SPES!% 17708

The Variable-Rate Reservoir Limits Testing of Gas Wellsby T.A. Blasingame and W.J. Lee, Texas A&M U.

SPE Members

O-apyrlght 19SS Society of Petroleum Engineers

TM8 papar wee prepared for prwantatlon at tha SPE C3aa Technology Symposium, bald in Dallee, TX, June 13-15, 19W,

Thla papar was aalacfad for preaantatlon by an SPE Program Committee following review of information contained in an ebatract submitted by theauthor(e). Ca’Nenta of the papar, aa preaantad, have not bean reviewed by the Society of Petroleum Englnaara and are subject to corractlon by theauthor(a). The material, as preaamad, doaa not nacwaarily raflact any pwltlon of the Society of Petroleum Englnaars, ita officers, or members. Paparapreaamad at SPE mwtings are subjeci to publication review by Editorial Orrmmittws of the Society of Petroleum Englnwra. Parmlsalon to copy Israatrictad to an ebatracf of not more than S00 words. Illuatrationa may not be coplad, The abatraot should contain consplcuoua acknowledgment ofwhere and by whom the papar Is presented. Write Put Watlona Manager, SPE, P.O. Sox 83W36, Rlchardwm, TX 7608S-SSSS. Telex, 730S89 SPEDAL.

The calculation of the adjusted time function forthis case is complicated because the gas properties

This paper presents a new method of estimating must be evaluated at the average reservoir pressure,drainage area size and shape from production data(bottomhole pressures and flowrates) for gas wells. ~, as explained by Fraim and Wattenbarger.4 However,

This method is rigorously based on theory presented it is not likely that ~ as a function of time will beearlier for the slightly compressible liquid case and knownrequl~es a computer program to iterate on gas-in-place . If values of F were known ahead of time the

umfil the gas flow and material balance equations are calculation of adjusted time would be straightforward

satisfied. and the new method could be done by hand. We havedeveloped a strategy to determine the gas-in-place, G,

The component equations and analysis technique and the average reservoir pressure function

are strictly applicable only after the initial simultaneously by iteration of the new gas flow

pressure transient has reached the outer boundary. equation and the gas material balance equation. This

The gas flow equation that we present is an method is only applicable to post-transient or

approximation for variable-rate, post-transient flow, boundary-dominated flow, although the results of the

but it has provided excellent results for all of the verification cases suggest that the calculated ~cases investigated. These results imply, as was also relation may also be valid for transient flow.noted for the liquid case, that as long as the changesin flowrate do not dominate the influence of the outer In the “Development of the New Method” section,boundary, this method should give acceptable results,Currently, this method is only derived for single-

we present the new gas flow equation and discuss theiteration scheme used to obtain the gas material

phase gas flow and does not consider the effects of balance required to apply this method. This newgeopressured gas reservoirs or non-darcy flow. method is verified using data froitt reservoir

simulators for homogeneous reservoirs and for wellscontaining a vertical fracture. Three field examplesare included to compare our method to the resultsobtained using other analysis techniques. Finally, a

The purpose of this paper is to present a step-by-step development of the new method andrigorous method of estimating reservoir size and shape progransrtablealgorithms for the application-of thisusing variable-rate production data from a gas well.This method was initially developed for the slightly

method are included in the Appendices of this paper.

compressible liquid case.~ The new gas flow equationthat we present here is similar to the liquid equation NT OF THE NFU-

in form due to the linearization of the gas flowdiffusivity. equation. This linearization is The new gas flow equation, adapted from the

accomplished by the use of “adjusted” time and liquid equation presented earlier? assumes thepressure functions which account for the pressure following:

dependent changes in gas properties.2,3.- Radial flow into the well over the net pay

thickness;References and illustrations at end of paper.

2 The Variable-Rate Reservoir Limits Testing of Gas Wells SPE 17708

Homogeneous and isotropic porous medium;Uniform net pay thickness;Permeability constant (independent of pressure);Isothermal conditions;Real gas law;Laminar flow;Single phase flow; andAny rate schedule.

The new gas flow equation for variable-rate post-transient flow is

APa ‘ai-pawfBi~{ 4A fa

—=—q q

—,(1)s70.6 ~ ln(e~) +

24 ctiG

where

ta~q(~) d?

ta = . ,....,....................(2)q(ta)

Eq. (1) suggests that a graph ofApa/q versus fawill

be a straight line of slope

1m.— *. . . . . . . ● . . . . . . . . . . . . . . . . . . . ● . . (3)

24 ctiG

and intercept

b = 70.6 ‘~ In(-+t) . ................(4)e CArw

will result. Note that Eq. (1) is only valid forpost-transient flow, i.e., the time after the initialpressure transient has reached the outer boundary ofthe reservoir. This implies that the transient datawill not fall on the straight line predicted by Eq.(1) and that these data should not reconsidered inthe graphical analysis. These relations also assumethat the correct adjusted time, ta, can be calculated.

Since this calculation requires knowledge of the

average reservoir pressure, b, as a function of time,

this calculation must be iterative, since E is notknown.

The definition of adjusted pressure, pa, is

Pi PPa=- j fidp, ...,......................(5)

Pi P.

and the definition of adjusted time, ta, is

t 1 &

‘a = ‘icti J . ...............(6)o P(F)ct(F)

As noted above, we must know F to calculate theadjusted time functions in Eqns. (2) and (6). Since

F is not known, we must calculate it from the gasmaterial balance equation which is

P Pi5)-. - (1 - - . ..........................(7)

I z ‘i Li

A logical iteration scheme would consist of

estimating 6 from an initial guess of the gas-in-place, G (this estimate could just be somethinggreater than the last cumulative gas production),

This F relation could be used tc calculate theadjusted time functions, from which the Apa/q versus

ta plot could be made. The slope of this plot could

be used to estimate G, as suggested by Eq. (3) . Thisnew value of G could then be used for the nextiteration and this process could continue until someconvergence tolerance for G is met.

We provide an algorithm for the pertinant gasbalance calculations in Appendix A. We also providem algorithm of the “Golden Search” method, adapted

from Scales5, for the iterative solution of thisproblem in Appendix B. These algorithms are writtenin a generic format that can easily be programed inany computer language the reader chooses.

OF THE &WME~

In this section we will verify the new gas flowequation and the gas balance algorithm presented inthe “Development of the New Method” section. Theserelations will be verified using data generated withhomogeneous and vertically-fractured gas wellreservoir simulators. Both simulators were formulatedusing fully implicit flow coefficients and thegenerated results are considered accurate.

servoir s~

In this section, we investigated the applicationthe new gas flow equation and the gas balancealgorithm for homogeneous reservoirs. We modeledthree different rate cases: constant rate, constantpressure, and a linear rate decline. These rateprofiles are shown graphically in Fig. 1.

Systsm Properties for thelbmgeneous ReservoirSindation Ca8es

Reservoir properties

Geometry - Single well centered in a bounded circularreservoir

Drainage area, A 40.OacresReservoir drainage radius, re 745 ft.

Net pay thickness, h 30.0 ft.Wellbore radius, rw 0.2 ft.

Reservoir permeability, k l.O.mdReservoir porosity, @ 0.3 (fracjion).

ITotal we Volumet Vp= W 15.7 x lo”ft.s

-U.

SPE 17708 T.A. Bla$ingame andW.J. Lee 3

Gas-in-place, G 4.0306 BscfReservoir temperature, T S60 F

Initial reservoir pressure, pf 4800 psla

Gas gravity, yg 0.605 (air = 1.0)

Initial gas FVF, B, 0.6929 rb/Mscf

Initial gas viscosity, ~i 2.3480 X 10-2 CP

Inftial gas compressibility, Cgf 1.4B0 x 10-4 psia-l

_tion sC&&@

Constant rate case q = 1600Mscf/DConstant pressure case pti = 1000 psia

Linear rate decline case q(t) =3 x 103 - 2.7tMscf/D(t tn days)

The results of these simulations were analyzed tntwo ways. First, the volumetric average reservoirpressure in the radius of investigation, as calculatedby the simulator, was used in the adjusted timecalculations. Second, the gas balance algorithm was

used to obtain ~ and make the adjusted timecalculations. These results are shown in Fig. 2 alongwith the analytical (liquid) solution for this

case?’7 Note that not only are all of the calculatedresults the same, but also these results exactlyoverlie the analytical solution, except at very earlytime. This disagreement at early time is caused bynumerical dispersion that occurs at early time inreservoir models. The new method should only be usedto analyze post-transient, boundary-dominated data.

However, these results suggest that the ~ relationcalculated using the gas balance algorithm can beaxtrapo;ated into the transient flow regime andaccurate adjusted time calculations can be made.

The Apa/q versus ~a graph for these cases is

shown in Fig. 3. As expected, all of the data lie onthe same trend. These results show that the new gasflow equation is valid, but perhaps, more importantly,the gas balance algo’”ithmwas shown to give the same

results as those calculated using ~ from thesimulator.

The gas balance algorithm also includes theproduction that occurred due to the difference in thetransient rate profile and the post-transient rateprofile extrapolated back into the transient flowregime. This “extra” gas production, not modeled bythe post-transient flow equation, must be accountedfor or the gas-in-place estimate will be incorrect.We simply extrapolate the post-transient flow equationback into the transient flow regime and calculate thedifference between this relation and the trueproduction. This procedure is outlined in detail inAppendix A.

The key point is to choose the correct start ofstabilized flow so that only stabilized data are usedin the calculation of the slope and intercept of FkCIipa versus fa data. From a practical standpoint, the

start of stabilized flow can be identified as the

inflection point where theApa versus time data start

sp a steep incline on a log-log graph. An example ofthis criteria is shown in Ffg. 4. The start of;tabilized flow can also be estimated, in some cases,From a log-log plot of rate vs. time at the pointihere the rate data start down a steep decline. An~xample of this criteria is shown in Fig. 5.

To investigate the method in homogeneous“eservoirs, two cases for the start of stabilized flowtiereconsidered. The first case used criteria for thestart of stabilized flow for a well centered in a

bounded circular reservoir published by Earloughers.rhis gave a time of 413.3 hr. The second case usedthe time at which the pressure at the outer boundaryiropped 1 psi from the initial reservoir pressure.rhis time was 161.2 hr.

An error analysis of these two cases is presentedin Table 1. This analysis showed that the first case

‘tstab =413.3 hr) gave the least error in the estimate

~f gas-in-place, G. The maximum error in the gas-in-place, G, for this case was -1.53%, for the constantwellbore pressure simulation run. These resultssuggest that the criteria published by

Earlougher8should be used to predi:’ the start ofstabilized flow.

Table 2 shows ths complete analysis for thetstab=413.3 hr case. This table includes the

graphical results and “extra” transient gas productionas calculated by the gas balance algorithm. Themaximum error in the shape factor, CA, was 6.36%.

This relatively high error was also noted for the

liquid case.l In that case, as in this, the error inCA can partially be attributed to the exponentiation

of the calculated intercept of the Apa versus fa data.

However, we feel that this error is acceptable.

AWel1 Conhumo A vertical Fr@mi. .

In this section we investigated the applicationthe new gas flow equation and the gas balance.alaorithm for a well containing a vertical fracture. .We modeled only the constant wellbore pressure rateprofile, but we considered two different dimensionlessfracture conductivity (Cr) cases. The first case was

for Cr=lO, which is considered to be a typical

fracture treatment. The second case was for Cr=500,

which is considered to be essentially infinitefracture conductivity (no pressure loss down thefracture). This case was considered because of the

availability of analytical solutions9,10 and

interpretation techniques.10,11 The rate profiles for

these cases are shown graphically in Fig. 6.

45

4 The Variable-Rate Reservoir limits Testing of Gas Wells SPE 17708

Systa Properties for the H@raulioally Yraotured GesR%ll Sismlation Cases

Geometry - Single vertically fractured well in thecenter of a bounded square reservoir

Drainage area, A 80 acresDistance to wellDreinage boundary, Xe 933.7 ft.

Fracture half-length, Xf 186.7 ft.

xf/xe ratio, xf/xe 0.2 (1/5)

Reservoir fracture shapeFactor, Cf (Cr =~case) 7.3980

Fracture conductivities used Cr = 10, 5(JO

Net pay thickness, h 65 ft.Reservoir permeability, k 0.05 mdReservoir porosity, fP 0.16

Total pore volume, Vp = .#hA 36.2x 106ft.3

Gas-in-place, G 10.285 BscfReservoir temperature, T 1580 FInitial reservoir pressure, pi 4800 psia

Gas gravity, yg 0.65 (air = 1.0)

Initial gas FVF, Bi 0.62756 rb/Mscf

Initial gas viscosity. Pi 2.482 X 10-2 CP

Initial gas compressibility, Cgi 1.296 x 10-4 -1psia

Flowing bottomhole pressure, pwf 1000 psia

The results of these simulations were analyzed

using the gas balance algorithm to obtain F and makethe adjusted time calculations. These results areshown in Fig. 7 along with the analytical (liquid)

solutions for these cases. 9,10,12 Note that theseresults exactly overlie the analytical solutions,except at very early time. This error is similar t~that observed for the homogeneous reservoir case.Note also that the lowest curve on Fig. 5 is theanalytical solution for the Cr=lOOO case. Since we

simulated the Cr=500 case, the error between these two

cases is expected at early times. Recall again thatour new method should apply only for post-transientboundary-dominated data. However, these fractured

well results suggest that the calculated F relationcan be extrapolated into the transient flow regime andaccurate adjusted time calculations can be made fromthis relation.

We again considered howto choose the best.pointto call the start of stabilized flow. For thesefractured well simulations, three cases wereconsidered for the start of stabilized flow. The

first case used the literature criteria for the startof stabilized flow for a well centered in a boundedsquare reservoir. This gave a time of 6802 hr. Thesecond case used the literature criteria for the startof stabilized flow for a well centered in a boundedsquare reservoir with an infinite conductivityvertical fracture of dimensions xf/xe=0.2. This gave

a time of 12245 hr. The third case used the time atwhich the pressure at the outer :uundary dropped 1 psi

from the initial reservoir pressure. This time was930.4 hr for the Cr=10 simulation run and 861.8,hr for

the Cr=500 simulation run.

An error analysis of these three cases, presentedin Table 3, showed that the third case (Cr.lO,

tstab=930.4 hr) gave the least error in the estimate

of the gas-in-place, G, of 0.04%. However, we feelthat for general use the literature criteria (withoutfracture characteristics) should be used to predictthe start of stabilized flow. The maximum error inthe gas-in-place, G, for this case was -2.05%, whichis comparable to the homogeneous cases.

Table 4 shows the complete analysis for thet~tab=6802 hr case. This table includes the graphical----results and “extra” transient gas production that wascalculated by the gas balance algorithm. The maximumerror in the reservoir/fracture shape factor, Cf, was

less than 0.9%. The maximum error in the xf/x~ ratio

was -15.3%. These parameters were estimated u-sing a

correlation proposed by Blasingame and Lee11 that isbased an theory for an infinite conductivity verticalfracture. Again, W* feel that these errors areacceptable.

In this section we will apply our vewmethod tothree field cases. We will compare the results of ournew method with the original analysis of each example.

le 1

Field Case--I.cnv-Pexmmbility Fractured Gas W%ll. The

data for this example are from Fetkovich, QU.13

Fetkovich, et al. analyzed these data using ratedecline curve analysis with ordinary time. Later,

Fraim and Wattenbarger4 analyzed this data using tyPecurve analysis with adjusted pressure and adjustedtime.

This well is a hydraulically fractured gas wellcompleted in +,heOnondaga chert in West Virginia. Thegas-in-place, G, was estimated to be 3.36 Bscf usingmaterial balance. Fraim and Wattenbarger adjustedthe constant bottomhole pressure from 500 to 710 psiato represent the lifetime average BHP. The flowrateprofile for this case is shown in Fig. 8.

System Properties for Ex=ople Well Aof Fetkovioh, M al aud Fraim andUhttenbarger

Geometry - Single hydraulically fractured well in abounded reservoir

Gas gravity, Yg 0.57 (air = 1.0)

Reservoir porosity, O 0,06 (fraction)Water saturation, Sw 0.35 (fraction)

Reservoir net pay, h 70.0 ft.Reservoir permeability, k 0.0805md (a)

0.0768md (b)Skin factor, s -5.520 (a)

-5.079 (b)

I

4a

‘ SPE 177QB T.A. Bhstnguse andW.J. Lee 5

dellbore radius, rw 0.364 ft.

teservotr temperature, T 1600 F[nitial reservoir pressure, pi 417S psia

Flowing bottomhole pressure, Pti 710 psia

[nitial gas FVF, Bi 0.70942 RblMscf

[nitial gas viscosity, Pi 2.167 X 10-2 Cfl

[nitial gas compressibility, Cgi 1.870 x 10-4 ‘1psia

[nitial gas-in-place, G 3.360 Bscf (a)3.036 Bscf (b)

B- result of Fetkovich, St al.3- result of Fraim and Wattenbarger

hll of the gas properties in this work were estimated

Jsing the current industry correlations.14-16

The gas balance algorithm was successfully~pplied to the data for this case. The results arepiven below.

lesults for Example 1:

m = 805640x10-5 psi/Mscf/D/hrb = 1,3094 psi/Mscf

GPext = 2.6715x104 Mscf

G = 2.6281 Bscf (A=88.03

R,adialFlow Aualvsis

BiPik= 70.6 — ,n{2.246A)

bh cArw

acres, xe=979.1 ft)

fO.70942)(0.021b7= 70.6 ~ ln(~~)(1.3094)(70) . .

~ ● 0.01841J

Using the reservoir drainage area, A, we canestimate the fracture half-length, Xf. This

calculation is

1/2Xf = (0.01841)(3.83x106)

= 36,05 ft

The results obtained using the new method arecompared to those obtained by Fetkovich, et al% andFraim and Wattenbarger in the table below.

Reservoir Fetkovich!a) Fraim and(a)&dU@fZ~~ ~

G, Bscf 3.360 3,0345 2.6281A, acre 112.6 101.6 88.03k, md 0.0805 0.0768 0.0423 (a,b)s -6.52 -5.08 -5.52 (a,b)Xf, ft 176.7 (C) 113.7 (c) 36.05 (d,e)

Xelxf 6.27 9.25 27.2

Cf 7.0535 (e) 7.4038 (e) 7.6094 (d,e)

= 0.05276.md

pnalvsis for an Inflnlte wetlvltv Vertical. . . .

,.

2Cfxf 2.246—=A kh

EXP(‘)70.6 Bi~i

(2.246)s

EXP((

0.0805)(70.0)(1.3094

(70.6)(0.70942)(0.02167)‘)

* 0.002506

From the correlation presented in Appendix C we

a- these results assume a radial flow modelIb - used Fetkovich, U value of skin

c- Xf = 2rwe-s

d - used Fetkovich, et. als value of permeabilitye - estimated from the correlation in Appendix C

These results indicate that there has beensignificant stimulation (productivity improvement) inthis well. The assumption of the radial flow modelmay be incorrect for this case. We sought to verifythis point by using a fractured well simulator on theresults derived from the radial flow analysis and thenew method presentedin this work. We choose to usethe Fraim and Mattenbarger results because they usedadjusted pressure and adjusted time in theircalculations. Also, we used the permeability from theFetkovich, QQ work in our results because it wasobtained from a pressure buildup test and should beconsidered the correct value.

The results of these simulations and originalrate data are shown in Fig. 8. Note that thesimulation using our results passes through all of thedata, except at very early time. The simulation usingthe results of Fraim and Wattenbarger only match theproduction data at late time. This graph verifiesthat our analysis of this well, considering it to bevertically fractured, is valid. This graph does notdispute the radial flow analysis of Fetkovich, mand Fraim and Wattenbarger, it merely proves radialflow parameters may not give an accuraterepresentation of a vertically-fractured well.

estimated the following parameters by interpolationi The results of our method aDDlied to the oriainal

Cf = 7.6094 ‘data and the simulation cases” are shown in Fi~. 9.Note that the simulation using our results lies

.47 -.

6 The Variable-Rate Reservoir Limits Testing of Gas Wells SPE 17708

.—

.

exactly on the data trend. The simulation using the G (Bscf) * 0.09S264 0.14133results of Fraim and Wattenbarger lie considerably off Bi (rb/Mscf) 6.4667 6.9232of the data trend. A final comparison of the resultsis shown in Fig. 10. Fig. 10 is a log-log plot of the pi (Cp) 1.170 x 10-2 1.166 x 10-2data in Fig. 9. This graph could be used for typecurve matching on a constant rate type ”curve. We Cgi (psia-l) 2.647 X 10-3 2.816 X 10-3attempted to match these data on a curve presented by

Gringarten10 for swell with an infinite conductivity *- estimated from decline curve analysisvertical fracture. However, the ’bestmatch was pastthe xf/@0.05 curve (this is the last case on the Example 2--This is well 103-050 from Ref. 17.

type curve). The production data for well 103-050 are shown inNote that these data do essentially followIn general, we suggest that the type curve matching ::$x;;nentia, decline

This observation served asof the Apalq versus ~a data on a constant rate type the basis for the expone~tial decline curve analysis

curve should be valid. We state this knowing that the of Blasingame, @ al.17 The gas balance algorithm wastransient and transitional data maytibe accurately successfully applied to the unsoothed data for this

modeled by the Apa/q and ~a functions. However, the well, TheApa/q versus ~a data are shown in Fig. 12.

results shown earlier in Figs. 2 and 7 suggest that There are a few outlying data, but a well defined

the type curve matching of Apa/q versus fa data is linear trend is nhserved. The calculated resultsindicate about 2$::difference in the gas-in-place, G,

valid. between the ;/resent work and the results ofBlasingame, St al. The difference in the permeabilty-thickness product, kh, was 23.2%. The kh product for

les 2 and ~ this well was not estimated originally by Blasingame,&t_iL due to the lack of an initial pressure

Field Cases--Low Pressure Gas Wells in Northern estimate. We choose to correlate this well withTennassee. The data for these examples are from similar wells in the reservoir anti, based on several

Blasingame, W_al.17 These data were analyzed using parameters, we formulated what we feel is a reasonable

exponential decline curve analysis using adjusted estimate of pi. We then used this estimate and the

pressure and ordinary time in Ref. 17. decline curve analysis results of Blasingame, et al.to estimate kh. The results using our new method for

These we’lls are in the Monteagle formation in data from this well are given below.northern Tennessee. The reservoir is an ooliticlimestone that is typically encountered at about 1200 Results for Example 2:ft, Initial reservoir pressures are usually less than350 psia and the flowing bottomhole pressure of wellsin this reservoir is assumed to be 50 psia. The m= 1.2352x10-4 psi/Mscf/D/hr

production data for these wells were obtained from = 1.3064 psi/MscflD

sales tickets and some of these wells were :pext = 0,0 Mscf (all post-transient data)

periodically shut-in due to seasonal gas demand. G = 0.12741 Bscf (A=40 acres - assumed)

Bystun Properties for thewlls in the R.iu.iial F~OW AnalvsisR&mteagle Fonn=tion

Bi/Likh ❑ 70.6— ,n 2.24~,

ervoir Prooertia b( cArw

Geometry - Single well centered in a bounded squarereservoir ~)

Shape factor, CA 30.8828 = 70.6 ,nt~2.246)(1.74x10:l}(1.3064) (30.88)(0.25)

Drainage area, A (well spacing) 40 acresGas gravity, Yg 0.63 (air = 1.0) G 59.38 md

Skin factor, s (assumed) 0.0Formation temperature 83.20 F The results obtained using the new method are

Flowing bottomhole pressure, pwf 50 pSia (est.) compared to those obtained by Blasingame, at al. in

Reservoir geologythe table below.

Oolitic limestoneAverage depth of reservoir 1200 ft. Reservoir Blasingame,

~ ~ Jhis Work

‘ G, Bscf!&I1 prooerties0.095264 0.12741

kh, md-ft 45.61 59.38

Well Q&Q.t!Q m:

pi (psia) 400 375

.-4a

SPE 17708 T.A. Blasingame and W.J. Lee 7

The results using our new method compare“easonably well with those obtained by Blasingame, &~ using exponential decline curve analysis. Our newnethod is the more general model, as opposed toiecline curve analysis, and we would expect the newnethod to give more consistent results. The majortheoretical difference in these results is thatllasingame,@ al. used adjusted pressure but ordinary:ime, whereas we used both adjusted pressure and]djusted time in our analysis. This should account~or the majority of the differences in the results ofLhe two methods.

Example 3--This is well 1479 from Ref. 17.

The production data for well 1479 are shown in‘ig. 13. Note that these data indicate that this wellras shut-in approximately once a year. The solid lineindicates the best fit exponential decline that was#seal by Blasingame, et al. to estimate formation]roperties. Though this is a reasonable approach tothis problem, we would like to apply a method that#ould at least partially account for the flow datainxnediatelyfollowing the shut-in periods. This wouldillow us to use more of the available data for]nalysis, rather than isolate the data that we thinklie on the correct trend, Since our new method istalid fc- any rate schedule such that the changes in“ate do ot dominate the influence of the outer>oundary, we should be able to apply our method to~his data.The gas balance algorithm was successfully applied

to the unsoothed data for this well. The Apa/q

{ersus fa data are shown in Fig. 14. There is

considerable scatter in the data, but the solid line,>redicted by the gas balance algorithm, does establishin average trend through the data. The scatter inLhis data is due to two factors. First, we only havei single bottomhole pressure estimate and because ofthe shut-ins this is not a constant bottomhole]ressure case, Though this does breach our need fortrue bottomhole pressures and rates, we see in Fig. 14Lhat this effect is not devastating and aninterpretable trend does result. The second factor is,that, inxnediatelyfollowing the shut-in, the flowrate:hanges do dominate the influence of the outer

]oundary. This means that, by definition, our method#ill not correctly model these points.The calculated results indicate 10.2% difference in

the gas-in-place, G, between the present work and the‘esults of Blasingame, et al. The difference in thetermeabilty-thickness product, kh, was 25,1%. The‘esultsfor this well are given below.

!esults for Example 3:

m= 9.4061x10-5 psi/Mscf/D/hrb = 1.3930 osi/Mscf/OGpext = OeO’Mscf

G = 0.15729 Bscf (Az=40 acres - assumed)

Bdial F1OW Analy&i_s

BiPikh = 70.6 —

b ln(g&)

f6.92323)(0.01166N 70.6 ~ ln(~~~)(1.3930) (30.88)(0.26)

= 59.42 md-ft

The results obtained using the new method arecompared to those obtained by Blasingame, et al. inthe table below.

Reservoir Blasingame,~ ~ Jhis Work

G, Bscf 0,14133 0.15729kh, md-ft 44.52 59.42

The results using our new method compare quitewell with those obtained by Blasingame, St al- usingexponential decline curve analysis. Recall thatBlasingame, ei alt analyzed these data using adjustedpressure and ordinary time. This,could, and likelydoes, account for the majority of the difference inthe results. Since we used both adjusted pressure andadjusted t~,li~ we consider our results to be correct.

RYANO CONC-

We have developed a new method for estimating gasreservoir size and shape from variable-rate productiondata. Our new method is based on a rigorous gas flowequation, presented in this work, coupled with the gasmaterial balance equation. This method should beconsidered accurate so long as the reservoir outerboundary is being felt by the pressure response.

This new method was verified using numericalsimulations of gas well performance in homogeneous andvertically-fractured reservoirs. These simulationsshowed that the results of our new method were almostidentical to the analytical solution for a particularcase,

The major conclusions of this work are sunxnarizedbelow.

1. Gas reservoir drainage area, A, can reestimatedfor both homogeneous and verticitlly-fracturedreservoirs using our new method.

2. The reservoir shape factor, CA, can be estimated

for homogeneous reservoirs. Also, the reservoir/fracture shape factor, Cf, and the fracture half-

length, Xf, can be estimated for vertically-

fractured reservoirs.

3. The new method was successfully applied to threefield cases and the results compared very well withthe original analysis for each case.

A = Reservoir drainage area, ft2(m2)

I.-

0 The Variable-Rate Reservoir Limits Testing of Gas Wells SPE 17708

—

—

b . 70.6 ‘+ ln(e~~), intercept constant

Awin the post-transient flow equation,

psilHscflD (kPa/stdm3/D)

B = Gas formation volume factor, RB/Mscf

(res m3/std m3)

Cf = Pore space compressibility, psi-l (kPa-l)

C9= Gas compressibility, psi-i (kPa-l)

co = Oil compressibility, psi-l (kPa-l)

Ct . cgSQ + COSO + CWSW + cf. total

compressibility, psi-l (kPa-l)

Cw = Water compressility, psi-l (kPa-l)

CA = Reservoir shape factor, dimensionless

Cf = Reservoir/fracture shape factor,

dimensionless

wkfCr ❑ —, dimensionless fracture conductivity

wxfk

G = Original gas-in-place, Mscf (stdm3)

Gp = Cumulative gas production, Mscf (std m3)

= GP(tstab)- ‘Ptrn’‘pext “extra” gas production

during transient flow not modeled by thepost-transient flow equation, Mscf

(std m3)

tstab

G J qtrn(~)dr~ cumulative gas productior~ptrn = o

during transient flow as modeled by thepost-transient flow equation, Hscf

(std m3)

h = Net pay thickness, ft (m)

‘iPa = -J ‘~ dp, adjusted pressure, psia (kPa)

Pi Po~

Pai = Initial formation adjusted pressure.

psia (kPa)

Pawf = Flowing bottomhole adjusted pressure,

psia (kPa)

APa = ‘ai-pawf’ adjusted pressure drop,

psia (kPa)

Pi = Initial formation pressure, psia (kPa)

Pwf = Flowing bottomhole pressure, psia (kPa)

kh(pi-pwf)pD . dimensionless pressure

141.2qBp ‘

kh(pai-pawf)paD = dimensionless adjusted

141.2 qB~ ‘pressure

5 = Average reservoir pressure, psia (kPa)

q . Gas flowrate, Mscf/D (stdm3/D)

‘trn = Gas flowrate during transient flow as

modeled by the post-transient flow

equation, Mscf/O (std m3/D)

rw = Wellbore radius, ft (m)

r; = rwe-s, effective wellbore radius, ft (m)

s = Skin factor, dimensionless

s = Gas Saturation, fraction9

so = Oil Saturation, fraction

Sw ❑ Water Saturation, fraction

M = Objective function in the gas balance

algorithm, [psi/Mscf/D]2 ([kPa/std m3/D]2)

k = Effective formation permeability, md ta tl= picti J dt, adjusted time, hro #(F)ct(6)

kf= Fracture permeability, md

I *

~aq(r) dr

m. 1— , slope constant in the post- fa = , Plotting function for variable-24 ctiG q(ta)

transient flow equation, psi/Mscf/D/hr rate tests, hr

(kPa/std m3/D/hr)

f.“w

‘D = 0.0002637.~2, dimensionless time

*ictfrw

t aDE 0.0002637.--&2, dimensionless adjusted

~ictirwtime

‘Dxf= 0.0002637+-, dimensionless time based

Wictixfon fracture half-length

t aDxf= 0.0002637~2, dimensionless adjustedtiictixf

time based on fracture half-length

t Start of stabilized flow, hrstab=

tol = Convergence criteria for the gas balance

algorithm, Mscf (std m3)

T = Reservoir temperature, ‘R

‘P= #hA, reservoir pore volume, ft3 (m3)

w = Fracture width, ft (m)

Xe . Distance to the reservoir boundary in a

vertically-fractured well, ft (m)

‘f= Fracture half-length, ft (m)

i = Gas compressibility factor

m

Y = 0.577216, Euler’s constant

Yg = Gas gravity, (air=l.0)

P = Fluid vfscosity, CP (Pa-s)

+ = Porosity, fraction

p = Gas density, lb/ft3 (kg/m3)

.

a = Adjusted (normalized) variable

cal = Calculated

i = Initial

obs = Observed

The authors wish to acknowledge the inspirationand interest of M.L. Fraim towards this work.

1. Blasfngame, T.A, and Lee, W.J.: “Variable-RateReservoir Limits Testing,” paper SPE 15028presented at the 1986 SPE Permian Basin Oil andGas Recovery Conference, Midland, TX, March 13-14, 1986.

2. A1-Hussainy, R., Ramey, H.J. Jr., and Crawford,s “The Flow of Real Gases Through Porous

~~~~~,” m (May 1.966)624-36; -, AIME, 237.

3. Lee, W.J. and Holditch, S.A.: “Application ofPseudotime to Buildup Test Analysis of Low-Permeability Gas Wells ifithLong-DurationWellbore Storage Distortion,’”U (Dec. 1982)2877-87.

4. Fraim, M.L. and Mattenharger, R.A.: “GasReservoir Decline Curve Analysis Using TypeCurves With Real Gas Pseudopressure andNormalized Time,” u (Dec. 1987) 671-682.

5. Scales, L.E.:~~, SPringer-Verlag IncH~w York (198~).

Linear ODtl

6. van Everdingen, A.F. and Hurst, W.: “TheApplication of the Laplace Transformation to FlowProblems in Reservoirs”, -, AIME (1949) 186,305-324.

7. Matthews, C.S. and Russell, D.G.: PressurQ~S Henry L. DohertySeries. SPE, Dallas (1967) 1.

8. Earlougher, R.C. Jr.: s in Well Test~, Henry L. Doherty Series, SPE,Richardson, TX (1977) S.

9. Gringarten A.C., Ramey, H.J. Jr., and Raghavan,R“ “Unsteady-State Pressure DistributionsC~;ated by a Well With a Single InfiniteConductivity Fracture,” ~ (Aug. 1974) 347-360.

10. Gringarten, A.C.: “Reservofr Limits Testing forFractured Wells.” paper SpE 7462 presented et the1978 ‘SPEAnnual Technical Conference andExhibition, Houston, Oct. 1-3.

11. Blasingame, T.A. find Lees W.J.: “proPerties ofHomogeneous Reservoirs. Naturally FracturedReservoirs, and H,ydraulfcallyFracturedReservoirs from Decline Curve Analysis,” paperSPE 15018 presented at the 1986 SPE Permian BasinOil and Gas Recovery Conference, Midland, TX,March 13-14, 1986.

12. Cinco-ley, H., Samaniego-V.t F.! and Dominguez-A., N.: “Transient Pressure Behavior for a WellWith a Finite-Conductivity Vertical Fracture,”~ (Aug. 1978) 253-264.

13. Fetkovich, M.J., Vienot, M.E., Bradley, M.D.,and Kiesow, U.G.: “Decline-CurveAnalysis UsingType Curves--Case Histories,” & (Dec. 19g7)637-656.

14. “Theory and Practice of the Testing of GasWells,” Energy Resources Conservation Board,Calgary, (1975).

51

10 The Variable-Rate Reservoir Limits Testing of Gas Wells SPE 17708

lo

16, Lee, A.L., Gonzalez, M.H. and Eakin, B.E.: “TheViscosityof Natural Gases,” U (Aug. 1966) 997-1000; ~, AIME,237.

16. Sutton, R.P: “CompressibilityFactors for High-Molecular-Weight Reservoir Gases,” paper SPE14265 presented at the 1985 SPE Annual TechnicalMeeting and Exhibition, Las Vegas, NV, Sept. 22-25, 1985.

17.WB~asingame, T.A., Poston, S.W., and Hedberg,.,: “Evaluation of Reserves for a Number of

Small, Partially Shut-In Gas Fields in NorthernTennessee,” paper SPE 16853 presented at the 1988SPE Annual Technical Meettng and Exhibition,Oallas, TX, Sept. 27-30, 1987.

I Nowwe have assembled all of the components of ouriteration scheme. We still must define an objectivefunction to be minimized. A logical choice of anobjective function is the sum-of-squared-residuals

(SSR) between the observed Apa/q versus ~a data a~d a

linear least squares fit of the data. The equation ofthe fitted data is

AP&)cal = b +mta ........................(A-5)

where

b= 70.6B# ln(-#&) ,..,...,..........(A-6)

A - OF THE~FVAT~ SCHFME F~GAS IN PLACE F~POST-TR&QIFNT GAS F~-- and

IIn this appendix, we will derivean iteration 1m=—

$cheme to determine the gas-in-place, G, using the new. ......,...,.,..,.........,.,.,(A-7)

lost-transient flow equation and the gas material 24 ctiGoalance equation.

The sum-of-squared-residuals (SSR) formula is given asThe post-transient flow equation is given as

!ipa ‘ai-pawf Bi~i fa—=—q q

a 70.6 ~ ln(~~) + —,(A-l)

Aw 24 ctiG

#here

fa = 0— . . . . . . . 0 . . . . . . . , . . . . . . . . . . . . (A-2)q(ta)

Eqns. (A-1) and (A-2) serve as the fundamentalrelations upon which our new method is based. Theserelations are straightforward and relatively simple tocalculate. However, because our definition ofadjusted time, ta, is based on using the average

‘SR ❑ ; (A~)obs, i ‘~)cal, i)2 ““””””””””(A-8)ial

At this point we must address the fact that theseequations only model post-transient flow and that thegas-in-place, G, estimate ignores the effect oftransient production which is not modeled by Eq. (A-1)or its equivalent Eq. (A-5). The rate profile thatwould have existed if the post-transient model wereextrapolated into the transient flow region is

qtrn ‘Ah =!?_l# ....................(A-9)

‘A)cal aq

The total production, Gptrn, during transient flow is

obtained by integrating Eq. (A-9)

reservoir pressure, ~, to evaluate the required fluid I tstab

properties, we must determine, by iteration, ~, as a ! = J qtrn(T)dr‘ptrn o......................(A-1O)

function of time. The definition of adjusted time,ta, is

The error that Eq. (A-1) would yield in gas-in-place

t 1 dtis the difference between the observed production and

ta =/.4ictiJ . .,.............(A-3) the production predicted by Eq. (A-1O), up-to theo #(b)ct(E) start of stabilized flow, tstab. This volume 1s shown

graphically in Fig. A-1 and is expressed as

liecan determine the gas-in-place, G, from the slope‘pext = ‘p(tStab) -Gptrn . . . . . . . . . . . . . . . ..(A-11)

of a plot of Apalq versus fa, but we need another

relation to yield ~ as a function of time. Hence, we Adding this “extra” production to the gas-in-place, 2.

use the gas material balance equation given below. predicted by Eq. (A-7) gives

6 Pi-=_(l-5).

I .— ....................(A~12)(A-4) G = 2; m ~ti

+Gpext..........................

z z+ GI

52

SPE 1;?08 T.A. Blasingame andW.J. Lee 11

#here Eq. (A-12) should give the correct gas-tn-pldc8,t, if the correct start of stabilized flow is chosen.

B -AN~ITHM FOR THE S-OF THE

: POST 1~ GAS FLOU~.

In this appendix, we will develop an algorithm forLhe solution of the problem stated in Appendix A. Thejeneral minimization routine to be employed is the

iolden Search method.5 This method is well known andlas given consistent performance on the cases in thisitudy.

The steps, per iteration, to ca)culate the gas-in-~lace, G, are expressed in the following algorithm

~: GASBAL(G,SSR}*

1 ti~ [from Eq. (A-4) using current G]L

2 get~ [by interpolation]3 w ta [from Eq. (A-3)]

4 W fa [from Eq. (A-2)]

5 tib, m, SSR [from linear least squares onAPa---)ob5 versus la data for t ?tstab]a

6 w.lc Gpext [from Eqns. (A-9) to (A-11)]7 m the new gas-in-place, G, [from Eq. (A-12)]

k - All other pertinent data are carried in “COMMON”type blocks of storage

The following algorithm, adapted form Ref. 5,Oerforms the Golden Search method minimization~rocedure for the gas material balance problem

;all for Algorithm 2: m GOLDEN(GMIN,GMAX,tol,G)

Algorithm 2: GOLDEN (al,bl,tol,G)

1 ~r = 0.6180342 c1 = al + (l-?)(bl-al)3 dl = bl - (l-~) (bl-al)4 ~GASBAL(cl,Fc)

5 ~GASBAL(dl,Fd)

6 f& k=l,2,. ..repeat? tiFc<Fd Lb.en8 ~ ak+~ = ak9 bk+l = ‘k

10 ‘k+l = Ck11 Ck+l = ak+l + (1-7)(bk+l-ak+l)12 Fd = Fc

13 M GASBAL(Ck+l,Fc)

16 ~ ak+~ = Ck17 bk+l = bk18 Ck+l = ‘k19 ‘k+l = bk+~- (l-r) (bk+~-ak+l)20 Fc m Fd

21 @ GASBAL(dk+l,Fd)22 ~ G = dk+l

IX C - ~FOR THE EST~lON Qf

This appendix is provided as an addition to the

correlation presented by Blasingame and Lee.11 Thesedata have been expanded to include xf/xe ratios from

1/15 to 1/50, These later data were-taken from a

graph presented by Gringarten14 for an infinite con-ductivity vertical fracture.

‘f/xe

1/11/1.5

1/21/31/5117

1/101/151/201/251/301/351/401/451/50

Cf Cfx;/A

1/21/31/41/6

1/101/141/201/301/401/501/601/701/801/90

1/100

3.15245.43966.35007.03567.39807.49887.55287.58087.59637.60597,61217.61637.61917.62097.6219

0.78810.60440.39690.19540.073980.038260.018880.008420.004750.003040.002120.001550.001190.0009410.000762

.53

.

TASLE 1

ERROR ANALYSIS FOR THE END OF TRANSIENT FLO14 -NONOGENEOUSCASES (G = 4.0306 Bscf)

1. Time for less them 12 error fOr toA >0.06 (Circul.xr

hr.

case

Constant Rate 3.981Constant Pressure 3.969Linear Rate Oecl ine 3.9B0

reservoir) = 413.3

ErrOr. ~

-1.23-1.53-1.26

2. Time for outer boundary pressure to drop 1 psi . 1S1.2 hr.

case ~~

Constant Rate 3.972 -1.4sConstantPressure 3.954 -1.90LinearRate Oecl$ne 3.974 -1.40

TABLE 3

ERROR ANALYSIS FOR THE ENO OF TRANSIENT FLO!4 - FRACTUREO

‘fHELL CASE (-- = 0.2. Cr . 10 and 600, G = 10.2B5 Bscf)

1.

2.

3.

“e

Tim for less than 1% error

Case

Cr . 10

Cr . 600 *

limafor less than 1!4 error

~fconductivity fr&cture, -- u

‘e

case

Cr = 10

Cr = 600 ●

TABLE2

RESULTSFOR THE 80UNOE0CIRCULARRESERVOIRSIMULATIONCASES

G .4.0306 Bscf (40Acres)CA . 31.62 (boundedcircularreservoir)

The resultsbeloware basedon a time to reach stabilizationof 413.3 hr(tOA = 0.06).

~:

Case ‘pext m bvr A

Ncf i 1031 ~ lGWMJXQ M.cCISl

ConstantRate 1.1720 7.0761x10-5 0.56937 39.61

ConstantPressure 4.3314 7.1029x10-6 O.S6993 39.42

LinearRateOecline 2.4072 7.0799XI0-6 0.66960 39.50

:

Case

ExoctConstant RateConstant PressureLinear Rate Oecl ine

G, Bscf Error ,S CA Error,%

— — —

4.031 --- 31.62 ---3.981 -1.23 33.63 6.363.969 -1.53 33.03 4.463.980 -1.26 33.41 5.66

TABLE4

RESULTSFOR TNE VERTICALLY-FRACTUREORESERVOIRSIMULATIONCASES

G . 10.285Escf (80Acres)

Cf = 7.398(InfiniteFractureConductivityCase. ~ = 0.20).

for toA >0,05 (square reservoir) = 6802 hr..

Xe * 933.5ft Xf * 186.7 ft

cddMuAx Em&4EKm

10.478 1.BB

10.496 2.05

for toA > o.og (squarereservoir,infinite

0.2) w 12,246hr.

kMMWLBMhQLMCcW

10.623 2.31

10.642 2.60

Timefor outerboundarypressureto drop 1 psi.

~k.imJIC~ ErrOr. De CQO.Lr

Cr = 10 930.4 10.289 0.039

Cr = 600 * B61.8 10.291 0.05B

* The Cr ● 600 case is considered equa I to the infinfte fracture

conductivity case.

The resultsbeloware basedon a timeto reachstabi1izationof 6802 hr(tOA . 0.05)

~:

Case G m b Aixext

5k MfWWLCIM!&QM&@M3.Cf x 10

Cr = 10 1.9283 3.12G7x10-6 1.2637 B1.50

Cr = 600 2.0304 3.1233x10-5 1.2361 01.64

Case G, Bscf Error,X Cf Error,% Xflxe Error,%

—.. — ——

Exact 10.285 --- 7.398+ ---Cr = 10

0.20 ---10.478 1.B8 7.463* 0.879 0.1695‘ -15.3

Cr = 600 10.496 2.05 7.449* 0.689 0.1768* -11.6

+ Infinitefractureconductivitycase.

2Cfxf

* ValUO estimatedby correlationof _ with Cf and !f for the infiniteA ‘e

fractureconductivitycase.

64. S= 17708

. . ●

t, hr

F@. l-mm hltiorkz * ON h0m3sG+%J c M4uvofr 41ml*n 22U8.

-1

e: 3-Ji!:: Oata For All Cases:

s z— . COnitint R*tc

-m - Constant Pressure

a - Linear Rate Decline

1-

+x,,, .’ stab,!,,., Flow

a I I , I , 1 , , , I t , , , I , , 8 , I 1-0 Itsama 20080 30000 4mBam 5

f,,hr

*. a-hp,{q W. i. for fhwtWmOOMWUs mcotvolr sfmulafion ~.

103 41 st,rt of Sttbl liztd Flow

-1

1s)2I , , ,~ ~-3 I“’’u’’’ l’’’’ l[’’’i’’’’ i’” ‘1’”

10 b lB 1s33

10-2 1 ls#t. W

10+

!4!4. ~ vs. t fOI tlw Const#nt PmwJm, homoq4n90us rozavolr slmuktlon cue.

1(1)Cr.lo-AnalyticalSol”tlc,”

10 (.?)C,*1O- NWrlcal Solutlon

(3) Cr.soo - Numarlcal Solutlon

(4) Cr.lOOO-Antlytlcal Solution /

‘,aj“//’,..1 (2)” (3)7(4) Stablllzd Flow 1

10-2 , I , t ,~e-s I I,a-4 , .&2 I I

1 102 I

toxf 0’-~lolf

St+rt of Stablllstd Flowm --l

,Q-l I , 1 , I

1 1‘& ie~ taE 1OB ~GIB

to or $0

Flq. 2.-c.0mm+rl&n&19 Mw nmbod to tb4 *Mlytkaf ,OfUooil 10? StW ~s r44Mv01r

10

, (3)

(1) Constant Rtto Ctio(2) COnKait Pra!sureC*S*(3)Line!rR#taDoclfnoCcte (2)

~t (1)~:&l;-.$3

Startof Stiblliz@d F1OW

10-1 , , , ,

10-3l“’’I’’’’ 1’’’’ [’’’’ 1’’’’ 1’’”1 ‘“r

10-1 10 103 I

10-2 1 dt, hr

1 ‘#

Pig. 4-dpJq w. t for On ho m0gM9Qu8 ruerwolr slmu18tf0n 3w9*.

102

-i

‘Q2*’,1,,,1,,,,1,,,,1, ~1,,1,,1 102 to~

la-l 10t, hr

103 1 ‘#

Start of Stibil138d Flow-i

flu. 6-Me hhlo 10s for tha vertlully fmctumd mnwoh shnumtlon cans.

1 d I 1I I It , ,,, ,, , , r ,,, ,, , , , ,, ,1, , a , , ,,h

1 1$3 102 103 104

t, deyt

F@ d-R4t0 hlstcwy fcf th8 WNt In ZX9MS4C 1Fb. 7-COmPutWn OF on now nmhod to the nulytkd 801u21.20S for th8 VUUUIIY fnctumd rssw.

v0irUmu19f10n3nn.

E4 SPE”17708

XNP

i,, hr

FIo. 9-&p.lqW.ifortfnWI InErunPn1.

‘“~0 zae 400 600 me 1!300 1am 1400

f day$

Ff9. 1 f-_tO hUfOIY fO? tho WOII 1“ ~p!4 2,

‘e’~Nate tha effcct$ 0{ oar IQdtc shut- ins

10 I 1 I , I I 1 I Ia 200 400 6s0 80s 1000

t, days

f%. 12- ffat* IIbtw for me well In 2xsmph 3.

‘“’rrrrla

1

0 1“’’l’’’’l’’l@ll#al[Ma 11{1,,,{1,,❑ 10909 20000 30Lfaa 4aea0 Saaae 69000 7

$, hrFf9. 12-AP./q v8. i,lm W-wdllnmMtioz.

o

0

00

c)

1 I 1“’’l’uull(tcc~~ll$lll ,I]ltll5000 10900 1s000 zamaa 2500% 30000 35(

[,, hr

F1.J.14-,tp,/qn. i, Iortfmwul k.ti.mpl.a.

I

ITrue flowrate proffle

la~

~ - 61s t$roductlon during tr.3ns lent flow not

5# modeled by thw oost-transtont flow equatfon

&

1a3— : L‘trn

- Flowntg profllm durtng trm$lent flow as

m.Qdel Bd by tha post-transient flow euuatlon

“l I

I‘“z~a SWa3 1W3wl isaaa Zaama 2s$30%

t, hr

,,.

89

0

66

‘PE 2?708