Energies 2016, 9, x doi: FOR PEER REVIEW www.mdpi.comjournal energies [612457]

Energies 2016, 9, x; doi: FOR PEER REVIEW www.mdpi.com/journal/ energies
Article 1
Characteristic Production Decline Patterns for Shale 2
Gas Wells in Barnett 3
Keqiang Guo 1, Baosheng Zhang 1,*, Henrik Wachtmeister 2, Kjell Aleklett 2 and Mikael Hö ö k 2 4
1 School of Business Administration, China University of Petroleum (Beijing); Fuxue Road 18, Changping, 5
Beijing 102249 , China; [anonimizat] (K.G.) 6
2 Department of Earth Sciences, Uppsala University; Villavägen 16, Uppsala 752 36 , Sweden; 7
[anonimizat] (H.W.); [anonimizat] (K.A.); [anonimizat] (M.H.) 8
* Correspondence: [anonimizat] ; Tel.: + 86-010-8973 3792 9
Academic Editor: name 10
Received: date; Accepted: date; Published: date 11
Abstract: This paper de rives the characteristics of the decline rates for shale gas production by 12
analyzing historical production data using decline curve analysis of 14 453 shale gas wells in the 13
Barnett shale play. The Hyperbolic model and the Stretched Exponential model are a pplied on the 14
well -by-well production data at aggregate -well and individual -well levels to derive the 15
characteristic parameters. Both the Hyperbolic curve and the Stretched Exponential curve display a 16
good fit to the data for both the aggregate and the ind ividual shale gas wells. The Hyperbolic model 17
performs slightly better than the Stretched Exponential model in this study. The first year rate of 18
decline for production of a shale gas well is around 60% and over the first two years is around 73%. 19
There is an increasing trend in initial production for new wells over the last decade . A supposed cut 20
off production rate of 133 mille cubic feet per day result in the estimated URR of about 1.4 -2.7 billion 21
cubic feet and a well life time of 10.7 -28.9 years , which is in line with other studies . 22
Keywords: Shale gas well -production ; decline curve analysis ; initial production ; URR ; Barnett 23
PACS: J0101 24
25
1. Introduction 26
Conventional petroleum supply is facing increasing challenge s in meeting demand. Recent oil 27
and gas production has become increasingly reliant upon unconventional resources, such as 28
hydrocarbons extracted from tight deposits, that cannot be produced at economic flow rates nor can 29
they recover economic volumes unless a special technique is used to stimulate production. Such tight 30
formations include tight sands, coal bed methane and gas -bearing shales. Shale gas accounts for the 31
bulk of the recent increase in unconventional gas production and has become the focus of intense 32
interest from business, p olicy makers and the wider public [1]. 33
Referencing the source shale formations, the surging hydrocarbon output witnessed over the 34
last decade in the U.S. is commonly referred to as the “shale gas boom” or “shale revolution” [2]. 35
Other countries have expres sed hopes for developing such domestic unconventional gas resources, 36
particularly shale gas, and are in various stages of planning and evaluation to lay groundwork for 37
potentially larger commercial undertakings in the future. This includes countries such a s China [3], 38
Poland [4], Mexico [5], India [6] and Australia [7]. However, future development of unconventional 39
gas resources still remains unclear as technological, economic and environmental factors remain 40
uncertain [8]. The U.S. experience is the only s ource of empirical data for shale gas production 41
behavior and it serves as a suitable foundation for any attempt to describe or predict future shale gas 42
developments. 43

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1.1. Shale gas regions 44
Methodologies for analyzing production patterns have been around f or many decades, and one 45
of the most frequently used approaches is the decline curve analysis [9]. Shale gas production is a 46
fairly new activity and as a result the availability of long -term production data is still rather limited. 47
To alleviate this, close r examination of the most prominent U.S. shale gas areas is needed. According 48
to the Annual Energy Outlook 2015 [10], the production of total dry natural gas in the U.S. rose by 49
35% from 2005 to 2013, with the natural gas share of the total U.S. energy con sumption increasing 50
from 23% to 28%. Figure 1 shows the historic and expected production of natural gas by different 51
sources in the U.S. In 2040, the U.S. natural gas production from shale is predicted to account for 52
about half of total domestic production . 53
54
Figure 1. US natural gas production by sources, 1990 -2040. Data Source: EIA 2014 [11]. 55
An EIA report [12] assessed 137 shale formations in 41 countries and claimed that shale 56
formations are much more extensive than traditional gas reservoirs and are present on every 57
continent (Figure 2). In this map, the EIA have estimated shale resources fo r the red areas; the yellow 58
areas are included in the study but resource estimates are not available; the white areas are not 59
assessed in the report. Thus, there are large areas omitted in the estimate figures, which means that 60
there may be more shale form ations globally. 61
62
Figure 2. Global shale basins. Source: EIA 2013 [12]. 63

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Technically recoverable shale gas resources were estimated at 7 201 trillion cubic feet (Tcf) for 64
the entire world embracing 32% of all global natural gas resources [12]. The U .S. and Canada are the 65
only major producers of commercially viable natural gas from shale formations to date. However, 66
about a dozen of other countries have sunk exploratory test wells. Chin a is the only nation outside of 67
North America that has registered commerc ially viable production of shale gas, although the 68
volumes contribute less than 1% of the total nation al natural gas production [13]. 69
Shale resources and production are found in many regions of the U.S., but only seven areas are 70
prolific. These are Marcell us, Haynesville, Eagle Ford, Fayetteville, Barnett, Woodford, and Bakken. 71
Figure 3 shows the range of shale plays in the U.S. lower 48 states by 2015 , and Figure 4 illustrates 72
the growth in natural gas output from different shale plays since 2000. The Barnett shale located in 73
the Fort Worth Basin (marked in Figure 3) is one of the largest onshore natural gas fields in the United 74
States. It consists of sedimentary rocks and the productive part is estimated to cover about 13 000 km² 75
near the city of Dalla s and at least 18 counties. 76
77
Figure 3. Map of shale plays in the U.S. lower 48 states. Source: EIA 2015 [14]. 78
79
Figure 4. U.S. dry shale gas production by regions. Data Source: EIA 2014 [11]. 80
1.2. Aim of this study 81
Discovered in the 1950s, the Barnett formation was not commercially viable until the 1980s, and 82
significant drilling activity did not begin until gas prices increased in the late 1990s [15 -16]. The 83

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Barnett shale play is sufficiently representative and of considerable analytic value because of its long 84
production history and important position in the history of US shale developments (see green part in 85
Figure 4). This study will focus on deriving representative production decline curves & analyzing 86
their attendant decline rates, the initial prod uction and the estimated ultimate recoverable resource 87
(URR) for shale gas production by observing and modelling historical producing behavior of a 88
number of shale gas wells in the Barnett play. Decline curve analysis models and statistical analysis 89
will b e applied on the well -by-well production data both on aggregate well and individual well levels 90
to find the distinctive parameters for shale gas production decline curves. Therefore, through the 91
study of the shale gas producing behavior in Barnett increase d understanding of issues surrounding 92
production in shale formations can be acquired. 93
2. Methodology and Data 94
2.1. Decline curve models 95
Decline curve analysis is a method to analyze production rates of individual wells to predict the 96
performance of future production by extrapolating a suitable decline function [17]. Many of the 97
existing decline curve models are heuristic and based on a framework derived by Arps [18], who 98
proposed tha t the curvature in the production -rate-versus -time curve can be expressed 99
mathematically by the hyperbolic family of equations. In produ ction decline curve models, the 100
decline rate ( 𝜆) can be expressed by using derivatives with production rate (𝑞) and ti me ( 𝑡) in arbitrary 101
units [19], as shown in Eq. 1: 102
/dq dtCqq 
. (1)
Where C is a constant and β is the decline exponent constant. Three general cases exist : when β = 0, 103
the decline is exponential (Eq. 2); When β = 1, the decline is harmonic (Eq. 3); When 0 < β < 1, the 104
decline is hyperbolic (Eq. 4). 105
0()
0 ()ttq t q e
, (2)
1
00 ( ) 1 ( )q t q t t   
, (3)
 1/
00 ( ) 1 ( )q t q t t  
. (4)
Where 𝑞(𝑡) is the production rate, and 𝑞0 is an initial production rate at time 𝑡0 from which 106
production begins to decline ( i.e. its peak rate of production). Originally, the Arps curves were a set 107
of mathematical equations with no physical basis other than that the equation gave a declining trend 108
that provided a good fit with empirical data. However, a connection to physics has been proved for 109
an exponential decline curve, which represents the solution to the flow equation at constant pressure 110
[20]. This methodology has been widely used for forecasting production or es timating reserves in 111
conventional oil and gas formations. 112
Due to particular reservoir conditions and seepage characteristics, production decline curves for 113
shale gas wells differ from those of conventional gas reservoirs. It has been claimed that the Arps 114
decline curves encounter problems when used in “unconventi onal” formations, particularly with 115
regards to overestimation of reserves. When traditional decline curve models are used on shale 116
formations, Arps’ values for larger than 1 are commonly obtained yielding infinite cumulative 117
production [21]. As the importa nce of shale gas has increased over the last decade new methods have 118
been proposed to model the behavior exhibited by long horizontal wells with multistage hydraulic 119
fractures in shale reservoirs. These methods include, but are not limited to, the Stretche d Exponential 120
(SE) model [22], the Power Law model [21], and Duong’s model [23]. 121
Two decline curve analysis models are used in the study – the Arps Hyperbolic model and the 122
Stretched Exponential model. The advantages of these decline curves lie in the stro ng empirical 123

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compliance they display and their ease of use. The down side of these analytical models is that they 124
do not provide estimates of reserves or impart insight to reservoir characteristics. Their key properties 125
are described in Table 1. 126
Table 1. Key properties of the Hyperbolic and the Stretched Exponential models. Adapted from Satter 127
et al. [19], Valko [22], Höök et al. [24], Kanfar [25] and Lei et al. [26]. 128

()qt
()Qt
URR
Hyperbolic
 1/
001 ( )q t t
11
0
00 1 1 ( )(1 )qQ t t     
 00 / (1 ) Qq
Stretched
Exponential
 expn
iiq D t
011,n
iq tQn n n
                 
01iqQnn
In Table 1, 𝑄(𝑡) is the cumulative production in the decline phase, 𝑄0 is the initial cumulative 129
production, and 𝑈𝑅𝑅 is the estimated ultimate recoverable resource which is the sum of 𝑄0 and 130
𝑄(𝑡) when 𝑡→ 𝑡𝑐𝑜𝑟 (Figure 5 ). The cut -off rate ( 𝑞𝑐𝑜𝑟) is the technic al or economic limits of production, 131
after which the well should be abandoned. For the Hyperbolic, modification of the parameter β can 132
alter the shape of the production rate curve and be used to determine what kind of decline curve is 133
suitable for fitting to empirical data; the value of the decline parameter 𝜆 governs how steep the 134
decrease will be. In the SE method, 𝑞𝑖, 𝐷𝑖 and 𝑛 are undetermined parameters; a large 𝑞𝑖 value 135
compensates for a small 𝑛 value; 𝜏 is equivalent to (𝑛/𝐷𝑖)1/𝑛 [22]. 136
137
Figure 5. The conceptual production curve. 138
2.2. Data handling 139
The study is based on monthly shale gas production data from the DrillingInfo database that 140
includes 14 453 shale gas wells from the Barnett shale play that commenced production in January 141
2000 with data culminating in October 2014. Only horizontal wells with separately reported 142
production are included. Figure 6 shows the number of included wells by vintage. For production, 143
the unit is thousand cubic feet (Mcf) as used in U.S. reported statistics . For conversion, 1 Mcf equals 144
28.32 cubic meters of natural gas. 145
The data set covers a maximum number of well production months of 174 (from April 2000 to 146
October 2014). However, the decline phase does not typically cover the complete production time 147
period. On average, for the shale gas wells investigated in this study, the initial cumulative 148

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production ( 𝑄0) accounts for about 7% of total well production, which means the decline curves 149
capture most of the production. The production of a well usually peak s shortly after being brought 150
on-stream and the decline curve describes the decline phase after the peak. Thus, the highest 151
production before the onset of decline is denoted “Initial Production (IP)” by convention, as 152
illustrated in Figure 5. In addition, months with zero production were removed before analysis, to 153
remove external events that affected production such as annual maintenance and scheduled 154
downtime. All of the other wells production data were completed by using this approach. 155
156
Figure 6. The original number of wells (14 453 in total). 157
The characteristic decline curves of shale gas wells were identified using several approaches. 158
The first one is the normalized value approach that allows a broad set of wells to be directly compared. 159
The producti on data are normalized in the sense that the initial production (IP in Figure 5) is set to 1 160
and the following monthly production figures are displayed as a fraction of this. Only the normalized 161
decline curves can be obtained in this way, while actual prod uction levels are lost through the 162
normalization procedure. The actual value approach was also used to acquire production parameters 163
such as the initial production and the estimated URR. 164
Characteristic decline curves can also be studied at either the aggre gate or individual well level. 165
The “aggregate decline curve” is fitted to the average monthly production of several wells, smoothing 166
fluctuations and representing an average behavior for a collection of wells. A different number of 167
wells are used for diffe rent months due to the limitation of the production data series. In contrast, the 168
individual well decline curves are derived from analysis of the production of each individual shale 169
gas well. 170
The number of observed data series should be long enough to avoi d over -fitting. According to 171
the characteristics of the shale industry and the data set, the period of two years (24 months) was set 172
as the minimum number of production months. Therefore, only wells with production data 173
exceeding 24 months are included dur ing curve fitting. 174
2.3. Goodness of fit 175
Computer software, such as Matlab, was used in the numerical analysis of the data set. The 176
coefficient of determination (R2) and the normalized root mean square error (N -RMSE) are used as a 177
measure for the goodness o f fit to assess how well the models described the data in the curve fitting 178
process. 179
For R2, it ranges from 0 to 1, and the goodness of fit improves as R2 moves towards 1; for N – 180
RMSE there is also ranges from 0 to 1, but the goodness of fit weakens as it m oves towards 1. The 181
boundaries that are introduced to exclude the poorest fits are R2≥0.8 and N -RMSE≤0.2 concurrently. 182
These limits may not be optimal or result exceptionally good fits, but the measure is primarily used 183
to simplistically exclude appalling fits. 184

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3. Results 185
3.1. Decline curves 186
3.1.1. Aggregate well decline curve 187
About 84% (12 211 wells) of the original 14 453 wells have production data exceeding 24 months 188
and these were normalized well by well. The aggregate normalized production of the first 4 years (48 189
months) and the number of wells used for each month are displayed in Figure 7. The reason that only 190
48 months production are averaged is that the tail of the production curve holds more uncertainty as 191
it is based on a declining number of wells after the 48 month point. The curves are very similar and 192
hard to distinguish on a regular scale, for this reason the curves are presented on a logarithmic scale, 193
as shown in Figure 8. There is almost no difference between the models for <20 months. 194
195
Figure 7. The aggregate decline curve. Month is on the x -axis, well number used for different months 196
on the left y -axis and normalized production on the right y -axis. 197
198
Figure 8. The hyperbolic and the stretched exponential decline curves fitted to average normalized 199
production data on logarithmic scale. 200
The decline rate slows down with increasing time. The annual decline over the first year of 201
production is 62.18% and over the second year 29.55%. Both curves are best fits with the goodness of 202
fit in terms o f R2 and N -RMSE. For the Hyperbolic model, the R2 value is 0.9981, and the N -RMSE 203
value is 0.0073; for and the Stretched Exponential model, the R2 value is 0.9996, and the N -RMSE 204
value is 0.0034. The parameter values of the curves are also shown in the box in Figure 8. 205
3.1.2. Aggregate well decline curve 206

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Individual curve fitting using Hyperbolic and Stretch ed Exponential models was also performed 207
on the 12 211 wells with data exceeding 24 months. After excluding poor fits based on the R2 and N – 208
RMSE limits, a usable number of 8 547 wells remained. This is about 70% of 12 211 wells or 59% of 209
the original 14 453 wells. 210
1. Hyperbolic decline curve 211
The distributions for the best fitted λ and β parameter values of the hyperbolic fits are displayed 212
in Figure 9 (a) and (b) respectively with key descriptive statistics attached in the textboxes. Some of 213
the λ values were extremely large (>4 000), causing issues for visualization in histograms. Therefore, 214
16 wells with λ values larger than 4 are excluded in Figure 9 (a). 215
In Figure 9 (b) , the largest value for β is 8.3; only about 16% (1 362 of 8 547) of the β values are 216
less than one; and values between 0.5 and 2.5 account for about 88% (7 556 of 8 547). Probability 217
distributions of both parameters can be described by the Generalized Logistic distributions according 218
to Chi -Squared tests (Purple line, see Appendix A. 1 for details). The Normal distribution (Red line, 219
Appendix A. 2) are also displayed as comparison. 220

(a)
(b)
Figure 9. (a) Distributions of λ values. λ values are on the x -axis and the probability on the y -axis; (b) 221
Distributions of β values. β values are on the x -axis and the probability on the y -axis. 222
2. Stretched Exponential decline curve 223
Some key descriptive statistics of the best fitted parameters for the Stretched Exponential decline 224
curves are shown in Table 2. 225
Table 2. Descriptive statistics of the SE curves. 226
qi Di n
Mean 186.8 2.3 0.3
Std. Deviation 625.2 2.3 0.3
Min 0.7 3.47E -7 0.04
25%(Q1) 1.8 0.6 0.1
50%(Median) 3.6 1.3 0.2
75%(Q3) 28.8 3.4 0.4
Max 7999.9 9.2 9.5
3.1.3. Comparison of decline curves 227
Based on the studied historical production profiles, the characteristic decline curves for shale 228
gas production are derived by using the Hyperbolic model and the Stretched Exponential model. 229
Both the Hyperbolic model and the Stretched Exponential model fit well to the aggregate and the 230
individual shale gas wells. Comparison by means of goodness of fit indicate that Hyperbolic curves 231

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are better fit than the Stretched Exponenti al for about 51% (4 366 of 8 547) of the wells according to 232
the R2 and for about 61% (5 198 of 8 547) of the wells according to the N -RMSE. 233
Figure 1 0 provides a summary of the different derived characteristic decline curves investigated 234
in this study, incl uding curves resulting from the median and mean of the estimated parameters for 235
the Hyperbolic model and the median of the estimated parameters for the Stretched Exponential 236
model. The three estimated parameters of the Stretched Exponential model are inter connected and 237
by taking the mean, the connection between the parameters will be lost. In Figure 1 0, the aggregate 238
decline curves (see Figure 8) are also displayed. The aggregate curves (Purple solid line and Purple 239
dotted line) for two models are declining more gradually than their corresponding mean, median or 240
mean lines. The Stretched Exponential median curve (Green dotted line) are declining steepest over 241
a medium to a long -term horizon while the hyperbolic median and mean curves (Green solid line 242
and Re d solid line) decline rapidly in the initial phase after which they flatten out. 243
244
Figure 10 . Summary of different derived typical decline curves (on logarithmic scale). 245
3.2. Decline rates and initial production 246
The average decline rates in different time phases and the initial production, i.e. peak production, 247
are also statistically analyzed based on the 12 211 wells. 248
3.2.1. Comparison of decline curves 249
The distributions for the first year (12 months after IP) decline rates and the decline rates over 250
the first two years (24 months after IP) of every studied shale gas wells are displayed in Figure 1 1 (a) 251
and (b) respectively. 252

(a)
(b)
Figure 11. (a) Distributions for the first year decline rates ; (b) Distribution s for the first two years 253
decline rate s. Rates are on the x -axis and the probability is on the y -axis. 254

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The average value of the first year decline rates for all wells started in 2000 -2012 is 62.2% and the 255
first two years decline rates is 73.4% (Table 3). The Generalized Logistic probability densi ty function 256
and the Normal distribution are listed in Appendix (A. 1 and A. 2). 257
Table 3. Average value of the first year and the first two years decline rates for new wells over years. 258
Year 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Total
First year 56.3% 54.0% 60.5% 58.1% 59.5% 61.9% 64.5% 64.2% 62.0% 63.3% 62.4% 61.3% 57.6% 62.2%
First two years 75.2% 66.9% 68.8% 69.3% 70.3% 73.1% 76.1% 73.9% 73.6% 75.2% 73.9% 72.8% 69.1% 73.4%
3.2.2. Initial production 259
The IP distribution from all wells that commenced production in 2000 -2012 are displayed in 260
Figure 12(a) . About 77% (9 367 of 12 211) of wells reach a peak monthly -production in the interval 261
from 20 000 to 100 000 Mcf. Figure 12(b) shows average IP for wells by age. There is a clear trend 262
towards higher IP levels. The Log -Pearson 3 probability density function and the Normal d istribution 263
are listed in Appendix (A. 3 and A. 2). 264

(a)
(b)
Figure 12. (a) Distributions for initial production, production are on the x -axis and the probability is 265
on the y -axis, monthly -production unit is Mcf ; (b) Average monthly IP from 2000 to 2012, years are on 266
the x -axis and production is on the y -axis, monthly -production unit is Mcf. 267
3.3. Estimated URR 268
For the studied shale gas wells, the mean cumulative production before the peak (IP) is 60.5 Mcf, 269
which is regarded as 𝑄0 in the 𝑄(𝑡) functions shown in Table 1. This is just a few percent (<7%) of 270
the ultimate production and rather insignificant compared to the production output in the decline 271
phase. The URR will be chiefly dependent upon the length of the de cline phase and the IP. 272
The IP mean value of 1 896 Mcf/d (Figure 1 2 (a) ) and characteristic decline curves from Figure 10 273
can be used, with assumptions on well life span, to estimate cumulative production. Results for 274
assumed life times of 10 -year, 20 -year , 30-year and 40 -year are displayed in Table 4. Due to the high 275
decline rates, production levels would be very low 10 years after the peak and could be below the 276
economic limit. Hence, assumptions of very long life spans (>20 years) for shale wells must be 277
properly grounded. 278
If the cut -off rate of production ( 𝑞𝑐𝑜𝑟, technical and economic limits, as shown in Figure 5) for a 279
shale gas well can be defined, the expected 𝑄(𝑡) at the time of the cut -off rate ( 𝑡𝑐𝑜𝑟) can be used to 280
estimate URR. According to Kaiser [27], who studied the economic limits of oil/gas field production 281
in Texas between 1993 and 2008, the last annual average production in all land wells was 215 Mcf/d. 282
However, condensate was included in the gas stream in that study. Correcting for this, the cut -off 283
rate for natural gas production will likely be less. Browning [28] found that the economic limit for 284
closing a well is 50 Mcf/d for dry gas in Barnett. Based on the average of those studies, a cut -off rate 285
at 133 Mcf/d was assumed, and y ields estimated URR and well life time as shown in Table 5. Results 286
for 215 and 50 Mcf/d are also shown in Table 5 as reference. 287

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Table 4. The expected cumulative production per well. Unit for q(t) is Mcf/d and for Q(t) is billion 288
cubic feet (Bcf). 289
Hyperbolic Stretched Exponential
Median Mean Aggregate Median Aggregate Median Mean Aggregate Median Aggregate
q(t) Q(t) q(t) Q(t) q(t) Q(t) q(t) Q(t) q(t) Q(t)
10-year 166 1.4 155 1.3 227 1.6 140 1.4 196 1.5
20-year 106 1.9 101 1.7 160 2.2 70 1.7 120 2.1
30-year 82 2.2 78 2.1 131 2.8 45 1.9 88 2.4
40-year 68 2.5 65 2.3 113 3.2 32 2.1 70 2.7
Table 5. URR and well life time per well using a cut -off rate. Unit for URR is Bcf, and for well life time 290
is year. 291
Hyperbolic Stretched Exponential
Median Mean Aggregate Median Aggregate
215
Mcf/day URR 1.2 1.0 1.7 1.1 1.4
Well life time 6.7 5.9 11.1 6.0 8.7
133
Mcf/day URR 1.6 1.4 2.7 1.4 1.9
Well life time 14.1 12.8 28.9 10.7 17.4
50
Mcf/day URR 3.0 2.7 >4.3 1.9 3.1
Well life time 64.0 60.8 >70.0 27.3 59.6
4. Discussions 292
4.1 Discussion of data 293
As one of the oldest shale plays in commercial exploitation, the Barnett shale has a relatively 294
long time series as well as abundant well and production data. This study indicates that most of the 295
shale gas wells in Barnett have a typical production pattern of peaking in a short time frame and 296
declining steeply after the peak is reached, while only few wells don’t show this characteristic 297
performance. 298
Twice “bad” wells were removed for curve fitting according to the l imits of data series length 299
and goodness of fit. It is assumed in this study that only wells with production exceeding 24 months 300
can show an unbroken production pattern (the first wells eliminated) and that only wells with good 301
fit in line with R2≥0.8 and N-RMSE≤0.2 can on behalf of a characteristic production pattern (the second 302
set of wells removed). Table 6 shows the change of the count and share of wells starting from different 303
years during analysis. 304
Table 6. The number of yearly new wells used in study and the share of original data. 305
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Total
Original 8 21 52 208 392 762 1257 2155 3672 626 1638 1415 1120 839 288 14453
Length 7 21 50 195 369 711 1158 2013 3393 600 1546 1308 840 0 0 12211
Share (1) 88% 100% 96% 94% 94% 93% 92% 93% 92% 96% 94% 92% 75% 0% 0% 84%
Good fit 5 14 30 128 272 531 836 1421 2354 413 1090 854 599 0 0 8547
Share (2) 63% 67% 58% 62% 69% 70% 67% 66% 64% 66% 67% 60% 53% 0% 0% 59%

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As shown in Table 6, about 59% of the original wells data are used to fit typical decline curves 306
for shale gas production. Each year, from 2000 to 2012, display a similar percentage with the largest 307
standing at 70% in 2005, and the lowest, 53%, in 2012. Wells starting in December 2012, 2013 and 2014 308
are excluded, because those wells will not have produce for 24 months by October 2014 – the end date 309
of the data set. 310
4.2 Discussion of decline rates 311
The decline rate of production is around 60% for the first year and after two years only 27% of 312
the initial production level remains due to production decline (Figure 1 1 (a), (b) and Table 3). These 313
high decline rates are important, as they necessitate a significant number of new wells being drilled 314
annually just to offset the decline in exist ing production. The results also verify the high decline rates 315
for shale wells found by other researchers. In comparison, they are significantly higher than the 316
average decline rates seen for conventional petroleum. 317
4.3 Discussion of initial production 318
As shown in Figure 12 (a) and (b), most of shale gas wells in Barnett have an average monthly 319
production ranging from 20 000 to 80 000 Mcf when they peak, which is about 658~2 630 Mcf/d. 320
Newer wells tend to have higher IP than older well s. This could be expla ined by technological 321
developments such as increased horizontal lengths of wells and increased number of fracturing 322
stages. 323
4.4 Discussion of URR 324
Table 7 present a summary of the estimated URR from other studies of the Barnett shale play for 325
comparison. Th e results from this study are in line with the EIA and other agencies. Among them, 326
the NETL/DOE study seems to be the most optimistic, and the USGS more pessimistic. 327
Table 7. Summary of average estimated URR per well in Barnett of other studies. 328
Reference Mean URR/Well (Bcf) Method of Estimation Publication time
U.S. EIA [29] 1.4 Decline curve analysis 2011
E. Berman [30] 1.3 Two -stage exponential decline based
on decline curve analysis 2011
USGS [31] 1.0 Decline curve analysis 2012
NETL/DOE [32] 3.0 N/A 2012
This study 1.4-2.7 Decline curve analysis
5. Conclusions 329
Both the Hyperbolic curve and the Stretched Exponential curve fit well both to aggregate and 330
individual shale gas wells. The Hyperbolic model is slightly better than the Stretched Exponential 331
model in terms of goodness of fit in this study, even where about 84% of β-parameter values of the 332
Hyperbolic were larger than 1. 333
Characteristic decline rates have changed little over time (Table 3). After the first year after IP, 334
around 60% of the production level has been lost due to decline. After 2 years, the production level 335
is only 25% of IP. These high decline rates are in agr eement with earlier studies [28 -30]. There is an 336
increasing trend in IP (Figure 12(b) ) that most likely reflects te chnological gains. 337
On average, shale gas wells were estimated to yield an ultimate production of 1.4 -3.2 Bcf (Table 338
4). However, the highest numbers imply very long well life spans (> 20 years). If a cut -off rate is used, 339
the URR estimate becomes more reasonable and ends up between 1.4 -2.7 Bcf (Table 5), and the well 340
life time ranges from 10.7 to 28.9 years. The results are in line with other studies (Table 7). 341

Energies 2016 , 9, x FOR P EER REVIEW 13 of 14
Acknowledgments: The auth ors would like to thank DrillingInfo for providing access to their extensive database, 342
without which this study would have been difficult to accomplish. The authors also would like to give many 343
thanks to the National Social Science Foundation of China (Gra nt No. 13&ZD159) for sponsoring this research. 344
This study has been supported by the StandUp for Energy collaboration initiative. We would also like to thank 345
Simon Snowden for proofreading and helpful comments. 346
Author Contributions: Mikael Höök and Baosheng Zhang conceived and designed the experiments; Keqiang 347
Guo and Henrik Wachtmeister performed the experiments; Keqiang Guo and Mikael Höök analyzed the data; 348
Kjell Aleklett contributed data; Keqiang Guo and Henrik Wachtmeister wrote the paper. 349
Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design 350
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the 351
decision to publish the results . 352
Appen dix A . Probability density functions 353
A. 1 Generalized Logistic Distribution 354
1 1/
21/
2(1 ),0
1 (1 )()
exp( ),0
1 exp( )k
kkzk
kzfx
zk
z

  
355
A. 2 Normal Distribution 356
21exp2
()
2x
fx


357
A. 3 Log-Pearson 3 Distribution 358
11 ln( ) ln( )( ) exp()ax c x cfxx b a b b          
359
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