.SPE 18855

Second Law Analysis of Petroleum Reservoirs forOptimized Performanceby F, Civan and D. Tiab, U. of OklahomaSPE Members

Copyright19S9, Societyof Petroleum Engineers, Inc.

This paper was prepared for presentationat the SPE ProductionOperations Symposiumheld in Oklahoma City, Oklshoma, March 1S-14, 1989.

This papar was selected for presentationby an SPE ProgramCommitteefollowingreview of informationcontaina in an abatract submittedby the author(a).Contentsof the paper,aa presented, have not been reviewed by the .Societyof PetroleumEngineerssnd are subjectto correctionby the author(s).The maferial, aa presented, does not necessarilyreflectany positionof the Societyof PetroleumEnglnaera,its officers,or members.Papars presentedat SPE meetingsare subjectto publicationreviawby EditorialCommittws of the SocietyofPatrofaumEngineers.Permissiontocopyiara@rict@to an abstractofnotmorethan 300 words.Illuatrationamayno!be copied.Tfw abstractshoufdcontainconspicuousacknowledgmentof where and by whom the papar ia praaantad, Write PubficationaManager, SPE, P,O. Sox SSS636, RichardWn, TX 750s3.3SSS. Telex, 7S09S9 SPEDAL.

ABSTRACT configuration, (b) the properties of the reservoirfIuids, and (c) the operating controls, including

Thesecond law of thermodynamics determines control of the available driving mechanisms, thethe practical limits of operating systems. rate and location of production of reservoirHowever, within these limits, conditions leading to fluids, and the pressure behavior. However, onlyoptimum performance can be found. the production rate can be externally controlled

once the locations of the wells are fixed. ManyIn reservoir analyses the conditions or authorsl-4 recognized very early the role of

variables which are controllable are somewhatlimited. These include well head pressure,

production rate on the ultimate oil recovery andconcluded that there exists a maximum rate of

production or injection rate, number of wells andtheir locations and the recovery technique. The

production that will permit reasonable fulfillmentof the basic requirements for efficient recovery.

efficiency of reservoir exploitation decreases as Increase in the production rate beyond this maximumthe entropy generation increases. Since entropy value will usually lead to rapidly increasing lossgeneration translates into irreversible loss of of ultimate recovery, and reduction in rate belowfluid energy,.the operating conditions of a this maximum will not substantially increase thereservoir should be selected to minimize the ultimate recovery of oil. Considerable controversyentropy generation over the productive life of the exists concerning the degree of efficiencyreservoir. attributable to rates, however, everyone recognizes

the importance of using efficiently the in-situIn this paper, it is shown how entropy reservoir energy.

generation function is calculated and used as aguideline for selecting the conditions leading to Versluyslhigh ultimate recovery. This is accomplished by

and Schilthiusz investigated the

calculating the cumulative entropy generation overreservoir energy changes that occur during the

the reservoir volume and the production period andcoiuse of production. Schilthius based his study

by determining the conditions minimizing theon an imaginary thermodynamic engine in which net

entropy generation. change in energy is equivalent to that in petroleumreservoirs. His analysis provided an explanation

INTRODUCTIONof the energy supplied by various sources,including the expansive energy of the oil and the

The amount of oil,and gas which may begas with which it is associated, both dissolved andfree, the energy supplied by water drive, and the

recovered from a reservoir is a widely varying energy of gravity. However, his approach did notquantity, dependent partly on the particular include the loss of fluid energy due toconditions imposed by nature on the underground irreversible proceeses, and is only applicable tostructural trap and on the properties of the macroscopic processes and volumetric reservoirs.contained fluids, and subject further to the Lacey and Sage3 applied thermodynamic concepts tocontrols exercised by the operator in itsdevelopment and operation. The most important

analyze the energy relations in a flowing well anddemonstrated the usefulness of these concepts.

factors which influence the recovery of oil are (a) Tiab et al.4 and Sarathi and Tiab5 used the conceptthe characteristics of the productive formation, of available energy and the transient state flowsuch as the permeability, porosity, and structural principles to investigate the effect of productionReferences and f~gures at end of paper.

2 SECOND LAW ANALYSIS OF PETROLEUM RESERVOIRS FOR OPTIMIZED PERFORMANCE SPE 18855

.-

rate upon the utilization of primary reservoirenergy of dry and condensate gas reservoirs. Bothstudies used the maximum reversible work functionas a working tool. Tiab and Duruewuru6 used theavailability function to describe the behavior oftheoretical work during rnultiphaseflow.

The efficiency of real thermodynamic systemsdecreases as the entropy generation increases.Therefore, to maximize the utilization of energy,systems should be designed and operated such thatthe entropy generation is minimized. Applicationsof this principle to a variety of engineeringsystems are given by Arpaci7, Mukherjee et ala,and Son et al.g. In this study, it is shown howentropy generation is calculated for the reservoirand the well, and how it is used as a guideline forselecting the optimum production rate path leadingto high ultimate petroleum recovery.

ENTROPY GENERATION IN RESERVOIRS

Several expressions of entropy production forvarious applications are available in theliterature.7*8*9 However, for the purpose of thisstudy a simplified case will be considered dealingwith the flow of a single fluid at isothermalconditions. Thus, the rate of entropy productionper unit volume by irreversible conversion ofmechanical energy to internal energy is given by

1 ++++.5=- (-Z:VV) ,................................(1)

T

+where T is t~e absolute temperature, z shear stresstensor, and v velocity of the fluid. For a porousmedium flow a local volume average of Eq. 1according to Slatterylo yields

= Evp/T ....- 0............................(2)

where E is the rate of dissipation of mechanicalenergy % the porous medium, and is expressed asfollows:

1Evp=-

/(-~?;)dV .......................(3)

P P

where Vp is the fluid volume in the por~ space.Considering the geometric irregularities of porestructure, the function ~p is derived in AppendixA.

5P .

where

PscfpPuj.............................(4)

@(l - Swc)B

fn denotes the friction factor as function ofthe por%us media Reynolds number which is given byEq. A.12. Combining Eqs. 2 and 4 yields

P5cfp@J3 =

....s..........,............(5)$41 - SWC)TB

The total entropy generation is calculated from

HtfST = dVdt .....,......,..............(6)ti Vwhere ti and tf denote the initial and final time,respectively, and V is the reservoir volume. Forradial flow, V=nr2h, Eq. 6 becomes

Htf reST = 2nh rdrdt ....................(7)ti rwor in differential form

a ()~sT . = 2vhr ...............,,.,..........(8)ar atFor steady state flow, Eq. 7 is simplY

!reST = hw,t rdr ........................(9)rwor

dST = 2whktr ..............................(lo)r

where rw is the wellbore radius and re the externalboundary radius and bt=tf-ti. Expressions for uand dp/dr for steady state and pseudosteady stateconditions for Darcy and Forchheimer equations arepresented in Appendix C. For non-Darcy flow, thepressure gradient during fluid flow is given by

dp v=- U + BPU2 .......,.......,..........0...(11)dr k

in which

u = q/(2mrh) ................................0 (12)

The volumetric flow rate equations for steady stateand pseudo steady state cases can be expressed,respectively, as

q = %@ . . . . . 0 , , . , , . . . . . , . . . . . . . . . . . . . . . . . . . . (13)

and

q= [1 - (r/re)21q5cB ........................(14)For Darcy flow simply set the turbulence factor(3=0.

Note that the transient state case will notbe considered since the production period at thetransient state condition is negligible compared tothe pseudo steady state condition.

Letting ST=O and P=Pwf at r=rw, then Eq. 8for pseudo steady state or Eq. 10 for steady stateand Eq. 11 are integrated simultaneously until theexternal boundary, i.e. r=re, using a n~ricalmethod for the ordinary differential equations. Inthis study the Runge-Kutta-Fehlberg four (five)Methodll is used.

ENTROPY GENERATION IN WELLS

For wells operating isothermally at constantterminal rates the rate of entropy production ispredicted by:

& P,cfv3 = = ...........................(15)

T 2DTB

where the rate of energy dissipation due tofrictional leases is given by Eq. A.6. Thevelocity term is expressed as

..%q

v ......?...................~D2/4 mD2/h

and B is the formation volume factor of the ffluid. Hence, the total entropy generation o~the well length is

ft~ fL

.(16)

owinger

HST= dldt .........................(17)tj Oor in differential form

a

0

ST .= ........................5.......(16;

Tt at

For steady state flow Eq. 17 simplifies to

ST =At

or

Ld k ........................,....(19)

o

dST = At ..................................(20)d!t

Since B is dependent on pressure, Eq. 18 or 20 mustbe solved simultaneously with the well pressureequation given by (Eq. D.7)

~+&(-pB2)..........(21):=mwhere L and H denote the length of well and depth,respectively. The wellhead pressure, pwh, and flowrate, qsc, are specified with respect to thesandface andS~O for !2=0. Eqs. 18 or 20 and 21are integrated simultaneously until the wellbore,~~~~~~:~i~f the Runge-Kutta-Fehlberg four -

.

SKIN EFFECT

The akin effect can be incorporated into theanalysis by the apparent well bore radius concept:

rw= rwe-s

........,....................,....(22)

SPE 18855 FARUK CIVAN AND DJEBBAR TIAB 3

287

where s>O for damaged wells and s

4 SECOND LAW ANALYSIS OF PETROLEUM RESERVOIRS FOR OPTIMIZED PERFORMANCE SPE 18855

2. Choose a production rate qsc. 5* Second law analysis of a dry gas well-reservoirsystem is presented. Although, for this particular

3. Let Pwh=(pwh)f be the abandonment pressure at8=0,

simple case a meaningful mathematical optimumterminal rate cannot be found the analysispresented in this paper can serve as means of

4. Integrate the well equations, Eqs. 18 and 21, calculating the total entropy generation, and howfromE=O to E=H and then, the reservoir equations, it relates to reservoir performance.Eqs. 8 and 11, from r=rw to r=re using the Runge-Kutta-Fehlberg four (five) mathod.ll Calculate 6. Wellbore conditions affect the value of thed(S~ell+STreseryoir )/dt and the average reservoir maximum entropy generated and, consequently, thefluid pressure, p, by Eq. C.1O. recovery factor. A positive skin causes the in-

situ reservoir energy to be consumed nuch fasteri.. Increase pwh by some amount Ap. than for a negative skin, for the ssme production

rate. inefficient use of the reservoir in-situ6. Repeat the steps 4 and 5 until the pressure at energy will result in low ultimate recovery.r=re becomes equal to Pe?i.

ACKNOWLEDGEMENT

7. Calculate the total entropy generation over thepseudo steady state ~eriod by The authors gratefully acknowledge the

National Science foundation - EPSCOR Grant RII-

mre2h

J

fii d(S~ell+STreservoir ) QXF)ct(F)8610676 for financial support of this work and Dr.

ST= d;W.A. Sibley, the EPSCOR Project Director for theState of Oklahoma. The support from the School of

% if dt B(;) Petroleum and Geological Engineering and the

...(24)College of Engineering computing facilities at theUniversity of Oklahoma is appreciated.

in which the integral is evaluated according toCivan and Sliepcevich21, and Eq. C.11 ia used to

NOMENCIMURE

express ti~ in terms of the average pressure. a coefficient of accelation, m/s2

8. print qsc and sT. B formation volume factor, standard m3/m3

9. Repeat eteps 1 through 8 until sufficient data% total effective compressibility, Pa-lof ST VS. qsc are generated.

10. Correlate or plot the recovery factor versusz mean pore diameter, m

the cumulative entropy production over the pseudo- D diameter, msteady etate production period, as shown in Fig. 2,and determine the optimum recovery factorcorresponding to reasonable terminal rate value. &

rate of dissipation of mechanical energy intubular flow, J/m3-a

It is evident from Fig. 2 that the wellbore%p rate of dissipation of mechanical energy in

conditions, i.e. skin factor, affect the rate of porous media, JJm3-s.entropy generation and consequently the optimumrecovery factor. For instance when a=5 the entropy fF Fanning friction factor, dimensionlessgenerated reaches its maximum early, which causesthe recovery factor to drop drastically. However f~when s=-5, the entropy generated reaches its

Moody friction factor, dimensionless

maximum much later, therefore the reservoir is fp Porous media friction factor, dimensionlessproduced at a high recovery factor for a longerperiod of time for the same production rate.

Flz gravityforce, N

CONCLUSIONS Fi inertical force, N

1.. A more rigorous formulation of the porous mediaP pressure force, N

momentum equation is presented. New, accuratecorrelations for the permeability and the non-Darcy Fv viscous force, Nflow coefficients are proposed based on the methodof dimensional analysis. h formation thickness, m

2. Nonlinear, steady state and pseudo steady state H well depth, msolutions arc presented for accurate analysis ofradial flow problama. k permeability, m2

3. A generalized pressure equation for flowing t length, mwells ie.derived.

L well length, m

4. Expression for the energy dissipation rate andentropy calculation for flow through circular tubes P pressure, Paand porous usidiaare presented.

SPE 18855 FARUK CIVAN AND DJEBBAR TIAB 5

Pe external boundary pressure, Pa

Pob overburden pressure required for Tiss andEvans13 correlation, Pa

Pwf flowing well pressure, Pa

q flow rate, m31s

qsc production rate at standard conditions,ztandard m3/s

r radius, m

e external boundary radius, m

rw wellborb radius, m

i entropy production rate, J/K-m3-s

ST total entropy production, J/K

sWc connate water saturation, fraction

tab abandonment time, s

tf final time, s

ti initial time, s

T temperature, K

u volumetric flux, m3/m2-s

v velocity, mjs

v volume, m3

Ppore volume, m3

w work, J

z gas.deviation factor, dimensionless

D non-Darcy coefficient, m-l

8 specific gravity of gas, dimensionlessc variable

e angle of inclination, degree

L1 viscosity, Pa-s

P density of fluid, kg/m3

T stear stress, Pa

Q porosity, fraction

1. Veraluys, J.: Energy Relationships in theOil Bearing Formation, The Oil Weekly, Oct.15, 1934, pp. 38-46.

2. Schilthius, R.J.: Active oil and reservoirEnergy, Trans., AIME, Vol. 110 (1936) pp.33-51.

3. Sage, B.H. and Lacey, W.N.: *EnergyRelationsin Flowing Wells, API Drilling and

4.

5,

6.

7.

8.

9.

10.

11.

120

13.

14.

150

16.

17.

Production Practices - 1935, AmericanPetroleum Institute, New York, NY, 1936, pp.107-115.

Tiab, D., Sarathie, S.P., and Chichlow, H.B.:Thermodynamic Analysis of Gas Reservoirs,ASHE Proceedings, Energy Tech. Conf. andExhibition, New Orleans, LA, Feb. 3-7, 1980.

Sarathi, S.P., and Tiab, D.: Effect ofProduction Rate on the In-Situ EnergyUtilization of Dry and Condensate GasReservoirs, SPE Proceedings, Middle EastTech. Conf., Bahrain, March 9-12, 1981.

Tiab, D., and Duruewuru, A.U.: ThermodynamicAnalysis of Tranaient Two-Phase Flow i.nPetroleum Reservoirs, SPE Production Engr.J. (Nov. 1988) pp. 495-507.

Arpaci, V.S.: Radiative Entropy Production -Lost Heat Into Entropy, Int. J. Heat MassTransfer, Vol. 30, No. 10 (1987), pp. 2215-2223.

Mukherjee, P., Biswaa, G., and Nag, P.K.:Second-Law Analysis of Heat Transfer inSwirling F1OW Through a Cylindrical Duct,Trans. ASME, J. heat Transfer, Vol. 109 (May,1987), pp. 308-313.

San, J.Y., Worek, W.M. and Lavan, Z.:Entropy Generation in Convective HeatTransfer and Isothermal Convective MassTransfer, Trans. ASME, J. Heat Tranafer,Vol. 109 (Aug. 1987), pp. 647-652.

Slattery, J.C., Momentum, Energy and MassTransfer in Continua, McGraw-Hill Book Co.,New York, 1972, 679 p.

Fehlberg, E.: Low-order classical Runge-Kuttaformulas with stepsize control and theirapplication to some heat transfer problems,NASA TR R-315, NASA, Huntsville, AL, 1969.

Blick, E.F. and Civan, F.: Porous MediaMomentum Equation for Highly AcceleratedFlow, SPE Reservoir Engr. J., Vol. 3 (1988)pp. 1048-1052.

Tiss, M. and Evans, R.D.: The Measurement andCorrelation of the Non-Darcy FlowCoefficients in Consolidated Porous Media, toappear iriJ. Petroleum Science andEngineering.

Tiab, D. and Donaldson, E.C.: Reservoir RockProperties, SPE Textbook Series, Dallas, TK,to appear.

Cornell, D., and Katz, D.L.: Flow of GasesThrough Consolidated Porous Media, Ind. Eng.Chem., Vol. 45, p. 2145, 1953.

Ikoku, C.U.: Natural Gas ProductionEngineer-, John Wiley & Sons, New York,1984, 517 p.

De Vries, A.S. and Wit, K.: Rheology ofGas/Water Foam in the Quality Range Relevantto Steam Foam, proceedings of 1988 SPE Annual

.l

6 SECOND LAW ANALYSIS OF PETROLEUM RESERVOIRS FOR OPTIMIZF!D P17RFOUCE SPE 1885:.--.-. -- .

Technical Conference & Exhibition, since the Moody and Fanning friction factora areEOR/Ceneral Petroleum Engineering, Oct. 2-5, related byHouston, Texas, SPE 18075, pp. 193-203.

f~ = 4fF18.

(A.7)Collins, R.E.: Flow of Fluids Through Porous

..................................,.

Materials, PennWell Publ. Co., Tulsa,Oklahoma, 1961, 270 p.

and O=pae/B.

Note that if Eqa. A.3, 4 and 7 are combined19. De Nevers, N.,: Fluid Mechanics, Addison- the Moody friction factor is defined by

Wesley Fubl. Co., Reading, Massachusetts,1970, 514 p.

20. Ahmed, N. and Sunada, D.K.: Nonlinear flowin porous media, Proc. ASCE J. Hydraulic

fH=+f;)fl~~;) ..........(A.8)

Div., Vol. 95, NY 6, 1847. The friction factor, fM, is correlated with respectto the Reynolds number

21. Civan, F. and Sliepcevich, C.M.: SolvingIntegro-Differential Equations by theQuadrature Method, Integral Methods in

PVD

Science and Enfzineering,Payne, F.R., et al.Re= ....................................(A.9)

(eds), Hemisphere Pub. Co., New York, pp. M106-113, 1986. (2) Porous Media Flow

APPENDIX AENERGY DIsSIPATION RATE

Eqs. A.5.or A.6 and A.? are applicable forcalculating the energy dissipation rate during

(1) Tubular Flowporous media flow, if the porous system isrepresented by an equivalent tube whose apparent

The rate of energy dissipation due todiameter is a volume average of irregular shaped

frictional losses is expressed by (De Vries and pore apace conducting the fluid, considering the

Wit17) effect of the irregularity on the flow pattern. Bythis definition the mean pore diameter ~ used

&wby Collins18 and the hydraulic diameter suggested

&=Pvr (Al)by De Nevers19 are not adequate for the purpose of

....,..............,..........l ,l this study. To include the effect of flow patternsAhmed and Sunada20 and Cornell and Katz15 utilizedan integral form of the Forchheimer equation,

where WL is the irreversible loss of work per unitmass of fluid. ,The frictional pressure loss and

considering the average properties of.fluids and

lost work during flow through a tube are related bythe porous media over the flow distance, and showed

(Ikoku16),that

()

1dp

()

Ap vp6w~

=+1 .......................(All)- ............................(A.2) ~PU2 L pufikdk f=~

Hence, the Reynolds number and friction factor forThe wall shear stress during tubular flow is given porous media can be written as according to Eq.by All,

D

()

dp Pufik%4 =.- .............................(A.3) Rep . _ .................................

d!i f(A.12)

u

where D is the tube diameter. The Fanning frictionfactor is defined by 1

()

Apfps ..............................(A.13)

=WfF =f3PU2 L

.................................(A.4),0.5pv2 Comparing Eqs. A.9 with Eq. A.12 yields an

Combining Eqs. A.1-4 yieldsexpression for the apparent diameter

4fl?D=~k .....................................(A.14)

% =- (o.5pv3) .,.........................(A.5) Since the actual velocity, v, end the apparentD. velocity, u, are related by

or, u

pscfMv3v= .............................(A. 15)

%Q(1 - Swc)

...............................,,(A.6)2DB Substituting Eqs. A.lfIand A.15 and P=Psc/B into

Eq. A.6 yields

.SPE 18855 FARUK CIVAN AND DJEBBAR TIAB .

P~cfp13u3

%=e(l. ,)B ..........................(A.16)Wc

APPENDIX BPOROUS MEDIA 140HENTUMEQUATION

For the purpose of entropy analysis of fluidflow through porous media, it is important that themomentum equation involves the effect ofirreversible leas of fluid momentum. A forcebalance for the fluid contained in the pore apaceof the reservoir formation ie expressed by

Fi=Fp-Fg-Fv .....................,.....(B.1)

where Fp, Fg, Fv and Fi are the pressv:e, gravity,viscous and inertial forces, respectively. Fp andFg are given by:

Fp = @tip ...................................(B.2)

where Ap is the pressure differential over adistance L in the flow direction, and

Fg . (pALPg. , . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . (B.3)

The viscous force can be expressed by thefunctionality

Fv =FV(P, u, U; z, L) ............,.........(B.4)

in which the mean pore diameter, ; is defined byEq. (A.14).

The dimensional anulysis method yields:

Fv

()

z Puz =fl . , .........................(B.5)puL L P

The inertial force is given by

Fi = @iLPa ......,...........................(B.6)

where the coefficient of acceleration, a, can beexpressed as follows

a = a(p, M, u, Ap, x, L) ....................(B.7)

Hence, the dimensional analyais method yields

aL

(7

~ - PAp .....0...0.......0......(B.8)

2= f2i p

Combining Eqs. B.1, 2, 3, 5 and 8 gives

-(:+P6)=*U+(3U2---Q--.-(B)Taking the limit as L+O, Eq. A.9 becomes

-( )dp B+pg =. u + DDU2 ...................(B.1O)d!t k

In Eq. B.1O, the permeability, k, and turbulencecoefficient, ~, are defined, respectively, by

()WAk s limit ...............................(B.11)L* f~()f~f!~ limit .....,........................(B.12)L* LConsidering the functional dependency of fl and,f2given by Eqs. B.5 and 8 it becomes clear thatpermeability and turbulence coefficient depend notonly on the porous media geometry, but also on theconditions of the fluid. This point has beendiecussed by Blick and Civan.12

APPENDIX CDARCY AND NON-DARCY FLOW EQUATIONS

I. Non-Darcy Flow

1. Pseudosteady State Flow

During pseudoateady state, the rate of changeof pressure with respect to time is constant and isgiven for a reservoir operating at a constantterminal rate of q9c by

ap qscB= ....,...................,......(Cl)at @cthnre2

in which B is the reservoir fluid formation volumefactor at the reservoir conditions, p pressure,redrainageradiua,h formationthickness,e porosity,and c total effective compressibility.

fLetting

ct=~ pap and using the chain rule and substitutingSq. C.1 and P=Pac/B into the equation of continuity

la .aP. (qp) = 2whQ .......O.................(C.2)r h at

yields

()()dq 2rR- 4Sc ..............S.........(C.3)z: re2Integrating Eq. C.3 gives the volumetric flow ratein the reservoir as

q ()r 2-=cl- %c .....00.....*............(C.4)B Yewhere Cl is a constant of integration. Thevolumetric flux is

u=q12nrh .................,...........,......(C.5)

For non-Darcy flow, Forchheimer equation appliea

dp p=- U + ppu ........,....................(C.6)dr k

To determine the constant, Cl, in Eq. C.4subi$titvteEqs. C.4 and C.5 in Eq. C.6 and let

lP=P=ch3. Hence, Eq. C.6 becomes I 11. Darcy Flow

:= B[:QX3+) If the flow of fluid is relatively slow thenits motion is governed by Darcys law; therefore,f3=0in Eqs. C.7 and C.14.+Bpsc~~7(&-3y]................c.7.For a volumetric reservoir, dp/dr=O at r=re, andEq. C.7 yields two solutions for the integrationconstant,

c1 = qsc ..................,.................(C.8)

2vrhBcl=qsc- I .............0..0.0...(C.9)k(3Psc r=reThe second solution yields negative reaulta.Therefore, only the first solution is consideredfor Eq. C.7 to determine the pressure distribution.

If the average reservoir pressure, p, isdefined by

2

J

e;=? prdr .,...,...,.,.....,.........(Colo)

.re w

then, Eq. C.1 yields the production time as uponintegration

1wrezh ~i v(~)ct(~)t.ti+_ d; .O..........(c.ll)qsc i (j)2. Steady State Flow

The equation of continuity, Eq. C.2,simplifies for steady state flow as

d~ (qP) = 0 .................,..,..,........(C.12)

SubstitutingP=Prc/B and integrating Eq. C.12yields

ql = qsc ..........,.........,.............(C.13)

in which q denote the actual volumetric flow ratein the reservoir and qsc the constant terminal rateat standard conditions. Thus, substitutingO=Osc/B, Eqs. C.5 and C,13 into the Forchheimerequation, Eq. c.6, yields

In the preceding equations, B=BO for 0:.1.For gas B=Bg where

APPBNDIX DPRESSURE EQUATION FOR FLOWING WELLS

The energy equation for a well of length Land de th H, flowing at a steady rate is given by

%Ikokul ~

dp dv 8W1+pv +pgsine+p_=O .............(D.1)dk dfi dt

On the other hand, one can write the followingreiutionships:

sine = H/L = dzjdt ..........................(D.2)

tiwl f@2=- v .,..............,.,....,,......(D.3)d!LD2

D = Psc/B ...................................(D.4)

q = qsc ,.....,............................l ,(D.5)

/()~D2V=q ................................(D.6)THence, substituting Eqs. D,2-6 and rearranging Eq.D.1 becomes

dp=

d~

+C):X++2BZB

()

2qec d +

%c~DZf4 dp

............(D.7)

For prescribed values of the flowrate, qsc, and thewell head pressure a numerical solution of Eq. D.7is obtained by means of the Runge-Kutta-Fehlbergfour (five) Methodll. For gases B ~ Bg given byEq. C.15 and for liquids B E Bo.

SI METRIC CONVERSION tiACTORS

bbl X 1.589873Cp x 1.0*ft X 3.048*psi x 6.894757lbm X 4.535924lbf X 4.448222Btu X 1.055056(F - 32)/1.8

E-01 = m3E-03 = PansE-01 = mE+OO = kPaE-01 = kgE+OO = NE+03 = J

=0 c

()P Tz * Conversion factor is exact.BS=E ............................(C.15)Sc PI

. .

E 18855

Table IData for dry gas well-reservoirsystem

Quantity FieldUnits S1 Units

Ct 1.0x10-2psia 6.893x10-zPaD 2.0 in 0.0508mH 5,000 ft 1.52x103mh 24 ft 7.32mk 5 mdarcy 5X10-15m2L 8,000 ft 2.44x103mPi 4,000psia 2.76x107Pap~b 1.45x106psia l.OxlOIOPa(p~h)f 2,000psia 1.38x107Pae 2,980 ft 9.1X103In

(640 ac-spacing)w 2.0 in 0.0508msWc 0.20 0.20T 702 R 390 KQ 0.15 0.15Yg 0.75 0.75

-.

WELL

(Pwh)i &f

(Pwh)f

SPE 1$85; .

RESERVOIR

i

---- -.

(Pwf)f

w RADIUS,r re

Figure 1. Pressure Distribution in Well and Reservoir System DuringPseudo-State Production at a Constant Terminal Rate.

1.0- wT

0.9qsc=0.01m3/s

(o.0305NMscf/d)0.8 q =0.1m3/s S=-5

(::305MNscf/d)0.7

0.6

~

s+);

0.55=5 /

I

10.40.0.2 -0.

0106 1017 ~018 ~019 ~020

CUMULATIVEENTROPYPRODUCTION,J/K

Figure 2. Recovery Factor Versus Cumulative Entropy Production as aFunction of Flow Rate and Skin Factor.