.SOCIETYOF PETROLEUIV1ENGINEERSOF AIME PAPER6200NorthCentralExpressway NUMBERDallas,Texas 75206

THIS IS A PREPRINT---SUBJECTTO CORRECTION

Analysis and Prediction ofGas Well Performance

By

SPE 3474

K. H. Coats andJ. R. Dempsey, Members AIME, InternationalComputer Applications;K. L. Ancelland D. E. Gibbs, Members AIME, PanhandleEastern Pipe Line Company

t)Copyrighl1971American Institute of Mining, Metdlurgicd, and Petroleum Engineers, k.

Thispaperwaspreparedfor the 46thAnnualFallMeetingof the Societyof PetroleumEngineersof AIME,to be heldin New Orleans,La.,Oct. 3-6,1971. Permissionto copyis restrictedto anabstractof not morethan300words. Illustrationsmaynot be copied. The abstractshouldcontainconspicuousacknowledgmentof whereandby whomthepaperis presented.Publicationelsewhereafterpublicationin theJOURNALOF PETROLEUMTECHNOLOGYor the SOCIETYOF PETROLEUMENGINEERSJOURNALisusuallygranteduponrequestto the Editorof the appropriatejournalprovidedagreementto givepropercreditis made.

Discussionof thispaperis invited.Threecopiesof any discussionshouldbe sentto the:~ocietyof PetroleumEngineersoffice. Suchdiscussionmaybe presentedat the abovemeetingand,withthepaper,may be consideredforpublicationin one of the two SPE magazines.

ABSTRACT INTRODUCTION

This paper describes a numerical Many gas wells exhibit pressuremodel for analyzing gas-well tests test behavior which is difficult ifand predicting long-term deliverability. not impossible to interpret usingField applications presented include conventional methods of analysis.an interpretation of a gas well test Difficulty of interpretation is fre-in a tight sand leading to an accurate quently encountered in low permeabilitylong-term deliverability projection. reservoirs and in layered reservoirsThe model presented numerically simu- with limited or incomplete crossflow.lates two-dimensional (r-z) gas flow In these cases, assumptions in con-and accounts for effects of turbulence, ventional analysis methods, such asskin, afterflow, partial penetration, complete (or no) crossflow andpressure-dependent permeability and negligible effects of afterflow orany degree of crossflow ranging from interlayer recirculation through thecomplete to none. Through a novel wellbore, are frequently invalid.treatment of the equations describingreservoir flow, skin and afterflow, This paper describes a numericalthe model simulates shutin at the well- model which accounts for many factorshead and then calculates afterflow and which are neglected in conventionalany subsequent circulation of gas methods of analysis. The modelthrough the wellbore from some layers numerically simulates two-dimensionalto others. (r-z), transient gas fiow in a cyiinderReferences and illustrations at end representing the drainage volume of aof paper. single well. The calculations account

2 ANALYSIS AND PREDICTION

for effects of turbulence, skin, after-flow, partial penetration, pressure-dependent permeability and any degreeof crossflow ranging from complete tonone.

Equations describing gas flow inthe reservoir, skin effect and after-flow are combined in a manner whichallows simulation of shutin at well-head rather than bottomhole; the modelcalculates afterflow and any recircu-lation of gas through the wellborefrom some layers to others. Thus, thecalculated results show the effects ofafterflow and recirculation on shapeof the pressure buildup curve.

Field applications presentedillustrate use of the model to predictthe long-term flow characteristics ofgas wells prim to ~~hnse+inn tQ Z3....-- -----pipeline. The wells selected forillustration have been tested withboth short and long-term tests toindicate the reliability of the method.An additional field application showsuse of the model to explain and repro-duce long-term (up to 600 days) gaswell buildups.

The method presented is equallyapplicable to simulation of oil welltests and performance and the slightlymodified equations for that case aregiven in the Appendix.

BRIEF DESCRIPTION OF THE MODEL

Equations comprising the model aredescribed in detail in the Appendix.Only a brief outline of the method ispresented here. The basic equation ofthe model is eq. (1) describing tran-sient, two-dimensional (r-z) gas flowin a cylindrical drainage volume*:

.* (1)

Horizontal and vertical permeabilities,kh and kv, are arbitrary functions ofr and z and formation volume is

GAS WELL PERFORMANCE SPE 3474

1pTS Mcf

9 = 1000 ZpsT CU. ft. (2)

Iquations (1) and (2) are combined,>xpressed in terms of a gas potentialmd written in finite-difference formEor the grid illustrated in Fig. 1.Phe result of these steps is a set ofQRxNZ difference equations in theWxNZ unknowns 0. ., i=l,2. ..,NR andj=l,2,....NZ. N~~nd NZ are the num->ers of grid blocks in the horizontalmd vertical directions, respectively.

The gas potential @ is defined3s

(3)

where f(p) is permeability at p dividedOy permeability at initial pressure.If permeability does not vary withpressure then f(p) is 1.0 and eq. (3)becomes identical with the real gaspotential [11.

The difference equations containan additional set of NZ unknowns, q ,!lt k....qNz representing the flow ra e(?lcf/D)Into the wellbore from eachlayer. NZ additional equations giveth= additionalskin effect as

a. -a =l,j S.q.33

pressure-drop due-to

j=l,2,. ..,NZ (4)

where S . is related to the skin factorfor lay~r j as described in the Appendix@ is gas potential evaluated at well-bore (bottomhole) pressure. Eq. (4)introduces the additional unknown O sothat we now have NRxNZ+NZ+l unknowns~,d~ 1.,MQvM7!+NZ @ IlatinnsCm&~ L..-..-..... -q-------$

The final equation describesafterflow or wellbore accumulation as

qj+q2+*o*+qNz = q + c% (5)where q is wellhead production rate andC is a function of 0, well radius anddepth as defined in the Appendix.Eq. (5) is simply a gas material balancemitten about the wellbore volume as asystem. This equation allows the flowsfrom the layers qj to be positive, zero

SPE 3474 Coats, Dempsey, Ancell and Gibbs 3

or negative and allows flow from theformation even if wellhead rate q iso. For each layer j in which thewell is not completed the correspondingequation of eqs. (4) is replaced by

.=0.3

The above equations form a systemof NRxNZ+NZ+l simultaneous, nonlinearequations in the same number ofunknowns. The unknowns are numberedin a manner such that the equationsform a basically pentadiagonal, bandmatrix of band width 2xNZ+1. Gaussianelimination is employed to solve theequations after linearization. Thisdirect (noniterative) solution elimi-nates almost entirely the convergencedifficulties encountered with iterativemethods in severely heterogeneous cases.We have treated with no computationaldifficulties cases of layer thicknes-ses varying from many feet to afraction of an inch (representing ahorizontal fracture) with correspond-ing layer permeabilities ranging from.001 md to several darcies.

The direct solution just describedyields values each time step for flowfrom each layer, bottomhole flowingpressure and pressure distributionthroughout the drainage volume. Thebottomhole pressure is converted to=.-..:-.--1-IL--2 --------- ..-:-- Al.-I*uwLKly we_L.L1ledu pLeaauLe uaJJ1y Lllc

Cullender-Smith equation [2].

Required input data for the model~~ d~s~ribed abQve ~~~ ~ and ~

~ asfunctions of pressure, porosit , khand kv as functions of layer and radius,~,m,~~~a~nwae.,,wa ..,611 nfitnmla+immyAGaauLG, WC=J.J. tiuLLLyA.=bAu.ainterval and production rate q as afunction of time. A slightly modifiedformulation described in the Appendixallows specification of bottomholeflowing pressure as a function of timerather than wellhead production rate.Wellhead production rate replaces 0as an unknown in this case. Turbulenceis simulated using transmissibilitieswhich are functions of flow rate.

FIELD APPLICATION 1

The well selected for this exampleis a completion at 6,550 feet. Thewell was badly damaged at completionand the test shown here reflects thecondition at that time. Reservoirparameters are shown in the tabulationbelow:

Thickness 10 ft.Porosity 11%

Water Saturation 44%

Permeability to Gas 20 md

Reservoir Temperature 607R

Gas Gravity 0.632

Casing 4-1/2

Tubing 2,,

Initial Pressure 2522 psia

The well was produced for threedays at a rate of approximately 475Mcf/D. The well was then shut in andthe pressure buildup was monitoredwith a bottomhole pressure bomb. Thesepressures are shown on Fig. 2.

A skin factor of 175 yieldedagreement between observed and calcu-lated drawdown prior to shutin. Thecorresponding calculated buildupportion of the test is shown on Fig. 2.

The badly damaged condition,coupled with the large volume of thewellbore resulted in an extended period

11-4= LA .-4=1 -,.. II ~~, ~ ~of aL LCLL.LUW l

--.-; -A evtarlaaa~GJ.AUU GALGLLUGU

at least until the end of the firstday of buildup. The capability of themocieito accurately account ror cnlse.-.LL:phenomenon is araDhically depicted on~..-......... ~-.~Fig. 2.

frhn 11~~~~~g~4~~~p.~ pc)~~~~p.lr(-jf &~.L.,G

buildup curve starts at approximatelyone day and extends to the end, orabout 2.5 days. This part of the curveis shown in detail on the small insertin Fig. 2. rhisportion can be plottedas a function of dimensionless timeand analyzed analytically; the resultis a formation permeability-thicknessproduct of approximately 200 md-ft.

The wellbore volume can bereduced by setting the tubing on apacker. This will result in a shorteneciafterflow period which would make ananalytical evaluation more reliable.This is illustrated on Fig. 3 as Case 1.,Theonly difference between the basecase and Case I is the reduced wellborevolume caused by setting the tubing ona packer.

. .

I

I

4 ANALYSIS AND PREDICTION OF GAS WELL PERFORMANCE SPE 3474

Another method to reduce the after-flev nerio~ WQUId be to remove the well-r-..-bore damage or skin. This can beeasily done with the simulator and theresult is shown on Fig. 3 as Case II.The drawdown rate was left the sameeven though it is possible to producea much larger rate with skin removed.The result is a nearly straight linewith very little character.

In summary, the test shown hereillustrates the ability to simulatethe actual performance of a weii inconsiderable detail. Analysis of datain this manner enables an engineerto account for all the factors affect-ing the pressures that he measureswithout having to wait for the well-bore effects to die out.

FIELD APPLICATION 2

This application illustrates the,~~eof ~h.ecimlll a+nY S=

-A..c----w .~ ~QQ~ ~Q

predict the long-term deliverabilityfrom a low permeability reservoir. Thetool is ideal for this applicationbecause it accounts rigorously for thenonlinearity in the gas flow equationwhich is necessary when large pressuregradients exist in the reservoir.

The available reservoir data aresummarized in the table below. Thetest data were taken over a six-monthperiod between the dates that the..-11

. ..2 ~ar,,m,ec~e~ ~~We.1..l. was drilled alLu a

pipeline. The data consist of threeshutin pressures and four flow tests.The shutin periods varied in durationfrQrnthree days to several months.

Zone 1 2

h, ft. 9 11

g, % 8.8 9.7

Sw, % .280 .21

k, md .15 .30

Depth, ft. 8609 8620

Temperature, R 642 642

&~ PW3.,;+,,uLav.L~ KOA.-a ~nA..

InitialPressure, psia 3290 3290

The well was initially perforated,stimulated, and tested. The stimula-tion was simulated with an increasedpermeability in the vicinity of thewellbore. The first pressure buildupof 2785 psia was observed after threedays of shutin. The weii was thenshut in for about 45 days and no knownpressures were taken. The well wasthen flowed for a single day, shut inseven days, and a pressure of 3090psia was observed. A very short-term4-point test was then taken and theweii was shut in for abo-utfa-urmonths.At the end of this four-month period,a pressure of 3290 psia was observed.The pressure behavior of the well,both calculated and observed isshown on Fig. 4. The flow test data,because of the short duration, are notshown here but actually were considered.The match shown on Fig. 4 was consideredadequate as the basis of an extendedprediction.

A simulation run was made assum-ing production into a 600 psi pipeiine.The results are shown as the predictedcurve on Fig. 5. The well has producedfor four years and the actual produc-tion is shown as the actual curve onFig. 5. This prediction representswhat would have been done had this toolbeen available several years.ago. Theprediction shown on Fig. 5 is veryadequate for any planning or economicevaluation that would have been neces-Szry very ezrly ~~ ~~~ ~if~ Qf tihewell.

FIELD APPLICATION 3

This application treats a gas wellwhich exhibited prolonged periods ofpressure buildup--one period in excessof 600 days. Conventional analysisassuming a single layer of radial flowfailed to explain the behavior in thata permeability sufficiently low to givethe extended buildup period would notallow flow at the observed rates. Thepurpose of the well pressure analysiswas estimation of gas reserves andlong-term deliverability.

Logs and core analyses from wellsin the field indicate gross and netp~~7s Gf ~beut zoo feet and ~00 feet:

respectively. Net pay horizontalpermeabilities range from .1 to 50 mdand porosities range from .03 to .14.The exterior radius for the welltreated here has been roughly estimated

SPE 3474 Coats , Dempsey, Ancell and Gibbs 51as 1000-1500 ft. Wellbore radius is reservoir. The permeable layer repre-~ in initial r~

6 ANALyS~S ArJD PREDICTION

here in a single working day or lessand at a cost of approximately oneminute of Univac 1106 computer time.This is not any more expensive thanthe standard analytic methods usedfor years.

In order to make the most effec-tive use of this capability, a changein test philosophy is needed. Toutilize analytic methods of transientpressure analysis, it is advantageousto maintain as nearly constant testrates as possible. However, thecapability of the simulator to handlemultiple transients makes it advan-tageous to introduce widely varyingpressures by testing at several differ-ent rates for shorter periods of time.To get full advantage of the capa-bility, the flow periods should beinterspersed with periods of pressurebuildup. This technique will intro-duce many transients which will helpdefine any reservoir heterogeneitybetter than a singie flow rake.

Finally, good turnaroundon adigital computer aids the trial anderror matching procedure. This turn-around is generally easily obtainedwith the model described here becauseof its 16-W- storage ad Cc)iliputlilcj tlrlerequirements. A problem using eightradial increments (layers) requiresless than five seconds of CDC 6600time for 50 time steps. We have foundvirtually no sensitivity to the numberof radial increments (NR) provided NRexceeds about eight.

REFERENCES

1.

2.

3.

4.

Ramey, H. J., Jr. and Crawford,P. B., The Flow of Real GasesThrough Porous Media, Trans-actions, SPE, Vol. 237, I-624,1967.

Katz, et al, Handbook of NaturalGas Engineering, McGraw-Hill, 1959.

Carter, R. D., Solution of Un-steady-State Radial Gas Flow,Transactions, SPE, Vol. 2251 I-549~1962.

Swift, G. W. and Kiel, O. G., ThePrediction of Gas-weli PerformanceIncluding the Effect of Non-DarcyFlow, Transactions, SPE, Vol. 225,1-791, 1962.

GAS WELL PERFORMANCE SPE 3474

5. A1-Hussainy, R. and Ramey, H. J. ,Jr. , Application of Real Gas FlowTheory to Well Testing andDeliv&ability Forecasting, Trans-actions, SPE Vol. 237, I-637, 1966.

6. Wattenbarger, R. A. and Ramey, H. JJr., Gas Well Testing withTurbulence, Damage and WeilboreStorage, Transactions, SPE, Vol.243, I-877, 1968.

7. Millheim, K. K. and Aichowicz, L.,Testing and Analyzing Low-Permea-bility Fractured Gas Wells,Transactions, SPE, Vol. 2431 1-19311968.

8. Adams, A. R., Ramey, H. J., Jr.and Burgess, R. J., Gas WellTesting in a Fractured CarbonateReservoir, Transactions, SPE,Vol. 243, 1-1187, 1968.

MATHEMATICAL MODEL DESCRIPTION

Substitution of b from eq. (2)into (1) gives 9

a( ( )P/Z-qv=@ :t (6)

where k and $ are permeability andporosity, respectively, at initialpressure and a is T~/1000 psT.Defining gas potential O as in eq. (3),we have

= LM$c=at (7)

where c is d[g(p)p/z]/dO, a single-valued function of 0.

An implicit difference approximaticto eq. (7) is

SPE 3474 Coats, Dempsey, Ancell and Gibbs 7I I

(ArTrAr@ + AzTzAz@)i,j,n+l - qi,j

(8)

Radial transmissibility for flowbetween ri-l and r~, (where {ri} areblock center radii) are defined by

..9T!UL1l Az . ~

T hi-l\2,jri-1/2,j = !tnri/ri-l (9a)

where the effective interlock permea-bility hi-l/2,j must be

kn ri/ri-lkhi-1/2,j = ri

fdknrk(r)

i-1

to correctly relate steady flow rateand pressure drop in the intervali-1 i

for the case of a givenpermeabl lty distribution k(r).Permeability is eq. (9a) is expressedas md x .00633.

Transmissibility for vertical flowbetween layers j-1 and j is defined by

The accumulation or capacity coef-ficient,

(VpC)i,j = CXIT(r~i+~,*- r~i_~,~),, .D. t-.L\P\g(p)~)i,j,n+l-(gtp- i,j,n

$.l,j @. - @.I,j ,n+l l,j,n (lo)

is a chord slope (with respect to 0)of the term representing gas-in-placein the grid block. The term r i-1/2for i=l is rl=r . The sink te~mq is the pro~uction rate from gridb~b~k i,j, Mcf/D. Each term in eq. (8)has units of Mcf/D.

Difference notation is

AT A@.r r r l,j,n+l

=Tri+l/2,j(@i+l,j - i,j)n+l

-Tri-l\2,j(0i,j - i-l,j)n+l

ATAO.z z z l,j,n+l

I=T (0. - 4.

zirj+l/2 l,j+l .)1,3 n+l

all(r2. 2= ml+l/2 - mi-l/2)kvi,g-1/2

.5(Azj + Az.]-1) (9b) -Tzitj-1/2 (0. - 4.l,j l,j-l)n+l

where the effective interlock permea-bility must be

I.

.5(Az. + Az,k

-~)vi,j-1/2 = zj I

fdz~(z)

j-1

mi+l/2 is the log mean radius(ri+~ - ri)/kn(ri+l/ri) and Zj isthe depth to the center of layer j.

I &@ = a. - 00l,j,n+l l,j,n

For clarity of presentation, ourremaining discussion will be pertinentto a system of eight radial grid blockextending from specified rw to re andfour layers. The term qi o is zeroeverywhere except for blo& at i=l(at the well). We number the gridblocks and variables 0. c linearlystarting with 1 at i=N*f]j=l, 2 at

8 ANALYSIS AND PREDICTION

L=NR, j=2, etc. , proceeding down firstmd then in toward the well. Thus thelinear index m is

m= (NR-i) x NZ + j (11)

and the eq. (8) canbe written in termssf m as

@ +aam,m-NZ m-NZ m,m-lm-l

-a O+am,m m m,m+lm+l

+am,m+NZ@m+NZ - m = bm (12)

wherebm is -(Vpc) . $I,j i,j,n/At andthe off-diagonal a coefficients aretransmissibilities. Eq. (12) writtenfor m=l,2,. ..,32 is a system of 32equations in the 36 unknowns {@m,m=l,32}and {qm,m=29~30 ,31,32}. The terms qm(m=29-32) are the flow rates into thewellbore from the four layers. Thetransmissibilities T

ri-1/2,j must be setto O for i=l in eq.(8) since these qterms account for flow into the well.AlSO, rNR+l\2,] for j=l,2,. ..,NZ arezero representing the closed exteriorboundary and the closed boundaries atZ=O and Z=H are represented by T

L_ -.zi,j-i/Z

= O for j=l, NZ+l.The additional potential drop at

the wellbore surface in each layer due,.

t~ Skiil iS

Q.-a=1

l,j SKIN2nAzj hjor in terms of m,

Qm- 0= smqm m= 29,30

j) qi,j

31,32 (13)

here hj is layer j permeability atthe well at initial pressure and O is

pthe value of

Jf(p)% dp at p = bottom-

hole wellbore pressure. If the well

GAS WELL PERFORMANCE SPE 3474

is not completed in a given layer thenthe corresponding equation of the set(13) is simply replaced by qm = O.

Counting the four equations (13),.W.e~,G.W>h s . ,a 2K ama.a+

.SPE 3474 Coats, Dempsey, Ancell and Gibbs 9I Ieqs. (10) and (16)) can be approximatedat the beginning of each time step by~~.~~~op~~ ~~ Q. . as determined fromtables of the f~fi~f?onsg(p)p/z and Pversus 0. For large pressure (poten-tial) changes over the time step 2 or3 outer iterations can be performedwhere the chord slopes are re-evaluatedand the 37 equations resolved. Wehave found on the great majority ofproblems that no iteration is necessary--i.e. the answer is not significantlychanged by iterating.

The above equations apply withminor changes to the case of single-phase oil flow. The potential forthe oil case is defined as

Jf(p) be(p) dp#= P. (17)and the right-hand side (capacity)coefficient involves the chord slopeof the function g(p) be(p) with respectto Q. The coefficient C in thecounterpart to eq. ..-nl.&:mm(14) LGAaLALLy well-bore oil volume to bottomhole pressurecan be easily derived for the two casesof a freely flowing or pumped well.

In the case of turbulent flowDarcys law is modified to [2]

(18)

for radial floy. Integrating thisequation for a constant flow q Mcf/Dfrom ri to rp yields

q = Trt(131-@2) (19)

Iwhere subscript t denotes modification Iof the transmissibility due toturbulence,

alkT hrt = r2 (3Mk (20)

km +1

*q(

. .

1

I

t 23 *... -------*ni

1=0

?.11

Fi:(.1 - Schematic of the PhysicalSystem

*

,/-/

. I 1al In

Ib, om

Fig. 3 - Field Application #1Analysis of Completion

+

$ .i {------- -- -------- -------- ------------

lW1 1 I I 1 I II IQ m m 40 v m RJ m wllm9,-

Fig. 5 - Fieid Application #ZPredicted and Actual Production

[LQ L----e

I Ial 10

Iln,om

.

~ig. 2 - Field Application I/lPressure BuilclupTest

I

B

~ . ..... --...

C4kti ~

t JAN I FE* I w i m 1 MAT I -

Fig. 4 - Field Application #2Test Analysis

I I I I I I I 1 I

4

1 I I I 1 I I 1 I

m.l lg. A.. -- T- L:-- 426 - ~ieid ~ppllca~lwn rt~Comparison of Observed vs.Calculated Pressure