rate-decline analysis for fracture-dominated shale reservoirs · rate-decline analysis for...

June 2011 SPE Reservoir Evaluation & Engineering 377

Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs

Anh N. Duong, SPE, ConocoPhillips

SummaryTraditional decline methods such as Arps’ rate/time relations and their variations do not work for wells producing from supertight or shale reservoirs in which fracture flow is dominant. Most of the production data from these wells exhibit fracture-dominated flow regimes and rarely reach late-time flow regimes, even over several years of production. Without the presence of pseudoradial and boundary-dominated flows (BDFs), neither matrix permeability nor drainage area can be established. This indicates that matrix contribution is negligible compared with fracture contribution, and the expected ultimate recovery (EUR) cannot be based on a traditional concept of drainage area.

An alternative approach is proposed to estimate EUR from wells in which fracture flow is dominant and matrix contribu-tion is negligible. To support these fracture flows, the connected fracture density of the fractured area must increase over time. This increase is possible because of local stress changes under fracture depletion. Pressure depletion within fracture networks would reactivate the existing faults or fractures, which may breach the hydraulic integrity of the shale that seals these features. If these faults or fractures are reactivated, their permeabilities will increase, facilitating enhanced fluid migration. For fracture flows at a constant flowing bottomhole pressure, a log-log plot of rate over cumulative production vs. time will yield a straight line with a unity slope regardless of fracture types. In practice, a slope of greater than unity is normally observed because of actual field operations, data approximation, and flow-regime changes. A rate/time or cumulative production/time relationship can be established on the basis of the intercept and slope values of this log-log plot and initial gas rate.

Field examples from several supertight and shale gas plays for both dry and high-liquid gas production, and for oil production were used to test the new model. All display the predicted straight-line trend, with its slope and intercept related to reservoir types. In other words, a certain fractured flow regime or a combination of flow types that dominate a given area or play because of its reservoir-rock characteristics and/or fracture-stimulation practices all produce a narrow range of intercepts and slopes. An individual-well performance or EUR can be derived that is based on this range if the best 3-month average or the initial production rate of the well is already known or estimated. The results show that this alterna-tive approach is easier to use, gives a reliable EUR, and can be used to replace the traditional decline methods for unconventional reservoirs. The new approach is also able to provide statistical methods to analyze production forecasts of resource plays and to establish a range of results of these forecasts, including prob-ability distributions of reserves in terms of P90 (lower side) to P10 (higher side).

IntroductionUnconventional reservoirs, especially wet shale gas and oil, are currently being pursued aggressively for new development in both the US and Canada. Forecasting production and estimating

reserves accurately for these resource plays have become more urgent and important than ever before.

Mattar et al. (2008) have discussed various techniques for production analysis and forecasting of shale-gas reservoirs, while Lee and Sidle (2010) have analyzed some of the commonly used procedures for forecasting and identifying key strengths and limitations of these techniques. The commonly used method for decline-curve analysis is Arps’ hyperbolic rate decline (Arps 1945), if adequate production data are available. Fetkovich type curves (1980) are not appropriate because the matched b value is usually greater than unity. For horizontal wells with multiple transverse fractures, numerical-simulation modeling is a highly preferred option. There are, however, some validity and applicability issues with these techniques.

The hyperbolic decline equation is conveniently used because it can have a “best fit” for the long transient linear-flow regime observed in shale-gas wells with b values greater than unity. Lee and Sidle (2010) showed that values of b equal to or greater than unity can cause the reserves to have physically unreasonable properties. To avoid this drawback, “stretched-exponential” models have been proposed recently (Valkó 2009; Ilk et al. 2008, 2009).

The current simulation models that are available in the mar-ket are still using conventional technologies for unconventional reservoirs. Some of the assumptions used in simulation modeling are inconsistent with the field-data observations such as pressure initialization and radial (or elliptical) transient flow. Field data indicate that the pressure transition through a shale-gas zone is at a disequilibrium state, while pressure initialization in simulation modeling is based on an equilibrium state with its fluid gradient. Newsham and Rushing (2002) showed that pressure gradients measured in the Lower Bossier can be as high as 63 kPa/m (3 psi/ft) or more than 25 times its saturated gas gradient or three times the lithostatic gradient. Spencer (1987) presents some pres-sure gradients in overpressured and gas-bearing shale plays in the Rocky Mountain region. The interpreted pressure profile across the active zone for wells in the Piceance basin shows a pressure gradient of approximately 36 kPa/m.

The concept of stimulated rock volume (SRV) that is supported by microseismic monitoring of hydraulic-fracture treatments is not confirmed by the observed radial flow from a numerical simulator, as pointed out by Mattar et al. (2008). These authors claim that the radial flow of fractured shale wells observed in a numerical simulator is more likely to be a false radial flow and may not be representative of the enhanced permeability of the SRV.

This paper introduces an empirically derived decline model that is based on a long-term linear flow in a large number of wells in tight and shale-gas reservoirs. On the basis of this model, a new method has been developed for production analysis and forecasting of unconventional reservoirs. This method also uses probability distributions of reserves in forecasting resource plays, to represent any uncertainty in reserves estimation.

Method DevelopmentLong-Term Linear Flow. Fig. 1 shows the production history of a vertical well producing from the Barnett shale. Production data show a half-slope line, indicating linear fl ow lasting for more than 5 years of production.

Long-term linear flow in a large number of wells in tight or shale gas has been observed as early as 1976 (Bagnall and Ryan 1976). Since then, many researchers have tried to explain what

Copyright © 2011 Society of Petroleum Engineers

This paper (SPE 137748) was accepted for presentation at the Canadian Unconventional Resources and International Petroleum Conference, Calgary, 19–21 October 2010 and revised for publication. Original manuscript received for review 29 September 2010. Revised manuscript received for review 27 January 2011. Paper peer approved 23 March 2011.

378 June 2011 SPE Reservoir Evaluation & Engineering

causes this phenomenon. Several authors have shown that this linear flow exists in any type of fractures whether they are infi-nite- (Agarwal et al. 1979) or finite-conductivity (Cinco-Ley et al. 1978; Cinco-Ley and Samaniego 1981a, 1981b), single or multi-stage (Bello and Wattenbarger 2010), hydraulic (Arévalo-Villagrán et al. 2001), or natural fractures (Arévalo-Villagrán et al. 2006). The difference among these models is the length of this transient linear-flow regime.

To prolong the linear-flow regime in any of those models, all one needs to do is enlarge the term xf

2/km in which xf is the effective fracture half-length of the fracture system and km is the matrix permeability. For any given shale-gas reservoir, matrix permeability is fixed. Thus, the connected fracture density of the fractured area must be increasing over time to support this fracture flow over the life of a producing well. This increase is achievable because of local stress changes under fracture depletion (Warpinski and Branagan 1989). Pressure depletion within fracture networks would reactivate the existing faults or fractures, which may breach the hydraulic integrity of the shale that seals these features. If these faults or fractures are reactivated, their permeabilities will increase, facilitating enhanced fluid migration. A fractured shale model that imitates this long-term linear flow and nonequivalent pressure distribution will be briefly discussed later and is the subject of another paper.

If a fracture flow regime (either linear or bilinear) is prolonged over the life of a well, the gas-flow rate q will be

q q t n= −1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

where n is one-half for linear flow, n is one-quarter for bilinear flow, and q1 is the flow rate at Day 1.

The gas cumulative will be

G q t qt

npt

n

= ∫ =−

−

d0 1

1

1( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

This gives

qGp

nt= −( )1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Field Applications. Field data from several shale-gas plays were used to test Eq. 3. These fi eld data were operated under the actual fi eld conditions that may violate some of the ideal assumptions used to derive Eq. 3. Some of the results are shown in Fig. 2 in which log-log plots for q/Gp vs. time in days were constructed.

These plots all give a log-log straight line with a negative slope, −m, and an intercept of a. That is,

q

Gpat m= − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

Note that the slope is negative but m is always positive. As shown later, m is always greater than unity for shale reservoirs. If m is less than unity, it may indicate a conventional tight well.

Equations for q and Gp can be derived as shown in Appendix A:

q

qt em

a

mt m

1

111

= − −−−( )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

PRODUCTION HISTORY FOR A SHALE GAS WELL

1

10

100

1 10 100 1,000 10,000Days

Rat

e, 1

03 m

3 /d

1

10

100

WH

P, M

Pa

Press. Corrected Rate

Measured Rate

Wellhead Pressure

Skin effect

Skin effect andSurface choking

Fig. 1—A log-log plot of field production and pressure data for a shale-gas well. Production data show a half-slope line, indicat-ing linear flow lasting for more than 5 years of production.

y = 1.1437x –1.1054

R 2 = 0.9861

a =1.1437m =1.1054

y = 1.2776x –1.1381

R2 = 0.9808

a =1.2776m =1.381

y = 1.3665x–1.1487

R2 = 0.9836

a =1.3665m =1.1487

0.0001

0.001

0.01

0.1

1 10 100 1,000 10,000

Days

q/G

p, 1

/Day

0.0001

0.001

0.01

1 10 100 1,000 10,000

Days

, 1/D

ay

0.0001

0.001

0.01

1 10 100 1,000 10,000

Days

q/G

p

(a) (b) (c)

, 1/D

ayq

/Gp

1

0.1

1

0.1

1

Fig. 2—A log-log plot of q/Gp vs. t for various shale-gas wells with more than 5 years of production. The data all form a straight line with a slope m and an intercept a. A slope of higher than unity is normally obtained, to account for any deviations from the ideal cases.


and

Gq

aep

a

mt m

= −−−

1 111( )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

It will be shown later that a can be eliminated from Eqs. 5 and 6 if they are rewritten in forms of dimensionless rate and time. This indicates that a and m are related variables, as shown in Fig. 3, obtained from various gas plays. Using this correlation to convert a into m, type curves for q/q1 vs. time in days can be constructed, as shown Fig. 4.

Fig. 4 shows that there is a maximum flow rate, qmax and cor-responding time tmax, for each m value of greater than unity. The parameters qmax and tmax as a function of a and m are found in Appendix B. Fig. 4 also indicates that tmax is normally less than 30 days. Thus, the well will flow at the maximum flow rate within the first month of production if there are no well constraints such as rate, WHP, or bottomhole-flowing-pressure limits.

During production, a well may deviate from fracture flow and show high decline rate because of factors such as water production or liquid dropout/holdup in the fracture system and wellbore. The values of q1 and q∞(the rate at infinite time, see the Individual Well Analysis subsection) are designed to account for this acceleration in rate decline.

Eq. 6 indicates that Gp is proportional to q1 for any given time t. Thus, a plot of cumulative gas against the gas rate at Day 1 of all wells with similar a and m would fall on the same straight line passing through the origin. The slope of this straight line is 1

11 1

ae

am x

mt−− −( ) in which tx is the prediction time such as 24 months (730

days) or 60 months (1,825 days), as examples. To reduce the data uncertainties, the best 3-month average rate q3ma is used instead of q1. Because the rate reaches maximum value within the first

30 days of production, q3ma will be equal to a monthly average of the first 3 months or first 90 days. The slope of the straight line in

this case will be 3 1 901 1

eam tx

m m− −( )⎡⎣ ⎤⎦

− −

.The characteristic of a linear relation between the best 3-month

average and cumulative production has been reported by Frantz et al. (2005) for Barnett shale wells. To validate this characteristic, a sample of 25 Barnett wells was used. Actual best 3-month aver-ages for both vertical and horizontal wells were plotted against their cumulative productions of 1, 2, and 5 years, as shown in Fig. 5. A very good match is obtained. See more details in the Analog Type Curves subsection of the Field Examples section.

In summary, if the value any one of the three parameters 3q1, qmax, or q3ma is known, reserves evaluations can be established for any individual well, group of wells, development area, or play with a given set of a and m. The q3ma is preferred because it uses a longer production period and, hence, reduces data uncertainties.

Fig. 4 gives a set of type curves for a typical field-data plot because they are based on the real time (days) and a real relation between a and m. To construct type curves in a dimensionless form, the curves in Fig. 4 were normalized using tmax and qmax. The dimensionless time tm and dimensionless rate q/qmax will be

tt

tm =max

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

and

q

qt em

mm

mtm

m

max

= − − −⎛⎝⎜

⎞⎠⎟

−

11

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

The general type curve is shown in Fig. 6 for an m range of 1 to 1.96. Note that a has been eliminated from the type curves (see Eq. 8).

a vs. m

a = 0.7364 m4.7954

R 2 = 0.9417

0

0.5

1

1.5

2

2.5

3

0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3m

Tight/Shallow Gas

Dry Shale Gas

Wet Shale Gas

a

Fig. 3—A correlation between a and m for various gas plays.

q/q1 vs. time

0.01

0.1

1

10

1 10 100 1,000 10,000Days

m~1.00m=1.05m=1.10m=1.15m=1.20m=1.25m=1.3

q max

q/q

1

Fig. 4—Type curves for q/q1 vs. time in days for m>1.

0.00

0.50

1.00

1.50

2.00

0 5 10 15 20 25 30

Cum Gas, 106 m3

Bes

t-3-

Mo

nth

Ave

, 106

m3

1 year 2 years5 years

Dots - 25 Barnett SamplesSolid lines - Eq. B-5 (P50)

Fig. 5—Best 3-month averages for a sample of 25 Barnett wells were plotted against their cumulative productions of 1, 2, and 5 years.


For comparison, the Arps rate/time relations (Arps 1945) were reconstructed in the same plot using dimensionless time tD=Dit and dimensionless rate q1D for their normal range of b from 0 to 1. Note that q1D is defined as the dimensionless rate at tD=1, not the same as normally based on qi. The results show that Arps’ curve of b=0.5 is equivalent to the new curve of m=1.96 at late time (tD>10), as shown in Fig. 6. For b>0.5, Arps curves give more-optimistic forecasts compared to the new approach.

Field ExamplesIn this section, several field examples from different unconven-tional gas plays were used to test the new approach. Examples were selected to illustrate the decline behaviors in various completion and production scenarios such as the cases of retrograde gas, newly developing or developed plays, and vertical or horizontal wells. For changes in flow regime, a and m values will be highlighted for each example in the following discussion. A comparison of results is also made with other methods.

Retrograde-Gas Case. Wet-shale-gas plays have become more desirable lately because of higher liquid contents and attractive liquid price. Unfortunately, very little is known about their fl ow characteristics and how the liquid dropout in the formation and fracture network behaves.

The following example contains production data from four horizontal wells from a new emerging play. The production rates from both gas and liquid were recombined and converted into

gas-rate equivalent and finally were normalized on the basis of the highest rate from these wells, as shown in Fig. 7. Rates on Wells 2 and 3 were reinitialized because of some significant downtimes.

Fig. 7a shows that the flow regime varies from bilinear to linear flow. Generally, wells with bilinear flow regime will have lower maximum flow rate compared to wells with linear flow. The lower deliverability rates and/or bilinear flow may be caused by liquid holdup in fractures or by fracture choking.

Values of a and m obtained from a retrograde shale gas have covered a large range of values; these vary from 1.02 to 1.67 and from 1.04 to 1.21, respectively, as shown in Fig. 7b for this example.

Bakken Oil. An abandoned vertical well that produced from the Bakken formation in the Viewfi eld pool, Saskatchewan, Canada, for more than 26 years is shown here as an example for oil pro-duction from a fractured shale or tight formation. Fig. 8a shows the well was under a transient linear fl ow (half slope) for all of its production life except for a few months at the start of production. The late-time fl ow that indicates a big rate drop in oil rate can be mistaken as BDF, but in reality it is caused by an unexpected increase in water production.

The normalized decline-trend plots for the well are shown in Fig. 8b. If q/Np is based on the total liquid produced, the plot gives a good fitted straight line (R2 of 0.9824) with a and m values of 1.7894 and 1.1477, respectively. The values are slightly higher if only the oil stream is used in the calculation. Thus, an increase

0.001

0.01

0.1

1

1 10 100 1,000 10,000

b=0 b=0.5 b=1

m=1.05

m=1.15

m=1.30

m=1.60m=1.96

q/q

max

or

q/q

1D

tm or tD

tm = t/tmaxtd = Dit

q1D → q at td = 1qmax → q at tm = 1

Fig. 6—A comparison between Arps’ and the new method’s type curves for dimensionless rate and time.

0.01

0.1

1

1 10 100 1,000Days

No

rmal

ized

Rat

es

Well 1Well 2Well 3Well 4

½ slope¼ slope

y = 1.0164x –1.0409

y = 1.4112x–1.1121

y = 1.18x–1.1128

y = 1.6655x –1.2139

0.001

0.01

0.1

1

1 10 100 1,000Days

q/G

p, 1

/day

Well 1Well 2Well 3Well 4

(a) (b)

Fig. 7—A condensate/shale-gas example. Flow regimes that change from linear to bilinear in this retrograde play may be caused by liquid dropout in the fracture. There is a wider range for a and m observed in this play compared to other types.


in water production will cause both a and m to increase in value (Fig. 8b). Because of the unexpected increase in water production at the late time, an overprediction of only 8% in the well’s EUR is projected.

Individual-Well Analysis. In this example, a step-by-step pro-cedure is shown in Fig. 9 of how to perform decline analysis on an individual-well basis. A vertical well taken from the Barnett shale has been selected for this illustration. The well has been in production for the last 5 years and operated with minimum water production.

1. Step 1: Data Check and Correction. The production-data histories such as flowing WHPs, gas rate, and water rate are plot-ted as shown in Fig. 9a. The well was initially choked back at the surface on the basis of the observed flat rate and high WHPs. A correction was made for the rate on the basis of average operating pressures. The wet-gas rate should also be corrected to the dry-gas rate equivalent if there is a high condensate/gas ratio.

2. Step 2: a and m Determination. A log-log plot of q/Gp (or q/Np for oil) is constructed to determine these values, as shown in Fig. 9b. Data are examined to determine which section should be

used to obtain the representative a and m parameters for the well. The R2 value is used to determine the best fit of the data. An R2 value of more than 0.95 is recommended.

3. Step 3: Rate Forecast. q1 Determination. To obtain q1, gas-flow rate should be plotted against t(a, m). From Eq. 5, we have

q q t a m= 1 ( , ), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

where t a m t emam t m

( , ) = − −− −( )1

1 1 ..The q-vs.-t(a,m) plot should give a straight line through the origin with slope of q1. However, this may not be the best selection in some cases because of the current wellbore operating conditions. In such cases, Eq. 9 becomes

q q t a m q= + ∞1 ( , ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

where q∞ is the rate at infinite time, so it can be zero, positive, or negative. In either case, the abandonment rate should be set on the basis of economics.

0

1

10

100

1 10 100 1,000 10,000Days

Liq

uid

Rat

e, m

/d3

Oil+WaterOil Only

½ slope¼ slope

y = 1.7894x –1.1477

R2 = 0.9824

y = 1.8316x –1.1488

R2 = 0.9963

0.00001

0.0001

0.001

0.01

0.1

1

Days

, 1/d

ay

Oil+WaterOil Only

q/N

p

1 10 100 1,000 10,000

(a) (b)

Fig. 8—An oil example of fractured shale with 26 years of monthly production data. Fracture-dominated flow regimes lasted for all of its production life. Rate drop at late time is caused by an increase in water production. (a) Linear flow lasted more than 26 years (9,500 days). (b) Normalized decline-trend plots for different fluid streams.

Step 1: Data Check and Correction

1

10

100

1 10 100 1,000 10,000Days

Rat

e, 1

03 m

3 /d

1

10

100

WH

P, M

Pa

Press. Corrected RateMeasured RateWellhead Pressure

Skin effect

Skin effect andSurface choking

Step 2: a and m Determinations

y = 1.4105x–1.1384

R2 = 0.9744

0.0001

0.001

0.01

0.1

1

1 10 100 1,000 10,000Days

Step 3: q1 Determination

y = 33.7x

y = 35.18x – 0.5207

0

10

20

30

40

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

t(a,m)

Step 4: Validation and Forecasts

1

10

100

0 365 730 1,095 1,460 1,825 2,190

Rat

e 10

3 m

3 /d

or

Cu

m. 1

06 m

3 /d

Rate DataMatched RateCum GasCal. Cum Gas

a =1.4105 d–1

(a) (b)

(c) (d)

Gas

Rat

e, 1

03 m

3 /d

m =1.1384

q/G

p, 1

/day

q1=35.18 103 m3/dq =–0.52 103 m3/d

Days

Fig. 9—Four steps for using the proposed approach.


Once q1 (and q∞) have been determined, gas-rate forecasts can be made by using Eq. 9 or 10.

4. Step 4: Reserves Estimate. Eq. 4 Gqa

tpm=⎛

⎝⎜⎞⎠⎟ is used for

cumulative calculation with q>qeco, whereas EUR is estimated by Eq. 11:

EUR ecoeco= q

atm

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

where qeco is the minimum economic rate, and teco is at q= qeco.

Analysis Comparisons. To compare the results, other meth-ods such as Arps’ hyperbolic with b greater than unity and the recently developed power-law-exponential method (Ilk et al. 2009) were selected. Texas A&M University has developed a rate/time spreadsheet on using the power-law-exponential approach (Ilk et al. 2008, 2009). Data from Example 3 from this spreadsheet was employed here for the purpose of comparison. The analysis results are shown in Fig. 10. The results from the power-law-exponential approach and Arps’ hyperbolic with b of 1.9 are included in Fig. 10 for comparison.

The reserves that were calculated on the basis of time limit of 30 years and economical rate of 2.83×103 m3/d (100 Mscf/D) are summarized in Table 1.

Table 1 shows that the new approach provides conservative reserves estimate compared with the other two methods.

Analog Type Curves. A fi eld example was used to illustrate how a developing play can be used as an analog for other similar plays.

This analog is in the form of type curves (P10, P50, and P90), which were generated by the new technique.

A set of 25 wells that includes eight horizontal wells and 17 vertical wells from the Barnett shale was selected for this study. In this study, there is not much difference between the vertical and horizontal wells except for their initial rate. For this reason, no attempt was made to separate them into two different data sets.

The following steps are proposed to establish the type curves:1. Establish a and m for each individual well. a and m were

determined for all 25 wells, as shown in Fig. 10b.2. Construct a probability plot to obtain the P10, P50, and P90

values for m (Fig. 11a).3. Establish an a–m relationship such (as shown in Fig. 3) if

sufficient data are available. Otherwise, use the correlation from that figure.

4. Estimate the best 3-month rate averages for both vertical and horizontal wells. The best 3-month rate averages are 0.51×106 m3/month (18 MMscf/month) and 1.10×106 m3/month (39 MMscf/month) for the vertical and horizontal wells, respectively. For com-parison, Frantz et al. (2005) obtain the best 3-month gas averages of 0.54×106 m3/month (19 MMscf/month) and 1.25×106 m3/month (44 MMscf/month) that are based on a data sample of 2,842 verti-cal wells and 246 horizontal wells from the Barnett shale.

5. Use Eq. B-5 to plot q3ma against Gp for various values of a and m to construct Fig. 12 for new well EUR prediction. Fig. 12a is for the EUR prediction based on the time limit of 30 years regardless of flow rate, whereas Fig. 12b is based on 30 years of production with a cutoff rate of 2.83×103 m3/d (100 Mscf/D).

6. Use Eq. B-3 to estimate the q1 values (P10, P50, and P90) based on the q3ma for both vertical and horizontal wells. The results for this example are shown in Table 2. The type curves, as shown in Fig. 13, were constructed on the basis of these parameters.

Conventional Tight Gas Wells. All examples illustrated thus far result in an m value greater than 1. However, m can be less than 1,as shown in Fig. 3. The cases in which m is less than unity in Fig. 3 are from a shallow and tight gas formation. A typical rate-decline analysis is shown in Fig. 14.

Notice the slightly concave-up shape of the rate plot, as shown in Fig. 14d. This concave-up shape, which is caused by a late time pseudoradial flow is characteristic for m<1. For m>1, a log-log plot of rate vs. time will display a concave-down feature (Fig. 4).

a and m

y = 1.1142x–1.0895

0.0001

0.001

0.01

0.1

1

1 10 100 1,000 10,000Days

q/G

p, 1

/day

a =1.1142m =1.0895

Rate at Day 1

y = 140.35x – 0.4779

0

50

100

150

0 0.2 0.4 0.6 0.8 1t(a,m)

Gas

Rat

e, 1

03 m

3 d

q1 =140 103 m3d

q∞=–0.48 103

Rate Comparison

1

10

100

1,000

1 10 100 1,000 10,000 100,000Days

Gas

Rat

e, 1

03 m

3 d

DataPower-Law ExpHyperbolicNew Approach

Cumulative Comparison

0.1

1

10

100

1,000

1 10 100 1,000 10,000 100,000Days

Cu

m G

as, 1

06 m

3 d

DataPower-Law ExpHyperbolicNew Approach

(a) (b)

(c) (d)

Fig. 10—Result comparison with power-law-exponential method and Arps hyperbolic with b=1.9.

TABLE 1—RESERVES COMPARISON BETWEEN METHODS

Time limit 30 Years

Rate limit 2.83 103 m3/d

Time limit Rate limit

Model Gp, 106 m3 Gp, 106 m3

Hyperbolic (b=1.9) 118 222

Power-law exponential

116 188

New approach 106 141


This feature distinguishes flow characteristics of a conventional (late-time pseudoradial flow with m<1) vs. unconventional (lack of late-time pseudoradial flow or m>1) reservoir.

DiscussionAbsence of Late-Time Pseudoradial Flow. Unsurprisingly, the m value for all shale-gas wells investigated in this study is greater than unity, or a log-log plot of rate vs. time will display a concave-down feature (Fig. 4). This fl ow characteristic, which indicates an absence of the late-time pseudoradial fl ow from shale gas reser-voirs, can be explained as follows.

Fig. 15 was constructed to distinguish between a conventional (tight) and shale-gas formation with an assumption that Point A, the wellbore location, has the same pressure in both conventional and unconventional formations. If Well A is located in a tight but conventional formation, its pressure will be in equilibrium with

its fluid gradients. As in the case of a gas-saturated formation, the pressure gradient should be approximately 2 kPa/m. A hydrauli-cally fractured well located in such a formation will be expected to pass through several distinct flow regimes such as fracture linear, bilinear, elliptical, and pseudoradial flows, and BDF, as discussed by Clarkson and Beierle (2010). An unconventional reservoir may not be working according to such a model because there is no matrix transient flow here to give rise to radial or elliptical flow.

If the reservoir is shale, the pressure gradient will be overpres-sured. This can be up to 63 kPa/m (3 psi/ft), as recorded in the Lower Bossier play (Newsham and Rushing 2002) and shown in Fig. 16. Spencer (1987) shows some pressure gradients in overpressured and gas-bearing shale plays in the Rocky Mountain region. The interpreted pressure profile for the MWX site wells of the Piceance basin shows that the pressure gradient based on mea-sured pore pressures in the active zone is approximately 36 kPa/m (1.7 psi/ft). The author of this study has also experienced pressure

Cumulative Normal Probability Chart Constants a and m for 25 Barnett wells

R a mP90 1.88 1.210P50 1.41 1.140P10 1.00 1.075

1

0.1

0.01

0.001

0.0001

q/G

p, 1

/day

1 10 100 1,000 10,000Days

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 19 20 21 22 2325 24 26 27 28

P10

P50

P90

SlopesLinear (Slopes)

0.01

0.10

0.50

0.90

0.99

Cu

mu

lati

ve N

orm

al P

rob

abili

ty

1 1.05 1.1 1.15 1.2 1.25 1.3m

(a) (b)

Fig. 11—A statistical approach to analyze production forecasts for a group of wells or a set of sample wells: (a) a probability plot for 25 wells and (b) a and m for all 25 wells.

EUR RANGES FOR BARNETT (30 YEARS)

0.0

0.5

1.0

1.5

2.0

0 20 40 60 80 100EUR, E6 m3

BE

ST

-3-M

ON

TH

AV

E, 1

06 m

3

P90P10

P50

Time limit only

EUR RANGES FOR BARNETT (2.83 103 m3/d)

0.0

0.5

1.0

1.5

2.0

0 20 40 60 80 100EUR, E6 m3

BE

ST

-3-M

ON

TH

AV

E, 1

06 m

3

P90P10

P50

EUR shifted becauseof rate cutoff

Rate limit with 30 years

(a) (b)

Fig. 12—EUR prediction based on time- and rate-limit constraints. Time limit would only give an optimistic prediction for EUR if the initial best rate is low.

TABLE 2—PARAMETERS FOR TYPE-CURVE GENERATION FOR BARNETT SHALE

Vertical well Horizontal well q3ma, 106 m3/month

0.51 1.10

P90 P50 P10 P90 P50 P10

a 1.88 1.42 1.00 1.88 1.42 1.00

m 1.21 1.14 1.07 1.21 1.14 1.07

q1, 103 m3/d

12.19 19.09 32.58 26.40 41.37 70.59


distributions across producing gas shales exceeding the lithostatic gradient, as in the Montney and other studied plays.

With this high pressure gradient in shale gas, the formation is thought to be in a dynamical equilibrium state in which hydro-carbons have been generated and leaked off the formation over geologic time. Under a normal production time scale, shale-gas formations can be treated as impermeable barriers that contain millions and millions of tiny isolated reservoirs (such as sandy laminates, microcracks, or isolated sand bodies); to produce gas

from this shale formation, these tiny reservoirs need to be con-nected by a fracture network, without which they will remain isolated and unproductive.

Because of such isolated characteristics of microreservoirs in shale-gas formations, capillary pressure or relative permeability curves should be applicable in each individual tiny reservoir, but they may not be applicable applied across the entire shale forma-tion. This can explain why the late-time radial- or elliptical-flow regimes are absent from all shale-gas wells studied.

Numerical-Model Comparison. To use the current conventional modeling for fractured shale, some assumptions or modifi cations must be made. This section proposes and briefl y discusses a frac-ture model consisting of a macrofracture located inside an SRV. To overcome the nonequilibrium issue, permeability within the SRV is assumed to have been anisotropically enhanced by stimula-tion so that the pressure within it is in a state of equilibrium and can be matched to the post-stimulation transient data. To prolong the linear-fl ow regime, the drainage area or SRV should be large enough to imitate the gradual increase in the SRV as fractures increase the connectivity.

A commercial reservoir simulator is used here for numerical-modeling work so that various fracture models in both conventional and unconventional reservoirs can be compared. Adsorbed gas

BARNETT VERTICAL WELL TYPE CURVES

0

2

4

6

8

10

12

14

16

0 12 24 36 48 60 72 84 96 108 120Months

Gas

Rat

e, 1

03 m

3 /d P90

P50P10

BARNETT HORIZONTAL WELL TYPE CURVES

0

5

10

15

20

25

30

35

0 12 24 36 48 60 72 84 96 108 120Months

Gas

Rat

e, 1

03 m

3/d P90

P50P10

(a) (b)

Fig. 13—Type curves for Barnett shale based on a set of 25 wells, for both vertical and horizontal wells.

(a) (b)

(c) (d)

Rate and Pressure Histories

0

1

2

3

4

0 365 730 1095 1460Days

0

1

2

3

4

Pre

ssu

re, M

Pa

Gas RatePressure

a and m Values

y = 0.6226x –0.9624

R 2 = 0.9865

0.0001

0.001

0.01

0.1

1

1 10 100 1,000 10,000Days

q/G

p

q1 Determination

y = 5.1312x

y = 5.3375x – 0.0424

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5t(a,m)

Results

0.01

0.1

1

10

Rat

e o

r C

um

, 103

or

106

m3 /

d

Actual RateCal. RateActual Cum.Cal. Cum

1/4 slope

Gas

Rat

e, 1

03 m

3 /d

Gas

Rat

e, 1

03 m

3 /d

Days1 10 100 1,000 10,000

Fig. 14—A rate-decline example from a shallow and tight gas formation.

AUnconventional

Shale Gas20–60 kPa/mConventional

Tight Gas≈ 2 kPa/m

Top reservoir

Bottom reservoir

Pressure

Depth

Fig. 15—Pressure-gradient distribution in tight and shale gas.


is not considered in the study but can be included in the model. Fig. 17 is a summary of the results in the form of normalized decline-trend plots. The q/Gp ratio of the proposed fracture model is shown as the blue line. If the stimulated region is much smaller, the red or BDF curve proposed by Bello and Wattenbarger (2008) is obtained.

To compare with the proposed analytical method, the thin white line with the matched a and m factors of 0.7785 and 1.0481, respectively, is produced. This line lies exactly on top of the blue line from the numerical simulation result.

The yellow curve shows one of the results of a normal reser-voir model where the virgin area surrounding the SRV is also in pressure communication. This yellow curve, which includes a late-time linear flow from the virgin region into the SRV, can also be obtained using the approach from Anderson et al. (2010). Another result is shown as a green line where both stimulated and virgin regions are the same or under a homogeneous state. Pseudoradial flow will be achievable to make m a value of less than unity. Fig. 14 illustrates a field example of a gas well producing from a tight reservoir.

Method Limitations. Downtime Effect. Because of some opera-tional conditions, a well may have to be shut in for some periods of time during its production life. If the downtime of each shut-in is signifi cant, rate initialization is required as in traditional decline-

curve-analysis methods. To use q/Gp, both rate and cumulative production should be initialized with pressure correction. Fig. 7 shows the example plots of rate initializations for Wells 2 and 3.

EUR Constraints. In most cases, without the presence of BDF in shale reservoirs, drainage area cannot be established for EUR prediction. In this method, the drainage area will be defined only when the fracture-expansion process is brought to an end. Thus, EUR is not based on the traditional concept of drainage area (BDF) but on the constraints of the latest trends of (q1, q∞) with both time and economic rate limits. Because of this, results are valid only where analog wells are similar in both interwell and fracture spacings.

In case of spacing differences, a numerical reservoir modeling (see the Numerical Model Comparison subsection) should be used for any EUR adjustment.

Latest Trends. During a well production, many factors can contribute to the deviation from a fracture flow to accelerate the decline rate such as water production or liquid dropout/holdup in the fracture system and wellbore. The values of q1 and q∞ (Eq. 10 and Fig. 9) are designed to account for this acceleration in rate decline in late time. An increase in water production will also cause both a and m to increase in value (Fig. 8).

Time Limit vs. Rate Limit. Fig. 12 shows a typical EUR predic-tion for an infill in the Barnett shale for both vertical and horizontal wells. The prediction based on a time limit alone would give an

≈ 36 kPa/m pressure distribution (measured

pore pressure)

≈ 63 kPa/m pressure distribution (acoustic

log response)

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

12,000

13,000

0 2,000 4,000 6,000 8,000 10,000 12,000 0 120 200 50

ft

Pressure, psig 1,000/div GR Acoustic

Sheliya1 .D45Mcadmsa7 .D45

Mcadmsa7

2

3

4

5

6

4

8

9

ft EXPLANATIONMEASURED PORE PRESSUREMWX-1 MUD-WEIGHT PRESSUREMWX-3 MUD-WEIGHT PRESSUREINTERPRETED PRESSURE GRADIENTFRACTURE GRADIENT IN SHALE AND MUDSTONEFRACTURE GRADIENT IN SANDSTONEFRACTURE GRADIENT IN COAL

M

DE

PT

H (

X 1

,000

)

2 3 4 5 6 7 8 9

PRESSURE (X 1,000)

10 20 30 40 50 60 kPaPSI

1.0

1.5

2.0

2.5

0.7% RO

0.8% RO

165°F (74°C)

200°F (93°C)ACTIVE ZONE

INACTIVE ZONE

0.8 PSI/FT

0.6 PS

I/FT0.433 P

SI/FT

(a) (b)

Fig. 16—Two examples of high pressure-gradient distribution across the producing shale gas: (a) the Lower Bossier play (New-sham and Rushing 2002) and (b) the Piceance basin (Spencer 1987).

Normalized Decline Trend Plots

y = 0.7787x–1.0481

R 2 = 0.9997

0.00001

0.0001

0.001

0.01

0.1

1

1 10 100 1,000 10,000Days

q/G

p, d

ay–1

Normalized Decline Trend Plots

y = 0.7787x-1.0481

R2 = 0.9997

1.E-05

1.E-04

1.E-03

Days

Bello et al. 2008Bounded SRV or BDF

ConventionalPseudoradial m<1

Anderson et al. 2010 Contribution from

Undisturbed Shale

Proposed ModelExpanding SRV

and Anisotropic k

(a) (b)

q/G

p, d

ay–1

100 1,000 10,000

Fig. 17—Normalized decline trend plots for various simulated fracture models in shale or tight reservoirs: (a) for all production data and (b) at late time.


optimistic value. Note that the EUR shifts to the left when the rate limit was imposed on the time-limit plot, as shown in Fig. 12b. The lower the initial rate is, the higher and more optimistic the value of the predicted EUR will be. This observation is especially applicable to vertical wells because of their lower initial rates. The recommendation is to use both time and economic rate limits for predicting EUR.

Conclusions1. A new approach that is easy and simple to use for predicting the

future rate and EUR has been developed for fracture-dominated wells in unconventional reservoirs.

2. The new approach is also able to provide a statistical method to analyze production forecasts of resource plays and establish a range of results for these forecasts, including probability dis-tributions of reserves in the form of P90 to P10.

3. A log-log plot of rate over cumulative production vs. time is observed to fit a straight line in all unconventional-reservoir cases studied so far. The slope and intercept are related to res-ervoir-rock characteristics (tight or shale), fracture-stimulation practice, operational conditions, and (possibly) liquids content.

4. EUR for the new approach is not based on the traditional concept of drainage area (BDF) but on the constraints of the latest trends (q1, q∞ ) with both time and economic rate limits.

5. Pressure initialization used in numerical modeling based on fluid gradients may be incorrect. Results from such numerical modeling may not be representative of the shale-gas flow char-acteristics. Some modifications of flow characteristics in both stimulated and undisturbed regions are needed.

Nomenclature a = intercept constant defi ned by Eq. 4, d−1

b = Arps’ decline exponent, dimensionless Di = Arps’ initial decline rate, d−1

G = original gas in place, m3

Gp = cumulative gas production, m3

kf = fracture permeability, 10−3 µm2

km = matrix permeability, 10−3 µm2

m = slope defi ned by Eq. 4 n = fracture time exponent, dimensionless Np = cumulative oil production, m3

q = rate, m3/d q1 = rate at Day 1, m3/d q1D = rate at tD=1, m3/d q3ma = best 3-month average rate, m3/month qeco = economic cutoff rate, m3/d qmax = maximum rate, m3/d q∞ = rate at infi nite time, m3/d t = time, days tD = dimensionless time defi ned by tD=Dit tDxf = dimensionless time based on fracture half-length tm = dimensionless time based on Eq. 7 tmax = time at maximum fl ow rate, days tx = prediction time, days t(a,m) = time function based on Eq. 9 xf = fracture half-length, m ε = decline rate defi ned by ε( )t

qGp

= , d−1

AcknowledgmentsThe author would like to thank the management of ConocoPhillips for their support and permission to publish this study. He is also grateful to Dev Mukherjee, Chris Clarkson, Sheila Reader, and Kevin Raterman for reviewing the paper; to coworkers in both the Houston and Calgary offices for peer review of the proposed method; and to Matt Maguire and his group for supplying the Barnett production data.

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Appendix A—General Equations for q and Gp

General equations are developed on the basis of q

Gt

p

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

ε( ) .

q

tGpε( )

= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)

Taking a derivative with respect to time, we have

dd

ddt

q ttGp/ ( )ε[ ] = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

Using q’= dq/dt and ε’(t) = ddε( )t

t, Eq. A-2 becomes

q t q t t q'/ ( ) '( ) / ( )ε ε ε− =2 , . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

or

d d dq q t t t t/ ( ) / ( ) ( )= +ε ε ε . . . . . . . . . . . . . . . . . . . . . . . . (A-4)

Integrating both sides from t=1 to t, we have:

ln ln( )

( )( )

q

q

tt t

t

1 11= ⎡

⎣⎢⎤⎦⎥

+ ∫εε

ε d , . . . . . . . . . . . . . . . . . . . . . . . . (A-5)

where q1 is the theoretical rate at t=1 day,

q qt

et t

t

=∫

11

1.

( )

( )

( )εε

ε d, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-6)

and

Gq

et t

p

t

=∫

1 1

1ε

ε

( )

( )d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-7)

If

q

Gpat m= − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-8a)

or

ε( )t at m= − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-8b)

Eqs. A-6 and A-7 can be written as:

q

qt em

am

t m

1

111

= − − −( )−

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-9a)

and

Gqa

ep

am

t m

= − −( )−1 1

11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-9b)

Appendix B—Specific Equations for t and qtmax and qmax. Fig. 4 shows there is a maximum rate flow qmax for each line where m is greater than unity. Time tmax, which gives qmax, can be estimated by

tm

a

m

max = ⎛⎝⎜

⎞⎠⎟

−1

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1)

and

q q am

em

m m am

max = ( ) − −−

1

11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-2)

Eq. B-1 was obtained by allowing the rate derivative of Eq. 5, dq/dt, to approach zero.

Best 3-Month Average, q3ma. Since tmax is less than 45 days (Fig. 4), the best 3-month average (q3ma) will be equal to the average of the fi rst 3 months of production and can be calculated by

qq

emaa

am

m

31 1

90 1

3

1

= − −( )−

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-3)

Eq. B-3 is obtained on the basis of Eq. 6 by calculating the cumula-tive gas of the first 3 months of production. Gas rate and cumulative gas based on this q3ma would be

q q at emam

am

t m m

= − − −( )− −

3 31

901 1

. . . . . . . . . . . . . . . . . . . . . . . . (B-4)

and

G q ep ma

am

t m m

= − −( )− −

3 31

901 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . (B-5)

Eq. B-5 is used to construct the solid lines in Fig. 3.

Anh N. Duong is a principal reservoir engineer and holds the title Reservoir Modeling Director at ConocoPhillips Canada, where he has worked since 2001. E-mail: [email protected] or [email protected]. He heads a subsur-face team that provides technical expertise in reservoir engi-neering and modeling to all ConocoPhillips Canada assets. These include oil sands, shale gas, and tight oil and gas res-ervoirs. Before joining ConocoPhillips, Duong worked 21 years for Gulf Canada Resources as a reservoir engineer. He holds a BSc degree (1978) in petroleum engineering from Petroleum University of Technology (Abadan Faculty). In late 1980s, Duong pursued but did not complete an ME degree in reservoir engi-neering at the University of Calgary.

Discussion of Rate-Decline Analysis forFracture-Dominated Shale Reservoirs

Martin Wolff, SPE, Occidental Petroleum Corporation

Duong (2011) presents a new forecasting approach using log-lograte/cumulative production (q/Gp) vs. time (t) plots for wells inunconventional reservoirs which often exhibit long-term linearflow. He also relates parameters of his new method to Arps curvesand discusses both linear and bilinear flow. Kupchenko et al.(2008) also discuss linear and bilinear flow and how they relate toArps parameters but note that rate and/or rate-derivative plots mayrequire sophisticated filtering to become interpretable because ofthe noisiness of field data. The following discussion presents an-other approach linked to Arps parameters which complements themethods proposed by Duong and Kupchenko et al.

A classic decline method, the linear-flow plot of cumulativeproduction vs. the square root of time (Nott and Hara 1991), sug-gests some advantages of using cumulative production with noisyfield data. These include the inherent filtering involved with cu-mulative rather than rate or derivative plots, as well as the ease offitting straight lines to Cartesian plots. Although such plots arenot as diagnostic as sensitive derivative plots, they may havesome advantages in allowing relatively robust and repeatableinterpretations once suitable time exponents are selected [possiblyfrom plots of q/Gp vs. 1/t as suggested by Eq. 3 in Duong (2011)].The following derivation shows how to construct generalized Car-tesian cumulative production vs. time plots for arbitrary b factors(=1).

Eqs. 1 and 2 define cumulative production Gp and instantane-ous rate q at time t for production declines following the Arpshyperbolic form with initial rate qi and initial decline, Di.

Gp ¼qi

b

ð1� bÞDiðq1�b

i � q1�bÞ: ð1Þ

q ¼ qið1þ b Di tÞ�1=b: ð2Þ

Substituting for q in Eq. 1 and rearranging, we can rewrite Eq. 2as

Gp ¼qi

ð1� bÞDi� qið1þ b Di tÞðb�1Þ=b: ð3Þ

When ðb Di tÞ� 1, the equation simplifies to the linearizedequation

Gp ¼ A tðb�1Þ=b þ B; ð4Þ

where

A ¼ qi

ð1� bÞDiðb DiÞðb�1Þ=b: ð5Þ

and

B ¼ qi

ð1� bÞDi: ð6Þ

Therefore, we can construct a Cartesian plot of Gp against tn

where n¼ (b–1)/b. This allows easily fitting a straight line whichcorresponds to Slope A. For the special case of b¼ 2, n¼ 0.5which is the classic linear-flow plot with Gp plotted against thesquare root of time. For bilinear flow with b¼ 4, n¼ 0.75.

When a range of hyperbolic decline exponents is used todefine a range of forecasts for a particular play or field, a set ofCartesian plots with varying n can be used to fit the data more eas-ily, rather than fitting both Di and b as required by Arps hyper-bolic forms. Figs. 1 and 2 show sample field data plotted with fitsfor b¼2 and b¼1.1 (which could represent a more-conservativeprojection, possibly for proven reserves). In this case, the b¼2plot shows a better fit from earlier time, suggesting some form oflinear flow; however, extended production histories are needed toconfirm how long this transient-flow regime will actually last indifferent situations.

As both Duong (2011) and Kupchenko et al. (2008) note, theArps hyperbolic form, with all its flaws but also with its well-understood parameters, remains the most common forecastingform used by the industry. Therefore, these Cartesian square-root-of-time plots may have application as simple, repeatable methodsfor forecasting with relationships to Arps decline factors.

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Linear-Flow Plot

y = 1.2325×104 – 8.0794×104

R 2 = 9.9942×10–1

0.0 5.0 10.0 15.0 20.0

Square root, Days

Cu

mu

lati

ve G

as, M

scf

Gas

Rat

e, M

scf/

D

EUR = 1.2 Bscf

Fig. 1—Field example of b52 linear-flow plot (time0.5). EUR5esti-mated ultimate recovery.

b = 1.1 Plot

y = 6.5763×105x – 9.7189×105

R2 = 9.9751×10–1

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

t 0.091, days

Cu

mu

lati

ve G

as, M

scf

Gas

Rat

e, M

scf/

D

EUR = 0.6 Bscf

Fig. 2—Field example of b51.1 plot (time0.091).Copyright VC 2014 Society of Petroleum Engineers

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2014 SPE Reservoir Evaluation & Engineering 1

References

Duong, A.N. 2011. Rate-Decline Analysis for Fracture-Dominated Shale

Reservoirs. SPE Reservoir Evaluation & Engineering 14 (3): 377–387.

SPE-137748-PA. http://dx.doi.org/10.2118/137748-PA.

Kupchenko, C.L., Gault, B.W., and Mattar, L. 2008. Tight Gas Production

Performance Using Decline Curves. Presented at the CIPC/SPE Gas

Technology Symposium, Calgary, Alberta, Canada, 16–19 June. SPE-

114991-MS. http://dx.doi.org/10.2118/114991-MS.

Nott, D.C. and Hara, S.K. 1991. Fracture Half-Length and Linear Flow in

the South Belridge Diatomite. Presented at the SPE Western Regional

Meeting, Long Beach, California, 20–22 March. SPE-21778-MS.

http://dx.doi.org/10.2118/21778-MS.

REE137748 DOI: 10.2118/137748-PA Date: 16-April-14 Stage: Page: 2 Total Pages: 2

ID: komathi.k Time: 09:44 I Path: S:/3B2/REE#/Vol00000/140019/APPFile/SA-REE#140019

2 2014 SPE Reservoir Evaluation & Engineering

http://dx.doi.org/10.2118/137748-PA

http://dx.doi.org/10.2118/114991-MS

http://dx.doi.org/10.2118/21778-MS

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