedure is used in testing the
alternative null-hypotheses. The results of the testing procedure are reported in Table 3.
A null-hypothesis that λ = 0 (calculated ) can not be rejected at the 5% level
whereas a null-hypothesis that λ = 1 (calculated ) is easily rejected. (The
critical value of with one degree of freedom.) Consequently, the test results
support a cost equation where the variables are linear in logarithmic form. This result is
consistent with results reported by Patrick and Chermak.
39.02 =χ
χ 70.5632 =
84.32 =χ
iitC βα +=
The panel nature of the data set allows investigation of well-specific effects. One
procedure for measuring this effect assumes that the cost differences across wells can be
captured in differences in the constant term. The fixed effects model introduces a specific
dummy variable for each well. The fixed effects model is written as:
, ititT X ε+
where the vector X includes all variables defined in Equation 6 and iα is an intercept
term specific to each well but constant over time. The fixed effects model allows
variations in X to affect costs in the same way across wells but for each well the level of
costs can vary. The random effects model captures cost differences across wells by
assuming that constant effects for each well are randomly distributed across wells. The
general random effects model is written as:
,iititT
it XC µεβα +++=
where it is assumed that the random effects term iµ has zero mean and is not correlated
with the random error term itε . Greene (2000) describes a Hausman procedure to test the
data to find statistical support for either the fixed or random effects model. The procedure
is a test for correlation of the errors specific to a given well. In the fixed effects model,
the constant term models differences across wells whereas the random effects model
includes a well specific error term. The null hypothesis is that the random effects model
is the correct specification and is distributed as with k degrees of freedom. The
Hausman statistic generates a =629.12 and compared to a critical value of 9.49 at the
5% level we reject the random effects model and conclude that the panel data support a
log-log fixed effects cost representation.
2χ
t +ln
24χ
t +
To fix ideas, the basic cost equation will take the following form:
ttti
itt epmRqC +++= ∑=
lnlnlnln 4321
22
1ββββα . (7)
where the fixed effects are captured by itα . The model is essentially that of Patrick &
Chermak except for the fixed effects modification. Different forms of the cost equation
are estimated to capture nonlinearity in output, the importance of depth of well and
constant effects caused by ownership, geological zones and reservoir pools. The results of
the estimation are reported in Table 4. The individual fixed effects dummy variables are
not reported, however, an F-statistic is reported at the bottom of the table testing the null
that all fixed effects coefficients are equal to zero. The first column of Table 4 shows
results reported by Patrick & Chermak, Model 1 is our corresponding fixed effects
model. Model 2 adds the variable depth which is not included in Patrick & Chermak.
Model 3 allows for non-linear output in the basic cost equation. Model 4 includes a
dummy variable to account for the fact that the 22 wells in the study are divided up in
ownership between two companies. Model 5 accounts for the five different geological
zones that define the area where the wells are located.ix Finally, Model 6 includes
constant effects resulting from different reservoir pools as defined by Alberta Energy and
Utilities Board.
The estimated equations seem to provide reasonable good fit to the data with
pseudo R2 of between 0.73 and 0.80x. All coefficients are of the expected direction, costs
are increasing in both quantity produced and age of the well, and decreasing in remaining
reserves. The estimated coefficients are reasonably consistent across models. This is
especially true of the estimated coefficients on remaining reserves (ln r), with a range
-0.84 to -0.98, (all of which are statistically significant at the 95 percent level). Except for
Model 3, which includes a non-linear output variable the coefficients on quantity
produced (ln q) ranged from 0.07 to 0.10. Introducing the square of the output variable
(Model 3) changes the sign and magnitude of the linear output variable however, the R2
statistic indicates that no real advantage in explanatory power of the model is obtained. A
range of -0.03 to -0.04 is observed for the coefficients on the time trend. The time trend
indicates that operating costs have a tendency to fall. The age of the well had a
coefficient that ranged from 0.05 to 0.16. In some of the models, the coefficients on time,
well age, and quantity do not hold their statistical significance at the 95 percent
confidence level however, all, but well age in Model 3 and Model 6, remain significant at
the 90 percent confidence level. Compared to the results obtained by Chermak and
Patrick (column 1), our results imply that the wells in this sample have a larger elasticity
of cost with respect to remaining reserves, and smaller elasticity of costs with respect to
quantity produced.
Models 5, 6 and 7 account for the fixed effect of ownership, geological zone and
reservoir pool, respectively; however, these factors are measured not to have a significant
effect on the cost equation. The fit, as measured by the R2, stays in a narrow range of 0.76
to 0.78. Moreover, the coefficients on production, reserves, time, and age, are altered
little with the addition of other variables.
The depth of well is found to be statistically significant (Model 2), and suggests,
as expected, that operating costs increase with the depth of the well. The addition of
depth has little influence on the coefficient on quantity, and decreases the coefficient on
reserves slightly. Both time and age of the well lose their significance at the 95 percent
level with the addition of depth. However, the fit of the model, as measured by the R2, is
not improved.
Model 4 includes a company dummy variable that is found to be significant,
implying that one company has higher operating costs than the other. This might be
explained by differences in accounting practices. The addition of this dummy variable
has little impact on the coefficients for quantity, reserves, or well age. The coefficient on
time again loses its significance with the addition of a company variable. Again the fit is
not improved with the addition of a company dummy variable.
Dummy variables for the different zones (Model 5) did not prove to add
explanatory power of the model. Only one of the four variables is significant at the 95
percent level, and only the coefficient on reserves remained significant at this level. The
results tend to suggests that operating costs are higher for zones other than the Belly
River zone. It is possible that this captures some of the same relationship that depth of
well does. In this sample, the Belly River zone (the reference zone) is the shallowest zone
and the Bow Island zone (Zone 1) (which is the only zone dummy variable that is
significant at the 95 percent level) is the deepest.
Dummy variables for the individual reservoirs were all significant at the 95
percent level (Model 6). This implies that different reservoir characteristics are important
in determining operating costs at the well level and that some reservoirs may have
characteristics that are more or less amiable to production. The addition of these dummy
variables lowers the statistical confidence on the variables quantity, time, depth and gas
extracted below the 95 percent level.
The fixed effects regression includes an F test with the null hypothesis that the
fixed effects are all equal to zero. The critical value for this test at the 99 percent level is
roughly 1.9. All of the tests are rejected soundly at the 99 percent level, with test values
ranging from 60.8 to 85.2. This implies that there exists some residual unexplained
variance in well operating costs that arises because of individual well differences.
V. Conclusions The primary focus of this work was an empirical estimate of operating cost for a
natural gas well. The results suggest that operating costs are increasing in quantity
produced, decreasing in the remaining reserves, and increasing with the age of the well.
These results confirm both what depletable natural resource theory has generally assumed
about operating costs, as well as the results obtained by Chermak and Patrick.
The results further found support for the notion that operating cost functions for
natural gas should be modeled at the individual well level. Using a panel data series, the
individual well effects were found to be non-zero across all models tested. This suggests
that ignoring the impacts at the individual well by aggregating up to the pool or the
region may not model costs effectively. Important differences in reservoir characteristics
and costs make aggregation difficult.
References
Berger, Bill D. and Kenneth E. Anderson, Modern Petroleum: A Basic Primer of the
Industry, (Tulsa: PennWell Books, 1992).
Brennan, Michael J., and Eduardo S. Schwartz, "Evaluating Natural Resource
Investments", Journal of Business, Volume 58, Number 2 (1985): 135-157.
Chermak, J.M., and R.H. Patrick, "A Well-Based Cost Function and the Economics of
Exhaustible Resources: The Case of Natural Gas", Journal of Environmental
Economics and Management, Volume 28, (1995): 174-189.
Chermak, J. M., J. Crafton, S.M. Norquist, and R.H. Patrick, "A hybrid economic-
engineering model for natural gas production", Energy Economics 21, (1999):
67-94
Cullen, Susan. Natural Gas From Wellhead to Burnertip (Calgary: Canadian Energy
Research Institute, 1993)
Greene, William H. Econometric Analysis: Forth Edition, (Prentice Hall: New Jersey,
2000).
Griffin, James M., and Clifton T. Jones., 'Economies of Scale in a Multiplant
Technology: Evidence from the Oilpatch,' Economic Inquiry, Volume 26, (1998):
107-122.
Helliwell, J.F., M.E. MacGregor, R.N. McRae, and A. Plourde, Oil and Gas in Canada:
The Effects of Domestic Policies and World Events, Canadian Tax Paper Number
83 (Canadian Tax Foundation, 1989).
Hobson, G.D., and E.N. Tiratsoo, Introduction to Petroleum Geology: Second Edition,
(Beaconsfield: Scientific Press, 1981).
Krautkraemer, Jeffery A., 'Nonrenewable Resource Scarcity', Journal of Economic
Literature 36, (December 1998): 2065-2107.
Kuller, Robert G. and Ronald G. Cummings, 'An Economic Model of Production and
Investment for Petroleum Reservoirs', The American Economic Review 64,
(March 1974): 66-79.
Livernois, John R. 'Empirical Evidence on the Characteristics of Extractive
Technologies: The Case of Oil', Journal of Environmental Economics and
Management 14, (1987): 72-86.
Livernois, John R., and Russell S. Uhler 'Extraction Costs and the Economics of
Nonrenewable Resources', Journal of Political Economy 95, (1987): 195-202.
Nind, T.E.W. Principles of Oil Well Production: Second Edition, (New York:
McGraw-Hill, 1981).
Sweeney, James L., 'Economic Theory of Depletable Resources: An Introduction,' in
Handbook of Natural Resource and Energy Economics, vol.III, Allen V. Knese
and James L. Sweeney, eds., (Amsterdam: Elsevier Science Publishers, 1993):
759-854.
Appendix
In this appendix, a description of operating cost is reported. Operating cost for
the well is made up of numerous categories within the company's records. Some
categories report expenditures sporadically throughout the year whereas others are
incurred annually or monthly. In Table A1, we present for descriptive purposes a
breakdown of expenditure by variable and quasi-fixed groupings. A variable expenditure
is defined as changing in response to the level of gas production and a quasi-fixed
expenditure is necessary for production to take place but independent of the level of
production. In building up a monthly aggregate expenditure index, the following
procedure is used. Monthly expenditure is reported directly in the index. For expenditure
that occurs sporadically throughout the year, the procedure is to apply the expenditure
equally to all future months until the expenditure occurs again, and the process repeated.
For annual expenditure, the total value is applied in each month. Of course, the actual
monthly expenditure index is not meaningful but nevertheless, the variation in the index
will capture fully expenditure changes that occur monthly and throughout the year.
Endnotes
i This is known in the literature as a ‘stock effect’ or a ‘degradation effect’. ii Aggregation is a problem because reservoirs are not homogenous. Different reservoirs have differing levels of porosity, permeability, pressure, and viscosity, among other factors. Even within the same reservoir differences may exist. Studies such as Livernois (1985) and Livernois and Uhler (1987) have shown that models that are built around individual reservoirs rather then regions are preferred. Chermak et al (1999) outlined four conditions for aggregatability from well to reservoir level (i.e. conditions under which a reservoir could be treated as if it operated like a single well). These are: homogeneity of a reservoir, identical production paths of the wells, identical time horizons of all wells, and identical production technology. iii Knowing the production function would make it possible to investigate characteristics such as economies of scale in gas pools. This may be useful in determining whether reservoirs should be operated in a unitized fashion, with one firm making production decisions for the entire reservoir. Griffin and Jones (1988) undertook such a study at the lease level and found evidence for unitization. iv Methane typically makes up 84 to 96 percent of the natural gas in a deposit, Hobson and Tiratsoo, 1981. More complex molecules are less common, though ‘wet’ gas reservoirs include heavier liquid hydrocarbons. Other gases also can be found in natural gas including carbon dioxide, hydrogen sulfide and helium.
v Production decline in petroleum reservoirs typically exhibits hyperbolic decline. In this case, using Nind (1981), the relationship between the rate of change in current production and time is:
)(
)()(btatq
ttq
+−
=∂
∂
where: q (t) is production at time t, a is the hyperbolic constant, b is the natural log of initial production (ln q0). Integrating to get the production at any time t as:
at
eqtq−
= 0)( if b = 0 (in this case a is the constant production decline rate), and
b
btaaqtq
1
0)(
+= if b ≠ 0
vi Silica sand, glass beads, and epoxy are often used as propping agents.
vii Well spacing requirements set the minimum distance between wells, and between a well and the lease boundary or property line. This promotes an optimal rate of production and limits damage from overproduction. In Southern Alberta well spacing requirements are such that there can be only one well in a legal subdivision (sixteen hectares), unless it can be shown that more are needed to drain the reservoir. This is in contrast to the 258 hectares, or one section, that is the norm in the rest of the province for well spacing. Cullen (1993) argues that this is because pools are generally "tighter", or less permeable in Southern Alberta. More wells are needed to drain reservoirs in a way that allows producers to generate sufficient revenues to make the ventures worthwhile. viii The wells are in three different regions (West Texas, East Texas, and Wyoming), ix The five different geological zones include Belly River (the reference case), Bow Island (zone 1), Medicine Hat/Milk River (zone 2), Medicine Hat (zone 3), and Basal Colorado (zone 4). x The fixed effects model does not provide a true R2 and is not bounded by 0 and 1.
Table 1 Annual Summary Statistics
Variable Name
Variable Symbol Mean Standard Deviation
Operating Costs, dollars
C 7465.5 9912.7
Gas Extracted, thousand cubic
meters
q 1687.3 2453.5
Reserves, thousand cubic meters
R 11967.6 20084.8
Well Age, months
m 86.0 111.0
Depth, feet
d 795.7 267.8
Table 2 Correlation Matrix: Annual Data
C Q r m d C
1
q
0.668 1
r
0.527 0.844 1
m
0.211 -0.078 0.099 1
d
0.508 0.319 0.381 0.477 1
Table 3
Test for Functional Form: Box-Cox Transform Number of Observations
608
λ
0.02
Log Likelihood at 0
-5560.86
Log Likelihood at 1
-6114.47
Maximized Log Likelihood
-5550.47
Wald Test λ =0a
0.39
Wald Test λ=1a
563.70
a The critical value for at the 5% level with one degree of freedom is 3.84 2χ
Table 4
Fixed Effects Cost Equation: various specifications Variable
Patrick & Chermak
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
ln q
0.48 (14.83)a)
0.09 (2.32)
0.09 (2.31)
-0.44 (-2.66)
0.10 (2.54)
0.07 (1.74)
0.07 (1.67)
ln r
-0.35 (-12.45)
-0.90 (-8.86)
-0.96 (-9.67)
-0.84 (-8.16)
-0.93 (-9.33)
-0.91 (-9.25)
-0.98 (-9.81)
l n t
-0.50 (-6.83)
-0.04 (-2.31)
-0.03 (-1.71)
-0.04 (-2.21)
-0.03 (-1.78)
-0.03 (-1.76)
-0.03 (-1.88)
ln m
0.13 (6.78)
0.11 (2.67)
0.06 (1.40)
0.16 (3.65)
0.09 (2.09)
0.07 (1.65)
0.05 (1.24)
ln d
- - 0.34 (5.87)
- - - -
ln q2
- - - 0.06
(3.30) - - -
Company
- - - - 0.18 (4.16)
- -
Zone 1
- - - - - 0.35 (5.68)
-
Zone 2
- - - - - 0.08 (1.39)
-
Zone 3
- - - - - 0.45 (0.55)
-
Zone 4
- - - - - 0.09 (1.36)
-
Pool 1
- - - - - - 0.61 (5.51)
Pool 2
- - - - - - 0.34 (2.97)
Pool 3
- - - - - - 0.25 (2.31)
Pool 4
- - - - - - 0.32 (3.44)
Pool 5
-
-
-
-
-
-
0.30 (2.55)
Adjusted R2 0.42 0.79 0.76 0.76 0.76 0.77 0.78 Fixed Effectsb) - 85.24 69.21 75.56 60.80 78.08 78.42 Model 1: Fixed Effects Log-Log equation comparable to Patrick & Chermak. Model 2: Fixed Effects plus depth. Model 3: Fixed Effects plus output squared. Model 4: Fixed Effects plus company dummy. Model 5: Fixed Effects plus geological zone. Model 6: Fixed Effects plus reservoir pool.
a) t-statistics in parentheses b) F-test that all Fixed Effects Dummy variables equal zero.
Table A1 Summary of Expenditure Categories
Variable Operating Costs Share of Costs
Quasi-fixed Operating Costs
Share of Costs
contract operator 19.6% lease and road* 2.8% Chemicals* 2.5% municipal property
tax* 6.7%
repair and maintenance* 1.9% freehold surface lease* 14.3% salt water disposal* 0.4% freehold lease* 0.6% trucking* 1.0% testing-pressure
survey* 2%
small tools and supplies* 0.3% analysis* 0.2% chart reading 0.6% production and Alberta mineral tax*
42.7%
AEUB administration fee* 3.2% * indicates expenditure that occurs annually or sporadically throughout the year