spe 116731 (ilk et al) power law exponential relation [wpres]

7/29/2019 SPE 116731 (Ilk Et Al) Power Law Exponential Relation [wPres]

1/51

SPE 116731

Exponential vs. Hyperbol ic Decline in Tight Gas Sands Understandingthe Origin and Implications for Reserve Estimates Using Arps' DeclineCurvesD. Ilk, Texas A&M University, J.A. Rushing, Anadarko Petroleum Corp., A.D. Perego, Anadarko Petroleum Corp.,and T.A. Blasingame, Texas A&M University

Copyright 2008, Society of Petroleum Engineers

This paper was prepared for presentation at the 2008 SPE Annual Technical Conference and Exhibition held in Denver, Colorado, USA, 2124 September 2008.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not beenreviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, itsofficers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission toreproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abst ractWhen tight gas sand reserves are assessed using the Arps rate-time equations, the decline behavior is typically defined in

terms of the Arps decline exponent, b. The original Arps paper indicated that the b-exponent should lie between 0 and 1.0 ona semilog plot. However, in practice we often observe values much greater than 1.0, especially prior to the onset of true

boundary-dominated flow. Unfortunately, the correct b-exponent is difficult (if not impossible) to identify during the early

decline period and (obviously) the selection of the wrong b-exponent will have a tremendous impact on reserve estimates,particularly when the b-exponent estimate is too high.

As an exercise to evaluate the b-exponent as a continuous function of time, we have used synthetic and field production

profiles. We then compare the computed b-exponent trend graphically to assess the "hyperbolic" nature of each case (recall

that the b-exponent should be constant for a given hyperbolic rate decline). The field data cases used in this study wereselected from a tight gas reservoir that has been previously evaluated on a per well basis using the production model based on

the elliptical flow concept. These cases indicate that only portions of the production history are matched by the hyperbolic

rate decline relation suggesting that using the hyperbolic relation by itself may not be appropriate for reservesextrapolations in tight gas reservoirs, or at least that great care must be used in creating production forecasts based on the

hyperbolic rate decline relation.

In addition to the hyperbolic rate decline relation we have also developed and employed a new "power law loss-ratio" rate

relation that has more generality than the hyperbolic rate decline relation. This new model tends to match production rate

functions much better than the hyperbolic rate decline relation for tight gas and shale gas applications, but we must stress that

at this time, the "power law exponential decline" rate relation is empirically derived from only tight gas/shale gas per-formance cases. We have applied the new model as well as the hyperbolic rate model to two synthetic (simulated) and field

(tight gas well) cases for production forecast.

Furthermore, the results of our synthetic performance cases do suggest that layered reservoir behavior can be accurately

represented by the hyperbolic rate decline relation. Unfortunately, as other studies have shown, multilayer reservoir per-

formance can be extremely difficult to generalize particularly when layers in transient and boundary-dominated flow are

i i ti H b li t d li l ti i ht b id d t bl h i f ti ti


2/51

2 D. Ilk, J.A. Rushing, A.D. Perego, and T.A. Blasingame SPE 116731

Introduction

Loss-Ratio, Exponential and Hyperbolic Rate Decline Functions:

Johnson and Bollens [1928] and later Arps [1945] presented the so-called "loss-ratio" and the "derivative of the loss-ratio"

functions as:

dtdq

q

D /

1 (Definition of the Loss-Ratio) .......................................................................................... (1)

dtdq

q

dt

d

Ddt

db

/

1(Derivative of the Loss-Ratio) .......................................................................................... (2)

Where Eqs. 1 and 2 are written in terms of "modern" variables ( i.e.,q, D, b). Eqs. 1 and 2 are empirical results based on

observations. For the case ofD = constant, Eq. 1 does yield the exponential decline which can be derived for the case ofpseudosteady-state (or boundary-dominated) flow in a closed reservoir containing a constant compressibility liquid and being

produced at a constant wellbore flowing pressure. The exponential rate decline relation is given as:

]exp[ tDqq ii = ............................................................................................................................................................... (3)

Ilket al[2008a] provide and alternate computation of the D and b-parameter using rate-cumulative data. The alternateD-

parameter formulation is given as:

dQ

dqD ......................................................................................................................................................................... (4)

And the alternate b-parameter formulation is given by:

DdQ

dqb 1 ................................................................................................................................................................... (5)

For reference, Blasingame and Rushing [2005] provide the derivation of the "hyperbolic" rate decline relation in complete

detail. The intermediate result of interest to this work is the expression for the decline parameter (D) for a hyperbolic rate

decline, theD-parameter is defined as:

btD

D

i+

=1

1..................................................................................................................................................................... (6)

Completing the derivation (see Blasingame and Rushing [2005] for details), the hyperbolic rate decline relation is given as:

bi

itbD

qq/1

)1(

1

+= ........................................................................................................................................................ (7)

Eq. 6 is relevant because part of our methodology will be to calculate theD and b-parameters as functions of time to illustrate

the characterof these variables. We recall that for the "hyperbolic" rate decline relation (i.e., Eq. 7), the b-parameter isconstant, and theD-parameter isprescribedby Eq. 6. Therefore, we can test how "hyperbolic" a particular data case may be

using theD-parameter computed using Eq. 1 (rate-time) or Eq. 4 (rate-cumulative). Similarly, we can then estimate the b-parameter using Eq. 4 (rate-time) or Eq. 5 (rate-cumulative). There will typically be minor differences in the "rate-time" and

"rate-cumulative" estimations of the D and b-parameters this is due to data quality (the rate-cumulative is always"smoother"), as well as the numerical derivative algorithm used (end-point effects).

Historical Perspective Exponential and Hyperbolic Rate Decline Functions:

As mentioned before the conditions for the establishment of the exponential rate decline is well-known constant

compressibility flowing fluid under pseudosteady-state (or boundary-dominated) flow regime in a closed reservoir with

constant flowing wellbore pressure operating conditions. On the other hand, the conditions under which a hyperbolic rate

d li t ll h b t i f d b t i it i t d ti th 80 l t 20 b f


3/51

SPE 116731 Exponential vs. Hyperbolic Decline in Tight Gas Sands Understanding the Origin 3and Implications for Reserve Estimates Using Arps' Decline Curves

Arps depletion-decline exponents (b-values) between 0.5 and1 can be obtained with a layered-reservoir description given (a)sufficient contrast in layer properties. A depletion-decline exponent (b) > 1.0 could not be maintained for any combination ofproperties investigated.

Except for the special cases in which (qrnax/Gi)R approaches unity or infinity, the composite-depletion b-value is always greater

than that of a single-layer system. Field and well rate/time data that exhibit higher-than-expected b-values (between 0.5 and1.0for gas reservoirs) suggest a layered reservoir system. Large initial percentage declines, not attributable to infinite-actingtransient production, followed later by small percentage declines, are a characteristic of high b-values and suggest a layered-reservoir response.

Clearly, Fetkovich et al[1990] believed that "high" b-values must be a product of reservoir layering. In addition, Fetkovich

et al[1990] also note that "infinite-acting transient production"yields "high b-values" which should NOT be interpretated

as boundary-dominated flow behavior (hence, not used for reserves extrapolations).

Fetkovich et al[1996] further considered the practical aspects of production performance from "layered no-crossflow" gas

reservoirs. In this work Fetkovich et alsuggest that "commingled" production (summed layered flowrates) can be modeled

using the hyperbolic flow relation in terms of total reservoir volume and the "best fit" qi and b-parameters. Fetkovich et al[1996] also stated the following "important characteristics" of layered, no-crossflow reservoir behavior:

Low-permeability, stimulated wells' production performance can appear similar to layered, no-crossflow reservoir responses ona semilog production curve. However, a log-log type curve be used to distinguish between the two. Further confirmation of no-crossflow can be made by measuring layer pressures and having some idea of the well's permeability level.

High value of the decline exponent, b: b> 0.5. This is reflected as an early rapid decline in rate followed by an extended periodof a low percentage decline.

Reservoir has an indicated unusually long producing life.

Thick reservoirs have a very high likelihood of exhibiting layered, no-crossflow behavior. (There appears be a strong correla-

tion of b with (reservoir) thickness.)

In a recent study, Rushing et al [2007] validated many of the points made by Fetkovich et al [1996] using a numericalsimulation study that was designed to assess the validity of estimating reserves using the hyperbolic rate decline relation (Eq.

7). The Rushing et al simulation study considered the combined effects of reservoir layering, heterogeneity (position-

dependent permeability), fracture half-length, and fracture conductivity as well as high reservoir pressures and tempera-

tures, and certain pressure-dependent properties. The "approach" used was to compare the hyperbolic regression and fore-

cast at specific time intervals (1, 5, 10, 20, and 50 years) compared to the "reference" cumulative production from thenumerical model at a specified production limit (in this case, 50 MSCF/D). This approach showed that, indeed, reserves esti-

mates based on the hyperbolic rate decline relation improve quite substantially with time from well over 100 percent error

in reserves estimates in the first year, to significantly less than 1 percent error at 50 years, for essentially all cases.Procedure and Development of the New Model:

Our approach in this work differs from that of Rushing et al[2007] in that we will estimate theD and b-parameters from the

rate performance data and evaluate the "applicability" of the hyperbolic rate decline relation (Eq. 7) at any particular time.

Not to prejudge our process, but we expect to observe "non-hyperbolic" behavior (i.e., b exponent is not a constant) for avariety of reasons such as multilayer effects (as suggested by Raghavan [1989], Fetkovich et al[1990], and Fetkovich et al

[1996]) should have a dominating effect on rate performance. We also expect the inclusion of transient flow data to

significantly affect theD and b-parameters. But in addition, we believe that there may be other mechanism(s) involved inparticular, in the case of low permeability/heterogeneous reservoirs, we believe that "contacted-gas-in-place" (CGIP)

increases in time, and we recognize that this phenomenon will tend to cause "non-hyperbolic" behavior. The intent of thispaper is not to address the "contacted-gas-in-place" concept specifically, but to recognize that this may be a controlling factor

in performance behavior at all times.

As a "non-hyperbolic" approach to reserves estimations, we can also consider the recently developed Ilk, et al[2008] method

which employs a different functional form for theD-parameter, where this form is given by:

)1(1

ntDDD

+= ......................................................................................................................................................... (8)


4/51


=

n

it

n

DtD

q

qexp

1

................................................................................................................................................ (9)

Which reduces in form to thepower law loss-ratio rate decline relation as defined by Ilk, et al[2008b]:

]exp[ nii tDtDqq = ................................................................................................................................................. (9)

Where:

iq = Rate "intercept" defined by Eq. 9 [i.e., q(t=0)] [this parameter has a different interpretation than qi in Eq. 7].

D1 = Decline constant "intercept" at 1 time unit defined by Eq. 8 [i.e.,D(t=1 day)].

D = Decline constant at "infinite time" defined by Eq. 8 [i.e.,D(t=)].

iD =Decline constant defined by Eq. 9 [i.e., nDDi /

1= ] [this parameter has a different interpretation thanDi in Eq. 7].

n = Time "exponent" defined by Eq. 8.

Valko [2008] independently proposed another form of Eq. 9, specifically as a statistical relation to be used for the rapidevaluation/analysis of a database of rate behavior. The form of Valko's result is:

])/(exp[n

i tqq = ......................................................................................................................................................... (10)

Where the iq and n-parameters in Eq. 10 are interpreted exactly the same as the corresponding variables in Eq. 9 however; in Valko's nomenclature, the iD -parameter would be defined as (1/)

n. We also note that Valko did not considerthe "long-time" behavior which would lead to the inclusion of the D-constant. We again note that Valko did not try todevelop a "rate-time" analysis relation but rather; he utilized the form given by Eq. 10 (which he references as a"statistical identity") as a means of evaluating a database of production data.

In this work we focus on the use of Eq. 9 (and its associated relations) as a "unifying" model one which provides datadiagnostics and is essentially a self-calibrating model (i.e., the key behavior is given by Eq. 8). We believe that Eq. 9 is ap-

propriate for reserves evaluation and will demonstrate its characteristics using the synthetic and field data cases evaluated inthe next section.

Illustrative Production Behavior

Orientation:

In this section we provide diagnostic validation of the hyperbolic and "power law loss ratio" rate decline models usingsynthetic and field data cases. The synthetic cases are designed to assess the validity of the hyperbolic rate decline relation in

terms of multiple layers and gas flow behavior. The field cases are sampled from a tight gas sand field study where each wellwas evaluated using modern production data diagnosis and analysis (elliptical flow model-based matching using time-

pressure-rate data) (Ilket al[2008b]).

As a prelude, we present the schematic plot of the rate (q) and theD and b-parameters versus time on a log-log plot in Fig. 1.

This plot serves a schematic or prototype from which we can establish our expectations of performance for the hyperbolic

rate decline relation (Eq. 7) and the power law loss-ratio rate decline relation (Eq. 9). We immediately note in Fig. 1 that, forthe hyperbolic relation, theD-parameter has a near-constant behavior at early times and a unit-slope, power law decay at late

times. As one may expect, for the "power law loss ratio" relation theD-parameter exhibits a power law decay behavior from

transient through transition flow, and then turns gently towards a constant value (i.e., D) at very large times.

Synthetic Case: Gas Multilayer Case

In this case we prepared a 3-layer reservoir simulation model (see Table 1 for system properties). This model was designed

to have at least one layer which exhibited transient/transition flow for decades (or even hundreds of years) the purpose ofwhich was to evaluate the application of the hyperbolic rate decline relation (Eq. 7) for a layered reservoir case. This effort

was developed to validate the observations of Raghavan [1989] and Fetkovich et al [1990] for the case of depletion in a

multilayered reservoir but specifically for the case of a well in a very low permeability layered reservoir systems, where


5/51


Table 1 Reservoir and fluid properties for the three layered gas reservoir (hydraulically fractured gas well)simulation case.

Reservoir Properties:Layer 1 net pay thickness, h1 = 10 ft

Layer 2 net pay thickness, h2 = 30 ftLayer 3 net pay thickness, h3 = 60 ft

Layer 1 permeability, k1 = 0.1 mdLayer 2 permeability, k2 = 0.01 md

Layer 3 permeability, k3 = 0.001 md

Wellbore radius, rw = 0.35 ft

Formation compressibility, cf = 110-9

1/psi

Porosity, = 0.06 (fraction)Outer boundary radius, re = 2980 ft

Initial reservoir pressure,pi = 10000 psiaWellbore storage coefficient, CD = 0 (dimensionless)Gas saturation, Sg = 1.0 (fraction)

Skin factor,s = 0 (dimensionless)

Reservoir temperature, Tr = 300oF

Fluid Properties:

Gas specific gravity, g = 0.65 (air = 1)

Hydraulically Fractured Well Model Parameters:

Fracture half-length,xf = 450 ftFracture conductivity,FcD = 2 (dimensionless)

Production Parameters:

Flowing pressure,pwf = 500 psia

Producing time, t = 7300 days

The result of this calibration is that we achieve a very good match of the boundary-dominated flow data (all models), but only

a "qualitative" match of the early data using the power law loss ratio model, and a very poor match using the hyperbolicmodel. The b-parameter defined by data as shown in Fig. 3 has a very distinct oscillation as one might expect, this

oscillation is due to the "addition" of the individual layer performances in the numerical simulation (recall that this is a"layered no-crossflow" case analogous to the work of Raghavan [1989] and Fetkovich et al[1990]). This behavior is quiteunusual by comparison to single-layer cases and the field cases we present in this work.

While we believe that our basis relation for the power law loss ratio ( i.e., Eq. 8) is generally applicable, there will always be

exceptions in this particular case, no simple relation is perfect (nor perhaps should it be given the degrees of freedom in a

multilayer system). In summary, as Raghavan [1989] and Fetkovich et al[1990] noted for the case of a multilayer reservoir,

we achieved a reasonable match using the hyperbolic rate relation further; we also achieved a reasonable match of thesedata using our new power law loss ratio relation (i.e., Eq. 9). The log-log and semilog plots of the rate and cumulative data

and the associated model functions are shown in Figs. 4 and 5, respectively. Finally, the comparison of gas-in-place

obtained using model-based analysis and the gas reserves estimated using the hyperbolic and power-law exponential raterelations are given in Table 2.

Table 2 Power law exponential and hyperbolic model parameters and reserve estimates (i.e., maximum cumulative gasproduction, Gp,max) for the three layered hydraulically fractured gas well simulation case. (G=100 BSCF fromnumerical simulation.)

Model iq or (qi)

iD orDi1

D(D-1)

n(d l )

Gp,max(BSCF)


6/51


fracture with finite conductivity) and we assume a constant flowing wellbore pressure to generate the synthetic flowrate

performance.

We use a numerical derivative (differentiation) algorithm to compute the D and b-parameters as a function of time, and wepresent these results on the "q-D-b plot" (log-log format) in Fig. 6. We note that there are apparent "end-point" effects (i.e.,

artifacts) in the computedD and b-parameters, which are caused by the numerical derivative algorithm in particular, at late

times. TheD-parameter exhibits a power-law trend (i.e., a straight line on a log-log plot) until the boundary effects begin todominate the rate performance. Once the boundary effects are fully established, we find that the character of theD-parameter

trend tends to deviate from our presumed power-law (i.e., straight line) behavior. We note that our simulation model

considers the case of a 2x1 rectangular shaped reservoir, and that the well is produced well into boundary-dominated flow.

We first use the hyperbolic model to approximate the D-parameter trend, and we note that early part of the data does notmatch with the hyperbolic model as should be expected because it is known that hyperbolic model is only applicable for

the boundary-dominated flow regime. We obtain a good match of the hyperbolic model with the D-parameter data trend

during full boundary-dominated flow conditions. For this case (and all of the cases in this work), we set the b-parameterequal to 1 (one) as a maximum value. The reserve estimation using the hyperbolic rate decline model (with b=1) givesalmost more than 4 (four) times the reserves used as input for the simulation. We present this example specifically to

emphasize that care must be taken when using the hyperbolic rate decline model and in particular, for production fore-

casting of gas wells. In conclusion, this example shows that using b=1 will yield an overestimation of reserves for thiscase in particular, but in general as well.

Next, we employ the new "power law exponential" model proposed in this work, this result (in q(t)) is derived from the

"power law loss ratio" relation for the D-parameter. The calibration of our new model consists of two steps. We first

approximate theD-parameter data trend with only a power law function (i.e., settingD=0 in Eq. 8) this can be considered

to be an "aggressive" estimate of reserves. In the next step, we adjust theD-term to obtain a best fit of theD-parameter datatrend (as seen in Fig. 1), theD-term only affects the late portion of Eq. 8. This process of calibrating theD-term provides a

lower bound for the reserves estimate, and not surprisingly (at least for this case), we obtain a result which is very close to thevalue of gas-in-place input into the numerical model. In Fig. 6 (i.e., the "q-D-b plot") we note that the matches of the data

functions and power law exponential model are excellent; the rate data are uniquely matched using the new "power law

exponential" model across all flow regimes.

The results for this case are summarized in Table 3. In Figs. 7 and 8 we present the rate and cumulative production data andproduction forecasts on log-log and semilog coordinates. Our final diagnostic is a comparison of the b-parameter data trend

versus time: As we noted earlier, the end-point effects caused by the derivative algorithm are quite apparent on the computed

D-parameter data trend. Computation of the b-parameter requires an additional numerical differentiation and therefore webelieve that the end-point effects are amplified, causing an even larger oscillatory feature in the b-parameter data function.For this reason we believe that we cannot effectively match the b-parameter data function using the power law exponential

model only early part of the computed b-parameter data set is matched with this model assuming D=0. For the case

whereD0, we note on Fig. 6 that our model does yield an average trend through the b-parameter data functions. Perhapsthe most important aspect of this exercise is that we clearly demonstrate that the b-parameter (data function) is not constant

as the rigorous hyperbolic rate relation requires.

Table 3 Power law exponential and hyperbolic model parameters and reserve estimates (i.e., maximum cumulative gasproduction, Gp,max) for the East Tx gas well simulation case. (G=2.55 BSCF from n umerical simulation.)

Model iq or (qi)

(MSCFD)

iD orDi

(D-1)

D(D-1)

n(d.less)

Gp,max(BSCF)

Power LawExponential

2.57x105 2.86 2.3x10-4 0.105 2.5


(D =0)2.57x105 2.86 0 0.105 13.9


7/51


In Fig. 9 we present the "q-D-b plot," and we immediately note that erroneous rates (caused by liquid loading) exist in the

latter part of the data set. In addition, the early-time rate performance data are clearly affected by well clean-up effects. As

in previous cases, we compute the D and b-parameters from the rate-time and rate-cumulative data using Eqs. 1 and 4,respectively. Although the computation using Eq. 4 appears to yield smoother results, we believe that the end-point effects

generated by the derivative algorithm may be more severe for this Eq. 4 (see Fig. 9). In this case we clearly observe the onsetof boundary-dominated flow at about 800 days. In terms of rate extrapolations, we first use the hyperbolic rate declinerelation forcing b=1 where the b=1 condition yields a very optimistic estimation of reserves for this case.

In our next step we match the D-parameter data trend using a straight-line trend (i.e., setting D=0 in Eq. 8). We then

calibrate the D-parameter to obtain the best match of the entireD-parameter data trend with the model (Eq. 8). Although

there is considerable scatter in the data, we obtain very reasonable matches of the flowrate data using the power law

exponential rate decline model (see Fig. 9) for both the D=0 and D0 cases. We compare the reserves estimates inTable 4, and as expected, applying Eq. 9 with the condition thatD0 yields a very close comparison with the result obtainedusing the model-based production analysis methods provided in (Ilket al[2008b]). In Figs. 10 and 11 we present the log-log

and semilog plots of the rate and cumulative data and the associated model functions, respectively.

Table 4 Power law exponential and hyperbolic model parameters and reserve estimates (i.e., maximum cumulative gasproduction, Gp,max) for Example Case 1 small w aterfrac wel l 1 (SWF1). (G=5.4 BSCF from Ilk et al [2008b]).

Model iq or (qi)

(MSCFD)

iD orDi

(D-1

)

D(D-1)

n(d.less)

Gp,max(BSCF)


1.7x104

0.33 3.0x10-5

0.3 5.4

Power Law

Exponential(D=0)

1.7x104

0.33 0 0.3 6.1

Hyperbolic(b=1)

6000 4.0x10-3 N/A N/A 26.0

Field Case: Example Case 2 Small Waterfrac Gas Well (SWF2)

This well is a hydraulically fractured gas well which was completed using a 40/70 proppant size in the hydraulic fracturetreatment. In Fig.12, we present the "q-D-b plot," and we immediately recognize the liquid loading problem on the rate data.

TheD and b-parameters are computed from rate-time and rate-cumulative data using the derivative algorithm. In this case an

obvious power law character is exhibited by the computedD-parameter data trend.

We first make use of the hyperbolic rate decline relation by forcing b=1 and we then approximate theD-parameter data trend

using the hyperbolic relation. In this case the early part of the data are not matched. In terms of forecasting and reserves

extrapolation, the estimation of reserves from the hyperbolic rate decline relation is very optimistic compared to results from

the model-based matching (see Table 5).

Table 5 Power law exponential and hyperbolic model parameters and reserve estimates (i.e., maximum cumulative gasproduction, Gp,max) for Example Case 2 small w aterfrac wel l 2 (SWF2). (G=2.3 BSCF from Ilk et al [2008b]).

Model iq or (qi)

(MSCFD)

iD orDi

(D-1)

D(D

-1)

n(d.less)

Gp,max(BSCF)

Power Law

Exponential3.2x10

41.24 1.5x10

-40.17 2.3


(D=0)3.2x104 1.24 0 0.17 3.8

Hyperbolic(b=1)

3600 6.0x10-3 N/A N/A 10.0


8/51


Lastly, when we compare the character of the b-parameter data function with our models, we observe very good matches upto

about 1000 days for theD0 case. We suspect that this disagreement could be due to the noise in the rate data and/or theend-point effects of the differentiation algorithm, or this feature could just be an artifact in the data.

Field Case: Example Case 3 Large Waterfrac Gas Well (LWF1)Using this example, we begin our review of the large waterfrac cases. This is a case of a hydraulically fractured gas well

where the well was completed using 20/40 proppant size. In Fig. 15 (the "q-D-b plot") we identify the liquid loading effects

by inspecting the rate behavior between 400-1000 days. Also in Fig.15 we observe that the computed D and b-parametersexhibit outstanding diagnostic character a near-perfect straight line trend is exhibited by the computed D-parameter data

and the computed b-parameter data trend converges to unity.

We obtain a good match of the hyperbolic rate decline model (b=1) except for the transient part of the data. Again, the

production forecast from the hyperbolic rate decline model overestimates the contacted gas-in-place estimate using themodel-based production analysis methods provided in Ilket al[2008b]. We use the power law exponential model first with

by settingD=0 and obtain a very good agreement of theD-parameter data trend and the model given by Eq. 8. The match ofthe rate data with the power law exponential rate decline model is excellent. Furthermore, we observe very good match ofthe computed b-parameter data with the model as well.

However, this production forecast yields twice and half more estimate of reserves than the contacted gas-in-place estimate by

the model based production analysis techniques in the work by Ilket al[2008b]. We adjust the value ofD and obtain an

excellent match of the data trends and the power law exponential model where the reserves result is consistent with the

contacted gas-in-place estimate obtained from the model based techniques. In Figs. 16 and 17, we present the log-log andsemilog plots of the rate and cumulative data and the associated model functions for this example case and Table 6

summarizes our results for this case.

Table 6 Power law exponential and hyperbolic model parameters and reserve estimates (i.e., maximum cumulative gasproduction, Gp,max) for Example Case 3 large waterfrac wel l 1 (LWF1). (G=3.0 BSCF from Ilk et al [2008b]).

Model iq or (qi)

(MSCFD)

iD orDi

(D-1)

D

(D-1

)

n

(d.less)

Gp,max

(BSCF)


3.8x104 1.64 4.3x10-5 0.14 3.0


(D=0)

3.8x104

1.64 0 0.14 7.4

Hyperbolic

(b=1)2000 2.4x10-3 N/A N/A 13.1

Field Case: Example Case 4 Large Waterfrac Gas Well (LWF2)

This example case is the second large waterfrac gas well where a smaller proppant size is used ( i.e., 40/70 sand) in the

hydraulic fracture treatment. In Fig.18 (i.e., the "q-D-b plot"), we observe an erratic rate behavior caused by liquid loading inthe latter part of the rate data. For this case we note that the behavior of the computedD and b-parameters is almost identical

to the previous case which suggests that these data are consistent.

We obtain outstanding matches of the computedD and b-parameters data trends with the power law exponential model. The

match of the hyperbolic rate relation is also good for the boundary-dominated flow portion of the data but the estimation ofreserves is almost five times higherthan the contacted gas-in-place predicted previously (Ilket al[2008b]). Once again, the

only difference in the production forecast the power law exponential model withD0 provides a conservative estimate of

reserves in agreement with the estimate from Ilket al[2008b]. Figs. 19 and 20 present the rate data and the related modelfunctions for this case in log-log and semilog coordinates, respectively. Table 7 summarizes the results for this case.

Table 7 Power law exponential and hyperbolic model parameters and reserve estimates (i.e., maximum cumulative gasproduction G ) for Example Case 4 large waterfrac wel l 2 (LWF2) (G=6 0 BSCF from Ilk et al [2008b])


9/51


Field Case: Example Case 5 Hybrid Waterfrac Gas Well (HWF)

Our last example is a "hybrid" waterfrac gas well case (Ilk et al[2008b]). In Fig. 21 we present the "q-D-b plot" and we

recognize that again we have a significant liquid loading problem at late times. In fact, the liquid loading distortion seemsmore significant than for the large waterfrac cases. The computedD and b-parameters reflect the effects of liquid loading,

but theD-parameter trend is essentially power-law, and the b-parameter trend, while erratic, is apparently converging to unity(i.e., b=1 at late times).

Considering theD-parameter data trend (i.e., our calibration tool), we could probably assumeD=0 however; as shown in

Fig. 21, when we do calibrate theD-parameter, we in term obtain a more conservative estimate of reserves. The log-log and

semilog data and forecast functions are shown in Figs. 22 and 23, respectively. In Table 8 we compare the model-based

estimate of contacted gas-in-place with the reserves estimate based onD=1.0x10-4 in short, this is an excellent comparison

of results (G=1.4 BSCF (model-based match) and Gp,max=1.4 BSCF (power-law exponential rate relation,D=1.0x10-4).

Table 8 Power law exponential and hyperbolic model parameters and reserve estimates (i.e., maximum cumulative gasproduction, Gp,max) for Example Case 5 hybrid waterfrac well (HWF). (G=1.4 BSCF from Ilk et al [2008b]).

Model iq or (qi)

(MSCFD)

iD orDi

(D-1)

D(D

-1)

n(d.less)

Gp,max(BSCF)

Power Law

Exponential1.38x105 2.79 1.0x10-4 0.11 1.4


(D=0)1.38x105 2.79 0 0.11 3.9

Hyperbolic

(b=1)3600 1.1x10

-2N/A N/A 5.6

Summary and ConclusionsSummary: Perhaps the most important conclusion of this study is that the hyperbolic rate decline relation (i.e., Eq. 7) has, at

best, a reasonable predictive capability in the case of reserves estimations for tight gas sands and then only for cases of

boundary-dominated flow behavior. As described in our methodology section, we use the continuous (rate-time and rate-cumulative-based) estimates of the Arps D and b-parameters to guide our fit of a data set with a particular mode we note

that this approach yields an excellent diagnostic capability (for orientation), as well as providing a direct analysis component

when theD-parameter is correlated with the proposed power law model.

In this work we present a new "power law loss ratio" formulation for the D-parameter, which in turn translates into theproposed "power law loss ratio" rate decline model (i.e., Eq. 9). Eq. 9 is derived from the observation of the power law

behavior of the D-parameter, and we have found that when properly constrained, this is an outstanding predictor of reserves

for tight gas and shale gas reservoir systems.

Hyperbolic Rate Decline Relation: (Eq. 7) In our experience, the hyperbolic rate decline relation is almost always an excessive predictor for reserves, with

errors easily on the order of 100 percent or more if this analyst is not careful. In this study we had the benefit of

a prior work where we had developed a model-based match of the production rate and pressure history and

we used these results as a "lower bound" of the reserves estimates for each case.

From the log-log plots of theD and b-parameters versus time, the hyperbolic rate decline relation is often weaklycorrelated with the production (particularly during transient flow, but on occasion for boundary-dominated flowas well). The cases we have selected for this work yield reasonable to good correlations with the hyperbolic

relation, but we have seen many cases particularly shale gas cases where the reserves are severely over-

estimated by the hyperbolic rate decline relation.

"Power Law Loss Ratio" Rate Decline Model: (Eq. 9) We have derived, validated, and demonstrated a new model using the observed behavior of the Arps loss ratio


10/51


Conclusions:

1. This work provides the development of a new diagnostic plot a log-log plot of the D and b-parameters versus

time, where theD and b-parameters are computed using rate-time and rate-cumulative data functions. The rate dataand all associated model functions (q, D, b) are also presented on these plots. These plots provide unique a

diagnostic insight (i.e., whether or not the data is "hyperbolic" in nature) as well as an analysis/calibration function.

2. This work provides a thorough investigation of the classical hyperbolic rate decline relation in the context ofsynthetic model cases and field data cases for tight gas reservoirs. While tuned to yield reserve estimates which are

"conservative" (based on the model-based analyses performed in another work), the hyperbolic decline relation

typically tends to yield high to excessively high reserve estimates.

3. This work provides a new "power law loss ratio"rate decline model (i.e., Eq. 9). This model is much more flexible it can be used to match transient, transition, and boundary-dominated flow data, and by the use of the decline

constant at "infinite time" (D i.e., as defined by Eq. 8), Eq. 9 yields an exponential decay at very large times. In

addition, the "power law loss ratio" rate decline model can be tuned graphically using a log-log plot of the D -

parameter versus time. The "power law loss ratio"rate decline model is the most substantive contribution of thiswork.

Nomenclature

Field Variables

b = Arps' decline exponent, dimensionlesscf = Formation compressibility, psi

-1cg = Gas compressibility, psi

-1

ct

= Total compressibility, psi-1

CD = Wellbore storage coefficient, dimensionlessD = Arps' "loss ratio," D-1Di = Arps'initial decline rate (hyperbolic model), D

-1D1 = Decline constant "intercept" at 1 time unit defined by Eq. 8 [i.e.,D(t=1 day)], D

-1D

= Decline constant at "infinite time" defined by Eq. 8 [i.e.,D(t=)], D-1

iD = Decline constant defined by Eq. 9 [i.e., nDDi /

1= ], D-1

FcD = Fracture conductivity, dimensionlessG = Original (contacted) gas-in-place, MSCF

Gp = Cumulative gas production, MSCFGp,max = Maximum gas production, MSCFh = Net pay thickness, ft

k = Average reservoir permeability, mdn = Time exponent defined by Eq. 8pi = Initial reservoir pressure, psiaptf = Flowing tubing (surface) pressure, psiapwf = Flowing bottomhole pressure, psiap = Pressure drop (pi-pwf), psiqg = Gas production rate, MSCF/D

qi = Initial production rate, MSCF/D or STB/Diq = Rate "intercept" defined by Eq. 9 [i.e., q(t=0)], MSCF/D

re = Reservoir drainage radius, ftrw = Wellbore radius, ft

s = Skin factor, dimensionlessSg = gas saturation, fractionSwi = Initial water saturation, fraction


11/51


References

Arps J.J.: "Analysis of Decline Curves," Trans. AIME (1945) 160, 228-247.

Blasingame, T.A. and Rushing, J.A.: "A Production-Based Method for Direct Estimation of Gas-in-Place and Reserves,"

paper SPE 98042 presented at the 2005 SPE Eastern Regional Meeting held in Morgantown, W.V., 1416 September 2005.Fetkovich, M.J., Works, A.M., Thrasher, T.S., and Bradley, M.D.: "Depletion Performance of Layered Reservoirs Without

Crossflow," SPEFE(Sept. 1990).

Fetkovich, M.J., Fetkovich, E.J., and Fetkovich, M.D.: "Useful Concepts for Decline-Curve Forecasting, Reserve Estimation,

and Analysis," SPERE(February 1996) 13-22.

Johnson, R.H. and Bollens, A.L.: "The Loss Ratio Method of Extrapolating Oil Well Decline Curves," Trans. AIME (1927)

77, 771.

Ilk, D., Rushing, J.A., Sullivan, R.B., and Blasingame, T.A.: "Evaluating the Impact of Waterfrac Technologies on Gas

Recovery Efficiency: Case Studies Using Elliptical Flow Production Data Analysis," paper SPE 110187 presented at the2007 Annual SPE Technical Conference and Exhibition, Anaheim, CA., 11-14 November 2007.

Ilk, D., Rushing, J.A., and Blasingame, T.A.,: "Estimating Reserves Using the Arps Hyperbolic Rate-Time Relation

Theory, Practice and Pitfalls," paper CIM 2008-108 presented at the 59th Annual Technical Meeting of the Petroleum

Society, Calgary, AB, Canada, 17-19 June, 2008a. (in preparation)

Ilk, D., Perego, A.D., Rushing, J.A., and Blasingame, T.A.,: "Integrating Multiple Production Analysis Techniques To Assess

Tight Gas Sand Reserves: Defining a New Paradigm for Industry Best Practices," paper SPE 114947 presented at the 2008Gas Technology Symposium, Calgary, AB, Canada, 17-19 June, 2008b.

Pratikno, H., Rushing, J.A., and Blasingame, T.A.: "Decline Curve Analysis Using Type Curves: Fractured Wells," paperSPE 84287 presented at the 2003 Annual SPE Technical Conference and Exhibition, Denver, CO., 05-08 October 2003.

Raghavan, R.: "Behavior of Wells Completed in Multiple Producing Zones," SPEFE(June 1989) 219-30.

Rushing, J.A., Perego, A.D., Sullivan, R.B., and Blasingame, T.A.: "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 Technical Conference and Exhibition, Anaheim, CA., 11-14 November 2007.

Valko, P.P.:Personal Communication, Texas A&M University, (June 2008).


12/51


Fig. 1 Schematic Plot: (Log-Log Plot) Hyperbolic and power law exponential rate decline and loss ratio models areillustrated for orientation pur poses.


13/51


Fig. 3 Numerical Simulation : (Log-Log Plot) 3 layered reservoir (gas well) definiti on of the D and b-parameters.


14/51


Fig. 5 Numerical Simulation : (Semilog Plot) 3 layered reservoir (gas well) empirical matches are shown us ing power lawexponential and hyperbolic rate models.


15/51


Fig. 7 Numerical Simulation: (Log-Log plot) East Tx gas well (SPE 84287) empirical matches are shown us ing power lawexponential and hyperbolic models.


16/51


Fig. 9 Example Case 1: (Log-Log Plot) Small waterfrac gas well (SWF1) (SPE 114947) defin iti on of the D and b-parameters.


17/51


Fig. 11 Example Case 1: (Semilog Plot) Small waterf rac gas well (SWF1) (SPE 114947) empir ical matches are shown usingpower law exponential and hyperbolic models.


18/51


Fig. 13 Example Case 2: (Log-Log Plot) Small waterf rac gas well (SWF2) (SPE 114947) empir ical matches are shown usingpower law exponential and hyperbolic models.


19/51


Fig. 15 Example Case 3: (Log-Log Plot) Large waterf rac gas well (LWF1) (SPE 114947) definit ion o f the D and b-parameters.


20/51


Fig. 17 Example Case 3: (Semilog Plot) Large waterf rac gas well (LWF1) (SPE 114947) empir ical matches are shown us ingpower law exponential and hyperbolic models.


21/51


Fig. 19 Example Case 4: (Log-Log Plot) Large waterf rac gas well (LWF2) (SPE 114947) empir ical matches are shown usingpower law exponential and hyperbolic models.


22/51


Fig. 21 Example Case 5: (Log-Log Plot) Hybrid waterfrac gas well (HWF) (SPE 114947) defin ition of the D and b-parameters.


23/51


Fig. 23 Example Case 5: (Semilog Plot) Hybrid waterfrac gas well (HWF) (SPE 114947) empir ical matches are shown us ingpower law exponential and hyperbolic models.


24/51

Slide

1/28

2008 SPE Annual Technical Conference and Exhibition Denver, CO 2124 September 2008

SPE116731 Exponential vs. Hyperbolic Decline in Tight Gas Sands Understanding the Origin and

Implications for Reserve Estimates Using Arps' Decline Curves (Ilk/Perego/Rushing/Blasingame)

D. Ilk Texas A&M University (23 September 2008)

D. Ilk, Texas A&M University

A.D. Perego, Anadarko Petroleum Corp.J.A. Rushing, Anadarko Petroleum Corp.T.A. Blasingame, Texas A&M University

Department of Petroleum EngineeringTexas A&M University

College Station, TX 77843-3116+1.979.458.1499 [email protected]

SPE 116731Exponential vs. Hyperbolic Decline in Tight Gas

Sands Understanding the Origin and Implicationsfor Reserve Estimates Using Arps' Decline Curves

Presentation Outline


25/51

Slide

2/28





Rationale For This Work

Overview of Decline Curve AnalysisDefinition of the Loss Ratio (Johnson and Bollens)Arps Exponential and Hyperbolic Rate FunctionsHistorical Perspectives

Development of the New Rate Decline ModelCharacterization of the D-parameterPower-Law Exponential Rate Decline Function

Illustrative ExamplesNumerical Simulation 3 Layered Gas ReservoirNumerical Simulation East Tx Gas Well (SPE 84287)Field Examples 5 Tight Gas Well Cases

Conclusions

Presentation Outline

(Ou

tline)



26/51

Slide

3/28






(Ration

ale)

ASSUMPTION: The Arps decline parameter, b, defines the declinebehavior when tight gas sand reserves are assessed.

REALITY: Difficult to identify the correct b-parameter duringthe early decline period selection of the wrong b-parameter greatly impacts reserve estimates.

b. (Log-log plot) Production forecast of a tight gas well.a. (Semilog plot) Production forecast of a tight gas well.



27/51

Slide

4/28






The primary objectives of this work are:

To evaluate the b-parameter as a function of time using syn-thetic and field production profiles for tight gas reservoirs.To provide a diagnostic understanding of the hyperbolic rate

decline relation in terms of the D- and b-parameters.To assess the applicability of the hyperbolic rate decline

relation to layered reservoir performance.To provide an alternative to the hyperbolic rate decline relation

that should be flexible enough to represent transient, transition,and boundary-dominated flow data.

(Ration

ale)

b. Decline curve behavior of a tight gas well.a. Various sandstone depositional sequences.

Overview: Loss Ratio (Definition and Behavior)


28/51

Slide

5/28





Overview: Loss Ratio (Definition and Behavior)

(Overview

)

Loss Ratio: (basis for exponential rate decline)

Loss Ratio Derivative: (basis for hyperbolic rate decline)

dtdqq

D gg/

1

dtdq

q

dtd

Ddtdb

gg/

1

]exp[ tDqq igig =

)/1()(1 bi

gi

gtbD

q

q+=

[From: Johnson, R.H. and Bollens, A.L.: "The Loss Ratio Method of Extrapolating Oil Well Decline Curves," Trans., AIME (1927) 77, 771.]

Overview:Arps' Rate Decline Functions


29/51

Slide

6/28





p

Case Rate-Time Relation Cumulative-Time Relation

]exp[ tDqq igig =

)/1()(1 bi

gig

tbD

qq

+=

)(1 tD

qq

i

gig

+=

Exponential: (b=0)

Hyperbolic: (0


30/51

Slide

7/28





Exponential Rate Decline (b=0): (various authors)Constant compressibility flowing fluid (i.e., black oil).

Boundary-dominated flow regime.Constant flowing wellbore pressure (i.e.,pwf= constant).

Hyperbolic Rate Decline (0


31/51

Slide

8/28





[From: Rushing, J.A., Perego, A.D., Sullivan, R.B., and Blasingame, T.A.: "Estimating Reserves in Tight Gas Sands at HP/HT Reservoir Conditions: Use and Misuseof an Arps Decline Curve Methodology," paper SPE 109625 presented at the 2007 Annual SPE Technical Conference and Exhibition, Anaheim, CA., 11-14November 2007.]

Hyperbolic Rate Relation for Tight Gas Systems: (model study)

x

y

Pressure

Monitoring

Point No. 1

Pressure

MonitoringPoint No. 2

Hydraulic

Fracture

Wellbore

X

X

x

y

x

y

Pressure

Monitoring

Point No. 1

Pressure

MonitoringPoint No. 2

Hydraulic

Fracture

Wellbore

X

X

Numerical Model Considers:

Reservoir Layering.kv/kh ratio.Fracture Length, xf.Fracture Conductivity, FcD.

Analysis/Validation Approach:Fit q(t) with Arps' hyperbolic relation.Compare reserves to model at30 years.

g y [ ]

(Overview

)

New Rate Equation: Ilk et al[2008]


32/51

Slide

9/28





q [ ]

(New

RateEquation)

)1(1 nitDnD

dt

dq

qD +

Observed Behavior of Decline Parameter(D):

Solving for Flowrate:

]exp[ n

iitDtDqq =

Solving for the b-Parameter:

nn

i

i ttDDnnDnb

+= 2)1( ][

)1(

New Rate Equation: Illustrative Behavior


33/51

Slide

10/28





"Power-Law Exponential Rate Decline": Discussion Use as a self-calibrating model that provides data diagnostics. "Power-law loss ratio" function exhibits a power law decay for transient

through transition flow, then becomes constant value at large times.

(New

RateEquation)

Illustrative Examples: Gas Multilayer Case


34/51

Slide

11/28





(Nu

mericalSim

ulationCase1)

Numerical Simulation: Gas Multilayer Case Objective: To apply the hyperbolic rate relation to a case of depletion in a

very low permeability multilayered reservoir system. Method: 3-layered gas reservoir is simulated this model is designed to

have at least one layer which exhibits transient/transition flow for aconsiderable period (for decades or even hundreds of years).

a. Numerical simulation: "Semi-log plot" 3-layeredreservoir individual layer rate and total rateresponses are shown.

b. Numerical simulation: "Log-log plot" 3-layeredreservoir total rate response is matched by ahyperbolic function at late times.

Illustrative Examples: Gas Multilayer Case


35/51

Slide

12/28





Numerical Simulation: Gas Multilayer Case The best match with the "power-law loss ratio" relation is obtained with

an average straight line balancing the earliest and latest time data. The hyperbolic formulation for the D-parameter is only valid for the

boundary-dominated flow regime.We assume that the oscillation in the b-parameter data trend is due to

the addition of individual layer performances.

a. Numerical simulation: "q-D-b plot" 3-layeredreservoir definition of the "D and b" parameters.

b. Empirical matches are shown using power-lawexponential and hyperbolic models.

(Nu

mericalSim

ulationCase1)

Illustrative Examples: Tight Gas Case


36/51

Slide

13/28





Numerical Simulation: East Tx Gas Well(SPE84287) Previously obtained model parameters are used to generate the synthetic

gas flowrate performance. Once the boundary-dominated flow regime is established, the trend for

the D-parameter deviates (somewhat) from power-law behavior. D

is set to 0 (aggressive reserves estimate) at first, and then adjusted to

obtain the best match.

"q-D-b" Plot: TG ExamplezD-parameter data trend

exhibits a power-lawbehavior essentiallya straight line.

zb-parameter data trendis not constant(contrary to hyperbolicformulation).

(Nu

mericalSim

ulationCase2)

Illustrative Examples: Tight Gas Case


37/51

Slide

14/28





Numerical Simulation: East Tx Gas Well(SPE84287) A good match is obtained using the hyperbolic rate relation during the

boundary dominated flow regime (b=1). The process of calibrating D

provides a lower bound for the reserves

estimate and we obtained very close agreement with the input G-value. Rate data are uniquely matched using the new "power-law exponential"

model (D0) across all flow regimes.

a. Semi-log plot empirical matches are shownusing power-law exponential and hyperbolicmodels.

b. Log-log plot empirical matches are shownusing power-law exponential and hyperbolicmodels.

(Nu

mericalSim

ulationCase2)

Field Examples: Small WF Gas Well(SWF1)


38/51

Slide

15/28





(F

ieldExamp

le1)

Discussion: Small Waterfrac Gas Well(SWF1)

Early-time rate performance data are affected by well clean-up effects. The effects of liquid loading are observed at late times. The onset of boundary-dominated flow is observed at about 800 days. D

is set to 0 (aggressive reserves estimate) at first and then adjusted to

obtain the best match.

"q-D-b" Plot: SWF1zD-parameter data trend

exhibits a power-lawbehavior.

zb-parameter data trend

is not constant(contrary to hyperbolicformulation).

zNote the end-pointeffects.



39/51

Slide

16/28





Discussion: Small Waterfrac Gas Well(SWF1) Hyperbolic rate relation (using b=1) yields the most optimistic reserves

estimate for this case. Reasonable matches of the rate data are obtained using the power-law

exponential model using both D=0 and D

0 cases.

Reserves estimates obtained by the power-law exponential model agreewell with the results from previous model-based PA study (SPE 114947).



(F

ieldExamp

le1)



40/51

Slide

17/28





Discussion: Small Waterfrac Gas Well(SWF2) Liquid loading effects are obvious in the latter portion of the flowrate data.

The onset of the boundary-dominated flow regime is observed. D

is set to 0 initially, then tuned to the latest data we obtain a very

good match of the D-parameter data trend with the power-law models.We observe a very good match of the flowrate data with the "base" power-

law exponential model (i.e., D=0).

"q-D-b" Plot: SWF2zD-parameter data trend


zb-parameter data trendis not constant(contrary to hyperbolicformulation).

zComputation of the b-parameter is severely

affected by noise.

(F

ieldExamp

le2)



41/51

Slide

18/28





Discussion: Small Waterfrac Gas Well(SWF2)

The hyperbolic rate relation (b=1) yields the highest reserves estimate. Excellent matches of data are achieved using the power-law exponential

model for both the D=0 and the D

0 cases.

The lower bound for the reserves estimate is 2.3 BSCF, which is con-sistent with our results from the model-based PA study (SPE 114947).



(F

ieldExamp

le2)

Field Examples: Large WF Gas Well(LWF1)


42/51

Slide

19/28





Discussion: Large Waterfrac Gas Well(LWF1)

The effects of liquid loading are observed between 400-2000 days. The D-and b-parameters exhibit outstanding diagnostic character a

near perfect straight line is exhibited by the computed D-parameter data. A good match is obtained using the hyperbolic rate decline model (b=1),

except at early times (i.e., the transient part of the data).

"q-D-b" Plot: LWF1zD-parameter data trend


zb-parameter data trend

is not constant(contrary to hyperbolicformulation).

zComputation of the b-parameter is somewhataffected by noise.

(F

ieldExamp

le3)



43/51

Slide

20/28





Discussion: Large Waterfrac Gas Well(LWF1) The hyperbolic rate relation only matches the boundary-dominated flow

portion of the data yields highest reserves estimate for this case (b=1). Excellent matches are obtained using the power-law exponential model

both the D=0 and the D

0 cases match well across all flow regimes.

Production forecast using D=0 yields a reserves estimate which is 2.5

times the model-based estimate of contacted gas-in-place in SPE 114947.



(F

ieldExamp

le3)



44/51

Slide

21/28





Discussion: Large Waterfrac Gas Well(LWF2) Erratic rate behavior caused by liquid loading is seen in the latter portion

of the rate data. The behavior of the computed D-and b-parameters is almost identical to

the previous case suggesting (to some degree) consistency of the data. Outstanding matches of the computed D-and b-parameters with the

power-law exponential model are observed.

"q-D-b" Plot: LWF2

zD-parameter data trendexhibits a power-lawbehavior essentiallya straight line.

zb-parameter data trendis not constant

(contrary to hyperbolicformulation).

zComputation of the b-parameter is signifi-cantly affected bynoise.

(F

ieldExamp

le4)



45/51

Slide

22/28





Discussion: Large Waterfrac Gas Well(LWF2) Estimation of the reserves using the hyperbolic rate decline relation is

almost five times higher than the contacted gas-in-place predictedpreviously using a model-based match (SPE 114947). Outstanding matches of the data are obtained using the power-law

exponential model, both the D=0 and the D

0 cases.

The D0 model provides the most conservative estimate of reserves.



(F

ieldExamp

le4)

Field Examples: Hybrid WF Gas Well(HWF1)


46/51

Slide

23/28





Discussion: Hybrid Waterfrac Gas Well(HWF1)

Severe liquid loading is observed at late times. The computed D-and b-parameters reflect the effects of liquid loading

however; the D-parameter data trend is essentially power-law. Good matches of the computed D-and b-parameters are obtained using

the power-law exponential model.

"q-D-b" Plot: HWF1

zD-parameter data trendexhibits a power-lawbehavior essentiallya straight line.

zb-parameter data trendis not constant

(contrary to hyperbolicformulation).

zComputation of the b-parameter is severelyaffected by noise.

(F

ieldExamp

le5)

Field Examples: Hybrid WF Gas Well(HWF1)


47/51

Slide

24/28





Discussion: Hybrid Waterfrac Gas Well(HWF1) The hyperbolic rate relation (b=1) yields the highest reserve estimate.

Reasonable matches of the rate data are obtained using the power-lawexponential model for both the D

=0 and the D

0 cases.

The power-law exponential model applied using D0 provides the most

conservative estimate of reserves (as is expected) this result is quitecomparable to the model-based results obtained in SPE 114947.



(FieldExamp

le5)

Conclusions:

Conclusions:


48/51

Slide

25/28





(Conclusio

ns)Conclusions:

Investigation of Hyperbolic Rate Decline Relation: While tuned

to yield reserve estimates which are "conservative" (based onthe model-based analyses performed in another work), thehyperbolic rate decline relation typically yields high to exces-sively high estimates of reserves in this work.

Conclusions:

Conclusions: (continued)


49/51

Slide

26/28





(Conclusio

ns)Conclusions: (continued)

Development of a New Diagnostic Plot: Log-log plot of the D-

and b-parameters versus time, where the D-and b-parametersare computed using rate-time and rate-cumulative datafunctions. These plots provide a unique diagnostic insight, aswell as an analysis/calibration function.

Conclusions:

Conclusions: (continued)


50/51

Slide

27/28





(Conclusio

ns)Conclusions: (continued)

New "Power-Law Loss Ratio" Rate Decline Model: Very flexible

model that can be used to match transient, transition, andboundary-dominated flow data. By the use of the declineconstant at infinite time (D

) we obtain an exponential decline

at very late times, which can provide a lower bound for reserveestimates.

SPE 116731


51/51

Slide

28/28

2008 SPE Annual Technical Conference and Exhibition Denver, CO 2124 September 2008SPE116731 Exponential vs. Hyperbolic Decline in Tight Gas Sands Understanding the Origin and



D. Ilk, Texas A&M University

A.D. Perego, Anadarko Petroleum Corp.J.A. Rushing, Anadarko Petroleum Corp.T.A. Blasingame, Texas A&M University

Department of Petroleum EngineeringTexas A&M UniversityCollege Station, TX 77843-3116

+1.979.458.1499 [email protected]

SPE 116731Exponential vs. Hyperbolic Decline in Tight Gas

Sands Understanding the Origin and Implicationsfor Reserve Estimates Using Arps' Decline Curves

End of Presentation

spe 116731 (ilk et al) power law exponential relation [wpres]

Documents