8/13/2019 SPE-001243-B

1/6

Application of Real Gas Flow Theory to WellTesting and Deliverability Forecasting

R Al-HUSSAINYJUNIOR MEMBER AlME

H J. RAMEY JR.MEMBER A/METEXAS A M Oll G STATION TEX.

As shown in Ref. 1 substitution of m p) in Eq. I leads to:

Eq. 3 looks like the diffusivity equation, bu t is stilln on -l in ea r b ec au se t he diffusivi ty d ep en ds u po n p re ssure,or m p). Eq. 3 also closely resembles th e equation forthe flow of an i de al gas. F ur th er mo re , th e dependenceof the diffusivity term upon m p) for th e real gas isvery similar to th e dependence of th e ideal gas diffusivityupon pressure squared. As is shown in Ref. I, this leadsimmediately to practical solutions for Eq. 3 which aresimilar to th e Aronofsky-Jenkins ideal gas flow solutions.E ngi ne er ing un it s will b e us ed throughout th e following.Fo r production of a real gas in an i de al radi al syste mat c on sta nt r at e with a closed o uter bounda ry :

The purpose of this p ap er is t o p ro vi de nec essar y engineering forms for use of th e real gas flow results, andto illustrate applications.Th e following considers flow o f real gases in an idealr ad ia l flow system. is assumed that: 1) formationthickness, porosity, water saturation, absolute permeability,temperature an d gas composition are constant; 2) gas

c ompressi bi li ty a nd d en si ty a re fun ct io ns of p re ssure asdescribed by the gas law pv = znRT 3) gas viscosi,tyis a f unc tion of p re ssure; 4) e ffec ti ve p erme ab il it y togas ma y be a function of pre ssur e to ac co unt f or liquidc on de nsat io n; 5) c on de nsat e is immobile; an d 6) theforma ti on h as n o di p.

3)

2) pm p) - 2 /L P Z P dp .

SUMMARY OF REAL GAS FLOW EQUATIONSCombination of t he c on ti nu it y e qu at io n an d Darcy slaw f or radial flow of a r eal gas yields the equation:

~ [r f P ~ Z P ) t . I)I f the l ef t- ha nd s ide of Eq. 1 is differenti ated, a terminvolving op /;orY arises whic h does no t occur in idealgas flow e qu at io ns. Thu s, mod ific at io n o f id ea l gas flowsolutions to fit real gas flow involves the assumptionthat p re ssure g ra di en ts a re small . An alternate procedureis to make a change of var iable in Eq. 1

W e d efine t he rea l gas p se ud o-pressu re m p) as:INTRODUCTIONA recent study of the flow of the transient flow ofreal gases in ideal radial systems showed that it ispossible to c on si de r gas phy sic al proper,ty de pe nd enc e

upon pressure by means of a s ub st it ut io n c alle d t he r ealgas pseudo-pressure. Th e substitution has th e advantagethat it enables engineering solutions for steady andtransient flow of re al gases that are more accurate andgeneral than t ho se p re vi ou sl y a va il ab le . Th e solutionsar e particularly i mp or ta nt f or the case of gas flow int ig ht , h ig h-pressu re forma ti on s under large drawdowns.E ve n f or more i de al flow c on di ti on s, t he rea l g as p se ud op re ssure c on ce pt to important rules fo r findinga ve ra ge ga s physiCliI p ro pe rt ie s p erti ne nt for o ld er welltest analyses. .

Ref. 1 provides a detailed study of t he t he or y b ehi ndth e r ea l gas flow c or re la ti ons to be used in this paper.

ABSTRACT

Original manuscript received in S

8/13/2019 SPE-001243-B

2/6

1.987 X 10-0 khTsc [ -)_ ( )] = 1 + + DT m p m p , n s qqpsc Yw 4)

gas viscosity, and slightly dependent on time for veryshort transiellit flows. However, D can be consideredconstallit ,as an adequate engineering approximation.where APPLICATION OF REAL GAS FLOW EQUATIONS

I 1PD(tD) = T [In tn+0.80907] , for 100 < tn

8/13/2019 SPE-001243-B

3/6

(- 5 92 10 qPuT cp CfJ.c,),r/)]m pw.) = m p) 7 X khT c Og 0 0.00266 kt(22)

The slope of a plot of m pw.) vs log o t yields the flowcapacity, while the m p) c an b e read on the straight line ata build-up time t of:cp fJ.c,),r, 23)t hours = O.0 02 66 k .

analogy is a geneml result. As flow approaches that ofan ideal gas, th e two methods and the real gas flowmethod yields the s ame answer.BUILD-UP TESTINGThe pertinent equations for analysis of pressure build-upin terms of the real gas pseudo-pressure are:Flow capacity = m pw.) = m p,) - 5.792 X 10 qPuTkh

(20)

19)

18)

.(21)drainagethe best

[ tp +t ]loglO-t-.- .

2.303 [ m p ~ ~ m p) ] ,

( )_ 0.000264 kt p

t D - - -;----;---- --P CP fJ.c )i Awhere A is the drainage area in square feet.

If producing time t is long enough t hat th eradius is stabilized hefore the well is shut in,approximation for pressure huild-up is:

an d the dimensionless producing time is defined as:

Or using standard conditions of 14.7 psia and 60F:q [ tp t ]m p .) =m p,) - 1,637 kh log,o-t .

Thus, a Horner- type plot of m pw.) should yield astraight line of slope b), and formation flow capacitycan be determined from Eqs. 13 or 14. Th e skin effect andnon-Darcy flow coefficient can be determined f rom Eq.15 if the real gas potential difference is replaced bym pl h ) = Pwp). Th e flow efficiency can he determinedfrom Eq. 17 if m pwp) is substi tuted for m pwt). Th epressure drop across the skin can be computed fromEq. 16 as for dmwdown.

In a completely analogous manner to previous build-uptheory, Eqs. 18 an d 19 indicate thiat the plot of m pw.)can be extrapolated to infinite shut-in t ime (a t ime faJtioof unity) to yield m p*). Th e Matthews- BronsHazebroeklOpressure correction charts can be used to correct theextrapolated m p*) value to m p), the average real gaspotential, if the ordinates of their pressure correctioncharts are changed to:

= m p) - m pwt) - 6 m p) kin. (17)FE = Juct / J h

8/13/2019 SPE-001243-B

4/6

or

28

29

q = 1.987 X 1 0 kh 2) (P-p , ) 27)p.B [I n 0.472 r,lr,,]Comparison of Eqs. 12 an d 27 leads to:

2 j j -Pwf) T [m p)-m p H/lB,J ,. p T

or

1 As a result of a cha nge of v ar ia bl e ca lle d the realgas p se ud o-pressu re , i t is possible to write approximatesolutions for the produc,tion of real gases from idealradial systems, which are analogous to the AronofskyJenkins ideal gas flow solutions.2. Th e a pp ro xi ma te s olut ions f or t ra ns ie nt flow of areal gas using th e r eal gas ps eudo- pr es sur e leads tomet ho ds for i nt erpret in g p re ssure d ra wd ow n a nd build-upt ests w hi ch ar e simil ar to current methods, except thatga s p hy si ca l p ro pe rt ie s e it he r do not appear in the engineering equations or appear at known pressures. Use ofthe r eal gas flow analysis indicates ther e a re many welltests which can be analyzed with current procedures, butthere ar e likely to be many others where current procedures a re i n error.3. Th e real gas flow a pp ro xi ma ti on c an b e used toestablish proper a ve ra ge v al ue s of physical propertiesfo r current procedures.Finally, it is emphasized that the real gas pseudopressure concept is pr in cip all y a c om pu ti ng device. Results of back-pressure, build-up an d drawdown tests ca n beconverted from m p) solutions to solutions in ter ms ofpressure or pressure s qua red t o p ro vi de f am il iar i nf or

m at io n a nd displays.

A = d ra in ag e a re a, sq ftB = f or ma ti on v ol um e faotor , res vol/std volb = slope of build- up or dr aw dow n plot, psi /cp-cycle

Again, th e average value of viscosity-formation volumefactor product would be used in de,termination of flowcapacity, no t i n d et ermi na ti on of dimensionless times.Extension of t he a ver age s to the tr ansi ent per iod isobvious in view of Eq. 4.It sho ul d b e e mp ha si ze d that th e l iq ui d c ase a pp ro xi mation recommended by Matthews, an d Tracy s method

of gas well b ui ld -u p analysis, will give e xc el le nt r es ultsin many cases where pressure dr aw do wn is not largean d p erme ab il it y is h ig h e no ug h that second-degree pressure gradients a re not i mpor tant. Nevertheless, it is r eco mmen de d t h at a real gas flow analysis is useful, evenin this case, to determine that the current approximatemet ho ds a re a pp li ca bl e. In any event, it is likely thatthe real gas flow methods outlined in this paper will bef ou nd s im ple r i n a pp li ca ti on than older methods once am as te r plot of m p) vs p is prepared for an y givenreservoir. This results mainly because it is no longernecessary to square pressures for plotting, and gas physicalproperties appear only in dimensionles s t imes. An additional benefit is that gas properties ar e always evaluatedat known pressures in the real gas flow method.

CONCLUSIONS

NOMENCLATURE

25

24)

26)

1.987 X 10- khT,,(p - P,et ) ,0.472r,p T p.z),,,,. In w

A similar r esult will h ol d if pressures a re expressed interms o f p, rather than Carter suggested th e use of anequation similar to Eq. 26 for determination of averageproperties. Since many gas reservoir flow tests ca n bea na ly ze d b y current theory, Eq. 26 may be of muchuse. Note however that th e p.z) Vg would be used only indetermination of flow capacity from th e usual equation-not in determination of dimensionless times. Dimensionless times should still be evaluated with physical propertiesat th e initial formation pressure before production.

Ma tth ew s liquid case analogy is used, an averagevalue of p.B g ) is required for flow capacity determination.A l iq ui d flow a na lo gy to Eq. 12 is:

flow c ap ac it y kh. Once this is done, t ca n be placedequal to 0.472 r an d e it he r Eqs. 4 or 12 used to generatea p lot of loglO [m jj) - m pw,)] vs log o q.

This is a na lo go us to t he f am il ia r b ack -p res sur e c urve .Th e m p) curve has many of 1he characteris tics of anormal back-pressure curve, b ut t he re a re some importantdifferences, also. At lo w r at es , th e n on -D arcy t erm Dqwill be negligible and the slope of th e curve will beunity. At h ig h v al ue s of flow rate, non -Da rcy flow m aybecome i mp or ta nt a nd th e slope of the curve will approach 2 n value of 0.5) in a fashion similar to thatde sc ri be d b y Carter et al. 12 However, th e m p ) backpressure curve is o nl y sli ght ly sen si ti ve to static pressurelevel, bec ause only th e constant D depends upon pressurethrough dependence upon gas viscosity Swift an d Kiel ).Thus, a s ingle, st ab il iz ed ba ck -pr es su re cu rve i n t er msof th e real gas pseudo-pressure difference ca n be used togenerate an entire family of back-pressure curves in termsof pressure s q u r e ~ o r used in te rm s of m p) wi th thegas ma,terials balance to pl Ovide a simpl e mea ns of forecasting gas well performance.

where we have neglected skin effect an d non-Darcyflow. Eq. 24 is compared with Eq. 12, the result is:

AVERAGE S PROPERTIESCurrent e ng in ee ri ng p ra ct ic es i n th e analysis of gaswell tests involves evaluation of an equivalent liquid flowsystem Matthews ), or modification of expressions forflow of an id ea l gas Tracy or C arte r ) . M at th ew s rec

ommended that gas properties be evaluated at an arithmetica ve ra ge p re ss ur e, whi le C ar te r f ou nd empirically thatevaluation of p hy si ca l p ro pe rt ie s a t an average pressureequal to the square root of th e average of squared press ur es gave r ea so na bl e a gr ee me nt w it h r ea l gas flow solutions. O ne i mp or ta nt result of t he p re se nt study is thatuse of the r ea l gas m p) leads to a solution f or the propera ve ra ge v al ue fo r g as p hy si ca l p ro pe rt ie s. Fo r example,it is not uncommon to assume that gas vsicosity a ndgas law deviation factor c an be regarded as constantat some average value f or the flow region. Fo r a b ou nd edsystem, this wo ul d lead to th e flow equation:

64 0 J OU R NA L O F P ET RO L EU M T E C H N O L O G Y

8/13/2019 SPE-001243-B

5/6

c = compressibility psi-1D = non-Darcy flow constant, (Mscf/Dt 'G p = cumulative gas produced, scfh = net formation thicknessJ = productivity index, (MscflD)/unit potential differencek = effective pe'rmeabilitym p = real gas pseudo-pressure (Eq. 2) psi'/cp

p = pressure, psiPn tJ) = van Eve,rdingen-Hurst dimensionle s pressure

dropq = gas rate, MscflDr = radial locationS = van Everdingen-Hurst skin effect, dimensionless

Sf = apparent skin effect, dimensionlessSq = fractional gas saturationT = temperature, ORt = time, hoursz = real gas law deviation factor, dimensionless

pv=nzRT = total porosity, fraction pore volume = viscosity of gas, cp

EfTpc[ .; in PrpsstlrP Build-up and Drawdown of Gas Wdls ,Jour Pet Tech (Feb., 1965) 223.7. Matthews, C. S.: Analysis of Pr ess ur e Build-up and FlowTes t Data , Jour Pet Tech (Sept ., 1961) 862.8. Russell, D. G.: Determination of Formation Characteris ticsfrom Two-Rate Flow Tests , Jour Pet Tech (Dec. , 1963)1347.9. Carter, R. D.: Solut ions of Unsteady-State Radial Gas Flow ,Jour Pet Tech (May, 1962) 549.

10. Matthews, C. S., Brons, F. a nd Hazebroek, P.: A Methodfor Determination of Average Prpssure in a Bounded Reservoir , Trans AIME (1954) 201, 182.II . Dietz, D. N.: Determina tion of Average Reservo ir P ressureFrom Build-up Surveys , Jour Pet Tech (Aug., 1965) 955.12. Car te r, R. D., Mi ll er , S. C. and Riley, H. G.: Dpterminationof Stab il ized Gas Wel l Per fo rmance from Short Flow Test s ,Jour Pet Tech (June, 1963) 651.13. Car te r, R. D.: Supp lemen ta l Append ix to Dete rm inat ion ofStabilized Gas Well Performance from Short Flow Tests ,ADI Doc. No. 7471, Library of Congress , Washington, D.C.

PPEN IXDRAWDOWN ANALYSIS EXAMPLE

I t can be assumed that wellbore storage effe'cts arenegligible, since the well is completed with a down-holepacker. The first step is to find m p vs p for this gas.

An isochronal flow test is per formed on a gas wellat two different: Taites. Given the reservoir data and fluidproperties below, determine the flow capacity, skin effectand non-Darcy flow coeff icient for this well. The wellis completed with tubing-annulus packer.

Reservoir and Gas DataPi = 2,300 psiah = 10 ftr w = 0.5 ftr = 2,980 ft (640-acre spacing)T = 130F = 0.1 fr BVS = 77 per cent PV

Gas Propertiesp psia z Viscosity (cp)400 0.95 0.0117800 0.90 0.01251,200 0.86 0.01321,600 0.81 0.01462,000 0.80 0.01632,400 0.81 0.0180

SOLUTION

Drawdown DataF low No . 1 Flow No.2

q= 1,600 Mscf/D) q=3 200 Mscf/D)Pm/ (psia) P (psia)

1,855 1,1051,836 1,0201,814 9541,806 9061,797 8601,758 7001,723 5391,703 387

0.2320.40.60.81.02.04.06.0

Flowing Time(hours)EFERENCES1. AI-Hussainy, R., Ramey, H. J., Jr. a nd Crawford, P . E.: The

Flow of Real Gases Through Porous Media , Jour Pet Tech(May,1966) 624.2. Aronofsky, J. S. and Je nk in s, R .: Unst eady Radial Flow ofGas Through Porous Media , Jour Appl Meeh (1953) 20 ,210.3. van Everdingen, A. F. and Hurst , W.: The Application ofthe LaPlace Transformation to Flow Problems in Reservoirs ,Trans AIME (1949) 186,305.4. Swift, G. W. and Kiel, O. G.: The Pred ict ion of Gas WellPerformance Including t he Effect of Non-Darcy Flow , JourPet Tech (July, 1962) 791.5. Tr acy, G. W.: Why Gas Wells Have Low Productivity ,Oil Gas Jour (Aug. 6, 1956) 84.6. Ramey, H. J., Jr.: Non-Darcy Flow and Wellbore Storage

ACKNOWLEDGMENTS

SUBSCRIPTS AND SUPERSCRIPTSD = dimensionlessd = drainagee = exterioT boundaryg = gasi = initial

1 hr = one hour on straight linem = basesuperbar, averagesc = standard conditions of pressure and temperaturesf = sand facet = total

w = well, inner boundaryw = well flowingwp = producing before shut-inw8 = shut-in well = extrapolated

avg = average

The authors gratefully acknowledge financial supportof the Texas A M U. and the Texas Engineering Experiment Station of Texas A M. The encouragement ofP. B. Crawford and R. Whiting is also appreciated.Portions of this work were done by Al-Hussainy in partialfulfillment of the requirements toward a graduate degreeat Texas A M.

MAY 966 64 1

8/13/2019 SPE-001243-B

6/6

5 0 0 --,- ----,- .., --,- _

10O. -;- -..L... --.J0 1 I

HOURSFIG. 2- pw, ) AND m pw,) vs LOG TIME FOR SAMPLE PROBLEM.

These two equations can be solved for q, = 1,600Msc flD , and q, = 3,200 MscflD to yield s = -0.03and D = 4.39 X 10-< Mscf/Dt . Thus, the skin effectis negligible and all of the resistance near the well iscaused by non-Darcy flow.

The difference in flow ca.pacities found above for thetwo rates is not significant. This sample problem wast aken f rom two computer solutions by Carter because itprovides a good example of an apparent skin effectcaused by non-Darcy flow. The true flow capacity usedby Carter in the solution was 50 md-ft. The differencebetween flow capaci ties determined above and the truevalue of 50 md-ft results because Carter approximalted t.heeffect of non-Darcy flow as a constant pressuTe-squareddifference for his solutions. In real gas flow, a betterapproximation would be a constant difference in m p).The n ~ x t step would be to substitute the values ofkh, sand D into Eq. 12 in the main text to provide ageneral equation for generation of stabilized deliverability

curves.

FLOW iO 13

[(435-279) 10s + Dq = 1.151 32X 10 - loglO

CO.1) (0.0176) 0 ~ ~ ~ ~ 4 l (0.77) (0.5)') + 3.23] = 0.657.The flow capacity determined from Flow 2 is 45.8md-ft, and the skin effect plus non-Darcy flow component is 1.36. Thus

Flow 1: s + Dq = 0.657Flow 2: s + Dq , = 1.36.

N FLOW {Cii NO.2 I>SIAFIG. I -m p ) AND 2 p//Lz) VS P FOR SAMPU;

DRAWDOWN PRODLEiI .

This can be done with the gas properties tabulated aboveand Eq. 2 in the main text. The quant ity 2 pip z canbe calculated and plotted vs pressure, as shown belowand on Fig. 1. Integration can be performed in a tabular calculation by reading mid-point values of 2 pip zfrom the graph and mUltiplying by f::,p. The computedm p), psi'lcp, is also shown on Fig. 1. This curve canbe used for fUiture tests with this well or other wellsproducing the same gas at the same formation temperature.Often, only gas gravi ty is available. In this case m p)can be found without integration from Ref. 1.

64 2 JOU I INA I. OF PETIIOLEUM TECHNOLUJa