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simultaneously. The approach appears to have definite advantages over Arps-type decline curve analysis, at least forunconventional gas applications.

Stretched Exponential DeclineIf we bear in mind that hyperbolic decline was introduced by Cutler 1924 because exponential decline gave “too conservativeestimates in the final stage,” then it is natural to consider it as a practical generalization of the exponential function. Such a specific

generalization is, however, rather unique to the petroleum industry. In physics, many processes manifest stretched exponential behavior, as first described by Kohlrausch in 1847, and then by a great number of authors (Philips, 1966). The approach foundfollowers even in the geophysical literature (Laherrère and Sornette, 1998).

A natural interpretation of the stretched exponential decay of a quantity is that it is generated by a sum (integral) of pureexponential decays with a “fat tailed” probability distribution of the time constants (Johnston, 2006.) Therefore, we can interpretthe stretched exponential production decline (SEPD) model (Table 1) as the acknowledgement of the heterogeneity: the actual

production decline is determined by a great number of contributing volumes individually in exponential decay (i.e., in some kindof pseudosteady state), but with a specific distribution of characteristic time constants. The family of the background distributionof characteristic time constants is known analytically, and has been visualized by Johnston (2006). The distribution is determined

by a parameter pair, ( n, τ ). Broadly speaking, the τ parameter is the median of the characteristic time constants. The nearer the parameter n is to zero, the larger is the tail of the distribution; that is, more elementary volumes have very large time constants.

TABLE 1—THE STRETCHED EXPONENTIAL PRODUCTION DECLINE MODEL

Defining differential equation of the model exp Rate expression as function of time 1 1 , Cumulative production as a function of time

EUR 1 EUR in terms of the model parameters

1EUR

11 1 , ln Recovery potential calculated from actual rate using two model parameters

One of the results of using the hyperbolic decline approach in analytical models and reservoir simulators is that the boundary-dominated flow regime is necessary for meaningful extrapolation. Since the traditional boundary-dominated state is veryquestionable in tight and especially shale gas, little use can be made of the consensus developed by Fetkovich and others in the80s. The stochastic interpretation, however, provides the advantage. that the distribution of time constants is an intrinsic propertyof a continuous unconventional gas reservoir, and hence it determines the totality of the production decline curve. In other words,we can extrapolate from early data, at least in some sense, and exactly that has been done in early attempts to apply SEPD (Valkó,2009; Ilk et al., 2008; Mattar and Moghadam, 2009.)

Some Properties of the SEPD ModelAs in the case of the formal treatment of hyperbolic and exponential decline by Arps, the SEPD model can be also derived from adefining differential equation with two parameters: in this case n and τ , while the third parameter, q0, appears as an “initial

condition” of the defining ordinary differential equation?Besides the more familiar gamma function, the new model incorporates the incomplete gamma function (Abramowitz and Stegun,0972). The recovery potential expression is derived by substituting the q and Q expressions into the EUR definition.

Compared to the Arps formalization, the new approach offers numerous advantages; among them the two most significant onesare the bounded nature of EUR from any individual well and the straight-line behavior of the recovery potential (rp) expressionversus the cumulative production.

For positive n, τ and q0, the model gives a finite value of the EUR, even if no cutoff is used in time or in rate. (Unfortunately,the Arps family of curves leads to an unbounded and physically impossible estimate of EUR for b ≥ 1.) Once the n and q0

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Fig. 2—Average monthly production increment (in Mscf) of the selected three month-groups yields clearer trends.

Fig. 2 shows the mean (per well) monthly production increment of all wells starting production in July 2004, 05, and 06. As wesee, the monthly mean is affected by the peak (occurring in the second or third month), by the time elapsed from start, and also byadditional effects related to the absolute time (seasonal demand, gas price, etc.) This latter phenomenon can be depicted from dipsappearing on all three curves at the same absolute time.

The explanation for the low first month value is that on average, the first month has only 15 producing days (assuming evenlydistributed actual starting dates).These three groups are only examples of the 230 month-groups for vertical wells and 80 month-groups for horizontal wells that

have at least 3 years of production history. Our selection of the 3-year criterion is not incidental; in some sense it is a compromise:should we select a longer time horizon, we would be forced to exclude exactly those wells we are really interested in. Should weselect an even shorter time horizon, the noise-to-signal ratio (due to season, demand, price etc.) would increase.

For each month-group we calculate the first-, second-, and third-year production. The results are show in Fig. 3 . Since eachmonth-group is represented by a hypothetical average well (three of them shown in Fig. 2), we anticipate some consistency.

As often happens with huge amounts of data, we still find great variability, obvious in Fig. 3.

Fig. 3—The x axis shows the index of the month-group. The y axis shows 1-, 2-, and 3-year cumulatives for the “average well” in themonth-group for vertical wells (left) and horizontal wells (right). Great variability is evident.

One of the problems associated with working with averages is that the number of active wells is not necessarily constant for amonth-group in the first 3 years. Nevertheless, we can considerably reduce the variability by constructing ratios of year-endcumulatives, and hence focus on relative decline instead of absolute level of production.

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Fig. 4—The ratio of 2/1 and 3/1-year cumulatives for the various month-groups show a more consistent behavior.

While Fig. 4 shows some promising consistency, it raises some questions, too. How is it possible that for some month-groups thethird year cumulative is more than 3 times the first-year cumulative? Obviously, those wells were not “declining,” not onlyindividually, but even as a group.

Nevertheless, both the vertical and the horizontal curves smooth out considerably (i.e., become essentially horizontal) for thenewer wells. The stabilization reflects two facts: the huge number of wells in these month-groups and the maturation of thefracturing technology.

Cumulative ratios are the integral counterparts of Johnson’s (1927) rate decline ratios and we favor them because they are morestable.

Since the 2/1 and 3/1 cumulative ratios behave relatively stably, we can calculate the weighted mean of the ratios, where eachmonth group value is weighted by the number of wells in the group. The results are shown in Table 2.

TABLE 2—WEIGHED MEANS OF CUMULATIVE RATIOS2/1 year cumulative ratio, r 21 3/1 year cumulative ratio, r 31

All (6,348 of 13,482) 1.563 1.985Vertical (3,542 of 3,724) 1.570 2.005Horizontal (2,806 of 9,758) 1.549 1.954

Now we solve two nonlinear equations to obtain the model parameters n and τ :

, . , . (1a) , . , . (1b)

The corresponding two equations for the Arps model are well-known, and we here show only the resulting b and D i parameters.

TABLE 3—MODEL PARAMETERS FROM DATA IN TABLE 2SEPD Arps

n , month b D i 1/month All

0.231

0.585

1.69

0.251

Vertical 0.203 0.246 1.82 0.292 Horizontal 0.247 0.776 1.58 0.240

The industry has known for many years that comparable approaches result in b > 1 for tight gas wells (Maley, 1985), though to ourknowledge this is the first time that such a large data set has been used to provide evidence.

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Comparison of the Two Decline ModelsFor comparison purposes we show what happens if ½- , 1-, or 3-year production data are available from a horizontal well. Themultiplier is the number by which we multiply the available cumulative production to extrapolate to a final month, shown on the x-axis.

Fig. 5—Multipliers to extrapolate the cumulative production to a given “final month”: red for SEPD and blue for Arps with parametersshown in Table 3 for horizontal wells. Extrapolation based on ½ - (circle), 1- (upper triangle), or 3- year (lower triangle) cutoff time.

All curves start from one. The red curves (SEPD) ultimately level off while the blue curves (Arps) grow beyond any limit. Thered and blue curves will very likely describe equally well all the available data; moreover, they will give nearly the same

prediction for the following 4 to 5 years. Therefore, a simple “quality of fit” argument cannot differentiate between them.For individual wells, even the 600-month prediction can turn out “better,” based on the blue curve, than the one based on the

red curve. The problem is that the fundamentally unlimited behavior of the Arps model with b > 1 acts as a “potential time bomb.”There is no reason other than tradition to stick with such a potentially dangerous model.

In the following we will work only with the SEPD model, on the basis that this choice is dictated by its realistic mathematical properties.

SEPD Master Curves for Horizontal and Vertical Wells in the Barnett ShaleFig 6 shows the two master curves (red for horizontal and blue for vertical) calculated from the definition

/ , (2)

Fig. 6—The master curves are constructed from that part of the SEPD model that does not include the q 0 parameter. The slight differencein shape stems from the difference in the model parameters n and . Units of both Δ Q m and Q m are months.

The actual construction is as follows: First the Qm,i values are calculated at t = i - 0.5, i = 1,2,... Then the difference series isconstructed:

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∆ , , , with , 0Finally, the ( ,; ∆,), i = 1,2, … points are connected and plotted as a curve.There is no significance to the actual value (level) of this curve; only its shape matters. Note, however, that the units of both Qm and Δ Qm are months.

Once a data set is available, we use the number of producing months ( i) and the cumulative production Q to “nail down” the q0 parameter for the well (or group of wells:)

/, (3)This simple “fitting procedure” determines the one free parameter— q0 — by satisfying the constraint that the cumulative

production calculated from the number of months should be equal to the observed cumulative production. In the whole procedurewe did not use any rate value, because rate is not measured and is certainly not reported to state regulatory agencies. Onlyincrements of cumulative production are used; i.e., the raw reported data.

From Fig. 6 it seems that the horizontal wells produce a greater fraction of their cumulative production in the earlier periods.This is somewhat anticipated, because the purpose of the multiple fractured horizontal well is to shift production to earlier periods;this, in turn, shifts the underlying distribution of characteristic times toward lower values. Nevertheless, the formation

characteristics dominate the underlying distribution and hence most of the difference between horizontal and vertical wells willshow up not in the n and τ parameters but in the q0 parameter.

Some Examples of the Results So FarAs a first example we consider our three July groups of Fig. 2. Shown on Fig. 7 are the production declines for a hypothetical“average well” in the three July groups (2004 red, 2005 blue, 2006 green).

Fig. 7—“Fitted” decline curves for the average wells in each July groups of 2004, 05 and 06.

For each group we determine the q0 from the last cumulative production using Eq. 3, and from that we obtain the solid lines. Thereis no other “fitting.” The actual monthly increments are surprisingly well described, even in the early part of the production.

A similar representation is shown on Fig. 8 for all the horizontal well month-groups between January 2004 and December2006, but each series is normalized by its own q0. The normalized series for horizontal wells (left) and should align along the redmaster curve of Fig. 6, the vertical wells (right), along the blue master curve.

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Fig. 8—The normalized month-group series aligns around the master curves for horizontal and vertical well groups. (Colored by startingmonth.)

The data seem to follow the initial sharp decline and then the subtle “hump” at intermediate times characteristic of the SEDPmodel with the horizontal well parameters provided in Table 3. One interesting feature of Fig. 8 is that there are no individual

point series (points with the same color) remaining on one side of the red master curve. Rather the individual series criss-cross the

master curve several times.A similar treatment shows the vertical month-groups and their master curve on Fig. 9. Interestingly, the data— though

admittedly with much more scatter— follows the vertical master curve on the right side of Fig. 9 in the sense that there is noevidence of the “hump” found with the horizontal wells.

So far we showed only point series, each averaged from several dozen wells starting in the same month. We have providedsome evidence that the month-groups follow the common trend depicted from the cumulative ratios for the two wellboreorientations. It still remains to see whether individual wells follow the group trend.

Making a similar normalized plot with individual well data would be impractical, because we would need to place an enormousnumber of points on one plot. For instance, currently there are 9,208 active horizontal wells, and that adds up to 278,652 monthly

production increment/ cumulative production pairs. Nevertheless, on Fig. 9 we attempt to visualize the data (horizontal wells to theleft, vertical to the right) for wells drilled during this millenium. We normalize each individual well series by its own q0 calculatedfrom the last cumulative production. Then we define “bins” of width 0.1 Qm unit (which is 0.2 months) on the x-axis and showonly the center of gravity of the points in the bin, together with limits of one standard deviation.

Fig. 9—Data from all horizontal (left) and vertical (right) wells new this millennium normalized and averaged over 0.1 unit bins generallyfollow the pattern presented.

Problem With Restimulated Wells

Old vertical wells represent a “problematic” group, as clearly seen from Fig. 10 .

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Fig. 10—(Left) Data from all inactive vertical wells of the last millennium (old) normalized and averaged over 0.1 unit bins; right,data from all active vertical old wells (last millennium) normalized and averaged over 0.1 unit bins. Data at left (inactive vertical wells)show a single decline trend. Data at right (active vertical wells) show two decline periods, the first one before restimulation, the secondone after restimulation.

Fig. 11—Data from the 10 individual wells started in July 1990. The re-stimulation around 2001 multiplied the EURs. Solid and dashedlines show current forecast by two methods.

Fig. 11 illustrates the dilemma for old wells, using the 1990 July group as an example. These 10 wells were restimulated around2001, using slick-water fracturing technology. To forecast production from these wells, we can use the current cumulative

production, but it is not clear whether we should assign it to the total number of producing months or to the number of monthsafter 2001. The two ways give rise to two different extrapolation options: the dashed line was obtained using the total number ofmonths; the solid line was obtained using only the number of months after 2001, effectively banking the production up to 2000 andadding a new hypothetical well for the incremental production starting in Jan 2001.

Repeating the procedure for all old active wells, we obtain EUR inf (EUR without any restriction on time or rate) with the totalnumber of months (first method) and EUR inf2 with the months only after 2001 (second method).

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Fig. 12—Comparison of EUR (in MSCF) obtained by two different methods for active old vertical wells, basically all of which were re-stimulated around 2001.

From Fig. 12 we conclude that the two EUR values correlate relatively well and hence either way of calculating EUR is tolerable.This is especially useful because old wells will eventually lose more and more producing months, an effect very difficult to capture

in a probabilistic way. As seen from Fig. 13, the actual number of producing months for old, inactive wells (blue) is less than thecalendar dates would suggest. This is quite natural. Active old wells (red) lose more and more producing months during their longcareers, too.

Fig. 13—The number of actual production months departs from the potential maximum because of downtime or abandonment of active(red) and already inactive (blue) wells.

Group-Data Controlled ForecastHow can we provide some evidence that the group-data controlled forecast represented by the two master curves lead toreasonable EURs? This seems to be an impossible task, because there is no control group for 40 or 50 years of production. In thissection we provide a proxy for forecasting the future. We pretend we are at a certain point in time, (½, 1, 2, 3, 4, 5 years after first

production), and forecast the “future” up to present day. Then we compare the prediction to what actually happened.As an example we select 4 horizontal wells from each July group. First we locate the median well (by cumulative at present

day) and then we pick the 2 nd and 6 th well above and below. The reader might suspect that in spite of our declared “objective” wayof selecting wells, we try to present some favorable examples. However, this time our goal is somewhat different and the examplesare actually not so favorable at all. By showing just a couple of the figures, we argue that the actual production histories of thewells are extremely hectic and that gleaning some kind of sophisticated individual decline behavior from a limited individual pastwould be a mistake.

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Fig. 14—EUR for four sample wells calculated from 5.5 years of production data (right) and from a half year of data (left).

Nevertheless, the procedure based on the “master decline” provides increasingly consistent EUR estimates, by allowingindividuality only in the q0 parameter that is solely determined by the latest cumulative (and the number of producing months).This is obviously because that the ( n, τ ) pair is controlled by the totality of horizontal well declines.

Table 4 shows the evolution of the calculated EUR inf (in Mscf) for 4 wells from the July groups of 2004, 05 and 06.

TABLE 4—CALCULATED EUR PER WELL, Mscf × 10July 2004 Well Number

Cutoff (year)1/212345

3063 255 5850 122222.052.041.941.881.821.80

1.811.95

1.941.962.00

2.04

2.182.242.342.332.312.32

0.971.532.142.472.712.84

July 2005 Well Number Cutoff (year) 13 3077 5561 11650

1/212345

1.862.001.721.741.731.70

2.912.522.132.011.931.88

2.082.122.052.112.152.17

2.052.252.362.452.492.52

July 2006 Well NumberCutoff (year) 1670 14226 2795 502

1/21234

3.182.802.171.871.86

2.492.472.202.092.07

1.921.921.992.012.02

2.012.162.192.112.20

We can conclude that the master curve approach sacrifices individual goodness of fit for overall consistency. The point is thatesthetics of individually picked examples does not substitute for statistical investigation. How well one can achieve a satisfying“goodness of fit” with a given combination of a model and parameter identification methodology does not say much at all aboutthe forecasting power in the statistical sense. We will return to this issue in the section “P90 Methodology”.

At this point we focus on the consequences of the time evolution of the q0 parameter.

Monitoring With Recovery Potential PlotDuring our time travel, at any point in time we can plot a straight line going from (0;1) to ;0 for each well. This is thetheoretical line the rp values should follow if the model were exact. The rp expression can be also calculated from q/q0 and plotted

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on the same graph. For instance, for Well 14155 of the July 2004 group, Fig. 15 shows the recovery potential plot after a halfyear (at the time only the solid points and the straight line can be seen; the empty circles are unknown) and after 5 years. Strictlyspeaking, both the straight line and the position of the points are changing from half to 5.5 years because q0 is updated.

Fig. 15—Recovery potential plot after half year cutoff (left) and 5 years cutoff (right). only the solid points and the straight line are

available at any given time. Production over time updates the n and parameters, changing the slope and the x-axis intercept of thecurve.

In case of Well 14155, the straight line already established itself in the first half year and afterwardschanged relatively little. The operator can draw various conclusions regarding the periods when the points are above and whenthey are bellow the line. In this case these conclusions can be drawn on the go, not only in hindsight. This is made possible by therelative stability of q0. We suggest that the recovery potential plot is a useful monitoring tool at hand, though admittedly this“hand-picked” example was a favorable one.

Another example (Well 9076 of the same July group), Fig. 16 , reveals that between half and 2 years the operator was able toimprove the well (i.e., q0 and hence EUR increased). The actions taken in the 5

th year did not lead to substantial increase of EUR, but rather a temporary acceleration of recovery. As seen from Fig. 16, the “future” points “smoothly land” onto the straight line.We have observed many such examples for the more prolific wells.

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Fig. 16—Recovery potential plot after a half year (top left), 2 years (top right), and 5.5 years (bottom), Only the solid points and the straightline are available at any given time.

In the case of extremely weak wells, the straight line position changes significantly with increasing cutoff time, and meaningfulconclusions about the EUR are more difficult to draw. This happens because these wells are still in an “adolescent” stage wherethe operator tries to improve the operating conditions (such as by liquid cleanup.) Desperate measures sometimes work, as isillustrated by Fig. 14, where the initially worst well, well 12222, became the best well, but by Year 5 well 12222 had joined the

regular decline mode trend.

P90 MethodologyOne of the most challenging tasks related to forecasting is to provide a P90 estimate of EUR, the focal point of new SEC reportingrules (Lee, 2009). In the previous section we developed a framework and now we apply it to arrive at some quantitative statements.

As we continue our time travel we design the following numerical experiment. For each well we take 6, 12, 18, etc. monthscutoff time and record the forecast value for the present day cumulative production . Since we have 13,482 wells and some of themhave several hundred months’ history, we get quite a large number of predicted values, in fact 103,542 individual predictions forthe 13,482 present day values. Some of the predictions are larger and some of them are smaller than the actual end value. We areinterested in the empirical cumulative distribution of the predicted (as a percentage of the actual) value.

Fig. 17 visualizes the empirical probability distribution resulting from the 103,542 predictions.

Fig. 17— Histogram of predictions as percentage of actual values.

The median seems to be about 100% and the median deviation is about 7%. This is promising, but the experimental cumulativedistribution shown in Fig. 18 reveals that in 50% of the cases, our prediction was optimistic. That is not surprising (actually itshows that in some sense we arrived at an unbiased way of estimating) but it is not what is required by P90.

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The requirement of P90 demands a procedure for which 90% of the predictions are conservative. Therefore, we have to apply a“safety factor” less than 1 that will bias our estimates toward the conservative side. With a little experimenting we arrive at safetyfactor 0.8.

Fig. 18— Empirical cumulative distribution of predictions (left) and with 0.8 safety factor (right), which shifts the distribution far enough to

the left to make 90% of the predictions conservative.

Multiplying all our predictions by 0.8 shifts the new empirical cumulative distribution to the left so that 90% of the predictions become conservative. Logically, the same safety factor can be applied to EURs obtained from the same “master curves” by thesame procedure (whatever additional actual time and/or rate limitation is incorporated.)

In Table 6 we report our reserves estimates for the Barnett shale, using the safety factor 0.8 and the SEPD parameters forhorizontal and vertical wells presented in Table 3. Now we have some quantitative justification to call the results P90 estimates,and repeating the construction of the empirical cumulative distribution periodically, say in every 6 months in the future, we cancontinuously monitor the validity of the hypothesis that our overall methodology still satisfies the P90 criteria.

TABLE 5—PRODUCTION FROM INACTIVE WELLS, Mscf × 10 6

Number of WellsGas Already

Produced, Mscf ×10 6 Vertical 366 245Horizontal/deviated 540 399

All 906 644

TABLE 6—RESERVES ESTIMATES, ACTIVE WELLS, Mscf × 10

Number of wells Already produced P90 EURinf* P90 future productionVertical 3358 2.06 6.19 4.13Horizontal/deviated 9218 5.26 20.1 14.8

All 12576 7.33 26.3 18.9*without time or rate limit

Of course reality may “play a trick on us.” It is possible, for instance, that a new restimulation technology will emerge, witheffects as dramatic as in 2001. Therefore, our claim is not so much that we can predict the future (nobody can) but that the

procedure provides a reliable and consistent evaluation methodology.

The Barnett shale example is unique in many ways. It includes a great number of wells, the technology spreads extremely fast(in the last 5 years vertical wells have given way to horizontal wells almost completely), and the decline trends seem to be quiteconsistent—at least within a completion type, most importantly for horizontal wells intersected by several propped fractures with

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the overall amount of sand reaching millions of pounds. Therefore we do not claim that repeating the analysis in another play willlead to similar consistency. Nevertheless, with some variations, the main concepts will be applicable.

ConclusionsTwo important points can be made from these investigations. First, a model family with good mathematical properties is betterthan a model family with less favorable mathematical properties, even if both can satisfactorily describe a couple of hand-pickeddata sets. In this respect we think that the SEPD model has definite advantages over the Arps family of decline curves, at least for

unconventional gas applications. Second, there are many ways to obtain model parameters, but in the era of increasing amounts ofopen and accessible data, we should try to process the data collectively rather than individually, especially if some theoreticalconsideration supports the existence of common trends in the data. While new aspiring models and fitting techniques will certainly

be offered again and again in the future, we place more hope in the transition to data-driven discovery, where the data amount (andthe open access to it) provide some transparent assurance of the repeatability of the results.

AcknowledgementThe authors are grateful to HPDI, LLC for providing access to their production data database. Contributions from our students,Bunyamin Can and Beau Clark, are appreciated.

Nomenclatureb = Arps' decline exponent, dimensionless

Di = Arps' decline constant, 1/month

EUR = estimated ultimate recovery, Mscf EUR inf = estimated ultimate recovery without imposing time or rate limit, Mscf

n = exponent parameter for SEPD model, dimensionless q = production, Mscf

q0 = production parameter common in Arps’ model and in SEPD, Mscf/monthQ = cumulative production, Mscf

Qm = normalized cumulative production, Q/q 0, months

, = normalized cumulative production for given number of months i, Q/q 0, month??rp = recovery potential, dimensionless

t = production time, months Δ Q = cumulative production increment, Mscf

τ = characteristic time parameter for SEPD model, month

ReferencesAbramowitz, M. and Stegun, I.A. eds. 1972. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, New

York: Dover Publications.Arnold, R. 1923, Two decades of petroleum geology, 1903-22, AAPG Bull. 7(6).Arps, J.J. 1945. Analysis of Decline Curves.. Trans. AIME: 160: 228-247.Cheng, Y., Lee, W.J., and McVay, D.A. 2008. Improving Reserves Estimates From Decline-Curve Analysis of Tight and Multilayer Gas Wells,

SPE Reservoir Evaluation and Engineering 11 (5):, 912-920.Cutler, W.W. Jr. 1924. Estimation of Underground Oil Reserves by Oil-Well Production Curves. Department of the Interior, U.S. Bureau Of

Mines Bulletin 228.Fetkovich, M.J., 1980. Decline Curve Analysis Using Type Curves. JPT 32 (6): 1065-1077.Fetkovich, M.J., Vienot, M.E., Bradley, M.D., and Kiesow, U.G. 1987. Decline Curve Analysis Using Type Curves: Case Histories. SPEFE .

2(4): 637-656.

Fourth Paradigm, The. 2009. Data-Intensive Scientific Discovery. Hey, T., Tansley, S., and Tolle. K.. eds. Seattle: Microsoft Research.Ilk, D., Perego, A.D., Rushing, J.A., and Blasingame, T.A. 2008. Exponential vs. Hyperbolic Decline in Tight Gas Sands —Understanding the

Origin and Implications for Reserve Estimates Using Arps' Decline Curves. Paper SPE 116731 presented at the SPE Annual TechnicalConference and Exhibition, Denver, Colorado, 21-24 September.

Johnston, D.C. 2006. Stretched Exponential Relaxation Arising From a Continuous Sum of Exponential Decays. Phys. Rev. B 74: 184430.Johnson, R.H. and Bollens, A.L. 1927. The Loss Ratio Method of Extrapolating Oil Well Decline Curves. Trans . AIME 77: 771.Kohlrausch R. 1847. Ann. Phys., Lpz. 12 393.Laherrère, J., Sornette, D. 1998. Stretched Exponential Distributions in Nature and Economy: "Fat Tails" with Characteristic Scales. European

Physical Journal B 2(4): 525-539.

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Larkey, C.S. 1925. Mathematical Determination of Production Decline Curves. Trans . AIME 71: 1322.Lewis, J.O. and Beal, C.H. 1918. Some New Methods for Estimating the Future Production of Oil Wells. Trans . AIME 59: 492-525.Lee, W.J. 2009. Reserves in Nontraditional Reservoirs: How Can We Account for Them? SPE Economics and Management 1 (1): 11-18. SPE-

123384-PA. doi:10.2118/123384-PAMaley, S. 1985. The Use of Conventional Decline Curve Analysis in Tight Gas Well Applications. Paper SPE 13898 presented at the SPE Low

Permeability Gas Reservoirs Symposium, Denver, March 19-22.Mattar, L. and Moghadam, S. 2009. Modified Power Law Exponential Decline for Tight Gas. Paper Paper 198 presented at the Canadian Intern.

Petr. Conf., Calgary, .

Phillips, J. C. 1966. Rep. Prog. Phys . 59: 1133.Rushing, J.A., Perego, A.D., Sullivan, R.B., and Blasingame, T.A. 2007. Estimating Reserves in Tight Gas Sands at HP/HT Reservoir

Conditions: Use and Misuse of an Arps Decline Curve Methodology. Paper SPE 109625 presented at the 2007 Annual SPE TechnicalConference and Exhibition, Anaheim, California, 11-14 November.

Shea, G. B., Higgins, R.V., and Lechtenberg, H.J. 1964. Decline and Forecast Studies Based on Performances of Selected California Oil Fields. JPT 16(9): 959-965.

Spivey, J.P., Williamson, J.R., and Sawyer, W.K. 2001. Applications of the Transient Hyperbolic Exponent. Paper SPE 71038 presented at theRocky Mountain Petroleum Technology Conference, Keystone, Colorado, May 21-23.

Valkó, P.P. 2009. Assigning Value to Stimulation in the Barnett Shale: a Simultaneous Analysis of 7000 Plus Production Histories and wellCompletion Records. Paper SPE 119369 presented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, 19–21January.