Contents
Introduction 1 ColllillUilY :l/II.llillll lilr 1{ di,,1 I.'IIJ\\' I(HI
Flow I.&lW~ IIII
1. Fluid Flow in Porous Media 2 Sillgl~Ph&l~ I:hlw of Slightly1.1 Inlr(>duction 2 C()\llpr~~~ibl~ Fluid~ I () 1.2 Thc Idc&l1 Rc~crvoir Modcl 2 Sillgl~Ph&l~~ G&I~ FI(IW I().:?1.3 Solutions 10 Diffusivily EI.JU&llion 3 Simull&ln~ou~ Flow of Oil, W ll'r, &llId (Ja~ I ().:?1.4 R&ldius of Invc iligalion 131.5 Principle of Superposition 15 Appendix B: Dimensionless Variables 1031.6 Homer's Approxim<ion 18 Inlr(>ducli(m I().~
RaJi&l1 Fill ' (If a Slightly Colllrr~.'i~ihl~ I'lllilll().~2. Pressure Buildup Tests 21 Radi&ll Flow With ColI~lalll BliP I(~
2.1 Introduction 212.2 The Ide&ll Buildup Te il 21 Appendix C: Van Everdingen and Hurst2.3 Aclual Buildup T~~I~ 23 Solutions to2.4 DcvialuJl1~ From A~~umplu)n i Diffusivity Equations 106in Ideal T~~I Th~OI)' 24 1lllrlklul'lillll II)(J2.5 Qualilaliv~ 8~havior of Fi~ld Te~l~ 26 ('(III:.lanl Rale al 1111I~r i('Ulldal)'.2.6 Effecls and Durdlion of Aflertlow 27 No Flow Al'ro~~ ()UI~r lillulldal)' 1 ltJ2. 7 ()el~mlillalu)n of Peml~,lbilily 2() ('oll~lalll Rail' al Imll'r 1illllJlllal)' ,
2.M W~III)&lIII&lg~ ,llIll Slillllllalioli .~() (illl:.I 111 Prl'~~llrl' al ()Ull'r li'llllillal)' 11172. () Pre~~urc l..cvel in Surrounding ('(III.'ilanl Pr~.'i.'illr~ al 1IIIIcr lillllllll I)'.FOml&lIU)n 35 No l low Al'ro~~ ()(lIl'r IilIUlIll,,'Y 11.'\2.10 Rcservoir Umil i Tesl 41
2.11 Modilic&llions for Ga ics 44 Appendix D: Rock and Fluid2.12 Modifications for Mullipha~e Flow 45 Property Correlations 119Inlr()(juclion II Y3. Flow Tests 50 Psclld(Il'rilical T~mPl..'I.llure alill3.1 Introduction 50 Prc~slIrc of l.il/uid IlyJrlll'iln)(llI~ IIY3.2 Pressure Drdwdown Tests 50 BubblclX)int Prcssur~ of ('l1Id~ Oil IIY3.3 Multirate Tests 55 Solution GOR IIYOil FOmlalU)n Volume Filclor 12()4. Analysis of Well Tests Compressibility of Und~~illural~d Oil 121Using Type Curves 63 COInpressibilily of Sillllral~J ('ruu~ ()il 1224.1 Introduction 63 Oil Viscosity 1244.2 Fundamenlals of Type Curves 63 Solubility of Ga~ in Wal~r 1244.3 Ramey's Type Curve i 64 Wal~r Fomtalion Volllm~ Fal lor 1254.4 McKinley's Type Curve i 68 Compr~ssibilily (Jr Wal~r ill4.5 Gringarten L't ill. Type Curve i Und~r~illurill~J Rl'~~lvlJir~ 12(Jfor Frdclurcd Wcll i 71 Cl)mpr~~sihilily or Wall'r ina S,IIUrall'd Rl'~l'lv(lir 1265. Gas Well Testing 76 Wall'r Vi~l'o~ily 12X5. I Introduction 76 P~l'ullocrilicill Prol~11i~~ or GilS 12X5.2 Ba iic Theory or G&I i Flow (J.I~I..IW I)~vialioll F l l(lr (/I 'aclllr) ..in Rescrvoir ; 76 anll GilS Foml lillli Vlllulll~ F.Il'llJr 12X5.3 FlowArtcrFlow Tc its 77 (jil~ Clllllrr~~~ihiliIY 1.115.4 Isochron&ll Tcsls 79 (ja~ Vi~c(l~ily 1315.5 Modifi~d I~ol'hrollal Tc~t~ HJ 1:llnllalilln ( )lIlrrl'~~ihilily 1.~25.0 U ie of P i~ulilipres~ure inGas Well Tesl Analysis M5 Appendix E: A General Theoryof Well Testing 1346. Other Well Tests 896. I IlIlr()UllCli'llI HI) Appendix F: Use of SI Units in0.2 1111~rl~r~lIcl.' l.'~lillg X') WellTesting Equations 1386.3 Pulse Tc iling 916.4 ()rill~tcm Tcst~ 97 Appendix G: Answers to6.5 Wirclinc Fontlalioll Te~l~ 9M Selected Exercises 148Appendix A: Development of Differential Nomenclature 151Equations for Flow Bibliography 154
r
in Porous Media 100 Author Index 156Introduction 100 Subject Index 157Continuity Equation forThreeDimcn~u)nal Fk)w I(X)~Ul~;c ,
r
Introduction
I'
This textbook explains how to use well pressures and dis~llssions of pressure buildup tests; pressurenow rates to evaluate the formation surrounding a drawdown tests; other now tests; typecurve analysis;tested well. Basic to this discussion is an un gas well tests; interferen~e and pulse te~ts; andderstanding of the theory of fluid flow in porous drillstem and wireline formation tests. Fundamentalmedia and of pressurevolumetemperature (PVT) principles are emphasized in this discussion, and littlert:lation~ for fluiJ ~ystL'm~ of practil;:al iI1tere~t. Thi~ I:ffort i~ made 10 bring lhe intended audi~1cebook contains a review of these fundamental con undergraduate pelroleum engineering students tocepls, largely in summary form. Llie frontier~ of tile subje~t. Tliis role is tilled mu~11One major purpose of well testing is to determine better by other publications, such as the Society of11I~ bililY ofa format )n to pr()du~e reservoir Iluius. Pelroleum Engineers' monographs on welliestingl,2Furtller, it is importallL to determine the underlying und Alberta Energy Re~ourcL's and Con~ervationreason or a well's productivity. A properly de~igned, Board'~ gas well testing manual.3executed, and analyzed well test usually can provide Basic equations and examples use engineeringinformal ion about formulion pemleabilily, exlent of unil~. However, to ~mooth lile expecled transition towellbore damage or stimulalion, re~ervoir pres~ure, lhe Inti. System of Units (SI) in the petroleum inand (perhaps) reservoir boundaries and hetero dustry, Appendix F Jis~usses lhis unit system andgeneities. restates major equations in SI units. In addition,The basic test method i~ Lo create a pressure answers to examples worked out in the text are givendrawdewn in the well bore; this causes formation in SI units in Appendix F.nuids to enter the wellbore. If we measure the flowrate and the pressure in the well bore during Iteferencesproduction or the pressure during a shutin period I. Mallhews, .S, and Russell, .G.: Pressure i/dup and Row
following production, we usually will have sufficient ~ests 1' ~Vells. onograph eries. P~. Da~las 1967) . .
infor tion to characlerize the tested well. 2. Eilrlullgher. R:C. Jr.: AtI ,,('t'S II ell Test AnalysIs,? a Monograph enc£, PE, }illla£ 1977) .ThIs book beglJ}~ wllh Ii dl~cus~lon of basIc 3. 171('uryuntlJ'rut't;('euflhe It ,.,;,,}: fGus ~~/('Ils, laird :dilion,
equations that describe the unsteadystate Ilow of I Pub. ECRIJ7534, ncrgy RI:£our~c£ nd Conservalion oard,
fluids in porous media. It then moves into Calgary, tla. (1975).
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'
;
Cllapter 1
Fluid Flow in Porous Media
1.1 Introduction
In this initial chapter on nuid now in porous media, oil), we obtain a partial differential equation that\\'c hcgin with a discussion of the differential simplifies toCqllat on~ t hat are u~~d most often to model un a2 p J ap cf>JlC ap
~tcady~tate now. SImple statements of these a:z+a= ka' (1.1)cqllations are provided in the text; the more tedious r r r 0.<XX>264 t
mathcmatical details are given in Appendix A for the if we assume that compressibility, c, is small andin~tructor or student who wishes to develop greater independent of pressure; permeability, k, is constantlInderstanding. The equations are followed by a and isotropic; viscosity, Jl, is independent ofdi~cll,~sion of some of the most useful solutions to pressure; porosity, cf>, s constant; and that certainthese equations, with emphasis on the exponential terms in the basic differential equation (involvingintcgral solution describing radial, unsteadystate pressure gradients squared) are negligible. Thisnow. An appended discussion (Appendix B) of equation is called the diffusivity equation; the termdimcnsionless variables may be useful to some 0.OOO264klcf>Jlc s called the hydraulic diffusivity andreadcrs at this point. frequently is given the symbol '7.The chapter concludes with a discussion of the Eq. 1.1 is written in terms of field units. Pressure, ,radillsofinvestigation concept and of the principle p, is in pounds per square inch (psi); distance, r, is in
~
f sllperposition. Superposition, illustrated in feet; porosity, cf>, s a fraction; viscosity, Jl, is inmlilt i\vell infinite reservoirs, is used to simulate centipoise; compressibility, c, is in volume persimple reservoir boundaries and to simulate variable volume per psi [c=(I/p) (dpldp)]; permeability, k,rate production histories. An approximate alter is in millidarcies; time, t, is in hours; and hydraulicnative to superposition. Horner's pseudopro diffusivity, '7, has units of square feet per hour.dlldiml time, completes this discussion. A similar eqllation can be developed for the.adialnow of a nonideal gas:1.2 The Ideal Reservoir Model I a a cf> aTo .dcvelop a~alysis and design techniqu~s fo~ \~ell a (~ r £) = 0.000264 k at ( ' ), (1.2)tCStlllg, we first must make several simplifYing r r JlZ Za~sumptiOJ1S bout the well and reservoir that we are where Z s the gaslaw deviation factor.nlOdcling. We Ilaturally make no more simplifying For simultaneous now of oil, gas, and water,assllillptions thall are absolutely necessary to obtain I a ap cf>c apsimple, useful solutions to equations describing our a(r a)= O()(X)2~ at' (1.3)sitllal ion but we obviously can make no fewer r r r. ,assllmptions. These a~sumptions are introduced as where c, is the total system compressibility,Ilccdcd, to comhine (I) the law of ~onservation of c =S c +S c ,+S c +c. (1.4)mass, (2) Darcy's law, and (3) equations of state to (0 0 W M g P, fachieve our objectives. This work is only outlined in and the total mobility ~, is the sum of the mobilities'his cllapter; detail is provided in Appendix A and the of the individual phases:Refercnces. k k kConsider radial now toward a well in a circular .~,'= (.£ + :.:.I. + ~). (1.5)re~crvoir. If we comhine the law of conservation of Po Jlp, Pwma~~ and Darcy's law for thc isothermal now of In Eq. 1.4, So refers to oilphase saturation, Co ton\lid~ of small alld constant compressibility (a highly oilphase compressibility, ,,>, ' and c M' o water phasc,satisfactory model for singlephase now of reservoir S and c to p,as phase; and c f is the formationI
d
lill..
.
.lI.~
h~ cc,
~ ~
;.;.
FLUID FLOW IN POROUS MEDIA 3
compressibility. In Eq. 1.5, ku i~ the effe\:live per al1u where Jl and YI are BI.'S~I.'I fun\:tion~. (Totalmeability to oil in lhe presence of the other phases, ~ompre~sibililY, CI' is used in all equalion5 in lhi5and 1J.0 is the oil viscosilY; k and p. refer to the gas chapter becau~e even formalions thaI produce aphase; and k wand p.w refer to tte water phase. singlephase oil contain an immobile waler pha5e andBecause the formation is considered compre5sible have formatioll compre5~ibility.)(i.e., pore volume decreases as pressure decrea~es), The reader unfamiliar with Bessel function5 ~houldporosity is not a constanl in Eq. 1.3 as it was assumed not be alarmed at this equation. It will nor beto be in Eqs. 1.1 and 1.2. necessary to use Eq. 1.6 in its complele form to, ',' calculate numerical values of Pw/; instead, we will1,3 Solutions to Dlffu~lvlty Equation use limiting forms of the 50lutlon in mo~t comThis section deals with useful solutions to the dif putations. The most imporlant facl about Eq. 1.6 i5fll~ivity equation (Section 1.2) uc~~ribing Ihe Ilow of that, unu\.'r the a5~umptioll~ nwdl.' in il~ dl.'v\.'lopnl\.'lll,a slightly compressible liquid in a porous medium. it i~ an exaci sohllion to Eq. 1.1, It ~ometime~ i~We also have some comments on solutions to Eqs. called the van EverdingenHurst constantterminal1.2and 1.3. rate solution.2 Appendix C discusses this solutionThere are four solutions to Eq. 1.1 that are par more colllpletely. Because it is exacl, it serves a5 alicularly useful in well resting: the solution for a standard wilh which we may compare more usefulbounded cylindrical reservoir; the solution for an (but more approximate) solutions. One such apinfinite reservoir with a well considered to be a line proximate solution follows.source with zero well bore radius; the pseudosteady tate solution; and the solution that includes well bore Infinite Cylindrical Reservoir With LineSource W~ storage for a well in an infinite reservoir. Before we Assume that (I) a well produces at a constant rate,discuss these solutions, however, we should sum qB; (2) the well has zero radius; (3) the reservoir is atmarize the assumptions that were neces~ary to uniform pressure. Pi. before prodllction begins; anddevelop Eq. 1.1: homogeneous and isotropic porous (4) the well drains an infinite area (i.e.. that PPi asmedium of uniform thickness; pressureindependent , CX». Under those conditions. Ihe solution to Eq.rock and fluid properties; small pressure gradients; 1.1 isradial flow; applicability of Darcy's law (sometimes qBp.
(
' 948 ~p.CI,2
)
alled laminar flow); and negligible gravity forces. P=Pi+70.6~ Ei k (1.7)We will introduce further as~umptions to obtain Isolutions to Eq. 1.1. where the new symbols are p, the pressure (psi) atdistance, (feet) from the well at time I (hours). andBounded Cylindrical Reservoir ~ uSolution of Eq. 1.1 requires that we specify two Ei( x) = ~ ~dl',boundary conditions and an initial condition. A x Urealistic and practical solution is obtained if we the Ei function or exponential integral.assume that (1) a well produces at constant rate. qB. Before we examine the properties and implicationsinto the well bore (q refers to flow rate in STB/D at of Eq. 1.7, we must answer a logical question: Sincesurface conditions, and B is the formation volume Eq. 1.6 is an exact solution and Since Eq. 1.7 clearlyfactor in RB/STB); (2) the well. with wellbore radius is based on idealized boundary conditions, when (ifr w' is centered in a cylindrical reservoir of radius, e' ever) are pressures calculated at radius, w from Eq.and that there is no flow across this outer boundary; 1.7 satisfactory approximations to pressuresand (3) before production begins, the reservoir is at calculated from Eq. 1.67 Analysis of these solutionsuniform pressure. Pi. The most useful form of the shows3 that the Eifunction solution is an accurate.desired solution relates flowing pressure, Pwf' at the approximation to the more exact solution for timesand face to time and to reservoir rock and fluid 3.79x 105 <PIJ.CI'I~.lk<I<948 <plJ.c,,;/k. For timesproperties.Thesolutioni~1 lcss than 3.79xlo5 ~/'C ,~.lk, Ihe assllmption of
qBIJ.
[
21 3 ll.'ro well ~ize (i.e.. a~~umillg the well 10 be a linePwf=Pi 141.2 2 . + In' eO SOllrcl.' or ~ink) lilllil~ IIII.' uc\:uracy of IIII.' c'llluliull; ,IIkh 'eO 4 times grealer than 948 <P1J.(.'/,;lk. Ihe reservoir's~ 2 ht)lllldari\.'s hl.'gill Iu afft.'\:1 III\.' prl.'SSllrC distrihlili0l1+2E e ,I/JJ1(u,,'eO) 1. ..(1.6) ~11..III.t.' rl.':il.'.rvoir, ;u tllulillt.' rt.' ;t.'rvuir i~ 110 lulIgt.'r

l
a2rr2/_. \~12/~ \ 1
[Jl
(a 'O)J2
1
(a »)j 1111.ll1lteactll1g.n lIe nfI
.
l
.
fi
.
fhI
.
hfl urt ler sImp I Icatlon 0 t e so utlon to t e owwhere. for efficiency and convenience. we have equation is possible: for x<0.02. Ei( x) can beintroduced the dimensionless variables approximated with an error less than 0.6070 by'eO='e/rw Ei(x)=ln(I.78Ix). , (1.8)and To evaluate the Ei function, we can use Table 1.1 for2 0.02 <XS 10.9. For xsO.02, we use Eq. 1.8; and forto=0.OOO464kl/~p.CI'W' x>10.9. Ei(x) can be considered zero for apwhere the an are the roots of plications in well testing.In practice. we find thal most wells have reducedJl(an'eO)Yl(an)Jl(an)Yl(an'eO) =0; permeability (damage) near the well bore resulting