.

SPESPE 23442

Production and Pressure Decline Curves for Wet Gas SandsWith Closed Outer BoundariesS. Mohaghegh and T, Ertekin, Pennsylvania State U,

SPE Members

—.

Copyright 1S91, society of Petroleum Engineers, Inc.

This pepar wes prepared for preaantetlon et the SPE Eastern Regionel Maellng held In Lexington, Kentucky, October 22-25, 1991.

Thie paper was selected for presentation by an SPE Program Committ- following review of Information contained In an abstract submitted by the author(s). Contents of the paper,as presented, have not been reviewed by the SNety of Petroleum Englnaara end are aub]ect to corraotlon by Ihe author(a). Tha material, as presented, does not rmcessarily reflecteny position of the society of Petroleum Englneere, Ita officere, or membere, Pepera presented at SPE meatinge are subject 10publication review by Editorial Commillaes of Iho Societyof Petroleum Englnaara. Permieeionto copy is restricted !Gan abstract of not more Ihan 300 words. Illustrations may not tM copied. The abstract should contai,l conspicuous acknowladgmantof where and by whom the papw la presented. Write Librarian, SPE, P.O. Sox SWWS, Richardson, TX 7S0834S36 U.5A. Telex, 730989 SPEDAL.

ABSTRACT

A family of pressure and production decline cuwcs am gen-erated for wet-gas sands with closed outer boundaries. Wet-gassands are characterized as gas reservoirs which produce substan-tial amounts of water together with ~as. Production of waterintroduces complications when practicing engineers use declinexwves designed for gas resemoirs in which gas is the onlyRowingphase. This usually translates to over estimation of theproduction performance of the reservoir,

Irr this paper a series of pressure and production declinecurves which accounts for the water production in wet-gas sandsis presented. These decline cumes provide a simple way ofextracting valuable information from avail~bleda!a, using graphi-cal methods and simple calculations, The prcposed declinewves arc generated for a radiid system with closed outer boun-dary with one centrally loeuted well which fully penetrates theformation. The application of ‘he proposed decline curves isillustrated through a series of examples,

In the case of constant pressure inner boundary, productiondecline curves are generiited for different ratios of sandface pres-sure and initial formation pressure, and in the case of constantflow rate inner boundary, pressure decline curves are generatedfor different initial formation press;res, The proposed declinecu~es covers a broad spectrum of reservoir extents and haveproven to be unique for a wide range of fluid and formationparameters,

INTRODUCTION

Analyzing petroleum reservoir; with the aid of type curvesis a practical method available to petroleum engineers, Typecurve analysis has been used for more than 20 years in the oil

References and figures at end of paper.

industry and for more than 40 years in hydrogeology’. Applica-tion of this method to gas reservoirs has been investigated byseveral scientists including Carte#,’ and Fetkovich4, In gas reser-voirs as well as other petroleum reservoirs the focus of typecurve analysis is on single-phase flow. Application of the typecurves in analyzing wet-gas sands is the subject of this paper.Wet-gas sarrds are gas reservoirs in which simultaneous flow ofwater and gas phases takes place. These reservoirs may producesubstantial amounts of water with gas production. Modeling ofthe flow phenomena in such reservoirs requires the considerationof two separate flow equations. Combining gas and water flowequations into a single compact form is the first step toward thegeneration of the decline curves. This equation can then betransformed into a dimensionless form and necess~dimensionless groups need “to be identified, The identifieddimensionless groups then can be used in the construction of thetype curves,

The type curves presented in this work are getieratcd for aradiul system with closed outer boundary itmJ with a fullypenetrating well at the cxxrter, Constant pressure and constanttlow rate cases at the inner boundary arc invcstigitted, [n thecase of constant pressure inner boundwy production type curvesare generated for different mtios of wellbore to initial formationpressure, and different reservoir extents. Within a certuinwellbore to inithd formation pressure mtio, type curves are gen-erated for different initial pressures. When constant flow rateinner boundary is specified, pressure type curves are labeled forboth different initial fommtiort pressures and different rescwoirextents.

‘7trispaper should be considered iis ii conlinuittion of a pre-vio~s paper presented by (lie iiuthors$, In this puper a change inth( formulation of the problem has been introduced throughwh):h iI previous limitation which wtts restricting the usc of thetype curves to low pressure reservoirs is successfully removed,

FORMULATION

In a previous paped the methodology used in combininggas and wa!er transport equations inks a single expression was

261

Iutlined in detail. A summary of this methodology which alsoIctails the implemented changes to the previous approach areIrescntedin this section.

Starting with the gas and water equations which representhe two-phase flow in wet-gas sands, one can write:

‘[5’’)=+ [%1(1)

v, [1#.vr .$w

($s.IIw-1

(2)

(3)

(4)

Multiplying Equation 1 by B, and Equation 2 by B. andIssuming that porosity is not a function of pressure, and then~ddingthe two equations one obtains:

‘W?V’I+B4*)V’1=

(5;

After some mathematical manipulations, Equatio:~5 can be writ-:en as:

‘:v[(t)v’l+B.v[(=)v’l=o~~ ‘6)where total system compressibility, c,, is defined as:

Expanding the left hilnd side of Equation 6, one obtains:

which con be also expressed as:

(AB+L)V2P+?+V $%+I I

Using !he following identity,

Vlnu=$Vu (lo)

quation 9 can be written as:

(~+k)V2p+k1Vh(~)vws

&Vln(#)vp=Oq~f (11)w

n the linearization of the atmve expression it is assumed that:

k‘=apB@

(12)

L—=bpB.

(13)

vhere “a” and “b” are constants. Therefore, Equation 11 can bemitten as:

jy recalling that:

Vln(#)= Vln(ap)=Vlnp=~Vp (15)I

If one defines the total system mobility as:

h=k,+k (16

Zquation 14 can be written as:

V2P+:(VP)2=+R

Equation 17 can be rewritten in a more compact form as:

(17:

(18:

Finidly, when Equation 18 is written in one-dimensional radiamordinates gives:

(19

Equation 19 is similar to Equation 18 of Reference 5, Th(~iffwence is that in Equation 19 pressure squared approach ha!xen used instead of the pressure itpprowh,

Trimsforming Equation 19 into dimc,lsionless form yields:

la.—r~ a rD

where

ilA~–1 aApD‘D a rr2 = ‘“-a ID (2(

262

.

Production Type CurvesAh= R2-P2

(pi*-PM*)%(22)

Using Ihe formation and fluid data shown in Table 1, gasand water production decline curves were constructed. Figures 1

2.637Xl@~ Lb= (23) and 2 show gas production type curves for ~ = 0,1 and initial~ q rW2

formation pressure of 2000 psi and 6000 psi, respectively. Thend corresponding water production type curves are shown in Figures

3 and 4. The proposed type curves were extensively tested,’

[ 1against a wide range of fluid and formation properties. Figures 5

~. qwBB(24) through 8 show the agreements obtained in some of these

2rrklqh(p~-pti2) ‘W verification tests, Several other verification tests were conductedand in every case high quality matches were achieved, Gas andwater production type curves were generated and verified for

To construct the type curves for gas and water phases it is other ratios of ‘*eeessary io identify dimensionless gas production and water ~ and different initial formation pressures andreduction rates explicitly. These dimensionless groups can be are available from the authors upon request. The relative per-ientified as follows: meability characteristics that was used in the construction of the

proposed type curves are shown in set A of Figure 9. The14.24X1(+~w Z T uniqueness of the proposed type curves with respect to the rela-

‘= Ach(p; -pd2)(25) tive permeability characteristics will be discus~ later.

Following example demonstrates use of the gas and water pro-duction type curves,

282.S3qWXBW P

‘=~h(p/-pti2)(26) .Example #1

A simulation run was conducted to generate gas and waterI order to enerate the type cuwes which.characterize the pres-

fproduction data. The formation and fluid data used for this run

Jre drop, imensionless presure drop group must be identified.?ris can be achieved by substituting Equation 25 into Equation are shown in Table 2, Gas production type curve for ~ = 0,12 which in turn yields: and initial formation pressure of 4tXKlpsi were used to perform

typr: curve matching from gas flow rate versus time data. In Fig-~h(~2-p2) ure 10 a match point with coordinates

A~=14.24X1($~ Z T

(27)

[~=0.05 ,~= 2000(S)],@uN, md [qt. lXIO’,t=0.9]hUn@

The constants irr Equations 23, 25, 26 and 27 imply the usef the field units for all of the entries in these equations. is identified, Using the information provided in Table 2 and

relative permeability characteristics of set A in Figure 9 follow-ing values were obtained:

DISCUSSION OF RESULTS~ =0,022 Cp z,= 0.86

A one-dimensional, two-phase, radial numerical simulatof kr~= 0,1024 kW=0.12%

was used to generate the rate-time and pressure-time data neces-wry for the construction of the type curves. In the case,of pro- Substituting above values and information from Table 2 in Equa-duction type cuwes the rate-time data generated by the smurlator tion 25 gives:was used to calculate the dimensionless production rate and thedimensionless time groups, These calculated values of the 0.05= ~ “424)( lx’07 ) ( 530) (@’@=dimensionless groups were then plotted on Iog-lcg scale. It was k ( 0.1024) ( 85.6)( 38342-383,42)

found that for different wellbore to initial formation pressureratios it was necessary to generate separate production type Jrcurves, Thus, production type curves were generated for a broad k = 22.39 mdspectrum of ~ ratios, Itwas also observed that within a given

wellbore to initial formation pressure ratio type curves are unique Substituting above calculated permeability in Equation 23, poros-for different initial formation pressures, ity of the reservoir is calculated as 15,7%.

In the case of pressure decline curves, pressure-time data To demonstrate the compatibility of the corresponding watergenerated by the simulator was used to caI.ulate the dimension- production type curve to that of the gas, a second type curveless pressure drop and the dimensionless time groups. These matching for this example was also performed this time using thedimensionless values were then plotted on log-log scale. Pres- water production type curve with the water flow rate versus timesure decline type curves are generated for different initial forma- data, As it can be seen in Figure 11 a match point with coordi-tion pressures, nates

In both cases type curves are constructed for differentdrainage areas. In the constmction of type curves all of the pres- [

~=o,2, tD=2m1. [~YHWCand q. = 700, t = 0,91MtPld

sure and saturation dependent variables are evaluated at initialconditions,

263

s used. It should be noted that in both cases of gas and watermxluction type curves the value of dimensionless time and realime in match points arc the same, This is essentially true sinceKsth type cumes share the same abscissa. This observationhould be used as a guide line in obtaining a match point.Following the same procedure as discused before and usingEquation 26 the

rrmeability and porosity of the formation are

xdculated to be 3.48 md and 16.4%, respectively. The actualAres used in the simulation run were 23.5 md for the permea-bilityand 16,2% for porosity.

As it was mentioned earlier, existing type curves for gas“eservoti assume flow of single phase. This assumption creates;ignificsnt limitations on the use of these type curves when ini-,ial water saturation is higher than the formation’s critical watermturation. Use of single-phase type curves, when analyzing a‘eservoir which is producing under two-phase flow conditions,will result in the over estimation of the reservoir’s performancewhen all reservoir’s parameters are known, On the other hand,lse of the single-phase type curves will lead to under estimation]f reservoir properties if type curve matching is performed. Fol-owing example focuses on demonstrating these shortcomings.

Example #2

Using the same reservoir as Example 1, a single-phase type:urve3 was used to find the permeability and porosity of the for-nation. Type curves in Reference 3 are generated for differentdues of L, which is defined as follows:

(28)

For this example the value of L is calculated to be 0.5S. Figure12 shows the type curves matching performed using the single->hase type curve generated for k = 0.05. l%e match point hashe following coordinates:

[qDR= 0,033,IDR= 3WKJ1T@Jm’c [ 1and q = 1X1O’, t= O.01~mnd

Using Reference 3, permeability of the formation is calculated tox 3.47 md, Using the calculated permeability value, one can:alculate the formation porosity as 1.5%. Comparison of theseralues to actual input values of 23.5 md and 16,2%, highlights:he inadequacy of using single-phase type curves in analyzing theiata generated flom a gas reservoir under two-phase flow condi-tions, If performance prediction of the reservoir is the goal thenJse of single-phase type curves will result in overestimation of;he reservoir capabilities, For the above match point, if the inputvalues for permeability and porosity were used the single-phase!ype curves would predict a gas flow rate of 380 MMSCFDinsteadof simulated value of 67.7 MMSCFD, at t=O,Olday.

Pressure Type Curves

To construct the pressure type curves, formation and fluidpropeflies shown in Table 3 were used. Again pressure versustime data were generated using the numerical modelf. Dimen-sionless ressure drop and dimensionless time groups were calcu-lated anJ

fIottul on log-log scale. Relative permeability charac-

teristics o set A in Figure 9 was used in the construction of theproposed pressure drop type curves. Figurw 13 and 1,4 showths dimensionless pressure drop curves for initial fonnatlon pres-sures of 20(KIpsi and 6000 psi, respectively. In Figures 15 and16 results of some of the verification tests are displayed. Similartype curves have been constructed for other values of initial pres-sure and are available from the authors upon rquest. As it is

shown in Reference 5 for production decline curves and demon-strated here for pressure type cumes, the pro~ed type cumeshave been found to exhibit unique characteristics for differentsets of dative permeability curves. It should be noted that for agiven real time wet-gas sands which share the same propertieswith the exception of relative permeability characteristics, will bedisplayed at different locations on the type curve. These shiftscaused by the relative permeability characteristics arc shown inFigure 17.

Application of the proposed pressure diop type curve isdemonstrated in the following example,

Example #3

Using the data shown in Table 4, a simulation run was con-ducted and a pressure drop versus time data was generated,Using the pressure drop type curve constructed for initial forma-tion pressure of 4000 psi, a type curve matching was performed.Figure 18 displays the match point as:

Using infornwtion in Table 4 and set A of Figure 9 followingvalues are obtained:

Be= 0,0235 Cp Z+=0.865

q= 0.3951 & = 0.0123

Using Equation 27 permeability of formation is calculated to be4.56 md. Using the calculated permeability in Equation 23 onecan calculate the porosity as 9.07%. Actual input perrneabdityand porosity values were 4.41 md and 9.3%, respectively andagree very well with the values derived from type curve match-ing.

During the construction of the presure type curves fordifferent initial formation pressures, it was observed that all ofthese type cutwes were showing qualitatively the same behaviorand they could be collapsed into a single pressure type curve. Itwas found that this can be done with a proper shifting in thedimensionless time group. Pressure type curves for initial forma-tion pressure of 2000 psi was chosen to provide the base for thispurpose. Figure 19 shows the amount of shift in dimensionlesstime needed for any initial formation pressure. These shifts canalso be represented using the two following simple relationshipsfor two different pressure ranges:

Imrt =3.8574X10-Spi+ 0.882852 for R c 2350 psig (29)

and

1~~, = 4,3574X10Api-0.00507 for p,> 2350 psig (30)

Following example will demonstrate the use of the shiftingfunctions described in Equations 29 and 30,

Example #4

In order to demonstrate the use of pressure ty~ curvesdeveloped for initial formation pressu~ of 2000 psi, in analyzingthe data generated from a reservoir wuh dtfferent initial pressure,a simulation run was designed with initial pressure of 5183 psi.Formation and fluid properties for this run arc shown in Table 5.

.SPE 23442 ‘ S. MOHAGHEGH ANDT.ERTEKIN 5

As it can be seen irr Figure 20 following match point from

pi= 20CKItypc CUIVC is obtained:

Using information provided in Table 5 and set A of Figure9 following values are derived:

Pv= 0,0274 Cp q = 0.980

km=0.3951 ~ = 0.0123

since the shift only takes place in abscissa, no adjustment isneeded for dimensionless pressure drop. Using Equation 27 per-meability is calculated to be 9,1 md. Using the relationshipgiven in Equation 30, the value of dimensionless time is adjustedas follows:

rmm= ( 4.3574x10A) ( 5183.0) -0.00507 = 2,2534

Using the new value of ~ and the calculated permeability, poros-ity is calculated as 11.65% from Equation 23. Actual inputvalues for_permeability and porosity were 9.45 rrd and 12.85%,respectively.

SUMMARY

Major achievements of this study can be summarized as fol-lows:

). Gas and water transpost equations were combinedinto a single expression in the form of a pressuresquared expression. This expression wastransformed into a dimensionless form and necessarydimensionless groups were identified.

2, Using the identified dimensionless production rateand dimensionless time groups, water and gas pro-duction type curves were construct for differentwellbore to initial formation pressure ratios for agiven initial formation pressure.

3. Using the dimensionless pressure drop and dimen-sionless time groups identified in this study, pressuretype curves were constructed for different values ofimtial formation pressure.

4, Extensive tests were conducted to ensure the unique-ness of the proposed type curves within a wide rangeof fluid and formation properties.

It is envisa cd that the proposed type curves could serve asta practical toolorpracticing engineers in performance prediction

as well as formation characterization of the wet.gas sands.These type curves will be useful in avoiding the overestimationof the reservoir capabilities which could be caused by the usc ofsingle-phase decline models when simultaneous production ofgas and wateris encountered.

,.

B=

c=

k=

h=p=q=

::t=

T=z .

NOMENCLATURE

formation volume factor, $#-

comprcssibility, psi-lpermeability, mdreservoir thickness, ftpressure, psiflow rate, MMSCF/dayradius, ftsaturation, fractiontime, daytemperature, “R,gas compressibility factor, fraction

Greek

k = mobility, md/cp~. viscosity, cp$ = porosity, fraction

Subscripts

D = dixmc;onless propertiese=g= gas properties

Sc = standard condition1= total properties

w= water propertiesWf= Wellboreproperties

REFERENCES

1,

2.

3,

4,

5.

6.

Gnngarten, A, C. :“Type Curve Analysis: What It Can andCannot Do,” JPT, (Jan, 1987), 11-13.

Carter, R. D. :“Type Curves for Finite Radial and LinearGas Flow Systems: Constant-Terminal-pressure Case,”SPEJ, (Oct. 1985),7 19-728,

Carter, A. D, :“Characteristic Behavior of Finite Radial andLinear Gas Flow Systems - Constant Terminal PressureCase,” SPE/DOE Low Permeability Symposium, (May 27-29, 1981), Denver, Colorado.

Fetkovich, M. J. :“Decline Curve Analysis Using TypeCurves,” 48th Annual Fall Meeting of SPE, (Sept. 30- Oct.3, 1973), Las Vegas, Nevada,

Mohaghegh, S., Ertckin, T, :“Production Decline Curves forLow Pressure Gas Rescsvoirs Undergoing SimultaneousWater Productions,” SPE 21269, SPE Eastern RegionalMeeting, (Oct. 31- Nov. 2, 1990), Columbus, Ohio.

Kin% G. R. :“Numerical Simulation of the SimultaneousFlo@”of Methane and Water Through Dual Porosity coalSeams: Ph.D. Dissertation, The Pennsylvania State Univer-sity (1985).

26s

Tabk L Formatim and tluld dsta used

10 the ametructbsr of gaehmter

productbn type curvee for wet.gae esnds.

Snilid Psessurc(psi)

Wellbore Presure (psi)

Well Redius (ft)

PermeabUi~ (red)

Poroeiry (%)

Formedon Tbkhss (ft)

e3seGfaviLy (eiA.0)

Rewrvois Tcmpaeture @)

Wuer Viscoeiry (q)

2000

200

0.5

1.0

6

20

0.6

530

1.0

WeUtme PressureQA) 383.4

WeU Redius (ft) C.5

Mid Ge Semetion (%)

I30 I

Famaion TM-s (fr) I 85.6 Ib Gravity (sir=l .0) I 0.6 IRuervoir Tempcrerurc ~R) I 530 I

Table 3. Formmbo snd nuld data wedin tbe KwBtsuetloo of tbe PsW8MN

type curves fw wet*ae mods.

Speci6ed Gee RoW Rue fhfSCFD) 10.0

Well Redius (ft) 0.5

Pwmesbiiily (red) 1.0

Pcloaity (%) 6

FmmaticmThiCkOtsS (ft) 20

Ges Grtvity (eir=l ,0) 0.6

Reservoir Temperature @R) 530

Waler Viscosity (cp) 1.0

Tabk 4, Formation ●id 8Ukt d8ta used

h Rxwsspk #3,

Speci6wf Gse Pbw Rate (MSCFD) 14.39

Well Rutiw (ft) 0.5

Fawwdon ~CkSSCSS (ti) I 17.4

(3s5 Gravity (sir=l.0) I 0.6

Reurvoir Tcnspmrure ~) I 530

Water Viscosiry (cp)I

1.0

Table $. Formation met tluid data wed

h Ehmpk $4,

~

Speci5ed Gas FbW R@e (MSCPD) 21,7S

WeU Ra&ue (ft) I 0.5

Forrneth ‘fhic&I-S (h) 39,45

au Grevity (*1 .0) 0.6

Wster Viscoeiry(Cp) I 1.0

206

Figure1.Gas production typecurves for ~~ = 0.1 andp,=ZOOIptit.

Figure2. Gas productiontype curvesfw ~ = 0.1 andn = ~ PJf#.

1

;

Z

i!

o,-$:,“

Dimt?nsionless Time

Figurs 3. Water prmiuction t~ curvss fw ~. O.I and p, .2013 pdl.

(nQm

.

Figsuc7. Afpfdim Ofwaurpmduaiontypecurves10a rcscwoirwishfollowing ChamcteristixSet A

5-=S0%.P.=l@Opsi.w=lfXI pSi. k=13.7 d +=.6% h=352 fL

1

PI

u

~

si5

D-MuAollrcssmm?

Fip’e8. A@iasim ofwabcrpmduaiwsYFcavvcs wamsa’hr Wm fLltlLwingChaI’xmisti:s

Sm=SO%,P,=@M psi. x=fKKt F&. k-40.O md. 0=13.5%.h=73.5 fL

*,,,87

●*

;0s,.

.3

.,

.,

*.

SPE 23442 :

.

Set B

Set C Set DFigure9. Ftelativcpcnncahili!y’chamcwristics used in analyzing the Pcrfomance of k proposed

type Curws

544’2

Figure 10, Type curve matching using gas production type curve for Example #l,

Figure 11, Type curve matching using water production type curve for Example #l,

D=lga ● ,

:.-

107

q w

106 1111

1

P- “m“m“m‘m‘ ‘m“I 0.001

Fiwme 12, Twe curve matching using Carter’s) type cuwe for Example #2.. .

270

0,1

%R

0,01

I

Dimendohess Time “

Figure 13. Rcssure IYIK cutves rOfp,EZo@IPW.

i

F@R 14, Prcwre typs curves fnf p, = MM pif.

Figure IS, Appllc&IIon of prewwe Iype cwvcs to a rcwvok with following chsractctistics:

S,, .S0%, A=1783 Pd, qxE7MXl SC.PD, kmo, I ml, 4u8%, hm49 fl,

Dimeaa{oheas I’{me ‘

Figure 16, Application of prcmIre IypO cosvcs to a resrmoir with following chuactcnstics

S,, -60%, PI-5872@, e,m=29300 SCFD, k=23.1 rm!, 0=1 1,3%, h=44 fl.

9 Set Di

Figure 17, Sensitivity of the pmposcd IYIM curves 10 the relative pcrmcahllity curves given In Figure 9,

Flgusw 18. Type curve rrulchlng ucln~ prmwe Iyp+ curve for Example #3,

2?2

,,* SPE 2344’2

I

I I

1 ! 1 12000 4000

Pressure (psi)

Figun 19. Shift in tfJwhenpressuretype curve forpi =20C@Psis is USC4! as the base

3l!I

Jxmmdon&41 mu

Figure20, Type curvematchingusingpressuretypecurvefor Example#4,

279